Geometric theory of functions of a complex variable (Translations of Mathematical Monographs, Vol. 26)

TRANSLATIONS OF MATHEMATICAL MONOGRAPHS VOLUME 26 G. M. Goluzin Geometric Theory of Functions of a Complex Variabl...

Author: Gennadiĭ Mikhaĭlovich Goluzin

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TRANSLATIONS OF

MATHEMATICAL

MONOGRAPHS

VOLUME

26

G. M. Goluzin

Geometric Theory of Functions of a Complex Variable

TRANSLATIONS OF

MATHEMATICAL

MONOGRAPHS

Volume 26

GEOMETRIC THEORY

OF FUNCTIONS OF A COMPLEX VARIABLE

by

G. M. GOLUZIN

AMERICAN MATHEMATICAL SOCIETY Providence, Rhode Island 02904 1969

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Il3gaTenbcTB0 ,HayKa" f naBHa$ Pe,gaxi i iDK34IKO-MaTeMaTxgecxoA JIKTepaTypbi MOCKBa 1966

Translated from the Russian by Scripts Technics

Standard Book Number 821-81576-0 Library of Congress Card Number 70-82894

Copyright © 1969 by the American Mathematical Society Printed in the United States of America All Rights Reserved No portion of this book may be reproduced without the written permission of the publisher

TABLE OF CONTENTS Page

A note on the author ......................................................................................................

1

Preface to the second edition ...................................................................................... 3 Preface (to the first edition) ........................................................................................ 4 Introductory geometric considerations ................................................:........................5 Chapter I. The convergence of sequences of analytic and harmonic functions .................................................................................................... 11

§1. The convergence of sequences of analytic functions ............................. 11 §2. The condensation principle ..........................................................................14 §3. The convergence of sequences of harmonic functions ............................ Chapter II. The principles of conformal mappings of simply connected domains .................................................................................................... 23

§1. Univalent conformal mapping ...................................................................... 23 §2. Riemann's theorem ........................................................................................25 §3. The correspondence of boundaries under conformal mapping ................ 31 §4. Distortion theorems ...................................................................................... 47 §5. Convergence theorems on the conformal mapping of a sequence of domains ......................................................................................................54 §6- Modular and automorphic functions ............................................................ 62 §7. Normal families of analytic functions. Applications .............................. 67

Chapter III. Realization of conformal mapping of simply connected domains ....................................................................................................76 §1. Conformal mapping of domains bounded by rectilinear and circular polygons .......................................................................................... 76 §2. Parametric representation of univalent functions .................................... 89 §3. Variation of univalent functions ..................................................................99

Chapter IV. Extremal questions and inequalities holding in classes of univalent functions ...............................................................................110 §1. Rotation theorems ........................................................................................110 §2. Sharpening of the distortion theorems ......................................................118 iii

iv

§3. Extrema and majorizations of the type of the distortion theorems ........ 128

§4. Application of the method of variations to other extremal problems .............................................................. ......................................... 140 §5- Limits of convexity and starlikeness ...................................................... 165 §6. Covering of segments and areas ............................................................... 170 §7. Lemmas on the mean modulus. Bounds for the coefficients ................ 182 §8. The relative growth of coefficients of univalent functions .................. 190 §9 Sharp bounds on the coefficients .............................................................. 196

Chapter V. Univalent conformal mapping of multiply connected domains ......... 205 §1. Univalent conformal mapping of a doubly connected domain onto an annulus ....................................................................................................205 §2. Univalent mapping of a multiply connected domain onto a plane with parallel rectilinear cuts .................................................................... 210 §3. Univalent mapping of a multiply connected domain onto a helical domain .......................................................................................................... 216

§4. Some relationships involving the mapping functions .................... ........ 222 §5. Convergence theorems for univalent mapping of a sequence of domains ........................................................................................................ 228

§6. Univalent mapping of multiply connected domains onto circular domains. The continuity method .............................................................. 234 §7. Proof of Brouwer's theorem ...................................................................... 244 Chapter VI. Mapping of multiply connected domains onto a disk ........................254 §1. Conformal mapping of a multiply connected domain onto a disk ..........254 §2. Correspondence of boundaries under a mapping of a multiply connected domain onto a disk .................................................................. 262 §3- Dirichlet's problem and Green's function .............................................. 266 §4. Application to a univalent mapping of multiply connected domains .... 275 §5. Mapping of an n-connected domain onto an n-sheeted disk ................. 277 §6. Some identities connecting a univalent conformal mapping and the Dirichlet problem ..................................................................................283

Chapter VII. The measure properties of closed sets in the plane .................... 293 §1. The transfinite diameter and Cebysvev's constant ..................................293 §2. Bounds for the transfinite diameter ..........................................................300 §3. The capacity of a closed bounded set ...................................................... 309 §4. Harmonic measure of closed bounded sets ..............................................314 §5. An application to meromorphic functions of bounded form ....................321

V

Chapter VIII. Majorization principles and their applications .............................. 329 41. An invariant form of the Schwarz lemma .................................................. 329 §2. The hyperbolic metric principle ................................................................ 336 §3. Lindeldf's principle .................................................................................... 339

Harmonic measure. The simplest applications ....................................... 341 §5. On the number of asymptotic values of entire functions of finite 4.

order ..............................................................................................................351 §6. The hyperconvergence of power series .................................................... 356 §7. A nonanalytic generalization of the Schwarz lemma. A theorem

on covering of disks ....................................................................................360 §8. Majorization of subordinate analytic functions ........................................368

Chapter IX. Boundary value problems for analytic functions defined on a disk ........................................................................................................ 380

§1. Limiting values of Poisson's integral ......................................................380 §2. The representation of harmonic functions by means of Poisson's integral and the Poisson-Stieltjes integral .............................................. 385 §3. The limiting values of analytic functions ................. ............................... 393 §4. Boundary properties of functions in the class Hp ................................. 402 §5. Functions that are continuous on a closed disk ..................................... 409

Chapter X. Boundary questions for functions that are analytic inside a rectifiable contour ..................................................................................417

§1. The correspondence of boundaries under conformal mapping ................417 §2. Privalov's uniqueness theorem ..................................................................428 §3. On the limiting values of Cauchy's integral ............................................430 §4. Cauchy's formula .........................................................................................435 §5. Classes of functions. Cauchy's formula ..................................................438 §6. On the extrema of mean moduli ..................................................................441 §7. Approximation in mean and the theory of orthogonal polynomials ........448 Chapter XI. Some supplementary information ......................................................... 454 §1. Gluing theorems ...........................................................................................454 §2. Conformal mapping of simply connected Riemann surfaces .................. 461 §3. An extremum for bounded functions in multiply connected domains .... 467 N. The three-disk theorem ...............................................................................476 §5. Transformation of analytic functions by means of polynomials ............480 §6. On p-valent functions .................................................................................487

vi

§7. Some remarks on the Caratheodory-Fejer problem and on an analogous problem ...................................................................................... 497 §8. Some inequalities for bounded functions ..................................................514 §9. A method of variations in the. theory of analytic functions ................... 526

The scientific works of Gennadii Mihailovic Goluzin ............................................ 545 Bibliography ................................................................................................................. 549

Supplement. Methods and results of the geometric theory of functions ............. 563 Introduction ...........................................................................................................563

§1. Basic methods of the geometric theory of functions of a complex variable .........................................................................................................565

§2. Univalent functions in a disk and in an annulus .................................... 577 §3. Functions that are analytic in multiply connected domains ..................629 Bibliography for the Supplement ................................................................................651

Subject Index ................................................................................................................673

A NOTE ON THE AUTHOR

G. M. GOLUZIN (1906-1952) The author of the present book, Gennadii Mihailovic Goluzin, was born in 1906 in the town of Toriok. In 1924, he entered the Mathematics Division of Leningrad State University. From then until his death, he never interrupted his connection with the University. In 1929 he defended his graduation thesis, which was then published in Matematiceskii Sbornik. After his graduation from the University, Goluzin began teaching; in 1936 he brilliantly defended his doctoral dissertation and in 1938 he was appointed professor. From 1939 on, he held a chair on the theory of functions of a complex variable at Leningrad State University, where, with great energy and devotion, he taught young scientists and also supervised their scientific research. Over a period of years he conducted the basic course as well as a number of special courses, and also led a seminar on the geometric theory of functions of a complex variable. From the founding of the Leningrad division of the Mathematical Institute of the Academy of Sciences of the USSR in 1940, Goluzin simultaneously worked also at that institute, where he carried on extensive scientific activity. The outstanding scientific services of Goluzin were recognized by his being awarded the first prize of the University in 1946 and the Government prize in 1947. The present book is another result of Goluzin's scientific and pedagogical labors. Goluzin died January 17, 1952, about the time of the printing of the first edition, after a long illness. In one of his earlier works, Goluzin used Loewner's parametric representation for the class S of functions f(z) = z + c2z2 + . that are regular and uni1alent in the disk jzj < 1, to obtain the precise form of the so-called rotation theorem, that is, a sharp bound for I arg f (z) I in the class S. For many years, Mnathematicians had been unable to solve this difficult problem. I

In subsequent works, 'Goluzin continued to develop systematically the method Of parametric representations and devised a general method of investigating a broad class of extremal problems. 1

A NOTE ON THE AUTHOR

2

Goluzin developed his own variant of the method of interior variations, and with it obtained a number of profound results. His variational method became one of the basic contemporary methods for investigating univalent functions. Goluzin obtained extremely powerful results in the problem of estimating the order of growth of the coefficients of univalent functions. In particular, he found a solution, in a certain sense definitive, of the problem of the relative growth of the coefficients of functions in the class S: If I c,1 < A lna, where A l is a constant, a> 0, and n assumes values equal to the terms of some arithmetic progression, then I c I < A 2n a for all n, where A 2 is a constant depending only on A 1 and a. For the coefficients of functions in this same class, he found the inequality I en I < % en, which was the first advance in the problem of the coefficients since the well-known result of Littlewood (1925). To the pen of G. M. Goluzin belong a number of works in other divisions of the theory of functions of a complex variable and also in mathematical physics. We mention his investigations on conformal mapping of multiply connected domains onto domains of the canonical type and his work dealing with the solution of the Dirichlet problem for Laplace's equation in the case of domains bounded by circles or spheres. Even a simple listing of the numerous and valuable works of Goluzin would require an excessive amount of space. The reader can get an idea of them by reading the present book. Gennadii Miha lovic` Goluzin was an extremely unassuming man, who made a

vivid impression of personal charm and outstanding ability. June 7, 1966

Academician V. I. Smirnov

3

PREFACE TO THE SECOND EDITION The first edition of Goluzin's monograph was published in 1952, shortly after the author's death. In the last decade, an extensive literature has appeared on themes closely related to the content of this monograph, and many of these results were obtained in the works of Goluzin's pupils. A survey of this literature is given in a special supplement. The text of the book has undergone only slight modifications, which we shall 'point out. In the beginning of the book, the well-known theorems of Runge and walsh on the approximation of regular functions with polynomials have been omitted since these are easy to find in the literature on the subject. The theorem on the construction of a mapping function by means of orthogonal polynomials has also been omitted. Additional material has been included in Chapters IV and XI. Specifically, a few additional remarks have been inserted at the end of § 1 of Chapter IV. In §2 of that chapter, some modification has been made in the arrangement of the exposition: relations (3)-(5) from a 1951 article of Goluzin have been inserted before the formulation of Theorem 1, and a corollary to Theorem 3 of Lebedev, and Theorem 4 from a 1951 article of Goluzin, have been added. The material at the end of §3 of Chapter IV has been extended on the basis of other works of Goluzin 11946f, 1947, 1951d] and put in a separate section (§4). This has necessitated a renumbering of the remaining sections of the chapter. Theorems 3 and 4 of Chapter V, §3 have been moved to §6 of that chapter (where they are numbered Theorems 6 and 7) and their proofs are omitted. Chapter IX has been extended by the addition of the four sections 6-9, in which the articles by Goluzin in (1940, 1946a, 1950, 19521 are incorporated almost without change. All the additions not included in Goluzin's works are indicated by the angle brackets Three bibliographic lists have been added to the book. One of these corresponds to references made in the main text, another to the supplement. In addition, it complete list is given of Goluzin's works. 1965

V. I. S.

4

PREFACE (to the first edition) This book is based primarily on lectures given by the author at Leningrad State University in the course entitled "the geometric theory of functions of a complex variable " and, in part, in the course entitled "supplementary topics in the theory of functions of a complex variable." It includes an exposition of the following topics: the theory of a univalent conformal mapping of simply connected and multiply connected domains, conformal mapping of multiply connected domains onto a disk, applications of conformal mapping to the study of interior and boundary properties of analytic functions, and general questions of a geometric nature dealing with analytic functions. In addition to various general problems in the geometric theory of functions, many particular problems under study at the present time are also considered. However, the book does not pretend to expound all questions related to the geometric theory of functions or to treat all topics with the same degree of completeness. This would have been an impossible task for the author. Therefore, it is only natural that certain questions in the theory of functions have remained untouched, whereas the treatment of others has not been sufficiently complete. In many such cases, the author has given appropriate bibliographic references. The book is intended for a reader who has already mastered the fundamentals of the theory of functions of a complex variable as taught in most university

courses. G. M. Goluzin

5

INTRODUCTORY GEOMETRIC CONSIDERATIONS

The geometric theory of functions of a complex variable makes a study of analytic functions defined by some geometric property or other and a study of various geometric properties of certain classes of analytic functions. Therefore, it naturally rests on a number of general geometric concepts that are encountered in present-day mathematics. Here we propose to make some brief remarks about these concepts, in order of their occurrence, that are associated with the complex plane and that we shall need in our subsequent exposition.

Sets of points in a plane. We shall usually denote sets of points in the plane by capital letters. We shall denote by lower-case letters individual points in the plane and we shall note the complex numbers corresponding to them by the same letters. If a point a belongs to a set E, we indicate this fact by writing a E E. If every point of a set E belongs to a set F, we write E C F and we say that E is

a subset of F or that E is contained in F. Every point in the plane has a collection of neighborhoods. A neighborhood of a given point a is the set of all interior points of some disk with center at a (or sometimes any set of points containing such a set). A neighborhood is said to be sufficiently small if the radius of the disk is sufficiently small. The complex plane is supplemented with an ideal point o., called the point at infinity, which is thought of as lying outside every circle. A neighborhood of the point at infinity is the set of all points lying outside some circle. Such a neighborhood is considered sufficiently small if the radius of the circle referred to is sufficiently great. We note that the distinctive nature of the point o disappears if 'we map the complex plane onto the complex sphere by means of a stereographic projection.

A set of points is said to be bounded if it lies entirely inside some circle. A point a in the complex plane is called a cluster point or a point of accumulation of a set if every neighborhood of a includes points of that set other than a. A cluster point of a set may or may not belong to that set. A point in a set that is not a cluster point of that set is called an isolated point of the set.

6


An infinite set of points always has at least one cluster point. If an infinite set is bounded, this assertion constitutes the well-known Bolzano-Weierstrass theorem, and in this case all the cluster points are finite. On the other hand, if an infinite set is unbounded, one of its cluster points has to be the point at infinity. For any cluster point a of a set, there exists a sequence of points belonging to the set that converges to a. A sequence of points may also converge to the point at infinity. For a sequence to converge to a finite point, it is necessary and sufficient that the distance between any two numbers in the sequence, from some point on, be less than an arbitrary prenamed positive number.

A set of points is said to be closed if each of its cluster points belongs to it. Corresponding to any set E is the closed set obtained by inclusion of all the cluster points of E. We denote this closed set by E and call it the closure of E. The distance between two disjoint sets (that is, without points in common) is defined as the greatest lower bound of distances between two points, one belonging to one set and the other belonging to the other set. We know that if both sets are closed, this distance is positive and the greatest lower bound referred to is actually attained by some pair of points (one in each set). Another important property of closed sets is given by the well-known HeineBorel covering theorem:

If a closed bounded set E is covered by a set of disks such that every point of E lies inside one of these disks, then a finite number of these disks will also cover the set E. A closed set consisting of more than one point is called a continuum if it cannot be decomposed into two nonempty disjoint closed sets. We shall call a set consisting of one point a degenerate continuum. A point is said to be an interior point of a set to which it belongs if some neighborhood of it is contained in that set. A set consisting only of interior points is called an open set. Obviously, the complement of a closed set in the complex plane is an open set and vice versa. Let us recall the definitions of union, difference, and intersection of sets. Let E 1, E2, denote sets (either finitely or infinitely many). The set of points belonging to at least one of these sets is called the union of these sets and is or UkEk. The set of points belonging to all the sets denoted by E1 U E 2 U E 1, E2, - , is called the intersection of these sets and is denoted by E1 n E2 n ... or


7

(lkEk. If a set E is contained in a set F, then the set of all points belonging to F that do not belong to E is called the difference of these sets and is denoted by

f - E. Set operations have the following properties: The union of a finite number of closed sets and the intersection of an arbitrary combination of closed sets are dosed sets; the intersection of a finite number of open sets and the union of an grbitrary combination of open sets are open sets. Domains and curves. One of the basic geometric concepts in the theory of junctions of a complex variable is the concept of a domain. A domain is defined its an open set any two points in which can be connected by a broken line consisting entirely of points of that set (the property of connectedness). The boundary points of a domain are those points in the complex plane that do not belong to Sbe domain but are cluster points of it. If a domain is other than the entire plane, it necessarily has boundary points. The set of all boundary points of a domain is called its boundary. The boundary of a domain is a closed set . Those points in the plane that are neither interior nor boundary points of a domain are called exterior points of the domain.' Every exterior point of a domain has a neighborhood containing no points of the domain. The union of a domain and its boundary is called a closed domain. In contrast with this, a domain itself is sometimes called an open domain. A domain is said to be simply connected if its boundary consists either of a continuum or of a single point or if -the domain itself is the entire complex plane. Otherwise, a domain is said to be multiply connected. Specifically, it is said to be doubly, triply, , n-connected if its boundary consists of 2, 3, , n disjoint continua (possibly degenerate). All these regions are said to finitely connected and the continua (including the degenerate ones) are called boundary continua

The role of domains in the study of closed and open sets is clear from the, following theorem:

Every open set E in the complex plane is the union of finitely or countably many domains.

Another basic geometric concept in the theory of functions of a complex variable is that of a curve. 1) These concepts are meaningful for any open set.

8


A continuous curve is a set of points in the plane the rectangular coordinates x, y of which can be written as continuous functions

X=T (t), Y=+(t)

(1)

of a real variable t in some finite interval a < t < b. It is easy to see that this set is a continuum. However, the concept of a continuous curve is too general for our purposes. There are continuous curves that do not at all correspond to our intuitive idea of a curve as a one-dimensional figure. In fact, it is possible to construct a continuous curve

that passes through every point of a given square. On the other hand, if we require that the curve have no multiple points, it will possess a number of clear-cut properties. Such curves are called simple curves or Jordan curves. Thus, the continuous curve (1) or, more briefly, the curve

z'=z(t),

actcb,

(2)

is called a Jordan curve if, for any two distinct values t and t" in the interval [a, b) with t ' < t", we have z(t ') At ") and z (t") r< z(b). The points z(a) and z(b) may or may not coincide. In the first case, the curve is called a closed, in the second case a nonclosed Jordan curve. We have the following important theorem (due to Jordan):

A closed Jordan curve C partitions the plane (including o.) into two simply connected domains both of which have C as their boundary. One of these domains is bounded and is called the interior of C. The other contains oo and is called the exterior of C. The complement of a nonclosed Jordan curve C consists of a. single simply connected domain containing oo and having C as its boundary. 1) From nonclosed Jordan curves, one can construct continuous curves that are not of the Jordan type. On the other hand, even a Jordan curve is sometimes too general. Then, depending on our purpose, we introduce curves of more restricted types, for example, smooth, piecewise-smooth, and rectifiable curves. A curve (2) is said to be smooth if the function z (t) has a continuous non-

1) Whereas the proofs of the assertions made up to now are extremely simple and the reader can reproduce them himself, the proofs of Jordan's theorem and of a number of assertions that we shall make below present considerable difficulty. The interested reader should consult special texts. With regard to Jordan's theorem itself, we note that there are two new and comparatively simple proofs of it due to Vol'pert [1950] and Filippov [1950].


9

zero derivative z' (t) everywhere in [a, b] (a one-sided derivative at the two endpoints). The requirement of smoothness is obviously equivalent to the require&eat that the curve have everywhere a continuously turning tangent. A curve con$isting of finitely many smooth curves is called a piecewise-smooth curve. The Definition of a rectifiable curve will be given in Chapter X. Finally, the simplest type of continuous curve is an analytic curve. This is a curve defined by an qquation of the form z = z (t) for a < t < b , where z (t) can be expanded in a power series

z(t)=co+cl(t-to)+ ..., 0, about each value to in [a, b]. A continuous curve consisting of a unite number of analytic curves is called a piecewise-analytic curve. Let us now look at certain questions dealing with domains. Sometimes we make cuts along various Jordan curves in a domain. Let B denote a domain. To make a cut along a Jordan curve L contained in B means to delete from B all points of the curve L. We shall consider only cuts consistlag of finitely or infinitely many sets of analytic arcs. We shall consider the infinite case under the assumption that the endpoints of these arcs cluster only at the boundary of the domain. A. cut in a domain B is said to be transverse if it Connects two (not necessarily distinct) boundary points of the domain B, which serve as its endpoints and if all its other points lie inside B. It turns out that if a transverse cut in a finitely connected domain connects boundary points on different boundary continua but does not partition the domain into two parts, it decreases the connectedness by 1; on the other hand, a transverse cut in a simply connected domain always partitions that domain into two simply connected parts. (This is a characteristic property of simply connected domains.) Analogously, if a cut is a closed Jordan curve lying entirely in a domain B, it is called circular cut. A circular cut always partitions a domain B into two domains. In the case of a simply connected domain B, one of the domains bounded by a circular cut lies entirely in B (another characteristic property of simply connected domains). Finally, a cut that is an open Jordan curve lying entirely in a domain . except possibly for one of its endpoints does not partition B into two parts. We note that, in a finitely connected domain B, it is always possible to make a transverse cut connecting points of two given boundary continua (that do not degenerate to Do) and also a circular cut encircling a given bounded boundary continuum but no other boundary points. Moreover, if a closed bounded set E is orith c I

10


contained in a domain B, then one can make a circular cut in B that separates E from the boundary of B, that is, a cut that encircles E but does not touch any part of the boundary of the domain B. These results can be achieved by means of cuts consisting of a finite number of rectilinear segments.

CHAPTERI THE CONVERGENCE OF SEQUENCES OF ANALYTIC AND HARMONIC FUNCTIONS

§1. The convergence of sequences of analytic functions Many divisions of the theory of functions of a complex variable, and in particular that division known as the geometric theory of functions, employ in their proofs the basic properties of the convergence of sequences of analytic functions. Thanks to these properties, such proofs are fairly simple and elegant in comparison with the analogous proofs of real analysis. , denote a Let us give some definitions. Let {fn (z)}, for n = 1, 2, ;sequence of single-valued functions defined on a set E of points in the z-plane. This sequence is said to converge at a point zo E E if the sequence of numbers fn (zo) converges. A sequence of such functions fn (z) is said to converge on E if it converges at every point of E. In such a case, we may speak of the limit function f(z) = (z) defined on E. The sequence (fn (z)( is said to converge uniformly on E to a function f(z), which is finite on E, if, for every e > 0, there exists an N > 0 such that the inequality n > N implies Ifn (z) - f (z)! < for every z E E. On the other hand, if f(z) = 00 on E, the sequence 1fn (z)1 is said, by definition, to converge uniformly on E to -a if, for every M > 0, there exists an N > 0 such that the inequality n > N implies Ifn (z)j > M for all z E E. It is easy to show that a necessary and sufficient condition for uniform convergence of a sequence to a finite function is that, for every e > 0, there exist an N>0 such that, for all m, n> N and all z E :E, Ifm(z) - fn (z)I 0, there exists a 8 > 0 such that the relations z E E and I z - zoI < S imply If(z) - f(zo)I <e. For sequences of continuous functions, we have the theorem:

If a sequence f(z) of functions that are continuous on a set E converges uniformly to a finite function f(z) defined on E, this function f(z) is also continuous on E. To see this, let zo E E. Then, for given t > 0, there exists a number n such that, for all z E E, we have I fn (z) - f (z)I < e/3. Furthermore, there exists a number 8 > 0 such that, for all z E E satisfying the inequality I z - zoI < S, we have I fn (z) - fn (zo)I < E/3 (because of the continuity of fn (z) on E). Therefore, for z C E and I z - zoI R (that is, formulas (1) with the first formula

applying also to f (a)) and considering the corresponding function fo (z). This completes the proof of the theorem.

As regards uniformly convergent sequences of regular functions, we shall prove another theorem, which has numerous applications.

Theorem 2. Let (f, (z) I, where n = 1, 2, - - -, denote a sequence of functions that are regular in a domain B. Suppose that converges uniformly in the interior of B to a regular function f(z) / const. Suppose that each of the functions fn (z) assumes a given value a at no more than p (p > 0) points of the domain B. Then the function f(z) assumes the value a at no more than p points of B. Proof. Let us suppose to begin with that B does not include ce . Suppose that f (z) assumes the value a at p + 1 distinct points zk E B, k = 1, 2, - - -, p + 1. Around each of the points zk, k = 1, - - - , p + 1, let us draw a circle Yk C B sufficiently small in each case that the circles do not touch each other and there are no zero of the functions f (z) - a on them. Since f (z) J coast, this can be done. Under these conditions, there exists an m > 0 such that if (z) - al > m on all the neighborhoods yk. Since the sequence (f (z)} converges uniformly on the circles Yk , there exists an n such that I fn (z) - f (z) I < m on each yk , k = 1, - - -

-, p+l. Since f. (z)-a=(1(z)-a)+(f, (z)-f(z)) we conclude on the basis of Rouche's theorem, that the function fn (z) - a does indeed have zeros inside each circle yk since the function f(z) - a does. Consequently f (z) assumes the value a at no fewer than p + 1 points of the domain B, which contradicts the hypothesis of the theorem. If the domain B includes oe but is not the entire plane, we can choose a suitable fractional linear function to map it onto a domain not including and apply to the image functions the result just proved. Finally, the case in which the domain B is the entire z-plane is excluded because, in such a case, the function f(z) must necessarily be constant. This completes the proof of the theorem.

§2. The condensation principle In many branches of mathematical analysis in which existence questions are answered, it is important to have an affirmative answer to the question whether a given sequence of functions all defined on the same domain of definition con-

§2. THE CONDENSATION PRINCIPLE

15

rains a convergent subsequence. Several proofs in analysis are made more complicated by the fact that this is not always the case. Not every sequence of functions defined, continuous, and uniformly bounded on a given interval contains a conver-

gent subsequence, for instance the sequence (sin nq}, n = 1, 2, . For an extremely broad class of sequences of analytic functions, however, the answer to the above question is affirmative. As a preliminary to our discussion, let us adopt some conventions on terminology. A family of functions 2 = {f (z)l defined on a subset E of the z-plane is said to be uniformly bounded on E if there exists a finite positive number M such that the inequality If (z) < M holds for

every function in the family and every z E E. If the functions f(z) are defined in a domain B and are uniformly bounded on every closed set E C B, then the family l is said to be uniformly bounded in the interior of B. Uniform boundedness in the interior of a domain is a weaker requirement than uniform boundedness in a domain.

Theorem 1 (the condens.ation principle). Let {fn (z)}, n = 1 , 2, - , denote a sequence of functions that are regular in a domain B. Suppose that the sequence is uniformly bounded in the interior of B. Then this sequence {fn (z)} contains a subsequence that converges uniformly in the interior of B to a regular function.

Proof. The proof consists of four parts. 1°. Let us show first that the functions fn(z), n = 1, 2, , are equicontinuous on any given closed bounded set E C B. This means that, for every r > 0, there exists a S > 0 such that, for every pair of points z 1 and Z2 in E with Iz 1 - Z21 < S, the inequality I fn (z 1) - fn (z 2) I R and is continuous in k - a! > R, then the analogous formula 2x

it (z) -

- 2a Suz(z+22do

(3)

is valid in the domain I z - al > R. Since harmonic functions can be regarded as functions of a complex variable z, the definitions of convergence given in §1 can be applied to them. Furthermore, using the same line of reasoning as in §1 for the proof of the Weierstrass theorem, except that instead of Cauchy's formula we use formula (1) or (3), we can prove the following theorem.

Theorem 1. Let fun (z)I, n = 1, 2,. , denote a sequence of functions that are harmonic in a domain B. If fun (z) I converges uniformly in the interior of B to a finite function, then the limit function is also harmonic in B. Furthermore, by using the same line of reasoning as in §2, we can prove

Theorem 2. Let lun(z)}, for n = 1, 2, , denote a sequence of functions that are harmonic in a domain B. If lun (z)} is uniformly bounded in the interior of B, it contains a subsequence that converges uniformly in the interior of B. In contrast, Vitali's theorem in the form in which it was given in §2 for regular functions is no longer valid. For example, the sequence defined by , converges u2 k (z) = 0 and u2 k _I (z) = y (imaginary part of z) for k = 1, 2, on the real axis but nowhere else. However, we do have Theorem 3. Let lun (z)1 denote a sequence of functions that are harmonic and uniformly bounded in the interior of a domain B. If fun (z)I converges in some subdomain B' of B, it converges uniformly in the interior of B.

§1. THE CONVERGENCE OF SEQUENCES OF HARMONIC FUNCTIONS

21

Proof. Suppose that the sequence lun(z); does not converge in the domain B. Then it contains two subsequences that converge in B to distinct limit functions u(z) and u* (z) that are harmonic and that coincide in the domain B' . Consequently, their difference in B' is zero. But then, the difference of their harmonic conjugates v(z) and v" (z) must, by the Cauchy-Riemann conditions, be constant io B'. 1) This in turn implies that the corresponding analytic functions whose 'real parts are u(z) and u` (z) coincide in B'. But these analytic functions will ,-then coincide throughout the entire domain B. Therefore, the supposedly distinct ,harmonic functions u(z) and u* (z) must coincide in B. This contradiction 'proves the convergence of the sequence f un(z)} in B. That the convergence is ;uniform in the interior of B now follows from 3° and 4° of the proof of the condensation principle for analytic functions. We conclude this chapter with another important theorem. , denote a sequence of Theorem 4 (Harnack). Let fun (z)), n = 1, 2, functions that are harmonic in a domain B. Suppose that, for each z E B, the sequence {un(z)1 is a monotonic increasing sequence (that is, un+i(z) > u,, (z), ). Then, the sequence {un(z)j converges uniformly in the intefor n = 1, 2, rior of B either to a harmonic function or to the function with constant value +D.

Proof. We may assume that all the functions un (z) are nonnegative since otherwise we would consider not the functions un(z) but the functions un(z) -

ui(z) for n= 1, 2,---. Consider an arbitrary disk Iz - al < r contained in the domain B and a

larger disk k - at < r' also contained in B. Since, in accordance with (2), we always have r'$

r

(r') 9

z'+z-2a z' - z

r'2 r (r' - r):

whenever I z' - al r' and I z - at < r, it follows on the basis of formula (1) as applied to the function un (z) (taking R = r') that U. (a) r,

r C ttn (z) < itn (a) r. ± r

(4)

Now, let us set u (z) = limn_, un W. Because of the monotonicity of the sequence {un(z)1, this limit exists for z E B. Inequality (4) shows that, if 1) The conjugate functions v(z) and v'(z) may not be single-valued in B, but this fact is of no significance in the present case.

22

I. SEQUENCES OF ANALYTIC AND HARMONIC FUNCTIONS

u (a) is finite for a E B, then the functions u (z) are uniformly bounded in a then )u (z) ) sufficiently small neighborhood of the point a, and, if u (a) = converges uniformly in that neighborhood to +°. Consider the set of all points in the domain B at which the function u(z) is finite and its complement, the set of points- at which u (z) = From what has been said, both these sets are open. But obviously, this can be true only when one of the two sets is empty. If the function u(z) is everywhere finite in B, then the functions u, (z) are uniformly bounded in the interior of B. By Theorem 3, the convergence of )u,, (z)) to u (z) is uniform in the interior of B, and by Theorem 1, the function u (z) is harmonic in B. On the other hand, if u (z) = +o in B, then Ian (z)) converges to +° uniformly in the interior of B. We note that if the domain B includes the point 00, then we must, as usual, use formula (3). This completes the proof of the theorem.

CHAPTER II

THE PRINCIPLES OF CONFORMAL MAPPING OF SIMPLY CONNECTED DOMAINS

§1. Univalent conformal mapping

One of the basic questions in the theory of functions of a complex variable is the study of analytic functions based on the nature of the mappings produced by these functions. Suppose that a function t; = f(z) is regular in a domain B except possibly at certain points at which it has poles (in which case the function is said to be meromorphic in B). Then we. say that the function f maps the domain B in the z-plane univalently onto some domain B' in the C-plane if it sets up a one-to-one correspondence between points of the domains B and B' . The domain B' is then called the image of the domain B under the mapping C_ f (z), and B is called the preimage of B'. If t = f (z) is a univalent mapping of

the domain B onto B', the derivative f'(z) is nonzero at all finite points of the domain B at which f(z) is regular." To see this, suppose that f'(a) = 0 at some point a E B. Then, the inverse function could be expanded in a neighborhood of the point b = f (a) E B' in a series of fractional powers of (C - b) and hence there would not be a one-to-one correspondence in neighborhoods of the points a E B and b E B'. The nonvanishing of the derivative means that we are dealing here with a conformal mapping. Therefore, a univalent mapping is also called a univalent conformal mapping (and, in the case of simply connected domains, it is called simply a conformal mapping) of the domain B onto the domain B' : The function t; = f(z) defining a univalent conformal mapping is called a univalent function. Its inverse is a univalent function defined on the image.

1) However, nonvanishing of the derivative everywhere in the domain B does not ensure univalence of the mapping. For example, the derivative of the function C = f(z) = (z - 1)" with n > 3 is nonzero throughout the disk IzI < 1, but this function is not univalent in that disk. 23

24

II. CONFORMAL MAPPING OF SIMPLY CONNECTED DOMAINS

If a function ((z) that is univalent in a domain B is not regular in B, it obviously has only one pole (and that a simple one) in B. The first question arising in the study of univalent conformal mappings is whether a given domain can be mapped univalently into another domain. We mention certain necessary conditions for this. In the case of finitely connected domains, a necessary condition is that the connectedness of the two domains B and B' be of the same order.' Let us show this. We may assume without loss of generality that the domains B and B' both include oo and that the mapping in question maps oo into co. (we can always arrange for this to be the case by means of suitable fractional-linear transformations). Let us enclose the boundary continua of the domain B inside closed broken lines that have no multiple points, that lie outside each other, and that enclose each a single boundary continuum. The mapping then maps these broken lines into closed piecewise-analytic Jordan curves

lying inside B' and outside each other. If B' is of lower connectedness than B, then at least one of these curves will not include boundary points of the domain B'. Since the function inverse to the original function is regular inside such a curve and since the corresponding point describes a preimage C as it moves around that curve, the interior of the curve will correspond to the entire interior of the preimage C, which, consequently, must be completely contained in B. But this contradicts the fact that there are boundary points of the domain B within C. This means that the domain B' cannot be of lower connectedness than B. By reversing the roles of B and B', we can show that B' cannot be of higher connectedness either. In the same way we can show that it is impossible to map a domain of infinite connectedness univalently onto a domain of finite connectedness. Therefore, when we now pose the question of the possibility of univalent conformal mapping of various domains onto a given simply connected domain, we can

confine ourselves to an examination of simply connected domains. In order to know how to map a simply connected domain onto another simply connected domain, we need only know their mappings onto any of the standard domains, for example, a disk, since the required mapping can then be obtained by first mapping one of the given domains onto the disk and then mapping the disk onto the other domain. The question arises whether an arbitrary simply connected domain can be mapped onto a disk. It turns out that there are two cases when this is impossible, namely when the domain is the entire plane and when it has a single boundary

§2. RIEMANN'S THEOREM

25

point (that is, when it is a punctured plane). Let us show that this is true. (In the second case, we may assume without loss of generality that the boundary point is 00.) If such a mapping is possible we have in both cases a mapping function that is regular on the entire plane except possibly at oo and, in addition, bounded. Such a function would have to be a constant, and a constant does not perform the desired mapping.

However, we shall see below that, with the exception of these two cases, every simply connected domain can be mapped univalently onto a disk. In fact, there are infinitely many ways of doing this, that is, there are infinitely many distinct functions that carry out the desired mapping. This last is a consequence of the fact that there exist infinitely many distinct fractional-linear functions that map a disk conformally into itself. §2. Riemann's theorem The following theorem of Riemann answers the question posed at the end of the preceding section and thus is a geometric principle of formation of analytic functions. As a preliminary, we shall prove a simple lemma, which has numerous applications in the theory of conformal mappings. Lemma 1 (Schwarz). Let f(z) denote a function that is regular in the disk Izl 0. To prove this last, consider the family 11 of all functions C = f (z) that are regular in the domain B, that map B univalently onto domains contained in the disk ICI < 1, and that have the properties f(O) = 0 and f'(0) > 0. This set is nonempty since, for example, the functions f (z) = az, for 0 < a < 1 helong to it. Let us pose the following extremal problem: Out of all the functions belonging to the family 11, find the one such that f'(0) is greatest. Let us first prove that this problem has a solution. We note that there exists a positive number p such that the disk I z I < p is contained in the domain B and hence that, in this disk, the functions f(z) E J are regular and satisfy the conditions of the Schwarz lemma with R = p and M = 1. This yields f'(0) < l/p; that is, the numbers f'(0) are bounded above. Let a denote the least upper bound of all these numbers. From the example given above, of functions in 9, we have a > 1. Let (fn(z)(, where n = 1, 2, ... , denote a sequence of functions in 2 such that f" '(0) --+ a. To this sequence we apply the condensation principle, which asserts that this sequence has a subsequence that converges inside B to a regular function fo(z) and that satisfies the conditions f0(0) = 0 and ff (0) = a > 0. By Theorem 2 of §1, Chapter 1 (with p = 1), the function f0(z), since it is not identically constant, assumes in the domain B an arbitrary value at no more than one point; that is, f0(z) is univalent in B. Finally, since the inequality I f (z)I < 1 in B implies that Ifo(z)I < 1 in B (for If0(z)I to be equal to 1 in B is impossible by virtue of the maximum principle), we conclude that f0(z) E 11. Let us now show that the extremal.function f0(z) maps the domain B onto the entire disk ICI < 1. If this were not the case, there would exist a point a in the disk IC I < 1 that would not belong to the image of B, which we denote by B0. Consider the two-sheeted disk ICI < 1 with unique branching point at a (that is, the Riemann surface obtained by taking two copies of the disk ICI < 1, making,a cut in each of them along the portion of the radius extending from a to the circle ICI = 1 and then gluing them crosswise along the cuts). Let us map this disk onto the one-sheeted disk Itl < 1 in such a way that the point t = 0 and the direction of the positive real axis at that point is mapped into t = 0 and the direction of the positive real axis. To do this, we first make a fractionallinear transformation of the disk ICI < 1 into itself in such a way that the branch point a is mapped into the origin. Then, by means of the function VS we expand the two-sheeted circle thus obtained into the one-sheeted disk. Finally, by means

28


of a new fractional-linear transformation of the disk ICI < 1 into itself we establish the required correspondence of the coordinate origins. We denote the function that carries out this mapping by t = c(C) and its inverse byC =,i (t). Then, c(t;) can be extended in the domain B0 along an arbitrary path and hence it is singlevalued. The function 0(t), being a rational and hence single-valued function by construction, satisfies the conditions of the Schwarz lemma with R = 1 and M = 1, and it is not a linear function. Therefore, t = maps the domain B0 univalently onto a domain contained in the disk ti < 1. Also, '(0) < 1; that is, 0' (0) > 1. The composite function t = 0 (fo(z)) _ 0(z), c (0) = 0, c'(0) > 0, maps the domain B onto the same domain and hence it belongs to the family 19. But V(0) = 0'(0) fo (0) > f 0(0) = a, which contradicts the definition of a. Thus, the function C = f0(z) maps the domain B univalently onto the entire disk. It remains for us to prove the uniqueness of the function mentioned in the theorem. Suppose that there are two such functions fl(z) and f2(z)- Then, the function F(C) = fl(fZ 1(L;)), which is regular in the disk ICI < 1 must map this disk univalently into itself. Consequently, in accordance with the Schwarz lemma, IF(C)I < ICI in the disk ICI < 1; that is, I fl(z)I I f2(z)I in B. Consequently, I fl(z)I = I f2(z)I in B. Since the function f1(z)/f2(z) is regular in B and is of constant absolute value, it follows from the maximum modulus principle that fl(z) = f2(z), which proves the uniqueness. This concludes the proof of the theorem. We have a few remarks to make concerning Riemann's theorem. In the first place, we note, that, under the assumptions made regarding the

domain B, there exists a unique function '= F(z) that is regular in B, that is normalized at a finite point zo C B by the conditions F(zo) = 0 and F'(z0) = 1, and that maps the domain B univalently onto the disk with center at C= 0 (another form of Riemann's theorem). This is true because the function F(z) = fo(z)/ fo(zo) is such a function, where f0(z), with fo(zo) = 0 and f0 (z0) > 0, is the function mentioned

F(z) in Riemann's theorem and the radius of the disk onto which the function = Fl(z maps the domain B is R = 1/f' (z0). If there existed a second function with F1(zo) = 0 and Fl'(zo) = 1, that maps the domain B onto a disk ICI < R1, then by Riemann's theorem, we would have Fl(z)/ R1 = f0(z) and hence 1/R1 = fa (zo); that is, F1(z) = fo(z)/fo (z0) = F(z). This completes the proof of the uniqueness of the mapping function t; = F(z).

§2. RIEMANN'S THEOREM

29

The quantity R = 1/f o (zo) is called the con formal radius of the domain B at the point zo. Second, the proof of Riemann's theorem is based on one of the many extremal properties of functions that map a domain B univalently onto a disk. It turns out that a number of extremal properties of such functions hold not only in the class of univalent functions but also in the class of all regular functions that are normalized in a suitable manner. We shall now present two extremal properties of this bind.

a) Minimization of the maximum modulus. Let B denote a simply conpected domain that includes z = 0 and has more that one boundary point. Let 9R denote the set of all functions F(z) with F (0) = 0 and F '(0) = 1 that are reg¢lar in B. The quantity M (F) = Supze9 IF(z)I is minimized by a unique function gnat maps the domain B univalently onto the disk I z I 0, there exists a S > 0 such that, if a point z C B lies at a distance less than S from the boundary of B, the corresponding point C lies in the annulus 1 - t < ICI < 1. Specifically, we may take for S the distance between the boundary of the domain B and the closed set constituting the preimage of the disk ICI < 1 - t. Analogously, we can show that 1, then the corresponding if points of the disk ICI < 1 approach the circle points z C B approach the boundary of the domain B uniformly. For the next step in our investigation, we introduce the concept of accessible boundary points of a domain. A boundary point c of a domain B is said to be an accessible boundary point if it can be joined with an arbitrary interior point of the domain B by a continuous curve 1: z = z (t) for a < t < b that lies entirely in B except for the endpoint c = z (b). Such a curve l can be replaced by a Jordan curve or even by a broken line with no multiple points that consists of a finite or infinite number of segments, in the latter case with endpoints clustering only about c. This is true because the arc z = z (t) for b - tk _1 < t 0 for

36


k = 1 , 2, .. , where Ek > Ek +11 and where Ek -. 0 as k --+ oe is a closed set and

hence lies at a positive distance 8k from the boundary of the domain B. Therefore, it can be partitioned into a finite number of arcs sufficiently small that the distance between any two points of any one of these arcs is less than 8k. Then, the broken line consisting of segments joining successive points of this partition will lie entirely in B. If we apply this for k = 1, 2, , we see that the curve I can be replaced with a broken line consisting of an infinite number of segments. If we discard any loops, we obtain a broken line without multiple points. This broken line is a Jordan curve.

In connection with the concept of accessible boundary points, we have an important supplement to Jordan's theorem (cf. introduction), presented by Schoenflies:' All points of a closed Jordan curve are accessible boundary points of the domains defined by it. For continua of a more complicated type, there may not be an analogue of Jordan's theorem and its supplement. For example, the curve shown in Figure 2 and representing a spiral-shaped curve without multiple points that approaches the circle C asymptotically partitions the plane into three simply connected domains, and, for the two bounded domains, the

circle C consists entirely of inaccessible boundary points. Analogously for the domains shown in Figures 3 and 4 and representing squares with rectilinear cuts that Figure 2 cluster about the segment cd, all points of the segment cd (with the exception of the point d in Figure 4) are inaccessible boundary points. In these last two examples, the point e (see Figures 3 and 4) can be reached along two continuous curves lying in the domain, one on either side of the cut. To distinguish two boundary points at a point e, we identify the accessible boundary point of the domain B not only by its position in the plane but also by the continuous curve in B that joins this point with an interior point. Let 11 and 12 define respectively accessible boundary points at z1 and z2 . If z1 z2, these accessible boundary points are distinct. On the other hand, if zl = z2, we take the view that two curves 11 and 12 lying in B define at zl two distinct accessiblepoints if and only if there exists a neighborhood of the point z1 in which 11

1) See, for example, Caratheodory [19121.

§3. CORRESPONDENCE OF BOUNDARIES

37

and 12 cannot be connected by a continuous curve lying both in B and in that neighborhood. If two curves li and 12 that terminate at a single boundary point d

Figure 3

Figure 4

have common points in an arbitrary neighborhood of z 1, these curves necessarily define the same accessible boundary points. Therefore, accessible boundary points defined by a continuous curve and a broken line inscribed in that curve coincide, as shown above. This enables us to define accessible boundary points by means of Jordan curves only. We have the following theorem 1) on the correspondence of accessible boundary points: Theorem 1. Given a univalent conformal mapping of a domain B in the zplane onto the unit disk ICI < 1, to each accessible boundary point z1 of the dozi

main B we can assign a specific point Ci on the circle ICI = 1 such that, as z approaches zi along an arbitrary curve 11 defining z1, the corresponding point C approaches Ci, and also such that two distinct accessible boundary points of the domain B correspond to two distinct points on ICI = 1 and the set F of points on ICI = 1 corresponding to all accessible boundary points of the domain -B is everywhere dense on fI = 1. Proof. We shall carry out the proof in several stages. 10, Let us show that, as a point z approaches an accessible boundary point z 1 of the domain B by moving along a curve l is z = z (t), for a < t < b, that defines the accessible boundary point, the corresponding point C moves along a continuous curve A1: J= f(z(t)) in the disk 1 and approaches a definite point on the circle KI = I.

1) Koebe [19151.

38


Let us suppose the opposite. Then, there must exist on the circle IcI = 1 two distinct points S1 and S2 such that there is on Al a sequence of arcs with endpoints that converge to S1 and S2 and the inverse mapping function z =' converges uniformly on these arcs to z1. Since /(C) is bounded in the disk ICI < 1, it follows from Lemma 1, as applied to the function 40(C) - z1, that

(W) - zl - 0; that is, -0 (0 a const in ICI < 1, and this is impossible. Consequently, as C moves along A1, it approaches a definite point on IcI = I. 2°. Let us show that, as z describes two curves 11 and l2 defining the same accessible boundary point z1 of the domain B as it approaches z1, the corresponding point describes curves Al and A2 that terminate in a single point on the circle ICI = I. Let us suppose that the curves Al and A2 terminate at distinct points and S2 on the circle ICI = 1. In an arbitrary neighborhood of z1, the curves l1 and 12 can be joined by a continuous curve and hence by a Jordan curve contained both in the domain B and in this neighborhood. Therefore, as the neighborhood is constructed to a point, the endpoints of the corresponding Jordan curves in the

disk ICI < 1 approach `'1 and CZ. On these curves, -O(C) approaches z1 uniformly. In accordance with Lemma 1, we again obtain (W) __ const in ICI < 1, which is impossible. Consequently, S1 = CZ. 3°. Let us now show that two distinct points `'1 and S2 on the circle ill = 1 correspond to two distinct accessible points z1 and zZ of the domain B. Suppose that z1 and z2 are defined by the Jordan curves 11 and 12. We may assume that they issue from a single point of the domain B and that they have no other points in common in B. Let Al and A2 denote the corresponding curves in the disk ICI < 1. Let us suppose that they terminate at a single point S1 on ICI = 1. The closed curve Al U A2 partitions the disk into two domains, of one of which it constitutes the entire boundary. Suppose that corresponding to this domain in B is a domain B1. The curve II U 12 is mapped into its boundary. If the complex numbers z1 and z2 are distinct, there exists on the boundary of the domain B1 a boundary point a of the domain B that is distinct from z1 and z2 because, otherwise, the boundary of the domain B1 would be the open Jordan curve 11 U 12 and thus would not be bounded, in contradiction with the assumption of boundedness of B (see p. 35). In a sufficiently small neighborhood of the point a, all boundary points of the domain B 1 are also boundary points of the domain B. Consequently, as z C B approaches these points, the corresponding point C must approach the circle ICI = 1, that is, the point 1. In accordance with Lemma 3, 1

39


this yields f(z) - S1 in B, which is impossible. On the other hand, if the complex numbers z1 and z2 are equal, 11 U 12 is a closed Jordan curve which is mapped into the boundary of the domain B1. In this case, if the domain B1 has a boundary point distinct from points of the curve 11 U 12, this point must lie inside the curve 11 U 12. Consequently, the hypothesis of Lemma 3 is satisfied in a sufficiently small neighborhood of this point and we may again conclude that coast in B, which again is impossible. This proves that the entire boundary of B1 is the curve 11 U lZ. But then, in an arbitrary neighborhood of the point z1, the curves 11 and 12 can be joined by a continuous curve lying in the domain B1 and hence in the domain B, and the curves 11 and 12 do not define distinct accessible boundary points of the domain B. This proves that S1 CZ.

4°. Finally, let us show that the set F referred to in the theorem is everywhere dense on the circle ICI = 1. Let us suppose that the circle JC1 = 1 contains an arc y no points of which correspond to accessible boundary points of the domain B. Let Co denote an interior point of this arc and let IC,, {, for n = 1, 2, ... , denote a sequence of points of the disk I I < 1 that converges to CO. The corresponding points zn E B have at least one cluster point z0, which necessarily lies on the boundary of the domain B. Let {znkI denote a subsequence that converges to z0. Let us denote by Ink the straight line segment connecting znk with the closest point of the boundary of the domain B. This segment defines an accessible boundary point. From 1°, corresponding to this segment in the disk JCj < 1 is the curve An extending from the point Cn to some point on the circle !CI = 1 not on the arc y. Also, -O(C) approaches z0 uniformly on knk as k 00. On each of the curves Ank, we can choose an arc such that the sequences of endpoints of these arcs converge to the point Co in the one case and to a point on the circle ICI = 1 not on the arc y or to one of the endpoints of that arc in the other. Then, we can apply Lemma 1 to the function O(C) - z0, and we conclude that CI) - z6 in the disk IC` < 1, which is impossible. Consequently, the arc y described above does not exist and the set F is everywhere dense on the circle This completes the proof of Theorem 1.

Prime ends. To make a complete investigation of the correspondence of boundaries under univalent mapping of a domain B onto the disk I < 1 , we introduce the concept of a prime end. Suppose that we make a sequence of trans1

verse cuts Ck, for k = 1 , 2, ... , in the domain B. The cut C1 connects two distinct accessible boundary points of the domain B and partitions it into two


40

domains, one of which we denote by B I. The cut C2 lies in the domain B 1, connects two distinct accessible boundary points of the domain B that are not endpoints of C1, and partitions the domain B1 into two domains. Let B2 denote whichever one of these domains has no points of the curve Cl on its boundary. In general, the cut Cry+1 lies in Bn, connects two distinct accessible boundary points of the domain B that are not endpoints of Cn, and partitions the domain Bn into two domains. We denote by Bn+l whichever of these domains does not have points of the curve Cn on its boundary. The closed domains Bn, for n = 1, 2, . , which are nested in each other, have a nonempty intersection, which we denote by E. Let us assume that the cuts Cn are always such that E contains only boundary points of the domain B. We denote by Gn the domains in the disk ICI < 1 corresponding to the domains Bn. The closed domains Gn` also have a nonempty intersection. If this intersection consists of a single point Co on the circle C I = 1, the set E is called the prime end of the domain B corresponding to the point Co

1)

It is easy to show that a prime end corresponding to a point CO on ICI = 1 is independent of the sequence of cuts defining it. Let IC, ; and IC, I denote two sequences of cuts defining two prime ends E' and E" corresponding to the same point Co. Then the domains C' and Gn corresponding to the domains Bn and B" defined by these cuts have a unique common point Co. Consequently, each of the domains LR contains all C" beginning with some m; conversely, each of the domains Gn contains all G' beginning with some m. This property carries over

to the domains B' and B", but then, E" C B' and E' C k' for all n; that is E" C E' and E' C Ell. This means that the prime ends E' and E" determined by the cuts C' and Cn consist of the same points, and this in turn means that a prime end is completely determined only by the point Co corresponding to it. A prime end can also be characterized independently of the sequence of cuts C defining it. Specifically, let us show that the prime end E of the domain B corn responding to the point C on ICI = 1 is the geometric locus of all cluster points of sequences of points of the domain B corresponding to all possible sequences

1) (A prime end can be defined without bringing in the concept of conformal mapping, as was done in the definition of accessible boundary points (cf., for example, Carathcodory [1912]).

41


of points of the disk I CI < 1 that converge to Co. Suppose that corresponding to the points C. --' S0, where I' < 1 and 141 = 1, is a sequence of points zn E B. Since any one of the domains Gn considered abodt)ontains all t beginning with some m, any one of the domains Bn has the same property with regard to the Consequently, every cluster point of the sequence (znI belongs to all points $n, that is, belongs to E. Now suppose, conversely, that zo E E. Then z0 lies on the boundary of all the domains Bn. In the domain B, let us choose a sequence z0 such that zn E Bn, where n = 1, 2, .... Corresponding to of points zn .them in the disk ICI < 1 are the points t;n E Gn where n= 1, 2,.... It follows t ; that is, z0 is a cluster point of the set of points zn E Bn correthat n Co Isponding to the points n The question of the correspondence of boundaries under conformal mapping is completely answered in terms of a prime end (see Caratheodory [1913a]): Theorem 2. Under a univalent conformal mapping of a domain B onto the unit disk ICJ < 1, there exists a one-to-one correspondence between points of the circle ICI = 1 and the prime ends of the domain B. Proof. It follows from the definition of a prime end that to every prime end of a domain B there corresponds a unique point on the circle ICI = I. Let us prove the converse, namely, that for every point Co on ICI = 1 there corresponds a prime end of the domain B. Let us take two sequences (t n } and k 1, where n = 1, 2, ... , of points on I C I = 1 distinct from C. which converge from opposite sides to Co and to which correspond sequences of accessible boundary points zn and C.,'J zn of the domain B. We may assume that, for each n = 1 , 2, ... , the points and lie inside the arc (n (n on Cl I= 1 which contains Co (see Figure 5). Suppose zn and zn are determined by the curves In and In . We denote the corresponding curves in the disk ICJ < 1 by An and For every n, we choose on the curves A n' and A"n arcs with endpoints Cn' and Cn such that all their points are closer to the straight line passing through Cn and Cn than they are to the line passing through Sn - 1 and C _ 1 or to the straight line passing through Cn+1 and Cn+1 Let us connect by a straight line segment the ends of these arcs which lie in the disk Cl < 1. As we move from C, along An and from Cn along An to the first points of contact with this segment, we replace our original arcs with those just mentioned and we replace the straight line segment with the line segment joining the endpoints of these arcs, which lie in the disk ICI < 1. The cuts thus obtained in ICI < 1 are constricted, as n --, - , to the I

Cn+1

An.

42

11. CONFORMAL MAPPING OF SIMPLY CONNECTED DOMAINS

point Co and have as preimages in the domain B the cuts C,, determining the prime end corresponding to the point Co. This completes the proof of the theorem.

Figure 5

As a supplement to this, we have Theorem 3. Let E denote the set of boundary points of a domain B defined by the cuts CR as above. For E to be a prime end of the domain B, it is necessary and sufficient that E contain no more than one accessible boundary point of the domain B. Proof. It follows from Theorems 1 and 2 that, if E is a prime end, it cannot contain more than a single accessible boundary point. Suppose now that E contains no more than one accessible boundary point. If E were not a prime end, the intersection of the domains GR used to define E would contain an arc of the circle (1 = 1. On this arc, let us choose two points S1 and S2 corresponding to the admissible boundary points z1 and z2 of the domain B. Corresponding to the curves 11 and 12 defining these points are curves Al and A2 in the disk W < 1 that terminate at (1 and Portions of the curves Al and A2 in sufficiently small neighborhoods of the points S1 and S2 lie in C,,. This means that portions of the curves II and 12 in neighborhoods of the points z1 and z2 lie in B,,. This in turn means that the points z1 and z2 belong to all the BR and hence to the set E, which contradicts the hypothesis of the theorem. This completes the proof of the theorem. In Figures 3 and 4, the segment cd is a prime end of the domain in question C2.


43

since it is easy to construct a sequence of curves Cn such that the intersection of the corresponding domains Bn is just this segment. In Figure 4 the segment cd contains only the single accessible boundary point d; in Figure 3 it does not contain any accessible boundary point at all. Analogously the spiral-shaped domain in Figure 2 has the entire boundary circle as its prime end, which does not contain a single accessible boundary point. Special types of domain. Up to now, we have considered the question of the correspondence of boundaries of arbitrary bounded simply connected domains under univalent mappings. For domains of special types, we can draw a number of supplementary conclusions. Thus, it follows from the preceding theorems that, if all the boundary points of the domain B are accessible, then a univalent mapping of this domain onto the disk ICI < 1 defines a one-to-one correspondence between boundary points of the domain B and points of the circle ICI = 1 because, in such a case, every prime end of the domain B consists of a single boundary

point. Let us show that, in this case, the inverse function z = 0(C) is continuous in the closed disk ICI < 1 if we define this function at a point Co on the circle ICI = 1 as the limit of its values as ( approaches Co from the open disk JCI < 1. We need only show that 4(t) is continuous on ICI = 1. If a sequence {Sn1 of points Cn of the closed disk ICI < 1 that converges to Co, where ICol = 1, contains only finitely many points of the circle ICI = 1, then the relation limT(Cn)=p(Co) n-+oo

follows from the definition of ¢(40). On the other hand, if infinitely many of the %o lie on the circle ICI = 1, then these points themselves constitute a sequence of points Tnk -+ CO and we need to show that for them limp (Cnk) _ p*(Co)

Let c denote any positive number. Consider, in the disk IcI < 1, a sequence of points Cnk satisfying the conditions I tnk - Cnk I C T

,

I'f lànk) - 'P

IC

Since nk --. Co, there exists a K > 0 such that, for k> K, we have -q5(CO)I < c/2. It follows that, for k > K, (Snk) - (CO) I <E, and this means that limk_-'0(Cnk) the point Co.

This completes the proof of the continuity of -0(T) at


44

If the boundary of the domain B also has the property that, for any two distinct boundary points zI and z2, the complex numbers zI and z2, are not equal, then the function f(z) is continuous in the closed domain B. To show this, let zo denote a point on the boundary of the domain B. If zn E B, where n= 1, 2, ... , and zn _` z0 , then the corresponding sequence {cn } converges to a specific point Co on the circle ICI = 1 because, otherwise, the sequence ICn E would-have two subsequences that converge to distinct points on 141 = 1 and then the corresponding sequences of the points zn would converge to distinct points on the boundary of the domain B. It follows that Co is independent of the choice of sequence zn-a zU and this proves the continuity of f(z) on B.

If we apply what has been said to domains bounded by Jordan curves, we ob-

tain by virtue of the properties of such curves Theorem 4.1) A univalent con formal mapping of a domain B bounded by a

closed Jordan curve onto the disk ICI < 1 maps B bicontinuously onto the closed disk ICI < 1. Since the above considerations are of a local nature, we can conclude immediately that the following more general theorem is valid: Theorem 4 '. If a simply connected domain B has on its boundary a Jordan arc y, no interior point of which is a cluster point for the set of boundary points not belonging -to y, then the function C = f(z) mapping the domain B onto the disk ICI < 1 is continuous in B including the interior points of the arc y and it maps the arc y bicontinuously onto an arc y' of the circle ICI = 1. In the case of domains containing analytic arcs on the boundary, we have Theorem 5. If a simply connected domain B has on its boundary an analytic Jordan arc y, no interior point of which is a cluster point of the set of boundary points not belonging to y, then f(z) is regular at interior points of the arc y and at such points f 0(z) l 0; that is, the mapping is conformal. Proof. Suppose that the equation of the arc y is z = z (t) , where a < t < b.

Let to denote a point in the open interval (a, b). Let z0 denote z(to). The function z(t) can be expanded in a power series in a neighborhood of the point t = to. If we admit complex values for 1, this series defines a regular function in

1) Caratheodory [1913].


45

a sufficiently small neighborhood of the point to. Since z'(to) L 0 (by the definition of an analytic curve), we can choose a neighborhood sufficiently small that it will be mapped univalently by the function z = z(t). The part gt of this neighborhood lying in the upper or lower half of the t-plane is then mapped into the part g s of the neighborhood of the point zo lying in B and having a common boundary arc with y. The function C= f(z(t)) is regular in gt and it maps gt univalently into the part of the neighborhood of the point Co = f(zo) lying in the disk JCJ < 1 and adjacent to the circle JC1 = 1. By Theorem 4', f(z(t)) is then continuous in gt up to the real axis. By the symmetry principle, it can be extended in a neighborhood of to across the real axis and, in particular, it is regular at the point to itself and has a nonzero derivative at to. If t = t(z) is the inverse of the function z = z (t) , it is regular at zo and t'(zo) 0. Consequently, the function f(z(t(z))) = f(z) is also regular at zo and f'(zo) 31 0. This completes the proof of the theorem. In particular, if a domain B is bounded by a closed analytic Jordan curve, a 1 is regular in B and its inverse function f(z) that maps it onto the disk is regular in the closed disk 1cj < I. We conclude our investigation of special types of domain by noting that in

Chapter X, §1, we shall use the theory of functions of a real variable to investigate in addition the correspondence of boundaries under conformal mapping of domains bounded by rectifiable and smooth boundaries. l)

We also point out that the preceding theorems lead immediately to theorems on the correspondence of boundaries under univalent mapping of various domains onto each other. For example, a univalent mapping of one simply connected domain bounded by a closed Jordan curve onto another is single-valued and continuous, including their boundaries; in addition, if the boundaries of these domains are closed analytic Jordan curves, the functions representing the direct and inverse mappings are regular in the closed regions. In conclusion, we present another uniqueness theorem dealing with conformal mappings. We saw above that if all the boundary points of a bounded domain B Qre accessible, then a function z = '(C) that maps the open disk ICI < 1 1) Profound investigations of the correspondence of boundaries under conformal mapping have been made by the Moscow school of the theory of functions, which includes N.N. Luzin, I. I. Privalov, M. A. Lavrent'ev and others. A detailed survey of the corresponding results can be found in the book Thirty years of Mathematics in the USSR, pp. 324-332

(1948).

46


< 1. In this univalently onto the domain B is continuous on the closed disk case, the boundary Cof the domain B has the representation z = (eit) = z(t), for 0 < t < 2n, where z(t) is a continuous function, so that C is a continuous curve. Furthermore, since log[0(() - O(0)]/C is obviously a regular function in

Jt;J < 1 and is continuous on ItI'< 1, it follows that arg[O(i) - x(0)1/4 is a harmonic function in JCJ < 1 and is continuous on ICI < 1. This means that the increment in this argument as C moves around the circle JCJ = 1 is zero. But this means that, as C moves around the circle JCJ = 1 in the positive direction, the increment in arg[95(0 - 0(0)] is equal to the increment in argC, that is, equal to 227. We take as the positive direction around the curve C the direction in which arg(z - z0), where z0 = 0(0), acquires an increment of 27r when z completes one encirclement. By virtue of the continuity of the increment in dz (arg (z - zo))c =

z

)

zoJ

depending on z0 C B, the positive direction around C is determined independently of the choice of z0 and hence independently of the choice of mapping function 95 (0 -

Theorem 6. Suppose that all the boundary points of a simply connected bounded domain B are accessible. Then there is a unique univalent mapping of B onto the disk ICI < 1 such that three given points z1, z2, and z3 on the boundary C of the domain B indexed in order of their occurrence as one proceeds in the positive direction are mapped into three given points C1, CZ, and C3 on the circle JCJ = 1 similarly indexed. Proof. Suppose that the function (J' = O(z) maps the domain B univalently onto the disk KK'I < 1. Corresponding to the points z1, z2, and z3 on IC1 = 1 and t;3 arranged in order of occurrence as one proceeds are three points around the circle in the positive direction. If we now map the disk JC' I < 1 univalently onto the disk ICI < 1 in such a way that the points X1,.42, and C3 are mapped into 1, CZ, and C3 respectively (which we know is possible and is achieved by means of a fractional-linear function), then the composite function will yield the required mapping. The uniqueness of such a mapping is proved, as uniqueness is usually proved, by contradiction. This completes the proof of the theorem.

§4. DISTORTION THEOREMS

47

§4. Distortion theorems A considerable portion of the theory of conformal mappings deal with the investigation as to what restrictions the requirement that a mapping be univalent imposes on certain quantities connected with such a mapping (for example, the

modulus of a function or the modulus and argument of its derivative), that is, on the degree of distortion produced by that function at various points in a domain, etc. These restrictions are expressed in various inequalities, the simplest of -which will be given in the present section. Our starting point for the derivation of a number of inequalities dealing with univalent functions is Theorem 1 (the area theorem)'). Suppose that the function F(C) = C+ co, Q'1/C+ - = C+ Y-k=1 Rk/Ck is regular in the finite plane, has a pole at C= and is univalent in the domain ICI > 1. Then, ro

E

(1)

n=1

The name "area theorem" is due to the fact that this theorem is the analytic expression of the following geometric fact: under a mapping of a domain ICI > P, where p > 1, by a function w = F(C) , a part of the plane w = u + iv of positive area remains uncovered (the area principle). This geometrical interpretation indicates the course of the proof. Specifically, the circle ICI = p, where p > 1, is mapped by the function w = F(t;) = u + iv into the closed analytic curve w = F(pei°) = w(t) , for 0 < t < 277, which serves as the boundary of a domain of area S calculated from the formula 2x

S=

2n

w(t)+w(t) 2

uv'dt

tv

2is

2

rPeit + Pe-it

XL

2

+

2i

00

ane Int+anetnt

n=1

2p

rpest+Pe tt lj1 L

(t)TP1'(t)dt

00

n1 =

1) Gronwall [1914-19151.

Wane-int + naneint 29A

Co

9

'CP -

I

n=1

nlan12 pen

48


and this area is positive for arbitrary p > I. If we take the limit as p -i 1, we obtain (1). A similar theorem holds for the corresponding p-valent functions, that is, functions that assume no value in the domain ICI > 1 at more than p points.`)

An important consequence of inequality (1) is the fact that I all < 1, with equality holding that is, with al = et°' if and only if a2 = a3 = - = 0, that is, if F(C) = C+ e'1/C: This last function maps the domain ICI > 1 univalently onto the plane with a rectilinear cut from the point w = 2e"i "2 to the point w = -2e0,1'2. Suppose now that the function on

f (Z) = z + c2z2 + ...

,

Cnzn

(2)

n=1

is regular and univalent in the disk I zl < 1. Then, the function

fP(Z)=P f(ZP)=z+ Z'9" +...

(3)

(for p = 2, 3, ) is also regular and univalent in IzI < 1. The univalence follows from the fact that, if we had ff(z1) = ff(z2), where z1 and z2 are distinct num= f(zP) and hence zP = zP; that is, z1 = bers in I z I < 1, we would have f(zP) 1 I 2 2 iz2, where i]P = 1. But then, since the series (2) contains only terms with powers znp+1, where n = 0, 1, 2, , we would have ff(z1) _ iiff(z2); that is i] = 1, and hence, z1 = z2, which contradicts the assumption. It follows that the function

F(C)=

111 =C- 2 C + f° c l

is univalent in the domain ICI > 1. Then, as a consequence of the area theorem, we obtain the inequality Ice/21 _< 1, so that Ic21 _< 2 with equality holding, that is, with c2 = 2eia, if and only if F(C) = C+ e`°' /C, that is, if and only if

w =f (z) = (I + ealz)a

= Z - 2earz9 -I- ...

(4)

The function (4) maps the disk Izi < 1 onto the plane with a cut along the ray which issues from the point a-"`/4, and which would include the point w = 0 if produced

in the opposite direction. Thus we have obtained the result 2) 1) See Goluzin [1940] and Alenicyn [1947, 19501. 2) Bieberbach [1916].

49

N. DISTORTION THEOREMS

If the function (2) is regular and univalent in the disk izi < 1, then Ic2I < 2. This is a sharp inequality. With this bound for the coefficient c2, we can now easily obtain a covering theorem for the same functions. Suppose that the function (2) is regular and univalent in the disk I zl < 1 and suppose that c does not belong to the image of that disk under the function (2). Then c # 0 and the function

f1(Z)=Ceff (ZL-=z+C9+ 1 Z9+... is also regular and univalent in Izi < I. Hence

Thus, Icl > % with equality holding only when Ice + 1/cl = I1/c I - Ic2I = 2 and 1C21 = 2, that is, C2 = -2eò'. However, this will be the case only for the function (4). Thus, we have obtained Koebe's covering theorem: Theorem 2. If the function (2) is regular and univalent in the disk zI < 1, then the disk Iwl < % (but not always a larger disk) is entirely covered by the image of the disk IzI < 1 under that function. The circle Iwi = % contains points not belonging to the image if and only if f(z) is the function (4).1) Koebe's theorem can be put in a different form: If a function f(z) such that f(0) = 0 is regular and univalent in I zI < 1 and does not assume the value c in Izi < 1, then I

(0) l

4c,

This follows from the application of the preceding theorem to the function

f(z)/f'(0),

which does not assume the value c/f'(0) in Izi < 1. If the, function (5)

maps the domain ICJ > 1 univalently, if c does not belong to its image, then the function

1) In this formulation, the theorem was proved by Bieberbach [1916]. Koebe[1907] proved only the existence of such a disk as that asserted in the theorem. This covering also holds for suitable p-valent functions (see Goluzin [19481).

50


Ft (z)=

=z+(c-ao)zs+...

F(i)-c 11

is regular and univalent in IzI < 1 and, consequently, Ic - a0I < 2. From this we obtain Theorem 3. If the function (5) maps the domain ICI > 1 univalently, then the

entire boundary of its image is contained in the disk Iw - aoI < 2. Another important application of a bound on the coefficient c2 is the deriva-

tion of bounds on the modulus and argument of the derivative of a univalent function.

Suppose that the function (2) is regular and univalent in IzI < 1. Then, for arbitrary z in IzI < 1, the function I(+ b (C) = f f(f'I(z)

(1C)

_f

Z)

I I)

= C -}- r2' -}- .. .

is also regular and univalent in ICI < 1. Calculation shows that

s=2 ((1-z2)f,(())-22). z Consequently, since I02I < 2; we have zf' (z) f, (z)

2rs

1-rs

c 14r-rs'

r= I z

(6)

If we replace the modulus with the real and imaginary parts in this inequality and note that

at (f' ((z)) = r d_ log I.f' ( z )

f (z))) = r Fr argf' (z),

we obtain the two inequalities

4±rs,

14-r'rdr 4 1

a

rs Car

(7)

4

arg f' (z)

rs .

If we integrate with respect to r from 0 to r, we obtain the inequalities (11+

),
0, that belongs to all the domains in Bn, we define the kernel of this sequence of domains as the largest domain containing z = 0 such that an arbitrary closed subset of it belongs to all the domains Bn from some n on. By "largest domain" is meant the domain containing any other domain possessing this property. If such a disk does not exist, the kernel of the sequence of domains B1, B2,... is defined to be the pointz = 0. We shall say that the sequence of domains B1, B2 , ... converges to the kernel B, and we shall denote this by writing Bn --.. B, if every subsequence of these domains has B as its kernel. In particular, if a sequence of simply connected domains B1, B29 ..., Bn, ... that include z = 0 converges to the limiting domain B (also including z = 0) in the sense that all boundary points of the domains Bn from some n on are arbitrarily close to the boundary of the domain B and all points of the boundary of the domain B are arbitrarily close to the boundaries of the domains B,,, then this sequence has the domain B as its kernel and it converges to that kernel. This applies also for a sequence of domains Bn that include z = 0 and satisfy the condition B1 C B2 C B3 C ... or for a sequence of domains Bn that

§5. CONVERGENCE THEOREMS

55

contain a neighborhood of the point z = 0 and satisfy the condition B1 3 B2

B33

.

Theorem 1 (Carathi6odory).1) Suppose that we have a sequence of functions fn(C), where n = 1, 2, . , that are regular in the disk ICI < 1. Suppose that z fn(0) = 0 and fn '(0) > 0 for n = 1, 2, .. . Suppose that, for each n, the function fn(C) maps the disk CI < 1 onto a domain Bn.

For the sequence ;fn(C)} to converge in ICI < 1 to a finite function, it is necessary and sufficient that the sequence }Bn} converge to the kernel B, which is either the point z = 0 or a domain having more than one boundary point. When convergence exists, it is uniform inside the disk ICI < 1. If the limit function f(C) i const, it maps the disk ICI < 1 onto the kernel B, and the sequence ton(z)} of inverse functions On(z) converges uniformly inside B to the function d7(z) inverse to f(C). in ICI < 1. Let us consider Proof. 1°. Necessity. Suppose that fn(S) -+ first the case f(C) - 0. We shall show that then B is the point z = 0 and that B

n

.-. B. Let us suppose that this is not the case. Then, there exists a disk

'I zi < p, p > 0, that belongs to all the domains Bn. Consequently, by the Schwarz lemma as applied to the disk IzI < p, we obtain I0, (0)l < 1/p, where 0n(z) is the inverse of the function fn(C). This means that fn (0) > p. But then, according to inequality (10) of §4 as applied to the function fn(S)/fn(0) in the disk ICI < 1, we have p ICI/(1 + IC!)z < I fn(S)I. Consequently, the sequence of the functions fn(C) does not converge to f(C) - 0, which contradicts the assumption. Since the same reasoning can be applied to every subsequence of domains among the B1, B2, ... , we have B n _, B. Now, let us suppose that f (C) J 0. Then, it follows from the inequality

fn(0)

IIICD'CIfn(C)ICfn(0) (1

(1

III

I)y

that the numbers fn(0) are bounded and that the functions fn(C) are uniformly

1) Caratheodory [1912]; Bieberbach [1913]. Markusevic [1936] showed that the conditions of 'Theorem 1 are also necessary and sufficient for convergence in mean of }fn(z)} to

f (z), that is, necessary and sufficient for 11m

n - ,:.

S

I fn (z) -f'(z) I' da = 0,

where fn(z) is taken equal to 0 outside the domain 8

.

56


bounded inside the disk Kl < 1. Therefore, in accordance with Vitali's theorem, the sequence of the functions fn(4) converges uniformly inside the disk ICI < I to the function f(C), which is univalent in'KI < 1. Consequently, the function z = f(C) maps the disk ICI < 1 univalently onto a domain B containing z = 0. Let us show that B possesses the property that an arbitrary closed domain B" is contained in all the domains Bn from some n on. Let S > 0 denote the distance from B" to the boundary of B. Let us cover the z-plane with a grid of squares of side 8/2 and let us consider the domain B, constituting the union of all (closed) squares containing points of B". Then, B" C B, and B, C B. Let A" and A" denote respectively the preimages of the domains B" and B, under the mapping z = in ICI < 1. Then, A" C A,, and A, is contained in the disk ICI < 1. Let S1 > 0 denote the distance from B" to the boundary of the domain B". Then, on the boundary of the domain A,, we have I f(C) - zol > S1 for any point zo C B". On the other hand, there exists an N > 0 such that, for n > N, we have l fr(y) - f(C)l < S1 on the boundary of A, since the sequence of the functions fn(C) converges uniformly inside the disk lCl < 1. Consequently, for every n > N, Rouche's theorem can be applied to the function

f, (Q - z0 = V n

(W + VP - Zn).

In accordance with this theorem, fn(C) - z0 has a zero in the domain A,. This means that the image of the domain A, under the mapping z = f(C), where n > N, contains an arbitrary point z0 C B". Thus, every closed domain contained in B and hence every closed subset of B is contained in all the domains Bn from some n on. Let us show that B is the largest domain possessing this property. Let B' denote any domain other than B that contains z = 0 and possesses this property. Then, from some n on, each function 0n(z) is bounded on an arbitrary closed subset of the domain B' and the set of such functions is uniformly bounded. It follows that the condensation principle can be applied to the sequence lon(z) 1. Let us choose from the sequence 10n(z)l an arbitrary subsequence lonk(z)1 that converges uniformly inside the domain B' to a regular function O(z)

such that 0(0) = 0 and 0'(0) > 0. Since 1

cp' (0) =urn p, (0) =1im n-m n-.mfn (0)

_

1

f (0)

it follows that 0'(0) > 0, that is, that OW is nonconstant and, consequently,


57

univalent in B', Let us show that c(z) is the inverse of the function f(C). Let Co denote an arbitrary point in the disk IQ < 1. Let Co (C B) denote the point f(C0). Let t denote a positive number such that the circle IC- CoI = e is con1. Let m denote a positive number such that 1g) - z0I > m tained in the disk on that circle. On the other hand, for k > K, we have I fnk(S) - f(01 < in on the circle K - coI = c. We again conclude from Rouche's theorem that the functions fn zo for k > K each have a zero, which we denote by Ck, in the disk '?- CoI <e. We have fnk(Ck) = zo, and C. = 0n (z0) for k > K. As k -- oc, the Ck -+ O(z0), so that O(z0) belongs to the disk JC- CoI <e. Since a is an arbitrary positive number, we see that O(z0) = Co. Thus, we have shown that z0 =

f(r;o) implies Co = /(z0); that is, O(z) is the inverse of the function f(c). Since this line of reasoning can be applied to any convergent subsequence of the sequence {on(z)} and the limit function is always the function O(z) inverse to

f(r;), the sequence {0n(z)I itself converges in the domain B' to the function 0 W. The function J = 0 (z) maps the domain B ' onto a domain A. Since ICbn(z)I < 1 in Bn, it follows that JO(z)1 < 1 in B '; that is, A is contained in the disk ICI < 1. This last fact enables us to show that B' C B. If z0 C B' and SD = o(z0), then ICoI < 1 and, consequently, z0 = f(Co) C B; that is, every point in the domain B' belongs to the domain B also. Thus, B is the largest domain containing z = 0 that possesses the property that any closed subset of it belongs to all domains Bn from some n on; that is, B is the kernel of the sequence of domains B

n

.

It follows that B has more than one boundary point.

Let us now take an arbitrary subsequence Bni, BnZ, .. , of the domains Bn. If we apply the conclusion just obtained to the corresponding subsequence fn1(0, fn2(J), of the functions fn(S), which converges to the function f(4) , we see that the function z = f(C) also maps the disk 1 onto the kernel of the subsequence JB }, and therefore this kernel coincides with B. This shows that --* B, so that we have proved the necessity of the condition of the theorem and have also proved the supplementary conclusions mentioned in the theorem. B

n

2°. Sufficiency. Now, suppose that Bn _., B and suppose that the kernel B either consists of the single point z = 0 or is a domain having more than one boundary point.

sy

First, let us suppose B consists only of the point z = 0. If f'n (0) did not converge to 0, there would be a subsequence ifnk(S)l such that, for each k,

58


f ' (0)I > E, e > 0, and

>(1+IC ),

throughout. the disk I I < 1. Consequently, the kernel B would not consist of a

single point. But if f'(0) -. 0, then, for all ( in the disk ICI < 1,

Ifn(t)ICIfn(0)I(l.l Consequently, fn(C) -+ 0 in ICI < 1. This proves the convergence of the sequence }fn(()}.

Now, let us look at the case when the kernel B is a domain having more than one boundary point. In this case, the numbers Q0) are bounded, because otherwise we would have a subsequence {fnk(0)} that approaches oo and we would conclude from the inequality

'

Iflk()I:Ifnk(0)I(1+Icl

I{I)I.

that the images of the disk 141 < 1 under the mappings z = fnk(c), from some k on, contain an arbitrary given disk I zI < R, which would contradict the convergence of 1B.I to B. It follows from what we have proved that the functions fn(0 are uniformly bounded inside the disk ICI < 1. Let us suppose now that the sequence if,,(C) I does not converge at the point Co such that ItOI < 1. Then, there

exist two subsequences }fn,(01 and If ,,(0} that converge in ICI < 1 to two distinct functions f+(C) and f«+(C). If neither f+(0 nor f+«(0 is identically equal to zero, then, from the proof of necessity as applied to the sequences and }f ,,(()}, we obtain the result that the functions z = f+(0 and z = map the disk ICI < 1 onto the kernel of the sequences }B.,I and {BR,, }, that is, onto the domain B. Note that we have f+(0) = 0, f; (0) > 0; f++(0) = 0, f;+(0) > 0. Therefore, by virtue of the uniqueness asserted in Riemann's theorem, we have f+(C) and this contradicts our assumption. On the other hand, if one of the functions f. (C), is identically equal to zero, the other cannot be identically equal to zero or they would again be identically equal to each other. By applying the necessity of the condition of the theorem, we conclude that the kernel of one of the sequences {B,, }, }B,,,,}


59

consists of the point z = 0 and that the kernel of the other sequence is some domain, so that the sequence {Br) does not converge to a kernel. Again, we have reached a contradiction. Thus, the sequence {fY)I must converge in the disk ICI < 1 to a finite function. Furthermore, in accordance with Vitali's theorem, the convergence is uniform inside the disk CI < I. This completes the proof of the theorem. Theorem 1 gives the conditions for convergence of univalent functions only in.the open disk ICI < 1. For the convergence of univalent functions in the closed disk I CI < 1, we give the following theorem, confining ourselves to domains of the Jordan type. Theorem 2.1) Let {Bn } n= 1, 2, ... , denote a sequence of simply connected domains each including the point z = 0 and each bounded by a Jordan curve. Denote the boundary of Bn by Cn. Suppose that the sequence 1B n' converges to a domain B (its kernel) bounded by a Jordan curve C. Let Ifn W I denote a sequence of functions fn(C) such that, for each n, fn(0) = 0, f'(0) > 0, and fn(O maps the open disk C1 < 1 onto the domain Bn. For the sequence 1fn(C)} to con1 to a function z = f(0 that vanishes at verge uniformly on the closed disk 0, has positive first derivative at 0, and maps the open disk ICI < 1 onto the do-

main B, it is necessary and sufficient that for every t > 0 there exists a number N > 0 such that, for n > N, there exists a continuous one-to-one correspondence between the points of the curves Cn and C such that the distance between any point of Cn and-the corresponding point of C will be less than E. f(C) uniformly on ICI < 1, then, by conProof of the necessity. If fn(C) sidering the points zn = fn(C) and z = f(C) as corresponding points on the curves C and C for each value of C on the circle ICI = 1, this establishes a one-toone continuous correspondence between Cn and C with the property stated in the theorem.

Proof of the sufficiency. We first prove that under the condition of the theorem, the set of functions fn(C) is equicontinuous on the circle ICI = 1. Let us suppose the opposite. Then, there exists a S > 0, a sequence of positive integers nk, k = 1 , 2, ... , increasing without bound as k increases, and two sequences of points on the circle 1 such that when k and

1) Rado [1922-1923]. In the case of domains with arbitrary boundaries, the question has been thoroughly investigated by Marlcusevic [1936].

H. CONFORMAL MAPPING OF SIMPLY CONNECTED DOMAINS

60

C'

k

r 0, and I fnk (Ck) _-fnk (Ck) I > 8.

We may assume that the sequences Irk I and I converge to a common limit a such thaattI al = 1 because, if this is not the case, we can shift to a subsequence Ifnk(( )I of functions satisfying this condition and keep the same notation for it. Analogously, we may assume that the sequences Iak I and 16k }, where

ak = fnk(Sk) and bk = fnk(converge, let us say, to points a and b. The

points ak and bk lie on Cnk and the points a and b lie on C. On each curve Cnk, we denote by ak the arc akbk corresponding to the arc Ckk , (which is constricted to the point a as k -+ o) and we denote by rk the remainder of the curve C. Since we are assuming that, for sufficiently large k, the distance between ak and bk exceeds S, there exists for such k a point ck on ak at a distance no less than 8/2 from ak and bk. We may assume that ck --+ C, where c E C because otherwise, we could shift to a subsequence Ifnk(C)M We have I c - al > 6/2 and Ic - bI > 8/2. Consequently, c is an interior point of one of the arcs into which the points a and b partition the curve C. We denote this arc by a and the remainder of the curve C by r. Let a' denote an arc on C that includes c and that is contained, together with its endpoints, inside the arc a. Let 77 denote a positive number that is less than the

distance from c to C - a' and less than the distance from a' to r. Let K denote a positive number such that, for k > K, there exists a continuous one-toone correspondence between Cnk and C such that the distance between corresponding points of Cnk and C is less than 77/2 and

Ia-ahI K, this arc lies outside Bnk and z0 lies in the interior of Bnk. The limiting values of tk(z)l as z approaches boundary points of the domain Bnk that lie in the disk i z - z0l < rl/4 do not exceed Ek since there are no points of the arc rk in the disk l z - z0l < q/4. Lemma 2 of §3 is now applicable. In the present case, M = 2 and r = Ek. Therefore, for k > K1, we have l

l

7k z0) l C m Ek`Zm-1.

But as k -+ o, the sequence fik(zO) converges to O(z0) - a, where O(z) is a function mapping the domain B onto the disk l zl < 1. Consequently, q(z0) - a = 0; that is, ¢(z) = a for all points z C B in the disk l z - cl < 77/2 and hence everywhere in B, which is impossible. This contradiction proves the equicontinuity of 1. The functions fn(C) are also uniformly the functions fn(c) on the circle bounded on ICI = 1. Reasoning as in subsections 2° and 3° of the proof of the :condensation principle, let us show that the sequence (fn(0 l contains a subseltence that converges on some set that is everywhere dense on the circle 1. Such a And then let us prove that this sequence converges uniformly on oice can be made from an arbitrary subsequence of the sequence (fn(C){. This Obables us to show that the sequence Ifn(O} itself converges on Kl = 1. Specifically, if the sequence (fn(C)I did not converge at some point on !, = 1, we could choose two subsequences of it that converge uniformly on the !8 rcle 1 and consequently in the disk Kl < 1 to two distinct functions that Ste regular in the open disk ICI < 1 and continuous on the closed disk Out this is impossible because, by Theorem 1, the sequence ffn(o I converges in `!


62

ICI < 1 to a unique function. Having proved the convergence of the sequence (fn(01 on the circle ICI = 1, we again keep in mind the equicontinuiry and uniform boundedness of the functions fn(s) to show, just as in subsection 3 of §2 of Chap. ter I, that the sequence (fn(0j converges uniformly on the circle ICI = 1 and hence on the closed disk ICI < 1. Obviously, the limit function is f(C). This completes the proof of the theorem. §6. Modular and automorphic functions

Among the different functions that map simply connected domains onto each other, of great theoretical importance are the functions that map certain circular polygons, that is, domains whose boundaries are arcs of circles, either onto a disk or onto a halfplane. In the present section, we shall study the nature of such mappings.

Let us consider first the case in which the domain B is the interior of a regular circular triangle with zero angles inscribed in the circle Izl = 1 (see Figure 6). The sides of this triangle are obviously orthogonal to the circle I zl = 1. Suppose that a function z = halfplane NO > 0 univalently onto the domain B in such a way that the points 0, 1, and oa are mapped into the

vertices of the triangle. Let C= µ(z) denote the inverse of that function. Without knowingly the explicit expression for the functions A(C) and EL(z), we can, by using the principle of symmetry, ascertain the essential properties of these functions in a purely geometrical manner. We denote the sides of the triangle by L1, L2, and L3

Figure 6

and the segments of the real axis corresponding to them by L, L2, and L' . The function ft (z) , which is regular in B and continuous on B except at one of the vertices, assumes real values on the boundary of the domain B. Consequently, in accordance with the symmetry principle, it can be extended past the sides L1, L2, and L3 into the domains (which we name B1, B2, and B3 respectively) obtained from B by an inversion about the circles of which L1, L2, an d L3 are arcs. The domains B1, B2, and B3 are also circular triangles with zero angles inscribed in Izl = 1. Each of them is adjacent to B along one of the sides

1) An explicit expression for the function k( t) will be given in § 1 of Chapter III.

§6. MODULAR AND AUTOMORPHIC FUNCTIONS

63

L1, L2, L3. This is true because under these inversions, an arbitrary circle is mapped into a circle and the circle I zl = 1, being orthogonal to all the circles across which the extension is made, is mapped into itself. In the domains B 1, B2, and B3, the function 4=µ(z) is regular and it maps each of these domains onto the halfplane (4) 0 along segments L11, LZ and L' respectively, we obtain a Riemann surface onto which the function C+ Et (z) maps the domain B ' bijectively. On the boundary of B ', the function C= t (z) is real. The domain B' is a circular hexagon with zero angles inscribed in the circle JzJ = 1. The sides of this hexagon are orthogonal to that circle. Let us repeat this procedure with the domain B'; that is, let us extend the function Et(z) past the six sides of B' into the domains obtained from the domains B1, B2 and B3 by inversions about the corresponding sides. The function µ(z) is regular in each of the six newly obtained domains and it maps them onto the halfplanes ` (c) > 0.Let us think of these halfplanes as being distinct and joined along suitable segments L 1, L 2' and L 3 of the real axis with the halfplanes NJ (C) < 0 that we already have. Consequently, the function C= Et (z) is regular in the newly obtained domain B ", which is a circular duodecagon. It maps B " objectively onto the Riemann surface consisting of seven upper and three lower planes connected in a suitable manner along L1, L2' and L. If we continue this extension process indefinitely, we get, on the one hand, a "modular" grid in the z-plane composed of an infinite set of circular triangles with zero angles all contained in the disk I zl < 1 and having sides orthogonal to the circle Izl = 1 and, on the other hand, a Riemann surface lying over the c-plane and consisting of an infinite set of halfplanes connected with each other along

suitable segments Li, LZ and L3 of the real axis. Let us show that the domain B' consisting of the union of all triangles constituting the modular grid coincides with the entire disk I zl < 1. To do this, it will be sufficient to show that the set of vertices of the triangle of the grid is everywhere

dense on the circle Izl = 1. Let us suppose the opposite, namely, that this circle contains an arc y that does not include any vertex of any of the triangles constituting the grid. Then there exists a sequence i l n j , n = 1, 2, ... , of arcs of the

64

H. CONFORMAL MAPPING OF SIMPLY CONNECTED DOMAINS

grid, each arc resting on a y-containing arc of the circle I zl = 1, that converges to an arc L of that circle. Obviously, the arc L is orthogonal to the circle I zl = 1 and it rests on an arc y' containing y. Then, there are no points of the domain B* between y' and L. But this is impossible because, for sufficiently large n, the inversion about the arc In of the entire polygonal domain Bck> whose boundary contains this arc maps the point z = 0 into a point between y 'and L, and hence there are points of B' between y' and L. This shows that the set of vertices of the modular grid is everywhere dense on the circle Izl = 1 and hence that the domain B* coincides with the entire disk I zl < 1. Therefore, the function µ(z) is regular in the disk Izl < 1 and it maps that disk bijectively and conformally onto the infinite-sheeted Riemann surface mentioned above. The function µ(z) assumes every value other than 0, 1, and oe infinitely many times in the disk Izl < 1. On the other hand, it does not assume these three values anywhere in that disk. With regard to the points of the circle I zl = 1, these are all singular points of the function µ(z) because an arbitrary neighborhood of each such point contains infinitely many triangles of the modular grid and hence µ(z) assumes in the subset of this neighborhood contained in I zl < 1 all values other than 0, 1, and °°. Consequently, µ (z) cannot be extended outside the disk I zl < 1 and this disk constitutes its entire domain of definition. In this case, we say that the circle Izl = I is the natural boundary for µ(z). The inverse function A((), defined by any element, can be extended to the (-plane along an arbitrary path that does not pass through 0, 1, or oo. Here, IA(t) < 1 everywhere, though A(C) is obviously a multiple-valued function. The function µ(z) is called a modular function and the Riemann surface onto which it maps the disk I zl < 1 bijectively is called a modular Riemann surface. This name is due to the connection between µ(z) and certain quantities encountered in the theory of elliptic functions and called moduli. The modular function µ(z) possesses the remarkable property of invariance under a certain group of fractional-linear transformations of the argument. Speciffically, fractional-linear transformations composed of an even number of inversions of the type encountered above constitute a group of transformations of the argument under which µ(z) does not change its values. Single-valued analytic functions that remain unchanged under a certain group of fractional-linear transformations of the argument are called automorphic functions, Examples of automorphic functions with infinite groups are trigonometric and elliptic functions. Next to elliptic functions, a modular function is the simplest nonelementary example of an

§6. MODULAR AND AUTOMORPHIC FUNCTIONS

65

automorphic function with an infinite group. We obtain other very simple automorphic functions if we consider a mapping onto the halfplan e ` (C) > 0 of a domain B contained in the disk Izl < 1, where B is an arbitrary circular triangle all sides of which are orthogonal to the circle I zI = 1 and all interior angles of which are nonzero. If the interior angles of this triangle have values it/ml, rr/m2, an d

rr/m3, where ml, m2, and m3 are positive integers, then, just as in the case of modular functions, the function C _ 0 (z) , which maps the domain B univalently onto the halfplane ` (C) > 0, can be extended to the entire disk I zl < 1. This extension is regular in I zl < 1 except at points that are mapped into - and are poles of 0(z), and it maps the disk IzJ < 1 onto a many-sheeted Riemann surface lying over the entire Cplane. To prove the extendability of 0(z) to the entire disk I z I < 1, let us prove first that the vertices of the triangular grid, all of which now lie in the disk I zI < 1, have

no cluster points inside the disk Iz) < 1 (see Figure 7). Let us suppose, to the contrary, that zo is a cluster point of the set of vertices of the grid and let {zk i,

Figure 7

for k = 1, 2, , denote a sequence of distinct vertices of the grid that converges to z0. Since these points zk are vertices of triangles Ak, for k = 1, 2, .. , obtained from B by an even number of inversions, that is, by certain fractional-linear transformations z ' = Tk(z) , where k = 1 , 2, , of the disk IzI < 1 into itself included in the group of transformations that leave the function S6(z) invariant, it will be sufficient to show that the functions Tk(z) map the vertices of the domain B into a set of points that has no cluster points in I zI < 1. We may assume that the zk are such that the triangular grids with vertices at the points zk for different values of k have no points in common, since otherwise we could shift to a subsequence of the sequence {zk { for which this would be the

66

If. CONFORMAL MAPPING OF SIMPLY CONNECTED DOMAINS

case. Let a denote one of the vertices of the domain B and suppose that the disk I z - al < p, p > 0, lies entirely in a domain composed of triangles of the grid with vertices at a. The functions z ' = Tn(z), n = 1, 2, . . , map this disk onto disjoint disks Kn contained in I zI < 1. Since the functions 7ID- T. (a+pz)-Ta (a) =.Z-+- .. . pTn (a)

are univalent in Izl < 1, the disk Iwl < % lies entirely in domains onto which these functions map the disk I zl < 1. This means that, for each n, the disks of radius not less than I Tn(a)/4I p lie in Kn. Since these disks are disjoint and are contained in I z'I < 1, the sum of their areas is finite and, consequently, the series 10=1I Tn (a)I 2 converges. Then Tn(a) -. 0 as n -. 00. If a = T (a), the n n function z' = T. (z) is obviously obtained from the equation

Z' -an èten Z-a 1 - anz'

1 - az '

where a n is a real constant. Therefore Tn (a)

eian

1

so that 1 and --. 1 as n --+ oo. This proves that the sequence lan I has no cluster points in I zl < 1. The situation is analogous with the remaining vertices of the domain B. Since each of the points zk is the image of a vertex of the triangle B under one of the functions z' = Tn(z), it follows that the sequence lzk l also has no cluster points in

I z I < 1.

Let us show now that any point z in the disk I z( < 1 belongs to one of the triangles of the grid or lies on its boundary. Let us suppose the opposite, that is, that there exists a point z in the disk Iz1I < 1 that does not belong to any triangle of the grid. Let us draw a line segment from zi to some point of the domain B. We conclude that this segment contains points of an infinite set of triangles of the grid since it would otherwise be possible to get from B to zI by means of a finite number of inversions and the point zi would belong to some triangle of the grid. From this infinite set of triangles, we choose a sequence of triangles such that the sequences of their vertices converge to limits a, /3 and y, which must necessarily lie on the circle I zl = 1. If all three points a, 8, and y are distinct, the sequence of the triangles themselves must converge to a circular triangle with vertices a, 0, and y and with sides orthogonal to I zl = 1, that is, to a triangle 1

§7. NORMAL FAMILIES OF ANALYTIC FUNCTIONS

67

with zero angles, which is impossible. On the other hand, if only two of the points a, 0, and y are distinct, the sequence of these triangles converges to some arc of a circle orthogonal to the circle I z+ = 1, which again is impossible. Finally, if a, 0, and y coincide, the triangles must be constricted to a point. But this too is impossible since all these triangles contain points of the segment joining z 1 with some point of the domain B. This contradiction shows that the triangular grid entirely covers the disk I zl < 1. §7. Normal families of analytic functions. Applications The results obtained above on conformal mapping owe their generality in many ways to the principle of condensation of analytic functions. But these results in turn lead to considerable further development of questions on the convergence of analytic functions. To show this, we introduce the concept, proposed by Montel, of a normal family of analytic functions (cf. Montel [1936]). An infinite family Yk = If (z)f of functions that are regular in a domain B is said to be normal in B if every sequence of functions belonging to that family has a subsequence that converges uniformly inside B to a regular function or to m. A family is said to be normal at a point z0 E B if it is normal in some neighborhood of z0. If a family 911 = jf(z)} is normal in a domain B, it is obviously normal at each point z0 in B. Let us prove the opposite, namely that if a family 9R = If(z)1 of functions that are regular in B is normal at each point z0 E B, it is normal in the domain B. We may assume that B does not include DC. Let us take the set of all points of the domain B with rational coordinates and arrange it in a sequence Irk j, k= 1, 2, . , and let us denote by Pk the radius of the largest disk with center at rk in which the family 911 is normal. One can easily see that the subsequence {p,, k } can converge to 0 only when the corresponding points cluster r"k only on the boundary of B. Now, let (ff(z)I, n = 1, 2, - , denote an arbitrary sequence of functions of the family R. Since 911 is normal in the disk Iz - r11 < p1, the sequence jfn(z)j has a subsequence {fR 1(z)j that converges uniformly in Iz - r1I < p1/2 to a regular function or to oo. Furthermore, the sequence (f 1(z)} in turn has a subsequence jf2(z) j that converges uniformly in I z - r21 < p2/2 to a regular function or to w. If we continue the process indefinitely, the diagonal sequence (fR R(z)1, n = 1, 2, , converges in the entire domain B to a regular function or to - and the convergence is uniform in each of the disks Iz - rkl
0 onto a regular circular triangle with zero angles inscribed in the circle I zl = 1. The composite functions )f(z) = X (f(z)) , where f(z) E 9, can be extended to B along an arbitrary path since the only singular points of the function X(0 are 0, 1, and ac and f(z) does not assume any of these values in B. Consequently, the functions of are regular in B and they satisfy in B the inequality I0/z)1 < 1. They do depend, however, on the choice of branch of the multiple-valued function X(C). Therefore, let us normalize Of (z) in such a way that the point Of (ZO) will, for given z0 E B and an arbitrary function f(z) E fit, lie in the basic (regular) circular

1) Mantel [19271.


69

triangle or in one of the adjacent triangles. We apply to the functions 0 f(z) the principle of condensation of analytic functions. Now consider an arbitrary sequence [fn(z)j of functions fn(z) E R. Let us look first at the case when the set of numbers fn(zo) has a cluster point a other

than 0, -1, or e. Let jfn,(z)I denote a subsequence of the functions fn(z) such that fn, (zo) --, a as n' -. -. The sequence (difn (z)), where 0fn, (z) = X(fn,)) has a subsequence {(kfnk (z)) that converges uniformly inside B to a regular function O(z).

We have I(k(z)I < 1 in B. Here, equality cannot hold at any point of the domain B because we would then have O(z) = const = e" Y and, in particular, O(z0) = e`y, that is, I.0 fn (z0)l ---+ 1, which in turn can be the case only if [fnk(z0)j converges k

to 0 or I or 00. Thus, I¢i(z)I < 1 in B. Now, let E denote an arbitrary closed subset of B. On this set, JO(z)1 K, we have ,cf (z) - O(z)1 0 such that the image of the disk I zj < 1 under an arbitrary function of the family completely covers the disk JwI < p. Proof. Suppose that no p > 0 with the property described in the theorem exists. Then there must exist a sequence of numbers an that approaches 0 as , of the family in ques-. and a sequence of functions fn(z), for n = 1, 2, n .lion such that, for every n, fn(z) 3L an. The functions on(z) = fn(z)/an, for n = 1 , 2, ... , constitute in i zj < 1 a family of functions that do not assume the value 1 at all and do not assume the value zero at more than p points of the disk Izl < I. Consequently, in accordance with the generalized normality test, this family is normal in I zI < 1. Since n(0) = 0, it is also uniformly bounded in the interior of the disk Izj < 1. Let {onk(z)} denote a subsequence that converges uniformly in ' (0) _ the interior of the disk jzi < I to a regular function (k(z). Then, 9 nk 1/ank which contradicts the fact that ak -. 0. This completes the proof of the theorem. Theorem 7.2) For arbitrary c in the open interval (0, 21r), there exists a pF > 0 such that the image of the disk I zj < 1 under any function w = f(z) = z + that is regular in j zl < 1 covers some circular sector of radius pE with c2z2 + center at z = 0 and with central angle 2rr - e. .

1) Fekete [1925]. For p = 1, the theorem was proved by Caratheodory with an indication of the best value for P. For more on this, cf. end of §1, Chapter III. 2) Valiron [1927]. Valiron proved only the existence of p . Bermant and Lavrent' ev [1935] found by a different procedure the best values for pE depending on E. This value is given in terms of an extremum of an expression consisting of a function inverse to the modular function.

74


Proof. Obviously, we may assume that c < n. Suppose that no pE > 0 with the property described in the theorem exists. Then there exist number sequences {an I and ((3n ( that approach zero as n -+ oo such that the difference On = argon - arg an lies, for each n, between c and it and a sequence of functions fn(z) = z + . , for n = 1, 2, ... , that are regular in I zI < 1 and such that, for every n = 1 , 2, .. and every z in I z+ < 1, we have fn(z) L an and fn(z) An . if On < n/2, then, by dropping a perpendicular from the point an to the line segment connecting the points 0 and fan, we see that lien - and is greater than the length of that perpendicular, which is equal to I an' sin (S and therefore not less than Ianl sin a. Consequently, we have INn - ani > Ianl sin c; that is, Ian/( fan - an)I < 1/sin E. On the other hand, if On > n/2, it follows from this same construction that Ion - an' > Ianl. Consequently, we again have an/((3n - an)I < 1 < 1/slot. Let us now consider the functions on(z) _ [f(z) - an] /(on - an), where n = 1, 2,.... These functions do not assume the values 0 and 1 in the disk IzI < 1. Consequently, they constitute a normal family. Furthermore, the numbers On(o) _ -an/((n - an) are bounded. This means that the family is uniformly bounded in the interior of the disk IzI < 1. Let 10nk(z) denote a subsequence that converges uniformly in the interior of the disk I zI < 1 to a regular function O(z). Since On k(0) --. 0'(0), it follows that 1,(nk ark) q'(0), which is impossible since ark and (ink approach zero. This completes the proof of the theorem. Theorem 8.1) Let K denote an annulus R1 < IwI < R2 and let p denote a positive number. Then, there exists a positive number p such that the image of the disk IzI < 1 under each of the functions w = f (z) = z + c2z2 + that is regular in IzI < 1 either covers the disk IwI < p or assumes every value w in the annulus K at no fewer than p points of the disk IzI < 1. Proof. Let us suppose that the theorem is not true. Then, there exist sequences IanI and tonI such that an --, 0 as n - and INnI belongs, for each n, to the interval (R1, R2) and a sequence ifn(z)l of functions fn( z) = z + - . that are regular in I zI < 1 such that, for every z in IzI < 1, we have fn(z) # an for every n and fn(z) = An for no more than p - 1 values of z. But the functions

On(z) = [fn(z) - a]/((3n - an) do not assume the value 0 in IzI < 1 and they assume the value 1 in that disk at no more than p - 1 points. Consequently, 1) Montel [1933].


75

these functions constitute a normal family in I zj < 1. Since 0n(0) t

- an/(,8n - an) , 0 as n -+ or they are uniformly bounded in the disk

zl < 1. n(z){ contains a subsequence tonk(z)I that converges uniformly

I

It follows that in the interior the disk Izl < 1 to a regular function O(z) such that '(0) = 0. The function o(z) must be identically equal to zero because otherwise, we could show, by using Rouche's theorem, that the functions on (z) have, from some k k on, zeros in I zj < 1, which, however, is not the case. Consequently, 0' k(0) = an) --. 0, which is impossible. Thus, the positive number p mentioned in the theorem must exist, and the theorem is proved. As a consequence of this theorem, we obtain Theorem 9. 1) There exists a positive number p such that every function that is regular in I zj < 1 covers entirely some neighborhood f(z) = z + cZz2 + of radius no less than p with center at the origin. I:urthermore, we have Theorem 10. For the family of functions f (z) =.z + c2z2 + . . . that are

regular in Izl < 1 and that do not assume in that disk agiven finite nonzero value a, there exists a positive number p such that the disk I zI < p is completely covered by the image of the disk Izl < 1 under an arbitrary function in the family. This theorem also follows from Theorem 8 if we take for K an annulus containing the point a. 2)

1) Landau [1922]. 2) A more profound study of the modular function led Bermant [1944] to a number of

generalizations and sharpenings of Theorems B-10.

CHAPTER III REALIZATION OF CONFORMAL MAPPING OF SIMPLY CONNECTED DOMAINS

§1. Conformal mapping of domains bounded by rectilinear and circular polygons

Although in Riemann's theorem we had a geometrical principle for forming analytic functions (specifically, to every simply connected domain was assigned a univalent function mapping that domain onto some canonical domain), neither in this principle nor in the rest of Chapter I, did we touch on the question of more or less explicit construction of these functions. In the present chapter, we shall look at some questions of this nature. Let us now look at domains of certain particular forms, for example, domains bounded by rectilinear or circular polygons for which we can examine the analytic nature of the mapping functions rather extensively and can obtain appropriate explicit analytic expressions for them. The basic tool for studying these functions in the special cases mentioned is the principle of symmetry. Domains bounded by rectilinear polygons. Assuming that this case is already familiar to the reader, we confine ourselves to formulating the generalized conditions under which the final results hold and to deriving the results themselves. Specifically, suppose that a simply connected domain B in the z-plane does not contain o and is bounded by a rectilinear n-sided polygon with vertices at the points A1, A2, , A. Suppose that the angles at these vertices are respectively aa.l, rra2, . , Iran. Let us regard these vertices as accessible boundary points of a domain B. Therefore, if several angular boundary points fall at a certain point in the z-plane, there will be just as many vertices. We shall also consider the case when one or several vertices lie at cc . Here, if the sides of the corre76

§1. DOMAINS BOUNDED BY POLYGONS

77

sponding angle are parallel, we assume the value of the interior angle to be 0. On the other hand, if the sides diverge at a vertex Ak = oc but their extensions meet at a finite point A k forming an angle of rrak directed toward the vertex A k, we take nak = -nak . With these conventions, ak can also assume values from - 2 to + 2.

Of course, the cases ak = 2, - 1, and - 2 should be understood as limiting cases. Under these conditions, if the domain B is mapped univalently onto the halfplane -`- (C) > 0 and if points ak, k = 1, 2, - - , n, on the real axis of the C_ Plane are mapped onto the vertices Ak , then, in the case in which these ak are all finite, we can show by the reasoning usually employed herel) that the inverse function z = f(c) is determined by the Schwarz-Christoffel formula n

.f lt) = C J rl

(C - ak)ak

dC + C;

(1)

k=1

where C and C' are constants. In the case in which one of the vertices, let us the same formula holds for say Ako, is mapped by this inverse mapping into f(O except that now, the product constituting the integrand is over all k = 1, 2, - , n excluding k = ko. Of course, formula (1) remains valid when the domain B is mapped onto the disk I zl < 1. In this case, the points ak, the images of the vertices Ak , are points on the circle C1 = 1. We note that, in accordance with Theorem 6 of §3, Chapter II, we can choose only three of the points ak arbitrarily. The remaining points ak and also the constants C and C' are uniquely determined by the domain. 2) The absence of a general method for determining the remaining unknown constants decreases the practical significance of the Schwarz-Christoffel formula. Furthermore, it is only n exceptional cases that we can evaluate the integral (1) in closed form in terms k,of elementary functions. We cite as verysimple examples of the application of ;formula (1):

(a) Suppose that B is the interior of a regular n-sided polygon with center at

z = 0 and with a vertex at z = a > 0. Suppose that z = f (C) is a function mapping the disk IC1 < 1 onto the domain B. Suppose that f(0) = 0 and f'(0) > 0. Then,

1) Of course, we use here the results established in Chapter II on the existence of such an inverse function and its behavior on the boundary. 2) (We have Kufarev's method [1947b] for determining the points ak, which is convenient in certain cases.)

78

III. REALIZATION OF CONFORMAL MAPPING

the constants ak are easily determined from symmetry conditions. Specifically, we have ak = e2nkc/n for k = 0, 1, , n - 1. From (1), we obtain C

-

dC

f G) = C

v)2/41

(2)

where C is found from the condition f (1) = a. (b) Suppose that the domain B is the z-plane with cuts along the rays issuing from the points ae2rrkt/n for k = 0, 1, , n - 1, where a is a positive number, in the direction opposite to the point z = 0. Suppose that z = f(C) is a function mapping the disk I CI < 1 onto the domain B and that f(0) = 0 and f'(0) > 0. Then, we can again determine the constants ak from symmetry conditions. From (1) we have

fly)=C

-

0(l-{-

)i

dC=

(3)

2

n

and C=a Above, we have assumed that the domain B does not include oo. If the domain B bounded by an n-sided polygon does include 00, then, for the function z= that maps the disk ICI < 1 onto B in such a way that f(0) we obtain the formula

with analogous values of the constants ak . Here, the rrak are the values of the interior angles (with respect to the domain B). Domains bounded by circular polygons. Now, let B denote a simply connected domain in the z-plane with boundary consisting of a finite number n of circular arcs or straight line segments. In the latter case, the segments can be regarded as degenerate circular arcs. We denote the endpoints of the arcs, that is, the , An and we denote the values of vertices of the polygonal domain, by A1, A2, the interior angles (with respect to B) at these vertices by tra 1, ,rat, ... , nan. We may assume that the domain B is bounded. We can always arrange for this to be the case by means of a suitable fractional-linear transformation. The domain B can be mapped conformally onto the disk ICI < 1, Let us examine in detail the function z = f(0) inverse to one of the mapping functions.

§1. DOMAINS BOUNDED BY POLYGONS

79

, n, points on the circle I CI = 1 corresponding to the We denote by ak, k = 1, , n. The function f (C) is regular in the open disk C l < I vertices A k, k = 1, and continuous on the closed disk I CI _< 1. On an arbitrary arc y of the circle I CI = 1, the values that this function assumes between adjacent points ak all lie on some circular arc. Consequently, in accordance with the generalized symmetry principle, the function f(O can be extended past y into the domain ICI>1 in such a way that f(C) is regular (with nonzero derivative) on y. Now, let us look at the expression

- f' (C)

(4)

21f

which is known as the Schwarz invariant or the Schwarzian. Obviously, it is a regular function in I CI < 1 and also can be extended past an arbitrary arc y of the type described above into ICI > 1. Furthermore, one can easily show that it possesses the important property of invariance under an arbitrary fractional-linear that is, if transformation of the function f f (C) + d

then

If,, 6=1f, C}.

Let us examine the behavior of If, Ci at the points ak, k = 1, 2, - , n. Let us suppose first that the numbers ak do not include the numbers 0, 1, 2 and that no circular arc of the boundary degenerates into a straight line segment. The circles on which the boundary arcs lie, forming angles with vertex Ak, have a second point of intersection Ak. The function t = ((z - Ak)/(z maps .the part of a neighborhood of the point Ak that lies in B onto the part of a neighborhood of the point t = 0 in the t-plane that lies to one side of a line passing through t = 0. According to the symmetry principle, the function ((f(O - Ak)/ ak Ak))1 is therefore extendable from the part of the neighborhood of the point ak lying in ICI < 1 to the part of the neighborhood lying in S I > I. In particular, it is regular at the point ak itself and has a nonzero derivative at that point. Consequently,in a neighborhood of J= ak, we have rf (C) - Ak ak

f(C)-Ak)

- b1 (C - ak) + ...

,

bl # 0,

from which we get f (C) -AA

f (C) -Ak -

(C - ak)ak (Cl

C1 # 0.

80


If we set up the expression (4) for this last function and keep in mind that it is equal to (f, Ji we get, after some simple calculations, the expansion in aneighborhood of eJ = ak :

(1 -a"t)

if, C}=

1

{fC

(C-ak)9

k

(5)

+C-ak-}-...,

where Ck is an unknown constant. This expansion shows that If, C1 has secondorder poles at the points ak.1) Therefore, the function If, JJ, originally defined in the disk ICI < 1 and then extended to the entire plane, represents a meromorphic function with poles only at the points ak . The principal parts are determined at these poles from formula (5). The difference

G) =V, C}-

n1

Ck

(C-ak)' +C-ak)

(6)

k-1 has no finite singular points at all. Let us examine its behavior at oo. Since a suitably chosen function

A R)= f(C)+dEA const is real on each of the arcs y considered above and since f 1(0)

it follows from the symmetry principle that this function can be extended from the disk ICI < 1 to the domain ICI > 1 by taking f 1 (S) = f1 In particular, it is regular at the point oe; that is, in a neighborhood of S = oo, it has an expansion oo ,

f, (C) _ co + C + ... , 0 (by virtue of the conformality of f 1(C) at the point the corresponding expansion for If, J) has the form where c 1

if, C}= i; -F-...

= 0). But then,

(7)

This means that { f, I has a zero of at least fourth order at oo . But then the difference (6) has a zero at the point oo; that is, this difference is regular on the entire plane including oo . Therefore, it is identically equal to zero, so that we have the identity

1) We note that, if a finite point to, I to, > 1, is such that f(C o) = oo, then in a neighborhood of t o we have f (C) = b_ 1/(C- C o) + bo + , b-1 ji 0 so that If, t l is regular at t = t o.

1. DOMAINS BOUNDED BY POLYGONS n 2

( _ a)

81

C k

!.

(8)

k=I

Our assumption up to now has been that none of the ak is 0, 1, or 2 and that there are no straight line segments on the boundary of the domain B. However, if these assumptions are not satisfied, we can still show that (8) is valid by replacing the domain B with suitable circular domains and then using the theorem on the convergence of univalent functions (Theorem 1, §5, Chapter II). Thus, equation (8) is proved for an arbitrary domain bounded by a circular polygon. It represents a third-order differential equation in f(C). Its right-hand imember includes the unknown constants ak and Ck , for k = 1 , 2, , n. There are certain relationships between these constants. Specifically, since If, C1 has an expansion of the form (7) in a neighborhood of w, the coefficients of 1/c, 1/ and 1/C3 must be equal to 0 in the corresponding expansion of the right-hand member of (8). This gives us the three conditions

21 Ck=0,

k=1 n

(1 - ak) + yy

2

kt n

Ckak = 0,

(9)

k=1 n

2] (1 - ak)+ 2] Ckak=0,

k-1

k=1

that the constants ak and Ck must satisfy. Furthermore, it should be noted that three of the constants ak can be chosen arbitrarily on C1 = 1 by virtue of Theorem 6, §3, Chapter II. In the case of a circular triangle, this is sufficient to deter.mine all the constants ak and Ck . Similarly we can also show that, when the domain B is mapped onto the upper halfplane (c) > 0, the inverse function z = f(C) also satisfies the differential equation (8), where, this time, the ak are points on the real axis corresponding to the vertices Ak. The only difference is that if one of the vertices A k , let us say A k o, is mapped into oc, then the summation in (8) is over all k = 1, 2, , n excluding k = ko. Now, let us look at the nature of the differential equation (8). Since it is a third-order equation, its general solution has three arbitrary constants. On the other hand, if f (C) is a particular solution of equation (8), then, obviously, the I

Ill. REALIZATION OF CONFORMAL MAPPING

82

fraction (af, (C) + b)/(cf, (C) + d) which already contains three arbitrary constants

(a/c, b/c, and d/c), is also a solution. Consequently, this is the general solution of equation (8). Although we need to know only one particular solution of equation (8) to find the desired mapping, the finding of that one solution is difficult because of the nonlinearity of the equation. However, the problem of solving equation (8) can be reduced to solving a second-order linear equation, owing to a single remarkable property of the function If, C1.

Let w 1 and w2 denote two linearly independent solutions of the linear differential equation

w"-}-q(C)w0

(10)

with given q(t). It is easy to show that w1w2 - w2w1 = const = c (the left-hand member is the Wronskian of the two solutions). Therefore, if we set ri w2/w 1, we have C

(C) =Vii,

C} _

-

!-L

d,

2V; dC

Wt

2q

()

It follows that the ratio of any two linearly independent solutions of equation (10) is a solution of the equation {,q, C} = 2q (C).

(10')

This shows that, if we denote the right-hand member of equation (8) by R(C), set q (C) = R(C)/2, and solve the differential equation (10) with this q (C), then the ratio of the two linearly independent solutions yields a solution of equation (8). 1) Equation (10) with this R(C) belongs to the most intensively studied class of linear differential equations with variable coefficients, known as equations of the Fuchsian type. However, the use of this advantage is made difficult by the fact that we do not have a general method for determining the constants ak and Ck.

1) Also, equation (10' ) can be reduced to equation (10) in another way, specifically, by remembering that

-2

12 (I )

and setting 1/\(-i' = w. Then, if w1 is a particular solution of equation (10), the corresponding particular solution 711 of equation (10') is obtained from -1 = 1/w 1.

§ 1. DOMAINS BOUNDED BY POLYGONS

83

Example. In conclusion, we present a simple but important example when we can make a complete determination of the mapping function. Suppose that we are required to map the disk I C < 1 onto the domain B consisting of the interior of an n-sided circular polygon with center at z = 0, with one vertex at the point z = 1, and with all the interior angles equal to nq, where 0 < q < 2. Suppose that the mapping function z = f(C) is normalized by the conditions f(0) = 0 and f '(0) > 0. To do this, we consider a mapping of the sector I C1 < 1, 0 < arg C < 277/n onto the circular triangle bounded by the two straight line segments extending from the origin z = 0 to the points z = e2n`/n and the circular arc contained in the boundary of the domain B. According to the symmetry principle, the function executing this mapping will provide the required mapping of the disk 1, is regular in the disk I zi < 1 and has no zeros in the domain 0 < I zI < 1. Then the image of the disk Izi < 1 under the mapping t;'= F(z) completely covers the disk IC1 < 1/16 but not always a greater disk with center at C = 0. A more profound study of the function (21) leads to the following theorem on the covering of segments: Theorem 2.2) Suppose that the function F (z) = z q + . , for q > 1, is regular in the disk Izi < 1. Then the image of that disk under the mapping = F(z) completely covers some segment of arbitrary prenamed slope that contains the point = 0 and is of length no less than A = 8n2/F(%)4 = 0.45 - - . The number A cannot be increased without additional restrictions on F W.

§2. Parametric representation of univalent functions

In the present section, we shall give a parametric representation exhibited by Loewner for an important class of univalent functions.3) Thanks to this representation, we can find the complete solution of a number of extremal problems in the theory of univalent functions. This we shall do in the following chapter. Let B denote a simply connected domain in the w-plane including the point w = 0 but not the point w = -a. Consider the set of domains obtained from B by making a cut along any Jordan curve that is contained in B, that has one end point on the boundary of B, and that does not pass through the point w = 0. Let us call these domains the domains (s). If the domain B is unbounded (that is, if -0 is one of its boundary points), this cut can extend to 00 . Then, in contrast with the preceding situation, we understand by "cut" a curve defined parametrically as follows: w = w(t) for a < t < b, where w (t) is continuous in a < t < b, w (t) L w (t") for a < t' < t < b, and w (t) oo as t b. One can easily show that it 1) Caratheodory [1907]; see also Goluzin [.1948]. 2) Bermant [1944] and Goluzin [1948]. 3) Loewner [1922]; see also Goluzin [1939d, 1949c].

90


is possible to approximate an arbitrary simply connected domain B' contained in B and including the point w = 0 with domains (s) in the sense that a sequence of functions w = f(z), f (0) = 0, f' (0) > 0 that map the disk Izi < 1 univalently onto the approximating domains converges to a function mapping the domain I zI < 1 onto the domain B'. To see this, note that the domain B' can be approximated first by domains bounded by Jordan curves contained in B by mapping the disk Izi < 1 onto B' and considering the circular images in B'. Assuming B' bounded by a Jordan curve C contained in B, we connect the boundaries of the domains B and B' by a straight line segment contained in B but lying entirely outside B'. We continue along it from the boundary of B to a point c E C and then proceed from c along the curve C to a variable point d. If we let d proceed around C until it returns to the point c, then, in accordance with Theorem 1 of §5 of Chapter II, the domains (s) so obtained approximate the domain B'. By applying a diagonal process, we can obtain the existence of the required sequence of domains (s) approximating an arbitrary domain B' C B including the point w = 0. Now, suppose that we are given a domain (s) obtained from the domain B by making a cut L. By shortening the cut L, beginning with the endpoint lying in B, we obtain a family of simply connected domains BL depending on a real parameter t that varies continuously in the interval 0 < t < to. The domain Bo is the domain B with the complete cut L; Bto is the domain B without the cut;

and, for t' < t", the domain B,, is properly contained in Be,,. Let za,=== g(z,t),

g(0,t)-0, g' s(0,t)>0, 0 1, we see that the function w = h(z, t, t ") maps the entire z-plane with a cut along the arc Bt, t onto the entire w-plane with a cut along the arc St, t and along the arc S, ,tt symmetric to St , t about the circle I w I = 1. Since one of the arcs Bt , , t and St #.t if U S, is constricted to the point A(t) in both limiting operations, the other must also be constricted to the same

point. This is proved on the basis of the principle of condensation of analytic functions and the fact that, if the z-plane with a single excluded point is mapped univalently by means of a function w = 0(z), where ,(0) = 0 and 0`(0) = 1, then O(z) = z. On the basis of what has been said, under both limiting operations, the function h(z, t `, t") converges to z uniformly on each closed bounded subset of the z-plane that does not include the point A(t). In particular, it follows that A(t) is continuous throughout the interval 0 < t < to. What we are concerned with at the moment is only that, under both limiting operations, the arc Bt, t,, is constricted to the point A(t). If we denote the endpoints of this arc by eiaand a°Q, where a< < a+ 2rr, then the function (z) = log h (z, f" 0 (0) < 0, (6) (D is regular in the open disk I z I < 1 and continuous in the closed disk Izl < 1 and its real part is zero everywhere on the circle I z I = 1 except on the arc (a, 0), on which it is negative. On the basis of what has been said, if we apply the Schwarz formula to the function (6), we have in I z I < 1 log

h (z, t',

( (eò)) eie ± d9.

2n a

If we now replace z with the value [(z, t P

log f (z, t") f (z, t')

to

we obtain

eie ±f (z, t'

(q (e )) eie - f (z, t') d0.

2n

(7)

a

For z = 0,

this yields

Y - t'

((1? (eie)) dB.

2n S

If we now apply the integral theorem of the mean to the real and imaginary

(8)

§2. PARAMETRIC REPRESENTATION OF UNIVALENT FUNCTIONS

93

parts of the integral in formula (7), we obtain log f (z, t"

(2, t') r re+f l+ -L1eIe'f(,t')

reiB" "+f (2,

)

f(, e )l

eie

1

X 2n S it ((D (e`)) d9,

(9)

a

where 0' and

8

are points in the interval (a, /3 ). If we divide (9) by ( 8), we

obtain log f

(z, t") -logf (z, t')

t" - t'

_

r

ete'-f-f(z, t')

\e10'-f(z, t')1+

t')']. l (e,e"---f(z, 1el t'))J

(10)

Since a and /3 approach the same limlt, arg A(t), under both limiting operations that we are considering, the same will be true of 0' and 8". Consequently, by taking the limit in (10), we obtain the differential equation

dlogf_ of

that is,

X(t)+f

a (t) - f,

tf

kf, dt - f '+ l - kf' k - k (t) _ (t) 1

Thus, equations (3) and (4) hold throughout the interval 0 < t < to. This result shows that corresponding to every curve L contained in B is a function k (t) that is continuous and equal to unity in absolute value in a suitable interval 0 < t < to such that the function g (z, 0), where g(0, 0) = 0 and g= (0, 0) > 0, which maps the disk jzI < 1 univalently onto the domain B with cut L is defined as the limit

g(z, 0)=limg(z, t), 1-+0

where g(z, t) is that solution of the differential equation dg dt

k (t) z _ dgdz z1-}1-k(t)z

that satisfies the conditions g (0, t) = 0 and gZ (0, t) > 0. If the domain B is not the entire finite w-plane, the function

W=g(z, to)= lim g(z, t) 1-+10 maps the disk I z I < 1 onto the entire domain B. On the other hand, if the domain B has as unique boundary point w = oc, then, for any M > 0, the closed disk

I wI < M lies entirely in all the domains Br fort sufficiently close to to and, consequently, for such values oft on the circle Izi = 1. Therefore, we also have

94


I z/g (z, t)I < 1/M in I zI < 1 since, in this case,

g(z, t)-+oo, g(0, t) ->oo to when 0 < I zl < 1. This means that to has to be oo. Now, let us look at the particular case in which the domains B, subjected to the similarity transformation w' = e-tw, converge as t oo to a domain B' as kernel (in the sense of §5, Chapter II). Then, the functions w = e-tg(z, t) converge as t - oo to a function ((z) mapping the disk Izl < 1 onto the domain B`. For example, if the cut L is such that, as we move the point w along it toward oo, this point apas t

proaches asymptotically a ray issuing from the point w = 0, then the function g(z, t)

that satisfies equation (4) also satisfies the condition

lime tg(z, t)_(1

IiI=1.

1- 00

(11)

With regard to the inverse function z = g'1 (w, t), let us show that, for any curve L that extends to oo, we have, for arbitrary finite w, lim petg t (w, t)=w. (12) I -. 00

In accordance with inequality (10) of §4 of Chapter II as applied to the function

/3-le`g(z, t), we have in the disk Izi < 1 (1P+IZI)2Clg(x,

t)I(l_Pet

Z09.

Then, by setting z = g-1 (w, t), we obtain for arbitrary w E Bt

Set(w, t)I

(1-I-(w,

-w

petIg- ` (w, t)1

(1-I g' (w,

t)1)$.

(13)

The first of these inequalities yields Pet lg-1(w, t)l < 4lwl and consequently,

limIg'(w, t)

0.

t -.co

But then it follows from the two inequalities (13) that, for arbitrary finite w, lim pet g 1 (w'

t)

= 1.

(14)

t -. 00

Now, if (12) fails to hold for some finite point w0, there exists a sequence

{ tv } such that t1, - oa and Se'vg-1 (w, t,)/w0 - a

If we choose from the a subsequence that converges everysequence of the functions where in the w-plane to a regular function OW, we conclude on the basis of (14) that I 0 (w)I = 1 and, since 0(0) = 1, O (w) = 1. But O(w0) = a 1. This contradiction proves (12) for arbitrary finite w. Net"9-1 (w, tv)/w

1.

§2. PARAMETRIC REPRESENTATION OF UNIVALENT FUNCTIONS

95

Let us turn now to the function f(z, t). Since f(z, 0) = z, we conclude from what was said above that f (Z' t) is a solution in the interval 0 < t < to of the differential equation (3) that satisfies the initial condition f(z, 0) = z. If B has more than a single boundary point, then, in addition, we have in the disk I zI < 1 li m f I-*10

(x, t) = f (z, to) = g ' (g (z, 0),

to).

(15)

In particular, if B is the disk Iwl < 1, then g(z, t0) = z and it follows from what was said above that the function g(z, 0) can be represented as the limit g (Z' 0) = li m f (z, t), t - io

where f(z, t) is a solution of the differential equation (3) satisfying the initial condition. This was Loewner's result. On the other hind, if the domain B is the entire w-plane excluding the point oc, then, in addition to the differential equation (3) and the initial condition f (z, 0) = z, we have in accordance with (2) and (12) g(z, 0) = 1im pe`f (z, t). This result can be immediately generalized somewhat. Specifically, suppose that the simply connected domain Bp has no exterior points and is, in fact, the wplane with cuts along a finite number of Jordan arcs that do not pass through the point w = 0 and have no points in common with each other except possibly endpoints. Suppose also that at least one of these arcs extends to oo. Then the equation for the boundary L of the domain Bp can be represented as a single equation w = w (t), for 0 < t < 00, where w (t) is a piecewise continuous function of a real

parameter t such that, as t varies from 0 to o, the point w(t) successively describes all the boundary curves in such a way that, for any value of t, the curve corresponding to higher values of t bounds a domain B, of the same form as Bo. Under these conditions, all our reasoning above obviously remains valid and equations (3) and (4) with all the supplementary conclusions hold for the corresponding functions f(z, t) and g(z, t). The function k(t), however, admits a finite number of points of discontinuity of the first kind corresponding to values of t at which the point w(t) jumps from one boundary arc to another. These results are vivid examples of the reduction of geometrical problems to purely analytical ones. For example, the conclusion that we have just drawn can be formulated in the form of the following theorem, which will play an important role in what follows:

96


Theorem 1. Let B denote a simply connected domain without exterior points that includes the point w = 0 but not the point w = -. Suppose that the boundary of B consists of a finite number of Jordan arcs. Then there exists a complex function k(t) of real argument t for 0 < t 0 for IwI < 1, it follows from (21) that Iwnl < This means that all the w = wn(z, t) are regular func. IzI for all n = 0, 1, w (0, t) = e-t. Let us tions of z in the disk IzI < 1 and that show that the sequence of the functions w converges uniformly inside the disk IzI < 1 and hence in the interval 0 < t < to, where t0 < oo. From (21), we have awn

own

_ -- w

at u

Therefore,

I + kwn-1 - kttln_.,

n1

'

0w'_1

at - at

(wn - W.-I) I ±

kwn-1 - 27?1n-1 (1

- kwn 11)

kwn-.)

CIW.-'wn_tI I-IZI TIWn-1-Wn-SI(1 )IIZI = A I W. - Wn -.i + B I W.-1 that is,

al wn-wn-1

-

wn-a I,

W. - P1R-1

B I wn.-1 - wn-S Ir where A and B depend only on z. If we multiply this by a-At and integrate, we

at

`4

obtain, for 0 < t < t0 (where t0 < oo), t

wn - wn_1 I C eAto B

7FJn-1 - TQJn_9 dt. 0

Integration leads to the following inequality (with n = 2, 3,

wn-wn-1C

):

(eAtoBt)n - I

(t1-1)1

+

C-C(z),

from which the asserted uniform convergence of the sequence (wn} follows. The limit function w = f (z, t) is regular in I z I < 1 and continuous in 0 < t < oc. Obviously, it satisfies equation (18) and the conditions wlt=0 = z. Furthermore, I f (z, t) I < I z 1. Let us show that the function f (z, t) is univalent in I zI < 1. Let us suppose the opposite. Suppose that, for some II in the interval 0 < t 1 < 00 and for two points z I and z2 in the disk I z I < 1, we have f (Z 1, 11) = AZ 2, t I).

Then the functions f(z1, t) and f(z2i t) have the same value at t = t1. From an


98

easily proved theorem 1) on the existence of solutions of equation (18), it follows that f (z 1, t) = f (z 2, t) everywhere on the interval 0 < t < t 1. For t = 0, this yields z1 = z2. It remains for us to prove formula (19). To do this, we substitute into equation (18) the function w = f(z, t) and write it in the form

a(logf+t) at

-

1 - kj'

-

I-kfdt.

2kf

Integration yields

1ogL(Z)+t-

(22)

But the function etf(z, t) C S. Consequently, for I z I < 1, we have I f(z, t) I < MZe-t, where Mz depends only on z for z I < 1. This shows that the integral J 'o (k f/(1 - k/)) dt converges uniformly inside the disk I z I < 1. From this we conclude on the basis of (22) that, as t --, -, e° f (z, t) converges in the disk I zI < 1 to a univalent regular function f(z) such that f(0) = 0 and f' (0) > 0. This completes the proof of the theorem.2) In conclusion, let us show how to obtain the coefficients in the expansion of the function f(z) referred to in Theorem 2. To do this, we consider first the function f(z, t) for 0 < t < to, where to < uo. We define I

gio (z, t) =f (1'-1(z, t), t0)= et -'o (z -f- c2 (t, to) zA + ...).

This function is regular at the point z = 0. From gt0 (f (z, t),

t) =f (z, to)

we have agt0 of

of dt

agto

at

=0.

Consequently, in accordance with (17), we obtain the equation ag5o

at

_ agio z 1l + k (t) z dz

- k (t) z

1) One gets the result from a suitable inequality of the form at

where fl = f(z1, t), f2 = f(z2, t), and Mz depends on z for I z I < 1. 2) Equations (17) and (18) were studied by Kufarev [1946, 1947, 1947a, 1947c, 1947d, 19481. In the article [19431, he studied the question of parametric representation from a more general point of view.

§3. VARIATION OF UNIVALENT FUNCTIONS

99

in a neighborhood of z = 0. If we substitute into this equation the expansion for gto (z, t) and equate coefficients of like powers, we obtain the system of equations n-1 c., (t, to) _ (n - 1) cn (t, to) __f- 2 E Pt cµ (t, to) kn "(t),

=1

n = 2, 3, From this system, we obtain, in view of the fact that g,o(z, to) = z and hence

cn (to, to)=O, for n = 2, 3, ... , to

2e(n-1) e f e (n t)t

C. (t, to)

nl 1cµ (t, to) kn-µ it

eJ

This formula enables us to calculate successively the coefficients Furthermore, since gt0 (z, 0) = f (z, to), by setting

to).

f(z, to)=e-0o(z+c9(to)z9+...),

.f(z)=z+c9z9+...,

we obtain

Cn (t0) = Cn (0, t0), cn = Jim c n (to) = Jim cn (0, t0). io-oo to-o0

In particular, we obtain co

cg=-2S a'k(r)dt,

(23)

0 00

c.,

2 S e-9' (k9 ('t) + 2k ('t) c9 (t, oo )) d r 0

=

00

e'k (T) dt)a.

2 S e-"W (2) d2 + 4 0

(24)

0

These are Loewner's formulas, which we shall need in what follows.

§3. Variation of univalent functions In solving extremal problems of conformal mapping, we often use different variations of univalent functions. A fairly general type of such variations is given by the

Theorem. 1) Suppose that a function w = f(z) such that f(0) = 0 is regular and univalent in the disk I z I < 1. Suppose that the function w* = F (z, A) is a regular function of both z and A in the region r < z I < 1, 1 A I < Ao and that, for 1)

Goluzin [1946f].


100

each A in the interval 0 < A < Ao, it is univalent in the annulus r < I z I < 1. Suppose that, for every z in the domain r < z I < 1 and small A, F (z, A) =f (z) -}- Xq (z) -F 0 (XI).

(1)

If we combine the image of the annulus r < I z I < 1 under the function w* _ F(z, A) with the domain interior to the image of the circle z I = r, we obtain, for small A, a simply connected domain BA including the point w` = 0. For the function w* = r (z), where r (0) = 0, which maps the disk I z I < 1 univalently onto BA, we have in i z I < 1 I

f* (z) =f (z) + X q (z) - Azf ' (z) S (z) + Azf' (z) S ( z)

+

0(X$),

(2)

where S(z) is the sum of terms with negative powers of z in the Laurent expansion of the function q(z)/zf' (z) in the annulus r < I z I < 1. The symbol 0(A2) in equations (1) and (2) denotes a quantity whose ratio to A2

is uniformly bounded for all z belonging to an arbitrary closed subset of the disk

IzI 1 are mapped respectively into the disks I z I < 1 and > 1 since I z = 1 on I z' 1. But then, for small positive A, the function 00

21 a"'P" lz')

'cv*=F(zre"=1 , A) (4) maps a rather narrow annulus r' < I z' I < 1 univalently onto a doubly connected domain whose outer boundary is the boundary of the domain BA. Since the function (4) is regular in z' I < 1, we conclude on the basis of a familiar theorem that it maps the disk I z' I < 1 univalently onto BA. Thus, the problem of determining the function w"` = f *(z) which maps the disk I z I < 1 univalently onto the domain I

BA will be solved if we can construct functions 0,, W) satisfying conditions 1)4).

To construct the functions (kn (z' ), we substitute the function (3) into F(z, A) and expand the result formally in a series of powers of A:

-

00

E )p " z,

n=0

where ID 0 (z') = f W ), and, for n > 0, Do

D

1

do

(z.) = n! W

_

1

"EIXp

(s')

F (z 'e

A) I

n-1 a'v

da

k-0

(z')

-nl d)lnF(z'e(1 +XV'P. (z)), 1`)I a-0 n--1

_nl

a"

d

(s), 1`)à-a zf W)'P.(z) n-1

_ xf (z) [n! I

E ""

- d+ Tn(z)J

do F(z'e" dtn z7' (z') 1

(6)

In particular, for n = 1, this gives tilt

/

(Z) = z f (z)

[Fi(zO) Z, f, (Z,)

P, (Z')] .

-J

(7)

But, for r < Iz' I < 1, we have the convergent expansion F)

(Z" 0)

1, C(l)z k.

v k

Z'f' (z')

k--oo

-t -

-1

If we define T, (,Z'')

' C( )z'k l_ Y

k=-co

(8)

I

kco

(9)


102

we satisfy conditions 1) and 2) with n = 1. We also satisfy in part condition Iz, I < 1 since the 4): specifically, the function '1(z') is regular in the disk bracketed expression in (7) has no negative powers of z' and $1 (0) = 0. (We note that the function 01 (z') is precisely determined up to a purely imaginary constant term by the conditions listed.) Let us suppose now that the functions for v = 1, 2, , n - 1, are defined in such a way that'conditions 1) and 2) are satisfied and the (Dv(z'), for v = 1, 2, - -, n - 1, are regular in I z' I < I with 0v(0) = 0. Then, in r < I z' I < 1, we have the expansion n-1 0v(zI),

(z')

Fi

do n! dXn !

P. (z'e- 1

,X)

z-o^

z 'f' (z'1

If we take

-1

n (Z)

Ckn1 z k.

-1

C(k.)Z'k + Y h__-oo

(10)

k=-oo C(n)

i,

k--co

then the bracketed expression in (6) will contain no negative powers of z' and, hence, On W) will be regular in z ' I < 1 and On (0) = 0. Furthermore, conditions 1) and 2) are satisfied for On (z' ). (Here, and in general, (bn (z') is determined up to a purely imaginary constant term.) If we choose On (z ') successively, for it = 1, 2, , according to the rule given above, conditions 1) and 2) will be completely satisfied and condition 4) will be formally satisfied. We still need to investigate the convergence of the series appearing in condition 3), on which the satisfaction of condition 4) depends. To do this, we need to prove the I

Lemma. Suppose that the function O(A) has the expansion O(A) _ Yn=1an,1n the coefficients in which satisfy the inequalities Ian I < 1/(n + 1)2 for n = Suppose that the kth power of this function has the expansion [0(A)]k = F,°°_Iank)An where k is a positive integer. Then the coefficients in this last expansion , where C is an absolute satisfy the inequalities la(k)I < Ck/(n + 1)z, n = 1, 2, constant.

Proof. If the coefficients in the expansions of the two functions m

cp (X) = 2] a,A", nc1

satisfy the inequalities

00

4 (X) = f bnX' n_1

§3. VARIATION OF UNIVALENT FUNCTIONS B A 1an1 (n+l)s, bnIC(n+1)s, n= 1 ,

103

2,

and if co

u (A)

(4) _ CE Cn4n n=1

then cn1

alb,,-1+aibn-s+ ... +an_lb1 3,

AB

AB

AB

2AB (n_1)s'I- ... + ns2s(n+1)z [22 1

n_],e + ... + n+1 s 1

1

s

n+1J

1

8AB

).

n

(fl[n+1j+2y]

(n+ 1) 2 where C = 8 (2-2 + 3-2 +.

1

(

/

1

22

CAB 3! + ... = (n+ 1

1)s

If we apply this inequality successively to the

functions 95 (AA)k, we get the conclusion of the lemma.

Let us make a convention on notation. For any Laurent series X(z) _ cnzn we define

IIX(z)II=

n=

Icnl Izln. 00

When such majorizing series converge, they possess the obvious properties IIXI(z)+X9(z)11c11X1(z)II+11X2(2)11, (12)

Let us investigate the convergence of the series that the inequalities DM-1

v=1, 2, ...,

Let us show

(13)

hold in the annulus r < I z' I < 1/r for suitably chosen constants D and M. To prove the validity of (13) for v = 1, we need only take for D the maximum of the quantity 4 11 01 (z' )1I in the annulus r < I z ' I < 1/r.. Let us seek, by choosing M suitably, to satisfy inequality (13) for all v. Let us suppose that inequality (13) holds for v = 1, 2, - , n - 1 and let us find a bound for 11 On (z') 11. To


104

do this, we consider the function (10), which we denote by r(z'). We define

F (:, ).) = 21

AklzkXl.

(14)

k=-00, 00 I -0, 00

This series converges for r < I z I < 1 and 0 < A < A0. From (10) and (12), we have

n-l n

II Tn (z) II C I

Aklz

nl dkn

zf (Z)

kale

k

YWy (z/) l

y

I

k, I I

1

zf'(z')

II

d' jIAk!I

x=0

II

n-I kXlelklYEIx°II'Py(Z')Ilr

x

Iz'I

k, I

, n - 1, we obtain on the circle I z'

Then, by (13) for v = 1, 2,

0

r,

zI A'DM°-'

n

do

IITn(x')II 1, 1 n are equal to 0. Therefore, when summing with respect to k, 1, and j, we may assume that 1 + j > 2 and, obviously, that 1 < n. Furthermore, since I

-if

n-l

do

( \1 1v (v-)-1)`/ XV

l

_

11,

1

do

A=o

n-l .-1

1-0

is the coefficient of An-l in the expansion

rn-i cuix" V=I we obtain, by applying the lemma,

for z 0 on the circle I z'

r,

cJ/(n - 1 + 1)2. Therefore (15) yields,


IIT; (z) II C B

k,t,J

I Ak1 I rkMn

=B

1

I k J1DhC1

II

Aki (f+ 1)9

(n_1+1)

1)2

I k -JDICI

rkMn-l-I

J!((t-f 1)(n-1+01i

k. l.!

NM" I +

105

I Akr I (1- l)9

rkM-1-11 k jIDIC!

(16)

k, [.J

where N is independent of z . If in the last sum we drop the requirement that 1 + j > 2, this sum will be equal to /

IkIDC

AN I (1-I- 1)Q rkM-h! k. l

and this series converges for sufficiently large values of M since the series (14) is a regular function for r < I z I < 1 and I AI < A0. Therefore, this last sum in (16) with large values of M is of the order of 1/M2. Consequently, the right-hand member in (16) is, for sufficiently large M (though independent of n), less than DMn-1/2(n + 1)Z. Thus, for such M on the circle I z' I = r, we have II T.

In particular,

)II2( +1)s

C(klllz kc2D(17) n-1

k:.-oo Furthermore, since the majorant series of the second sum in (11) does not exceed the majorant series of the first sum, for Iz ' I < 1, it follows that (17) yields the inequality DMn-t

(z) II ` ( + 1)

(18)

on the circle Iz'

r and hence in the annulus r < I z I < 1/r. For M chosen independently of n, we conclude by induction that inequality (13) holds for all v in the annulus r < I z ' I < 1/r. This shows that the series converges uniformly in the annulus r< Iz ' i < 1/r if IAI < 1/M, and this proves the validity of condition 3) and hence of condition 4). Thus the functions 0v(z ') satisfy all conditions 1)- 4) and function (4) constructed from them does indeed map the disk W I < 1 univalently onto the domain BA. This function f *(z') is, for every z' in the disk I z ' < 1, an analytic I


106

function of A in a neighborhood of the point A = 0. We have the expansion DO

Ia (z) = 1

(19)

.=o

where (Do (z ') = f (z '). Furthermore, keeping (1) in mind, we obtain from formulas (7) and (9)

411(z')=q(z')-zf'(z')(S(z)-SCZI

)).

,

From (19), we obtain (2). This completes the proof of the theorem. Let us give some examples of the application of this theorem. Let us consider first the function

I in

w*=w+Aw(A+

L ryAkYplk

\1

(20)

k=1

wm are finite points in the

where A and A k are complex constants and w I, w-plane.

For arbitrary given p > 0 and for sufficiently small A > 0, the function (20) is univalent in the infinite domain consisting of the exteriors of the m circles I W - wk = p, k = 1, , in, because, for distinct points w' and w " in this domain,

I w* (w') - w* (w') I

m

= I (w' - w') (1 + A), + k I Ak m, u,' -f - task (tai' + t°'')) k=1

('

M

Wk) W - wk)

'W'I i wklklIIw'+W'I

Iw -w'I(1-A%-Xmax a' wk-1

W'-W

where the maximum, taken over all pairs w' and w", is obviously finite. Therefore, for sufficiently small A, the last expression in parentheses is positive, and hence w" (w") / w" (w "). Now, suppose that the function w = f (z), where f(0) = 0 and f'(0) = 1, is regular in the disk 1 z I < 1 and that it maps that disk onto the domain B. If the , wm are all exterior points to the domain B then, for any positive p points w l, less than the distance between any of the points wk and the boundary of B, we conclude that, for small A > 0, the function (A = 0) M

=f* (z) =J (z) +

k-

f

( z)

(z)

(Z),s

(21)


107

is regular and univalent in the disk I z < 1 and that it is normalized by\,the condition f* (0) = 1. On the other hand, if the points w1, all lie in the interior of the domain B, then, for sufficiently small A> 0, the function (21) is regular and univalent in some annulus r < I Z I < 1. If Z1, , zm are points of the disk jz I < 1 such that wk = f (z k), k= 1, 2, - , m, then (21) can be rewritten -

in

Akf (Z)' w* =f (z) + A k _ f (z) -f (zk)

The theorem that we have proved is applicable to this function. Since the function m

_IA

f (z)' zf (z) `k=1 k T77 (z) f (z) -f (Zk) 4 (z)

1

has only simple poles in the disk z I < 1 at the points zk, k = 1, )2 , k = 1, residues Ak f(zk) 2 /zkf , m, it is obvious that

, m, with

m

Ak r f (Zk) `9

S+ (z) _

l Consequently, the formula yields

k=1

in

J * (Z)

=J{

Zk

(zkf' (zb)) Z - Zb in

AjLf

(Z) + x I

k=1 T

() (Zk) - AZf' (Z) 11 Ak (Zkf (Zk) ll9 Z ZkZk k=1

m XZp

(z )

aAk

Zkf

(Zk) Q

(X9 )

1

(22)

Here, we have

f*

m

(0)

I +X

L Ak (zkf((zk) 19 + °

(23)

k=1 1

If we again wish to obtain a variational function f= (z) satisfying the condi-

tion f' (0) = 1, we need only divide (22) by (23). We then have in

f* (z) =.f (z) + A I f (z) k=1

XP Z) I Ak (1 zk f (zk) /l $ f (zk) - AJ (z) k-1 m

- aZ f' (z))

k1

f (Zk) Ak (.Zkf

I

Zk

119

Z - Zk

(Zk)

in

+ Xz! J (Z)

k-1

Ak k

(Zk)

Ì9

( Zkf' (Zk))

-*kZ

1 - 2kz + 0

(k2).

(24)


108

Formulas (21) and (24) are the desired variational formulas. 1) From these formulas we can easily obtain the corresponding variational formulas for the function

> 1 except for the pole which is univalent and regular in the domain C _ 00 and which does not vanish in that domain. To do this, it is sufficient to apply (21) and (24) to the function

f(z)= and set

(C)=

1

(We note that F, (')

0 in

+ ...,

=C+ao

I I

> 1.)

Simple transformations lead to the following formulas: in

F* (C) = F (C)

I Ak k=1

m F* (C) = F (C) + A

A k

)F (C) (C(C-Wk

Fwk

+ O (X9),

(25)

(C)k:) F((tk)

k-1 M

+ AF (C)

M

Ak (C,yF((Ck)l Je - CF (C) k=1

+ ),CF C

m

A(

I

F ltk)

Ak )s

(

l C

i

lr (Xi),

(26)

CkF' GO

where the wk in formula (25) are arbitrary fixed points all exterior to the image of the domain ICI > 1 under the mapping by the function w = F(C). As our next example, illustrating the theorem on the variation of a univalent function, let us consider a function w = f(C) that is regular and univalent in z < 1 and that does not assume in I z I < 1 certain given finite values al. am, where m > 1. Following the lead of the preceding variational formulas, we can construct univalent varied functions with the same property, that is, not assuming the values a 1i , am in I z I < 1. Specifically, if B is the image of the disk I z I < 1 under the mapping 1) Formula (24) was first stated by Schiffer [1943].

§3. VARIATION OF UNIVALENT FUNCTIONS'

109

w = f (z), then, for arbitrary wk either lying inside B or external to B, for arbitrary A and for sufficiently small A > 0, the function m

II (w - ak)

w*=w+AAwkM1

(27)

11 (w - wk)

k=1

will, as was shown above, map the portion of some neighborhood of the boundary of B that lies in B onto a doubly connected domain and it will map the boundary itself into the boundary of some simply connected domain Bx that includes w* = 0. Furthermore, the doubly connected domain referred to above will lie in BX. Since , am do not lie in B and are mapped by (27) into themselves, the points al, it follows that, for sufficiently small A, they will lie outside BX. Therefore, the function w* = f* (z), which maps the domain z I < 1 onto the domain Bk does not assume in I z I < 1 the values al, - , am . This function is constructed in I

the same way as before. In the case in which the wk all lie inside B, that is, when

Wk=f(zk), IxkI (1, k=1, ..,, nr, it is convenient to put (27) in the form m

I

zm*-zee-{-Xw(A0+

W A

,k),

k=1 M

77

(wk -a,)

11

Ao=A, Ak=A" m

(28)

I

11 (wk-w,) Then, we find the formula for f * (z): M

II (f (Z) - ak)

P (z) =f (z) + AAf (z)

Mk-1

IT (f (Z) --f (--k))

k=l

m

- AAzf' (z)

Akf (zk)

k_1Zkf (Zk)' (Z - Zk) m

+ AAz2f (z) J

Akf (Zk)

k=1 i kf' (Zk)' (1 -ZkZ)

(29)

CHAPTER IV EXTREMAL QUESTIONS AND INEQUALITIES HOLDING IN CLASSES OF UNIVALENT FUNCTIONS

1. Rotation theorems As we noted in Chapter II and as will again follow from Chapter V, extremal questions for various classes of univalent functions are of prime importance in the theory of univalent conformal mappings. When these questions are answered, it turns out that the requirement of univalence of the mapping influences the increase in various quantities characterizing to some degree or other the amount of distortion of the domain being mapped. In §4 of Chapter II, a number of extremal problems dealing with inequalities involving the initial coefficients of the mapping functions, and covering and distortion questions were solved by an elementary procedure. In the present chapter, we shall concern ourselves specifically with these questions, this time using more powerful methods, for example, the method of parametric representation of univalent functions and the method of variations. 1) For convenience, let us adopt some notational conventions regarding certain classes of univalent functions with which we shall be dealing in what follows. Specifically, let us denote by S the class of functions ((z) = z + czz2 + ... that are regular and univalent in the disk I z I < 1. Let us denote by I the class of

functions FW = C+ ao + a1(-1 + .. that are meromorphic (with pole at Do) and univalent in the domain ICI > 1. As a supplement to the sharp bounds on the modulus of a function belonging

to the class S and the modulus of its derivative that were established in §4 of 1) The reader wishing to become more familiar with extremal questions regarding univalent functions is referred to the author's articles [1939d, 1949c] .

110

§1. ROTATION THEOREMS

111

Chapter II, we shall give in the present section the corresponding sharp bounds on their arguments. In particular, we shall establish the final formula for the rotation theorem for the class S consisting of a sharp bound on the argument of the derivative. The parametric representation of univalent functions given in §3 of Chapter III will provide us with these bounds.') Let us denote by S' the subclass of the class S that consists of functions ((z) having a representation of the form

f (z)=

lim etf (z, t),

(1)

t-m where [(z, t) is a solution of the differential equation df dt

_ _f 1-F kf 1 -kf'

(2)

satisfying the initial condition

f(z, 0)=z

(3)

that, for fixed t, is a regular function of z in I z I < 1 such that If (z, t) I < 1 in that interval, f(0, t) = 0, and [' (0, t) > 0. Here, k = k(t) is an arbitrary piecewisecontinuous function that is equal to unity in absolute value in the interval 0 < t < no. In order to find a bound for any of the quantities that we shall consider below for the entire class S, it will be sufficient for us to do this for the subclass S' because, as was shown in §3,of Chapter III, an arbitrary function ((z) of the class S can be approximated by functions [, (z) such that [, (0) = 0 and [' (0) > 0 each of which maps the disk I z I < 1 univalently onto the w-plane with cut along a Jordan arc that goes out to no and does not pass through w = 0, and consequently by functions [, (z)/[n (0) E S'_. When this approximating procedure is carried out, the sequence of bounded quantities for the approximating functions converges to the same quantity for the function 1(z). A bound for arg [I (z)/z] and arg [zI' (z)/I(z)]. We divide the identity df (z t) at

- f (z' t) 1

k (t) f (z, t)

'

by I(z, t) and separate the real and imaginary±parts. This gives us the two identities t)I1-If(z, t)I4 dIP2, t)1=-If(z, at

1 1-- k (t) f (z, t) 1s

1) See Goluzin [1936] and a number of subsequent articles by the author in Mat. Sbornik, for example, [1939c, 1946c].

(4)

IV. EXTREMAL QUESTIONS

112

It follows from (4) that

dargf(z, t)

23(k(t)f(z, t))

at

1- k (t) f (z, t)

I f (z, t) I

(5)

decreases monotonically from

IzI

to 0 as

t

increases from 0 to o c. This enables us to take as our independent variable the quantity I f (z, t) I instead of t. Now, from (5), we have 12' k f12 dt

I di argf 1

(where we write for brevity f and k instead of f(z, t) and k(t)) or, eliminating dt with the aid of (4) and keeping in mind that arg f = arg erf,

1I 2d

I dt arg e f I

l

l 1,

If we integrate this with respect to t from 0 to - or, consequently, with respect to I f I from I z I to 0, we obtain, by virtue of (1) and (3) argfZZ) I

(6)

Here, the argument is defined by the formula 00

arg

f i Z ) _ dr argf Z, t)

Consequently, it represents that branch of this multivalued function that approaches

0 as z---- 0. Inequality (6), which we have proved for the class S `, holds also for the entire class S by virtue of what was said above. Let us show that the bounds for arg Lf(z)/z] given by inequality (6) are sharp. To do this, we show that there exists a function k (t) of absolute value I which is continuous in the interval 0 < t < 00 and for which f (z, t) defined by equation (2) and condition (3) will also satisfy the condition (k (t) f (zo, t)) _ - I f (zo, t) I for arbitrary t such that 0 < t < oo and given zo, where Izol < 1. If (7) is satisfied, then from equations (4) and (5), which are equivalent to equation (2), we have for z = zo

dlfl

if 11 at - Ifl1+IfP,

dargf

-

21f1 1+lfj'.

tt

If we integrate the first of these equations, taking If(zo, 0)I = Izol, we obtain

(7)'

(8)

113

1. ROTATION THEOREMS

IffI'=eel-I go Zo18

I

(9)

From this we determine I f I as a positive function o ft that decreases monotonically from I zo I to 0 as t increases from 0 too . Furthermore, by eliminating dt from equations (8), we have

2difI l dargf=- 1-if from which we get ar

g

f(zo, t)-lo I+Iz°I -log

zo -

g

1-IzoI

g 1-If1'

(10)

which, on the basis of (9), can also be expressed in terms of t. Then, k(t) is determined from

argk(t)-}-argf(zo, t)_- 2 Since all these operations are reversible, we have equation (10) for the function f(z, t) defined from the function k(t), which is a solution of equation (2) evaluated at the point z = zn Z. Here, as t -4 a the rieht-hand member approaches log (1 + Izo j)/(1 - Izo I) . Consequently, the upper bound for arg [f (z)/z] given by inequality (6) is the least upper bound. In the same way we can show that the lower bound for arg [f(z)/z] given by inequality (6) is the greatest lower bound.

Since the function

zC) -f (z) f (C+z l l -}-

g(,)= f'(z)(I-Iz12) =C } ...,

(11)

where z is an arbitrary number in the disk I z I < 1, belongs to the class S when f(z) does, by applying inequality (6) to this function with J_ -z, we obtain the following inequality: 1) (z) arg zf f(z)

1 2 2

1) Inequality (13) was proved by the author (see Goluzin [19361 ). That it represents a sharp bound for I z I > 1/\/Z was shown by Bazilevic [1936a1.


116

Here, we can take either of the signs ±. Now that we have found k f expressed in terms of I f I, let us substitute the result into (4) and (5). Then, from (4) we obtain t as a continuous monotonic function of if i and, consequently, I f I as a continuous and positive function of t. From (5) we then have a representation of arg [f(zo, t)/zo] as a function of t. Knowing arg kf, we find arg k(t) and thus k (t) itself. For the function (1) constructed from this k(t), we have, as we can see from the proof of inequalities (13), I

for I z

4arc sin z0

argf'(zo) = it

+ log

01

I

for IzoI,V-2

z0I' t-IZoI'

This shows that the upper bound for arg f ' (z) given by inequalities (13) is the least upper bound for arbitrary z in I z I < I. In the same way we can show that the lower bound given by inequalities (13) is the greatest lower bound. We note that, reasoning as above, we can prove the inequality arg f(2)') I

-

log (1 - I z I9),

(15)

which gives sharp bounds for arg Lz 2f ' (z)/f (z)Z] at an arbitrary point z in

z I < 1. Therefore, by shifting from f(z) to the function F(t) = 1/f(1/0 + for arg F '(c): const, we immediately obtain sharp bounds in the class

IargF'P

I 1 we have the sharp inequality I log F* G) I 1. Then T'TY' log

F(Cr)-F(C,) C -C n

1YT: log (1

} - Y1 C CY,/

(7)

§2. SHARPENING OF DISTORTION THEOREMS

121

for F(C) E E.1) Proof. Just as above, it will be sufficient to prove (7) for the functions F (C) = l/ f (1/C), where f (z) E S'. But, in the present case, we have from the first of formulas (2) F (CY) - F (C,.)

n y

IYTY' log Co

n

IYTY' 0

=1

n

00

kf"

1

2

kfY'

1 - kfY 1-kfY,

dt

kfY

2

TY

\

19

1 - kfY l

dt

V

and, consequently, n

F (C") - F (CY,) TYT:Iog CY-CY,

n

9

kfY

_< 2

TV I

V=1

M = 2

- kf

dt

Y

n

Y

'1 ' 1--M,kfY

Rj"'

1 - kfY,

dt !

From the second of formulas (2) applied to the last integral, this yields (7) and completes the proof of the theorem. As will be shown in §3, inequality (7) is sharp for arbitrary yr, and and determines the range of values for the summation on the left . In an analogous manner, we can prove the more general Theorem 3. Let n and n' denote two positive integers. Let yr,, v = 1, , n, , v = 1, , n, , n ' , denote n + n' complex numbers . Let and v'= 1, > 1. and complex numbers in the domain , denote n + n ' v r = I, n) . Let F denote a function belonging to E. Then, A6.d

J,TYT,'IogF(C")-F(C") V,

Y-1 Y'=1

CY' - C

1) The expression log F (C ) -F (C) with t and C in the domain I C I > 1 always refers to that branch (single-valued in this domain) of the multivalued function log

F (C)

-

F (C')

that approaches zero when at least one of the points

approaches

co. For I = C' , this quantity becomes equal to log FP(C) and then it can be understood as the value of that branch of the multivalued function that vanishes at

C = W.


122 n

17,T,' log 11 -

7;,1og

1

/

(8)

(If we write inequality (8) for F2(t;) E 12, replace n and n' with 2n and 2n', and set yn +y = - yy and Sn+v = - Sv for v= 1, , n and y.,+,, = Yv and ,,,}v = , n', 'we obtain the - Cv for v = 1, Corollary. Let n and n' denote positive integers. Let yy for v = 1, ,n for v' = 1, , n' denote complex numbers. Let Cv, for v = 1, . , n and tv, for v'= 1, , n', denote complex numbers in the domain C] > I. Let F2 denote a function belonging to 1 2. Then, and y,;

tJ`n1'

v

F,(Cv)-F2(Cv')Cv+Cr' 7riv log F. (Cv) + F. (Cr,) Cr l r'=1 `J1

c

"

7r7r, log

n1

CrCr, +

IX- -

CSC,, +

7,7" log C'XY, 1

-1

(9)

Theorem 3 and its corollary were first obtained by Lebedev [19511, who started with the area principle.) We mention some special cases of Theorem 3. In the first place, if n = n'= 1 and S I = I S I = P > 1, then (8) gives the sharp inequality (the subscripts are omitted for t;)

Ilog F(CC-C (C')

1.)

The applications of formulas (4) and (5) are effectively used.

E S. If we set

Theorem 4. Suppose that f(z) = In ((Ill

F(C)=

ICI>landlog F(C

;(C)'

«') C

00

E ak, t (:

ICI / l

k, 1=1

I,I

l

where the ak,t are rational functions of the coefficients ck, then, for arbitrary , n with n > 1), values of the complex variables xk (where k = 1, 2, n

n

,

ak,I xkxr

( 12)

k l9

k,1=1

k=1

In the case of real values of xk, equality holds in (12) for the functions f(z) _ z/(1 ±z)2. Proof. We can confine ourselves to functions f (z) E S'. In this case, the ak,1 have an integral representation (4) with the functions bk(t) subject to conditions (5). Therefore, n

oo

n

oo

n

E ak, txk,xt = - 2 E bk (t) b, (t) xkxldt = -2S (E bk (t) xk\2 dt k, 1=1

0 k, 1=1

0

h=1

/J


Y ak,1XkXI k, 1=I

00

2S 0

n

2

ao n

E bk (t) Xk dt = 2 S E b, (t) bt k=1

(t)xk_xidt.

(13)

0 k, 1=1

On the basis of (5), this yields inequalities (12). In the case of functions f(z) _ z/(1 ± Z)2, we have k (t) = ± 1 for 0 < t < so that the functions b k (t) for k = 1, 2, - -, corresponding to them are real. Therefore, equality holds in the conditional inequality (13) for these functions when the values of xk, k = 1,2, , n, are real. This completes the proof of the theorem. Schiffer [19481 proved the sharpness of inequalities (12) for arbitrary complex

values of xk. As a generalization of inequality (12), let us prove, under the same conditions,


124

M

n

J Ixkl'

I Yak, exkxe AMW

k=I1=1

k

k=1

x18 1

l=1

where in > 1, n > 1, and xk and x1' are arbitrary complex numbers (for k = 1,

2,---,m and l= 1,

n.)

Let us prove two more theorems for functions in the class

Theorem 5. For functions F (,'') E I and arbitrary n > 2, on the circle I C1 = p, p > 1, we have

, n, where

v = 1,

2

}(1-:)2(n_I)II

F (C,) - F (C,,)

H Y$Y

.

CY - CY

n

(14)

CYCyr

Y#Yr

In the derivatives, v and v

range over all pairs of values v, v' = 1, except coincidental ones (v = v' ).

,n

Proof. In (6), let us take for a v, two systems of numbers av v, and av v, satisfying the conditions of Theorem 1 and let us divide the resulting inequalities one by the other. We then obtain as one of the inequalities the following:

n

1 I,% -Y /

F(CY)

R

aY' Yr -

Yr

II 1

CY __ CVr

Yr Vr=I

Y, Yro1

aY' Y

V,

(15) 1

CYCY.

In particular, since the forms xi--+...+xng

+xjx8-,-...+xn_1xn.= (xl+...-}, xn)

X; + ... + x'R-n-1 (xtx9 + ... + xR_lxn) = 2 (n

I

1

)

I (xV

xY,)

Y$Y'

are both positive, we can set

= 1,

v,

v' = 1, ..., n,

for v=v',

1

n.

aY, Y _

-

for

1

n-1

v -A v;

We then obtain the inequality

11 V # V.

F (CY) - F (Cv,) CY - CYr

-

Ycl

I

\l

l

a

)2-

2

2

nil YY

n

(Y(, `

1

I

(16)

125


, n, this yields inequality (14). This completes the

with I v I = p, v = 1,

proof of the theorem.

Theorem 6. Let n denote an integer > 2. Let Ev denote distinct numbers , n. Let F denote a function belonging to I. Then, such that Ie, = 1, v = 1, on the circle k 1 = p, p > 1, I

max

(17)

n

p

where A is positive and depends only on n and cv, v = 1, 2,

, n.

, n, and Proof. We apply inequality (14) for Sv = ev C, v = 11 Denoting by M the maximum on the left-hand side of (17), we have

F ( t) I

f' 5'r

I

I = P.

2n(n t)M4(n 1).

Y#Y

Furthermore, on the circle I I = p, we have I=Arpn(n-t),

rjIE,C-e.CI=p'(n-1)TTIEv-`v V:;1-V'

Y#r

I1. 1r$r

_1 evev P

f

1 min

lI 1--

I0

Ekz l

I

r, where z re `s. Consequently, as z moves around an arbitrary circle I z where 0 < r < 1, in the positive direction, the function arg f (z) increases monotonically by an amount 2n; that is the point w = f (z) describes (one time around) a closed Jordan curve. But then the function w = f (z) maps the disk I z I < r univalently for any r in the interval 0 < r < 1. This means that f (z) E S. Now, for 0 < r < 1, we have

r

Ar

Ii I I - -8.,r I21n

(1 - r)2/a

If(Evr) I- n km1

where A is a positive number depending only on n and e, v = 1, - - -, n. This proves what was said regarding the sharpness of inequality (18). We note that, for n = 2, we obtain from inequality (14) the following inequality for F (C) E 1: F(C1)-F(ra) (19) C1-C,

where CI and C2 are arbitrary points on the circle ICI p, where p > 1. Inequality (19) is called the theorem on the distortion of chords under mapping of the domain CI > 1 by functions belonging to the class 1. This theorem, which" was proved by the author, l) has found numerous applications. Here, we mention the direct elementary derivation (given by Milin) 2) of inequality (19) and also of inequality (10).

Specifically, the function'() = log [F(O - F(io)], where ICO I > 1, is regular and univalent in I< I CI < I CO I, and its expansion in that annulus has 1) Goluzin [1946e]. 2) Lebedev and Milin[1951], inequality (11.1).


127

the form

log

F(C)-F(Co) C

+log(-Co)+ log (1-

_ - ` Cn

an (Co)

)

Co

Co

}- log (- Co)

n=l

-

Co

1

n=l

C

n

a (Co

Therefore, the area S of the bounded domain bounded by the image CP of the circle I z I = p, where 1 < p < 1 Co I, is 4\qual to 00

S- a\ Il

n I an p

2nl

fn

l

n (

1

1

10

)

0

CIO

log(1-Iol9}

11

n=1

n=1

ICoI/2n

If we now let p approach 1, this yields Co

nan(C0)I9c-log(1- IC )

n=1

Now we have, for 1 < I

1 < Co 1, 00

00

F(C)-'F(C0)

l I

log

t5:_

lan(Co)I

C - Co

I C In

n an

nIa_(Co)14' n-

nJC1 n

n-

(1

which, as we let

I Co I

log F (CC

_ _I C In

n-l

-

C Is) log (1

- C7

approach I I = p, yields

_F

(Co)

c- log (1- P9),

ICI=ICoI=P>1.

If we now replace the absolute value with the real part taken with a minus sign, we obtain inequality (19). We note that inequality (11) can be derived in the same way.


128

§3. Extrema and majorizations of the type of the distortion theorems

The method of variations also enables us to establish a number of results of the distortion-theorem type and to calculate completely the extremal functions. 1) Furthermore, in the investigation of extrema of certain quantities, we are able to ascertain their ranges of values. We shall stop here to go into only one theorem in detail.

Theorem 1. Let n denote a positive integer. Let y,,, v = 1, , n, denote n complex numbers, not all zero. Let S1, Cn denote distinct complex numbers in the domain ICI > 1. Let F denote a function belonging to E. Then, we have the sharp inequality n

n /

I\

j -1

7,7y, log (1

7Y7 109 .,--1

> 1 by the

Equality holds in (1) only for functions defined in the domain equation n

(1)

jjn

logF(C -C (C.)_-L7,

1

.=1

(2)

1og11-

y

> 1 onto the entire w-plane with a cut along

These functions map the domain the curve defined by the equation

(2]n 7, log (zap - F (C )))=const.

(2')

Furthermore, for I CI > 1, we have the inequality n

.,=1 .'

log F (C') - F (C,.) Cy - C,

n

1 under the mapping w = F (C) has an exterior point w0 . Then, let us form the varied function F. (0 E 10 by using formula (25) of §3 of Chapter III with m = 1 and with w1 replaced with w0. For this function, we have

(F,-Fy,)(1-AA1

n

IF*= at( E 7r7r' log

C - C' I

V, V-1

IF-X

7r7r'

( Al

(Fv -- W0)

(P"'

-

tD0)

. )+ (4)

[Here and in what follows we shall write for brevity, Fv instead of F(Cv).] Since w0 can always be chosen in such a way that the last sum is nonzero, we obtain IF. > IF for small ,A > 0 and suitable arg A 1. But this contradicts the extremality of the function F(J). Consequently, the domain B cannot have exterior points. Let us now turn to the varied function (26) of §3 of Chapter III with m = 1 and with t 1 replaced with CO , where ICI > 1. For this function we have a

Al

IFS =1F

F'2

17r7r'

v,r'-1 It

Al

(Fv - FO) (Fr'

7r7,' (C0 F 0

_

4

7,7" J - Al (C F v' Lr r' - 1 0

+

0

0

n

r,r'-1

7r7r'

r

v

C"F",

COC, -1 9

Y

F-F.

C,F,

A1(Ca

C''F'y'

C'FF

n

F

l9

FO)

1

toC,, -1 I+ 0 (A2).

Fy - Fr.

l

1

(Here, all the ratios representing indeterminate forms must be replaced with the ratios of derivatives.) If we replace the last term in the large parenthetical expression following $2 with its complex conjugate, we conclude on the basis of the arbitrariness of arg A 1 and the extremality of F (C), that the quantity in the parentheses following R after we divide by the common factor A 1 must be equal


130

to zero because otherwise, for suitable choice of A 1, we would have IF. > 1F, which would contradict the extremality of the function F(C). This leads to the condition n

it

(Fv - Fo)

00

-1 n

-}- N

TT,

v

vFr C2 1F. C, - Cr

F,

v

V r'_1

Cv"F"r"

Co-Cr,

- Fv, Cr,F;,

C. FY

-

r _1

TrTr'

Cc-I

Coc,, - 1

Fr-F,"

(5)

Since 4 in this equation is an arbitrary point in the domain I CI > 1, if we replace Co with C, we obtain the differential equation that the extremal function F (C) must satisfy. We denote the right-hand member of this equation by A (c). Let us show that A(C) > 0 everywhere on ICI = 1. To show this, let us consider a function t = z/i(Z) where 1/1(00) = 00 and 1/i' (oo) > 0, that maps the domain I CI > 1 univalently onto the domain t I > 1 with a small radial cut issuing from I

the point t = e'`B. This function is obviously obtained from the equation teie-}-

to=(1 +h) (Cete-+ -'Ceie)

+2h

with h > 0. From this we obtain, for small h,

1-Ceie+0("' COO)=1+h. By means of the function l/i(C), let us define the following function in the class Fl (C) _

(

F (C) - h (F (C) + CF (C) I,

0 w).

For this function, n

IF.=1F-h(Y

TvTv n

+

CYF'r+Y -Cr"F'r" C+Cy" C-C 7r7y

Fr - Fr.

,) + 0(h2),

r, r1 = 1

where r=

do.

Then, by virtue of the extremality of FW, we obtain, for all t on the circle

1

§3. EXTREMA ANP MAJORIZATIONS

r

131

C F' C+CY -Cy,F',,C

n

n

V, Y1= 1

V, V1

+C,,

IVIV

If we denote the left-hand member of this inequality by B (0, we have B on the circle I CI = I. On the other hand, on this circle we have n

A (C)-B (C)=12

TvTy'(1Y, 1

C"F',

-C"F'y,))

F, - F, ,

JJ

0

(6)

Let us show that the right-hand member of this equation is equal to zero. To do this, let us again consider the function F0 = e-`BF(e`B/') E E0 for different real values of 0. The quantity IF, attains its maximum on the interval 0 < 0 < 2rr at 0 = 0. Therefore, (dIF6/d0)I B=o=0, which, in expanded form is equivalent 1, we have to vanishing of the right-hand member of ( 6). Thus, on the circle 0.

It follows from this result that, if the rational function A (C) has zeros on the circle CI = 1, all these zeros are of even multiplicity because the increase in arg A (4) as C moves around a small circle with center at any of these zeros will, by virtue of the principle of symmetry, be equal to twice the increase in arg A (C) as C moves around an arc lying in ICI > 1 and this last increase is equal to an integral multiple of 27r. Therefore, the function VrjT?) is regular and real on ICI = 1. Turning now to the differential equation (5) (with CO replaced with C), we note that twice the sum in its left-hand member can be represented in the form n TY

( 1

Consequently, this equation can be written in the form n

C. I F, T' F = A (C) . V=1

If we solve it we get the following dependence of F(O on

j_

n

v.1

7, log

which holds in I CI > 1.

(Fy -F)

--

S

A (C) CC = C (C),

(7)


132

If we take an arbitrary boundary wo of the domain B and denote by Co the corresponding point on ICI = 1 under the mapping C = F -1(w), we conclude by passing to the limit that equation (7) holds (with wo instead of F) holds between 1. Also, since o and wo. In this sense, equation (7) also holds on

51 (C(C)) _ ( ae)) _

(1CC (-)) _ f (I i/A (C)) = 0,

1, where C= ee, it follows that the right-hand member of on the circle equation (7) and, hence, the right-hand side of the transformed equation

rsl lT,log Cy-C +7ylog(1 finI

(

C (C) -

V=1

(7, log G.,

- ) - 7ylog \ 1 -

11

(8)

will have a constant real part on ICI 1. But this means that the sum on the left side of (8) has a real part that is constant everywhere in the domain I C1 > 1 because it is a harmonic function in ICI > 1 and hence it cannot attain its maximum or minimum in I C1 > 1. From this we conclude that the left-hand side of (8) is a constant everywhere in I C1 > 1. Then, by considering its behavior at we conclude that this constant is equal to zero.

> 1 and Thus, for an extremal function, we have proved equation (2) in we have proved equation (2') for the boundary of the domain B. and sum over Let us now set C= Sv, in (2), multiply both sides by v' = 1, , n. We then obtain for the extremal function n

/

1p=-

log (I

1

(The sum on the right is, of course, real.) This proves inequality (1) with the supplementary conclusion regarding the equality sign. Furthermore, if we replace yv for v = 1, . , n in (1) with yve`B, where 0 is an arbitrary real number, we obtain the following sharp inequality, n

T-a" log F (Cy) - F (C,,)

e

r, .1=1

C'-Cy%

/

-

n

L+ 7y7y' V. V-1

log

CyC y

which, by the virtue of the arbitrariness of 0, implies inequality (3). But this also shows that the set of values of the quantity

§3. EXTREMA AND MAJORIZATIONS

WF

log F (C°) -

Gr V.

v = 1, 2, for given yv and 1, completely covers the circle

.

133

F (CY ")

Cy

r'=l

, n, and for F(C) ranging over the entire class

I wl=-

7Y7,'

log(1

-

V,

(9)

CyC,,)

n

WF,p= I ii,'1ogF(pC)

pCy(PC,,)

(10)

r'ml

with the same yv and Cv and with given p > 1. From what we have proved, as FM ranges over the entire class F,, the values W = WF.P cover the entire circle

7J,'log( 1

IW V, %'ml

-

1

P'CYCY,

)

But from the construction of the quantity (10), we can easily see that WF(PC)'

WFtC),p

where F(pC)/p E F, if F(O E I . This shows that the set of values of WF as

F(C) ranges over the entire class F, also covers the circle (11). As p ranges from 1 to oo, the corresponding set of circles (11) covers the entire disk

WI-

I n

7y7Y'log(1-C

l

V. V'= l

which, therefore, is the range of values for WF. This completes the proof of the theorem.

For n = 1, inequality (3) takes the form

IlogF'(C)I-1og(1-IClla),

(12)

and inequality (1) takes the form, with yl = e'a

ffI(e9=alogF(C))c-log (1-I

- 2 cac 2 .

I2

(13)

1

Here, equality holds in (13) with C= S1 only when Lcf. (2')] the function F(O maps the domain I CI > 1 onto the w-plane with cut along an arc of the curve defined by the equation ffi (e' log (w - F (Ci)) = const.

(14)

134


But equation (14) defines a logarithmic spiral with asymptotic point at w = F(CI) such that rays issuing from that point intersect the spiral at an angle 77/2 - a. In

the special cases a= 0 and a= rr/2, this spiral degenerates respectively into a circle with center at the point F(C1) and a ray issuing from F(C1). For a= 0 and a= rr/2, we obtain from inequality (13) the distortion theoreml)

for the class E:

I-

F'(C)I
1 onto the w-plane with cut along an arc of the circle with center at the point w = F(CI) and the greatest lower bound is attained only by a function F(C) that maps the domain I CI > 1 onto the plane with cut along a segment of the ray issuing from the point F(J1). Here, F(C1) obviously can be chosen arbitrarily. We give yet another application of Theorem 1.

Let us apply inequality (1) in the case n = 2, first for y j = e and Y2 - ei a and then for yj = ieà and y2 = -ie", where a is a real number. Then let us add the results. We obtain 1 (e2" log

F (C3 r

F(tl)log

8

=e(C.))c - 2 log [(1

(CQ) I 1. If we add and subtract these equations, multiplied respectively by 1 and i, we obtain the following two equations:

F(C)-F(C,)_(1_ C-C.

1

-e-2ia 1,2,

CC'

or 1

F(C)=F(CY)+(C-CY)(1-

Cc,)

-e - 2ia ,

v=1, 2.

In particular, from (16) with a= rr/2 and IC1I = IC2I = P, where p> 1, we again obtain the sharp inequality 1) Loewner [19191J.

§3 EXTREMA AND MAJORIZATIONS

135

F(C2)

(17)

C, - Cs

Here from what was said above, equality holds only for the function QiO F(C) +const,

=C-

argC1-}-argC2.

i and C1 4 C2, inequality (16) will not be sharp because, Furthermore, for a if it were, the extremal function F(C) would, as follows from the corresponding equations of the type (2'), map the circle I CI = 1 onto an arc of a logarithmic spiral with asymptotic point simultaneously at F(CI) and at F(C2) or ( for a= 0) onto an arc of the circle with center simultaneously at F(C1) and F(C2), which is impossible. With regard to the complete answer of the question that we have touched on here, we present the following theorem:

Theorem 2. Let a denote a real number and let C1 and t 2 denote distinct complex numbers outside the disk ICI > 1. Then, among the functions F(O E Y, the function maximizing the quantity legta iog

F

(18)

(L8) 1

(C )- F

maps the domain ICI > 1 onto the entire w-plane with cut along the curve defined by the equation

w _ F1±Fs+ F1-F2 (e(c+ioe to 2

2

+

C

(c+il)e-à

(19)

where t is a real parameter defining points of the curve. For a= 0, this will be a cut along the ellipse with foci at F1 and F2. For a= n/2, it will be a cut along the hyperbola with foci at F 1 and F 2, Proof. Following the same reasoning as in the proof of Theorem 1, let us show that extremal functions exist, that each of them maps the domain ICI > 1 onto a domain B without exterior points, and that each of them in I I > 1 satisfies a differential equation of the form efia

Cg F e

s) (F- F=

Q C(F-F1)

where Q(C) is a rational function that is real and nonnegative on I4I=1. When we solve this equation, we get the following dependence of w = F(C) on C:


136

which also holds on the circle IC I = 1. From this we conclude, just as above, that all boundary points of the domain B satisfy the equation

eialog( w-Fl- w-F9)=c+2lt, where c = const and t is real. When we solve this equation for w, we obtain a parametric representation of the boundary of the form (19). This completes the proof of the theorem. We mention without proof 1) that the quantity W- =log

F (Cl) -F (Cs) Cl - CA

has, for arbitrary C1 and C2, a disk for its range of values. But neither the radius nor the center of that disk can be expressed in terms of S1 and S2 by the use of elementary functions.

Theorem 3.2) For functions F(C) = C + a1/C+... E E, we have the sharp inequality F(C) -}- 1

-

I

-2

C A-

- 1. Here we use the notation

K (k) _ c

dx

(1- x$) (1- A'x')'

-

(20')

E (k) = SJ

1 - k9x9 1 - x9

dz.

0

Inequality (20) defines the range of values for the quantity F(0/C for arbitrary given C in the domain ICI > 1. Consequently, this range is a disk. Proof Let us look at the problem of maximizing the quantity IF = Rt (C)) (e91a F

1) Goluzin [1947 I. 2) Theorems 3 and 4 were proved by the author (see Goluzin [1943a1 ).


for given

F(C) = C+

137

>.h.nd for F(C) ranging over all the functions

where E aI/C+

Obviously, this maximum coincides with the maximum

of the quantity

rF = R (ells (F (Ci)

- an)) among all the functions F(0 E E. ( Here a0 is the constant term in the La urent expansion of F(C).) But I`" has equal values for nonconstant functions F(0 EE. Therefore, among the extremal functions of this last problem there are functions without zeros in Icl > I. Let F(C) denote one of them. By applying the varied function (25) of §3 of Chapter III, let us show that the image B of the domain ICI > 1 under the mapping w = F (C) does not have exterior points. If we then apply the varied function (26) of the same section, we get the following differential equation for F (C) in ICI > 1: C9F'4 Fe9i F1= Q (C),

where Q (0 is a rational function such that Q (C) > 0 on ICI = 1. If we now solve this equation, we conclude that the extremal function F(C) satisfies on J C I = 1 the equation Rt(e'a

F(C)-F(C1))=c, c=const.

(21)

Resting on this circumstance, we can construct an explicit expression for the function F(C). We need only consider the case 6 > I. Then, we have, on 16 = I. I4=(C-C1) (I-C,C)

IC-C,

C

If we divide (21) by I4- S1 I, we get

t(e ,a

F

C (F (C)

(C-C,)(1-C1C))-c

(22)

(C-C1)(1-C,C)

Since the function V (C-C1)(I-C1C)

is regular in CI > 1 and, obviously, continuous on I CI > 1 and since its real part on ICI = 1 is given by formula (22), this function itself in ICI > 1 is determined by the Schwarz formula (the integral is taken in the counterclockwise direction): e

C(F(C)-F(C1))

to

V

(C - C1) (1-C1C)

c

ttIt'I S

C'

C'+ Cdt'

(C'-C,)(1-C,C')C'-C C'

-{ ict

(23)


138

From this we obtain F(C). To determine the real constants c and c1 and also F(C) - ao, we use the normalization of F(O. Specifically, if we understand the radical appearing in (23) as referring to that branch that approaches 1 as V o and equate the constant terms and coefficients of 1/C in the expansions of the two sides of (23) about = we have e1'

t

dC'

c 2xl`,

C,

=1

(24)

k1

C7(C1-C')(1-C1C')

is (

2i

+'ao(Ct))=-

1C1-{-C1

IC'I=t

C'(C,

(25)

CC)(1-C1C')

Let us make cuts along the real axis of the C'-plane from 0 to 1/C1 and from C1 to 0o and if we take for YC' (C1_00 - C1C') that branch that is positive on the right-hand side of the cut (0, 1/r;'1). Since the integrands in (24) and (25) are regular in the cut planes, we can deform the paths of integration and, in particular, we can constrict them to the path extending from 1/C1 to 0 along the upper side of the cut and then from 0 to along the

lower side of the cut (0, 1/C1). As a result, we obtain ti

ele I

dC'

c

n

C1

JC'

(1 -C1C')

+ 1C

,

(26)

I

era

2i

2c '

II

C'dC'

C'+ + -ao- F (C1))=n

(27)

C'(C1-C')

where the radicals are understood to mean their positive values. By equating the. real (and imaginary), parts of the two sides of (26), we determine c and c I. We then obtain from (27) an expression for F (C) - ap : C1

`

,F (CI)-Qo=CI-

1

C1+2(e

9ia - 1)

C'dC'

O

VC'(C1-C')(1-C1C')

1

dC'

3/ C' (C1- C') (1- C1C')


139

If we make the substitution -xs SjS' in these two integrals and use the notation (20 ') for complete elliptic integrals, we obtain r1 +2C,(e-sta-1)r 1-E1C/

F(C,)-ao=Ci+C

T,

K(

For the maximum that we are seeking, we have IF=OR

(e2'%

E (1

(Ci

.+2Ct

CI)+2CI

1

K

_E

1

K \C,

From From this it follows that for all the functions F(C) _

+ aI/C+ E E and arbi-

trary given a, we have es;a

(F)+C-

-E(C1l

1

1

2C

(28)

1

K1C/

for t> 1. By virtue of the arbitrariness of a in (28), we obtain inequality (20). That this inequality also holds for complex C in the domain ICI > 1 follows from examination of the function The remaining conclusion regarding the range of values of F(C)/C is proved just as in Theorem 1. This completes the proof of the theorem.

Theorem 4. For functions F(0 E I in CF"(C)

P(C) +

41C 1--2 IC19-1

41CIO

E(ICI

> 1, we have the sharp inequality

c

4101'

/1-E\ICI/ Ir

KÌCII

I

(29)

ICI'-1 K11Cl/ Inequality (29) determines the range of values of the function V"(0/P(C) for arbitrary given C in ICI > 1. Consequently, this range is a disk .

Proof. When the function F(C) belongs to the class F ) 0 r FI ( C ) F 1±.ICLC I') + 2 (1 - I Ct Is) \

so does the function

=c -I

+...

for every CI in ICI > 1 and conversely. If we apply inequality (20) to FI(C) at the point C= ZI , we obtain inequality (29) for C_ CI . Conversely, from (29) we can obtain (20). To do this, note first that, if f(z) E S, then, by applying (29) to the function F(O ^ 1/f(11C), we obtain in I z I < 1 the inequality


140

z f " (z)

f' (Z)

2

' 4 - 2 1 z 1' + z f ' (z) + 4 E (I z 1) 2 f(Z) l -IzI' 1-IZ13K(IZU

4Iz12(1-K(Izl) Second, if f(z) = z + c2z2 +

(30)

E S, then, by applying (30) to the function f1fc+z

g(c)=

...Es

f-(Z)(1-Iz12)

(where I z I < 1) at the point t; = - z, we obtain in I z I < 1 the inequality

If(z)+c2z+1-Izj2-2

Finally, if F(C) = C+ a1/C+

(LO

(31)

E I and F(0 1 c in C1 > 1, then, by apply-

ing (31) to the function f (z) = 1/[F (1/z) - c] E S, we obtain inequality (20) in CI > 1. From this we get the conclusions on the range of values. This completes the proof of the theorem. §4. Application of the method of variations to other extremal problems

In all the cases considered in §3, the method, of variations led to complete solution of the extremal problems. This method can also be applied to various other extremal problems. However, final solution of the problem frequently depends on the determination of a certain number of unknown constants. We shall now stop to look at certain problems of this type. l) Let us now apply the method of variations to the investigation of the extremum of the average diameter of the boundary of the image of the domain > 1 under functions of the class . .2)

Theorem 1. Let F(0 denote a function belonging to I. Let n denote an integer > 2. Let PF denote the least upper bound of the product

Ia, - avI V$YI

(1)

over all systems of n points av, v = 1, -, n, located on the boundary of the image of the domain ICI > 1 under the function w = F(O. (The product (1) is 1) See Goluzin [1946f, 1947, 1951d]. 2) Goluzin [1947, 1949c].

§4. METHOD OF VARIATIONS

141

, n.) Let Pdenote taken overall pairs of distinct-numbers v, v' taken from 1, the least upper bound of the quantity PF over all functions F E Y'. Then P is attained by some function w = F (C) that maps the domain I ' > I onto the entire w-plane with analytic cuts and that satisfies the, differential equation [F1p12

C2

in

1

1

1

(F(C) - a") (F (C) - a",)

-n1 (n -

)

(2)

1.

Proof. In addition to the quantities Pp and P mentioned in the statement of the theorem, we define the following: for a given function F(C) E L_ we denote by PP ' F the maximum of the product [1 I F (C P) - F (C",. P)

(3)

V:f V,

over all possible systems of points C"P1v = 1,

, n, on the circle

( = P,

where p > 1, and we denote by PP the maximum of PPP. F over all F (C) E 1.

By virtue of the normality of the class 1, the maximum PP obviously is attained. Let FP(C) denote one of the extremal functions. We denote by C"'P, for v = 1, , n, the points on the circle p for which the quantity (3), set up for the function FP(C), attains its maximum PP. For the system of these points Cv, P, let us consider the extremal problem of Theorem 1 of §3. One of the extremal functions in that problem is obviously FP(C). It follows from the proof of Theorem 1 (equation (5) with y,,,, v = 1 for v v' and y v, = 0 for v = v') that the function FP(S) must satisfy in the domain > 1 the differential equation [CF (C)]9

I

V

(F(C) - F", P) (F(C) - F"',P) :Pl v

=n(n- I)+2

F' C'V.FF"P 1

11

(F",P - Fv'. P) (C - Cv,P) CV, PF".

P

v,P-Fv, p) (ft,, , - I)

(4)

where F,"P= FP(CvI P).

The right-hand member of this equation can be simplified further. Note that, if we replace one of the values Cv, P with Cv, Peta in the expression (3) written for FP(C) and for the values Cv, P, where v = 1, , n, we obtain a function of


142

a defined in the interval - rr < a< rr that attains its maximum at a= 0. Therefore, the derivative of the logarithm of this function vanishes at a= 0 and this leads to the condition Cv,P.F,

P

Fv,P-Fv'.P V1=1

_ o, v,

where the summation is over all v' = 1, , n excluding = v. Since equation holds for all v = 1, , n, we conclude that the numbers

this

n

v Cv, PFv, F, P V'=1 ,

v=1,

n,

-P

are all real. Therefore, the last term on the right side of equation (4) is equal to Cv, PFV. P v

v'

(Fv,p-Fv, P)(CCr,p-1)

Then, it can easily be combined with the preceding term. As a result, equation (4) takes the form 1

[CF' (C)]'

VI (F (C) - F, p) (F (C) - Fv',P)

=n(n-1)-2C 1 }, (FY vt-v'

Fv,

1)

) (C - Cv, p)'pFr'P (C-C,

p -1)

(5)

for w = FP(C) in ICI > 1.

Now, by virtue of the normality of the class 1, the condensation principle can be applied to the functions FP(C), treated as functions of the parameter p > 1. Thus, we can choose a sequence of functions that has the following properties as p --> 1:

1) FP(C) converge in > 1 to a function F(C) E 1, , n; 2) v = 1, , n, converge to some Cv, v= 1, Sv, P, , n, converge to av (points on the boundary of the image of 3) Fv, P, v = 1, the domain > 1, under the mapping w = F(J)); 4) 1) IF, (Cv,dI converge to certain finite numbers. This last is possible, because, for example, from the second of inequalities (15) of §3 for F (C) e I in I C1 > 1 we have (1 - 1/I CI ) I F'( C) I < 1. We note that the points av, for v = 1, , n, are all distinct because, otherwise, for the chosen p ---, 1

143


we would have PP --+ 0, which is impossible since it is easy to find a positive lower bound for PP that does not depend on p (calculating, for example, PP, F, where F (C) _ e . If, for the sample made, we take the limit in equation (5), written for F =

FP(C), as p -- 1, we obtain the following differential equation for F(C) in

ICI>1: [CF'V'2

\

1 Y#Y

(F(C)-a,,)

n

=n(n- 1)+C

I

C, 9,

cY = const.

(6)

Y=I

It is now easy to show that cv = 0 for v = 1,

- -

, n: We write equation (6) in inte-

gral form: 1

S

(F- aY) (FY#Y

a,,) dF

nn-1

A

I

cY

]

Y=1

the left-hand member of If some cv 10, then, in a neighborhood of the point which (7) is bounded whereas the right-hand member behaves like log is impossible. Thus equation (7) reduces to the form (1). We need also to show that the function FW that we have obtained is the extremal function in the problem mentioned in the formulation of the theorem. To do this, we note that, for given c > 0 and fixed pI sufficiently close to 1 but greater than 1, we have P > PP1 F - c. Furthermore, for a sequence p' -- 1, p' > 1, such that FP, (J) --+ F(C) we have, from some point on PP1,F - c > PPI,FPS 2c > PP, F , - 2c = PP, - 2c. Consequently, P > PP, - 2E. On the other hand, for these same values of p ' , we have PP, _> P. This shows that PP, --+ P; that is, the function F(J) exhibited above is extremal. > 1 onto the entire It remains to show that w = F(C) maps the domain w-plane with analytic cuts. The analyticity of the arcs constituting the boundary of the image follows from equation (2). We denote this image by B. That B does not have any exterior points follows from the fact, that, if it did, we could, without changing PF, add on to B a two-dimensional segment and then for the function w = F. (c), with IF'(-) I < 1, mapping I CI > 1 onto the modified domain B' we would have I F. (..) i < 1, by Lindelof's principle. But then the function w =

144


F' (01F*(-) E I would give PF a higher value than the function F (c), in contradiction to the extremality of F(C). This completes the proof of the theorem. The problem considered in Theorem l with n = 2 constitutes the problem of the extremum of the ordinary diameter of the boundary of the image of the domain 16 1 > 1 under the mapping w = F (C) E Y.. Its solution is well known: P = 4, and the extremal function is the function F (C) = £+ ??IC + const, where l1l = 1. This same result can easily be obtained by solving equation (2) with n = 2. The complete solution of this problem for arbitrary n is laborious. We note that the function F(0 = C(1 + rl/C")21n E E, satisfies equation (2) for arbitrary n > 2 also, for this function av = 22/net n"il" for v = 1, - , n. However, for the case n = 3, we can carry the solution through to the end.

Theorem 2. Let F(C) denote a function belonging to E. Then, for arbitrary points a1, a2, a3 on the boundary of the image of ICl > 1 under the mapping

w=F(C),

l (at-as)(at-aa)(as-aa)

12,

(8)

with equality holding for the function 2

F()=C(I+to

IT11=1.

(9)

Proof. Let us consider the extremum of the left-hand member of inequality (8) over the points a1, a2, and a3 mentioned in the theorem and all functions F(C) E E. We arrive at the problem considered in Theorem 1 with n = 3. One of the functions providing this extremum must, in accordance with Theorem 1, satisfy the differential equation (2), which for n = 3 can be written in the form F

(C)-a1+a,+a8 3

`2 [F (Q]9 (F(C)-a1)(F(C)-a9)

(F(C)-a.)=1.

(10)

Here, a 1, a2 , and a3 are boundary points providing the corresponding maximum for the left-hand member of (8). The fraction on the left side of (10) cannot be reduced further. To see this,

suppose, for example, that (a1 + a2 + a3)/3 = a1, that is, that (a2 + a3)/2 = a1. Then, since lag - a3 l < 4, we would have j al - a2 l < 2, la, - a3 1 < 2, and, consequently, P < 16, whereas for the function (9), which maps the disk lCl > 1 onto the w-plane with three rectilinear cuts extending from the point w = 0 to the points 22/3i71/3e2rrv`/3 for v = 1, 2, 3, the maximum of the right-hand mem-

145


her of (8) is equal to 12/3, which exceeds 16. But if the fraction cannot be reduced, the function F(C) must have expansions of the form

F(C) = av + c9( - C + ...

,

c2/:O,

for v = 1, 2, 3, on ICI = 1 corresponding to the points a 1, a2, a3. These expansions show that av , for v = 1, 2, 3, are the endpoints on the boundary of the image of the domain I z I > 1 under the mapping w = F(C) and the point (aI + a2 + a3)/3 is a multiple boundary point out of which issue the boundary arcs extending to av for v = 1, 2, 3. Since the extremality of the function F (C) is not lost when we add a constant to it, let us normalize the function in such a way that a1 + a2 + a3 = 0. If this extremal function has an expansion in neighborhoods of the points

,

then, by substituting it into equation (10) (with a1 + a2 + a3 = 0), we find that

ay= a0=0' a t= - - alas+alas+a9aa 4

a,a,aa. 6

(11)

Suppose now that x3 + px + a is a polynomial with zeros x= a 1, a2, a3. Then,

p = alai + alas + alas = - 4alr

q= - ala2as = - 6;. With regard to the left-hand member of (8), its square is the absolute value of the discriminant of the equation x3 + px + q = 0, which is equal to 4p3 + 27g2, so that we have (at - as)(at - as)(aa - aa) I9 44,al + 30.4al I < 4(431 a, Is + 311 w 19)

(12)

Now, let us find a bound for

Q=41 IaiI1+311as11

(13)

over all the functions + in the class Y.0 to which our exC+ aI tremal function belongs. Let us denote by C' the image of the circle ICI = p, for p > 1, under the mapping w = F(C). From inequality (4) of §7, we obtain, by constricting the curve C to the point w = 0,

SR14): 0, x > o.

C'

146

1V. EXTREMAL QUESTIONS

From this we get, just as in §7, for A> 0 and p > 1, 2x

I F(pei) Iado -- 0. dP

Let us apply this result with A = 3. We set F(C)3

C3/3 (I

c

c

3 3 cg=2a1, c8=2aQ..

For p > 1, 00

9 -2 (n- 2

3

r

\\

Zp.

n-2

If we take only the first two terms of the series, we get, by letting 3,

Iel11+31c911

that is, Ia112+31a2 12

Therefore, 31 a2 12 < 4/3 - I a 1 12. Consequently, (681

QC481a',18+33.4-31 1al11=38.4-4a11 4-Ia11 Since I al 1 < 1, so that the parenthetical expression on the right is positive, we

have Q < 33 4. Therefore, in accordance with inequality (12), we have (8) with equality holding for the function (9) with suitably chosen a1, a2, and a3 in (8). This completes the proof of the theorem. This theorem leads to a number of corollaries.- We present one of them that

deals with the class S. Corollary. Let f (z) denote a function belonging to S. Then, for an arbitrary triple of points c1, c2, c3 that lie on three rays issuing from w=0 forming equal angles with each other and that belong to the boundary of the image of the disk I z I < 1 under the function w = f (z), we have

cl c9 c3 >

(14)

4

with equality holding for the function

f(z)=

z

(1 +, za) a/a

III=1.

(15)


147

Proof. We may assume that all the cv are finite. Consider the function

F(0 = 1/f(1/t) E E0. Then, the points av = 1/cv, v = 1, 2, 3, lie on the boundary of the image of the domain I CI > 1 under the function w = F (C) and, in fact, on three rays issuing from w = 0 at equal angles with each other. From Theorem 2, we have inequality (8). On the other hand, for v v' (v, v' = 1, 2, 3, we have,

a,-a,

-I a,, I9-2cos23 Ia,a,, a,12+ Ia,,11+Ia,a,'I =31a,a,,I

and, consequeently, (a1 - a9) (a1 - a3) (a3 - a3) I i 3 V3 I a1aSa3

This last inequality, together with (8), yields I a 1 a2 a3 l < 4, which is equivalent to inequality (14). We verify directly that equality holds for the function (15) and the points cv = 2-2/3q-1/3e217vi/3, v = 1, 2, 3. Without going into detail, we mention the following. By using Schiffer's studies [19381, one can show first that equation (10) holds in the domain ICI > 1 for an arbitrary extremal function of the type mentioned in Theorem 2 and then that equality holds in (8) and (14) only for the functions (9) and (15) respectively.') Let us look at the following problem on the extrema of the coefficients in

the class S. Theorem 3. Among all the functions w = f(z) = X,°=1cvz' E S, the maximum of the quantity (16)

Tyco

for arbitrary given n > 2 and arbitrary given complex numbers y,, for v = 1, , n, where y,, X 0, is attained only by functions that map the disk I z I < I onto the entire w-plane with analytic cuts that extend to w = oe without finite multiple points and that form an angle less than n/4 with the corresponding radial direction at each point.

Proof. The existence of extremal functions is obvious. Suppose that f (z) is one of these. The function f (z) also maximizes the quantity

JI = t (e`,

T,cy)

r=1

(17)

1)(Theorems 1 apd 2 and the more general corollary of Theorem 2 have recently been proved by Reich and Schiffer [19641 with the additional assertion that the extremal functions are unique.)


148

where 0 is a real number, out of all functions of the class S. Let us suppose that the image B of the disk I z I < 1 under the mapping w = f (z) has an exterior point w0 . Let us form the varied function f (z) in accordance with formula (21) of §3 of Chapter III with m = 1. Let us denote by (00}n the coefficient of z' in the expansion of an arbitrary function 96(z) that is regular at z = 0 in a neighborhood of the point z = 0, taking AA I = h. We have "-1

f (E)

{f» (E)}" = c" + h

f ()

Tao

J + O (hs) = c" + h I

if (Ek

k=l


"-I

i,» = Jj + FR llh"=11 k=1

tyk

(hs).

J

°

+ 0 (hs),

°

`+

7"1 {f(E)k+,}

+1 }

Since the expression in parentheses is a polynomial in 1/w0 and the coefficient of 1/w0"'1 in that expression is equal to y / 0, this polynomial is, for the value of w0 that we are considering and wo arbitrarily close to w0, .also nonzero for the corresponding function f. (z) with small I h I and suitable arg h, J1- > Jf, which is impossible. Thus B has no exterior points. Let us now consider the varied function (24) of §3 of Chapter III with m= 1 and arbitrary z0 in the disk I z < 1. We/ have

tf# ()}" - "

+hf

fI(E)'

l f (E) - f (Zo)

-h

(

" -hf

'C

fo ` Zo

)

"

fo 1' (zof' (E)1 _f _ h-( fo ) ' Z 'A 1 zafo J 1 E- Z. f"

1,

zOE'f (E) 1

1- 2°E J" + Q (I h Is)

and consequently, n

Jf.=Jl+51 (eteh I17"(E)

-

n

eiBh Zofo (fo 1' f

n

((ef(zo)1"eieh Goofo )"Y1T"e" n

" Jzo`f' (E)1 + e'oh. (foo 1'"=f" J oE'f' (E)l) 1 Ezo J" 3 EJ + 0(1 hIs) Z' ( 1

r=1

1

The condition of extremality of the function f(z) requires that the inequality Jir. < Jf hold, that is, that


(ereh

7V

re fo 1' e h (zto

f (E) IRE) -f(zoJ.,

l

I1TVc,

149

fo 1' - e reh (z too /

a

z Ef

71{E - zo J,

+eie)i,(

T,l For small

IhI

1oE'

(E)

and arbitrary arg h, this can be true only if

ere

zT f (E)'

If

f(E)-f(zo)

}

v

n

l9

(r

f

(ere,(Yc. + et0T

/ 11

IzE

I zo)1Y-ereTV { i Eef'(E) }Y) =

0.

(18)

Since (18) holds for arbitrary zo in I z I < 1, we can by replacing z0 with z in equation (18), obtain a differential equation for f(z): n

ere (zf' (z)1'

J

f(z)

f (E)'

f(E)-f(z)

1

n (ere,(YcY

.=t

11

+

et0

JzEf (E)1 1

E-Z V

f zE'f (E) 11

ere

7°

l 1-3E lYl

(19)

It follows from this differential equation that the boundary of the domain B consists of a finite number of analytic arcs and that equation (19) holds for w = f(z) on the circle I z I = 1. Let us show that the right-hand side of formula (19) is negative for z I. We note that a function w = O(z) that satisfies the conditions t/i(0) = 0 and i/i' (0) > 0 and that maps the disk z I < 1 univalently onto the disk w < 1 with radial cut of length r issuing from the point w = e-`q6, can be determined from the equation I

werI-{- were - (1

zero -}-

h)

)+2h, sere

1

9

h - 4 T__8

For h, we obtain

yr(z)=z-hz i+zer ; +0(h9), 4+' (0)=1-h, h>0. w

Let us use this function to define a function in the class S: fk (z) =f ,+(((z)) =f (z) + hf (z) - hzf' (z)1 +

zzeil

erlp + 0 (h9).


150

For this function we have n

n

Jf.=Jf+hUt(e;e T+c+-e

1- Eet

0(h9). y

Consequently, by virtue of -the extremality of f (z), we have on I z I = 1 n

t (eie Y

n

+ efe

,

+-1

17+

v=1

{Ef' (E)

i

l

±E

< 0.

(20)

The difference between the right-hand side of (19) and the left-hand side of (20) is equal to n n ye+

ete

l.

+=1

But this quantity is equal to zero, as is shown by comparing its value for the function f (z) with its values for the functions e-"-"f (e àz) E S with real a. This proves that the right-hand side of (19) is negative for z I = 1. Let us now show that, if our extremal function w = f (z) does not map the disk I z I < 1 onto the w-plane with rectilinear cuts that would include the point w = 0 if extended in the opposite direction, then the real part of the quantity n

Q zm=

e;eI

J

f(E)°

+-I which appears in equation (19), is positive at all finite boundary points of the domain B. Let w0 denote an arbitrary finite point on the boundary of B. The function

-f( )(Z

0

f*(z)=wo1Of (z)=f (z)

'w0

(21)

belongs to the class S and for this function we have Jr. = Jf - ;7t (Q (two))

.

This shows that 92 (Q (w0 )) > 0. If 92 (Q (w0)) were equal to zero, the function (21) would also be extremal both for the quantity (17) and for the quantity (16).

This requires that Q(w0) be equal to zero. Let us write for the function f* (z) the equation corresponding to (19): n f*(E)*(E)

*(z)

R* (Z),

(22)

4. METHOD OF VARIATIONS

151

This R* (z) must again be negative on I z I = 1. But

_

f* (E)2

and, consequently,

f

P02

f (e), f (E) - wo

(z)

n

e"I

fs (E)'

} f* (E)-f* (z) = Q (f (z)) - Q (wo) = Q (f(z)) v=1 Therefore, if we replace f* (z) in (22) with its expression in terms of f (z), we J

7y

1

obtain

1'(Zf

\ww

1

If we divide (23) by (19), then, for wo

P

(23)

)QQf)=R*(z).

z

= 1, we have

=,t:- 0,

1-Y=

wo -f (z)1

117

-

where T is a real number. This last equation shows that all boundary points lie on the straight line passing through w = 0. Since this was excluded, the contradiction that we have obtained shows that 5 (Q (w )) > 0 at all finite points on the boundary of B. But then we also have l2 (1/Q (w)) > 0 and we see from equation (19) that Zf (Z)1°1< o

(24)

f (_) J J

at all points on I z 1 at which f (z) is finite. The additional conclusions on the boundary of the domain B follow from equation (19) and inequality (24). We can see this as follows. If the boundary B consists only of radial rays, everything is proved. On the other hand, if this is not the case, then Q (w) 0 at all finite boundary points w and it follows from equation (19) that none of these points can be a multiple or an angular point of the boundary. Consequently, the boundary of the domain B consists of analytic arcs that extend out to ° . Furthermore, it follows from inequality (24) that, for a suitable value of the argument, we have arg

zf (z) + < 71 f (z)

2

4

on the entire boundary of the domain. But the quantity on the left-hand side of this inequality measures the angle between the tangent to the boundary of B at the point w = f(z) and the ray passing through the points w = 0 and w = f(z). Consequently, this angle must be less than it/4. This completes the proof of the


152

theorem.

Let us now apply the method of variations to the investigation of a particular extremal problem for the class S. of functions

f

(z)=clz+c%Z%-{-...,

that are regular and univalent in the disk I z I < 1 and that do not assume m given , a,,, , where m > 1. Let us look at the problem of maximizing finite values a 1, for given n > I with respect to functions of the class Sa (see Goluzin [1946f]). Icn I -

Theorem 4. A function w = f(z) = c1 z + c2 z2 + C Sa for which I c I, where n is a given integer > 1, attains its maximum maps the disk I z I < 1 onto the entire w-plane with cuts along a finite number of analytic arcs and satisfies in the disk I z I < 1 the differential equation _

n.-1

R((1(z)) z2f''(z)9=n+ om l/ \Cv Z -v L..I (n - V) Cn V( )) l V - I

CCnv zv),

(25)

where m

m-3

k=1

k=-n-1

P(w)= II(w-ak), R(w)_ E bkwk, the bk being constants. In equation (25), the right-hand member is real and nonnegative on the entire circle I z I = 1, and the boundary cuts referred to are with a suitable parametric representation w = w(t) (t real), the integral curves of the differential equation R(n')

P (w)

(26)

dt

Proof. The existence of extremal functions is obvious. Specifically, the quantity I c I has a finite upper bound because, if f(z) = I' k=1 ck zk E Sa, we k have the well-known inequality I c I _< en for the function f (z)/c 1 =1 c k z and consequently, I c I < en I c 1 I . If we apply the familiar inequality If (z)/c 11 > I Z I/(1 + I Z I ) 2 to a point z on the circle I z I = r at which If(z)I < I a11, we obtain c 1 I< 4 I a 1 a s r -+ 1. ConseI C 1 I< ((1 + r) 2/r) I a I. It follows from this that quently, I

Icnl 0 and they exhaust all extremal functions in the sense of the characteristic of the images B. Suppose that f(Z) = 17_1 Ck Zk is one of the extremal functions such that cn > 0. Let B denote the corresponding image. By using the varied function obtained from equation (27) of §3 of Chapter III, when we replace w with f(z) and take w I = w2 = .. - = wn = 0, we can show, following the same reasoning as in the proof of Theorem 3, that the domain B does not have exterior points. Nov\We can easily obtain a differential equation for the function f(z). Since the functioa\(29) of §3 of Chapter III is a function in the class S. for arbitrary distinct z 1, j , z,, in the disk I z I < 1 and sufficiently small A, we have, setting AA = /Y in that formula m

t{g E

-CC.+h {(E) k-l

1J

! k=1

(f (E) - ak)

m

hI Akf(Zk) k=

(f (E}-f(n

'(E lZkf' (Zk)9 E - "Z n

m

+

+ 0 Q h 9) 1 -2kE n where (0())In denotes the coefficient of z in the expansion of the function O(z) about z =0. But since $2((f" 1, that is, that I c i does not necessarily increase when the domain B is I

broadened.

In conclusion, let us examine the following problem on disjoint domains: Let a, a2 , , an, where n > 2, denote n distinct finite points in the w-plane. Functions w = fk (z), k = 1, 2, , n, that are regular in the disk I z I < 1 map that disk univalently onto disjoint domains Bk including the points ak , for k = 1, 2, , n, respectively in such a way that fk (0) = ak for k = 1, 2, - , n, The question arises: can we speak of the maximum of the product rt

J_HIfk(0) I k=l

n? over all possible functions fk (z), for k = In the case n = 2, the answer to this question was given by Lavrent rev L19341 (on the basis of his variational method). It consisted in the sharp inequality

If (p)f;(0)Icla,-al I9,

in which equality holds only for the functions f 1(z) and f 2,(z) that map the disk I z I < 1 onto the halfplanes about the common boundary of which the points a1 and a2 are symmetric. The existence of extremal functions is easily proved. Specifically, since the function gk (z) = (fk (z) - ak )/ fk (0), belongs to S for k = 1, 2, , a, and does not assume in the disk z I < 1 the values (ak 1 - ak )/fk (0), where k 1 L k, it follows from Koebe's theorem that (ak 1 - a0 f; (0) > %; that is, I fk (0)I 4I ak 1 - akI . This proves that the quantity I has a bound independent of the form of the functions f k (z) for k = 1, 2, - , n. From this and the normality of the family of functions fk(z) that we are considering, we conclude that the least upper bound J0 of the quantity J is attained. I


158

, n, denote a system of extremal funcNow, let w = fk (z), for k = 1, 2, tions and let Bk denote the corresponding extremal domains. On the basis of Lindelof's principle ( p. 30 ), the domains Bk are such that the union of their closures Unk=1Bk coincides with the entire w-plane. Let us prove certain variational formulas that we shall need. Let n denote an integer greater than 1. For fixed points ak, where k = 1, 2, .. , n, consider the function n77

11 (w-a,)

w*=w+h v=1

kon,

1

n

II (w-w") V=1

v#ko

where A is a complex number and the wv, for v = 1, 2, points. This function can be represented in the form

.

, n, v / k0 are distinct

n

w*=w+h w+ xo+ I

au-1

v#ko

From this we easily see that, for sufficiently small the function w` is univalent on the entire plane from which sufficiently small neighborhoods of all points wv, for v = 1, 2, , n, v / k0 are removed. Therefore, if wv = fv(z,,), for v = 1, 2, , n, where z v I .< 1, then the function I

n

f k (z) =fk (z) + h

1I (fk (Z) - av) -I

1

n"

kon,

(30)

11 Vk (Z) -A (Z,))

V=1

v#ko

is, for k = k0, regular and univalent in the disk I z I < 1. For k / k0 , the function (30) has a simple pole z = zk in the disk z I < 1. Consequently, the theorem in §3 of Chapter III can be applied to it. As a result, we obtain the function I

H (fk (z) - av) f *k (z) =fk (z) 4-h

n

II

V=1 v= ko

`

(fk (Z)

-

-fv (zv))


159

n

17

1 (A (4) - av)

zfk (z)

V=1

1

n

Zk V; (zk))9

z

v=1 v#k, ko

(fk (zk) -fv (Z9))

n

17

+h

1 (fk (zk) - av)

i°fk (z) xk

1

n

(fk (zk))2

1

v=1 -,#k, ko

Zkz

-I T 0 (I h I9)

(31)

(fk (zk) -fv (zv))

which is regular and univalent in the disk I z I < 1 for small values of I A I: also , n, thus defined now map the f k (0) = ak . The functions w = f k(z), k = 1, 2, disk I z I < 1 for small I A I onto disjoint domains and f k(0) = ak ,. k = 1, 2, ; n. Equations (30) and (31) are the desired variational formulas. From the functions f k (z) for k = 1, 2, , n; k ko , let us now construct the varied functions (31) with z 0 for v = 1, 2, . , n, v / k0 , which, together with the function fk0 (z) constructed in accordance with formula (3), are also admissible functions of the extremal problem that we are considering. Now, we easily obtain the equation n v=1

(ak - av)

ko

J*_ ft Ifk (0)I=Jo 1+9Z I h k=1

y111 (ak -fv (zv)) of k n 7 11 (fk (zk) - av)

n

+21 Zk (f(zk))9

+ O (I h Is)

n

k-1

.

(fk (zk) - fv (zv))

k*ko

v=l

v$k, ko

By virtue of the arbitrariness of arg A and the extremality of the functions fk (z) for k =.I, 2, , n, this last relation can hold only if the expression in the large square brackets is equal to zero, that is only if n

k=1

11 (ak vl v$k

lI _ '

9 -/'ko

av)

(ak ---A (Z9))

n

11 (fk (zk) - av)

1

I

T

zk (fk (zk))9

k_ 1

v=1 11 v=1

k - k °

v#k. ko

= 0.

(fk (Zk) -A (zv))

(32)

160


Consequently, this condition must be satisfied for an arbitrary choice of the points z k , for k= 1, 2, - . , n in the disk I z I < 1 and for arbitrary k0 = 1, 2, - - - , n. It follows from (32) that the functions fk (z) are piecewise-analytic on the circle I z I = 1. To see this, let us fix all the points zk, k = 1, 2, - -, n; k k0, with the exception of one of them zk 1 = z (where 1 < k 1 < n). We see that condition (32) becomes a differential equation for fk 1(z): z9 (f"k, (z))9 = R (.fk, (z)),

where R (w) is a rational function. Our assertion then follows from the analytic theory of differential equations. Let us look at some particular cases. The case n = 2 can be carried through to its completion without difficulty and we shall not stop to do this. Now let us look at the case n = 3. In this case, equation (32) with 'k0 = 1 yields (a,- a9) (a, - aa) _L (a3 - a,) (a, - a,) (a, - f, (z8)) (a, f a (za)) (a9 -f2 (23)) (a8 -f8 (23))

1=1( a

(aa-a,) (as -f9 (za)) (aa -fa (za))

+

v2 z 9)- a,)

f, (z9) -fa (za)

3 s (f2 (z9))'

l

II (fa (za) - a,)

fa (za) - f9 (z9)

zi v a (za))a

- ()

If we multiply through by the difference f2 (z2) - f3 (z3) and decompose the first three fractions into partial fractions, we arrive at the form A (fs(za)) = A(fa (z3)),

(33)

where A is an operator defined by

A (f (z)) - (a, - a9) (a, - as) _L (a, - a,) (aa - as) a, -f (z) a9 -f (z)

a

(a8-al) (aa-a9) _l as -f (z)

T

II (f (z) -a,,)

y-1

za V, (z))9

Analogously, for k0 = 2, we obtain from equation (32) A (f, (z1)) = A (fs (za))

(34)

Since equations (33) and (34) hold for arbitrary Z1 , Z2, and z3 in the disk Iz I < 1, it follows that in the disk I z I < 1 we have A (fk (z)) = C = const for

161


for k = 1, 2, 3; that is, the functions w = fk (z) for k = 1, 2, 3 are each a solution of the differential equation a

II (ak-w)

k=1

z°w'9

_ (a,-a,) (a,-a,) + (a,-a) (a,-as) a, _W a, -w + (a, - a,) (a, - a+ C, a, - w

(35)

C=const. 1

Of cdurse, the constant C may depend on al, a2 and a3 However, we are not able 'to determine its value. Let us consider separately the two cases C / 0 and C = 0. In the first case, equation (35) reduces to the form

YP(w)w'

Z,

3

7J=±'1,

(36)

11 (ak - w) k=1

where p (w) is a polynomial of exactly third degree, and its zeros are distinct from a1, a2, and a3. For extremal domains Bk, the union UJ=1 Bk coincides with the entire w-plane. Therefore, the. point w = oe lies on the boundary of certain of these domains Bk , let us say, for example, on the boundary of B 1 . Let us 1 corresponding to it under the mapping w = denote the point on the circle I z f 1(z) by z = z 1. Then, in a neighborhood of z = z 1, we have

f; (A+ B-f-...) _'z fi

A t- 0.

3

f,2 Consequently, 1(A'-}-+...)=_71ogz_

fa

f,

z-z,+

z,

z,

c,(z-z1)", v-2

This means that, in a neighborhood of z = z 00

ray =fi (z) =

L b, (z -- z1) v--2

b_9 t- 0.

(37)

If the point w = oe were also a boundary point of some other domain Bk (k 1), the same would be true about the corresponding function fk (z). But this is impos-

sible since the domains Bk, k = 1, 2, 3, are disjoint. Thus, in the present case, the point w = oo lies on the boundary of only one of the domains Bk. The other


162

two domains must be bounded.

Let us turn now to the case C = 0. Let us suppose first that the points ak are vertices of an equilateral triangle, that is, that ak = aek, k = 1, 2, 3, where f = eZnl13. Then, equation (35) (with C = 0) reduces to the form a'ww'

3

z'

w8-a°

(36)

When we solve it, we obtain explicit expressions for the fk (z):

fk(Z)=askl -+ckz)3, ckz \\\

1

ck=const,

k=1, 2, 3.

For the functions fk(z) to be regular in the disk I z I < I and for the union U jc=1 Bk to coincide with the entire w-plane, we must have ck = e 1k for k = 1, 2, 3, so that we finally obtain 2

fk (Z) =

I +e i¢kz

l

\ I - etakz

k=1,2,3.

3

(39)

From this we conclude that each of the domains Bk is the domain between two rays with an angle of 2n/3 between them with vertex at w = 0, and with bisector passing through the point ak . Calculation then shows that J = (64/27)1 a 13 or, what amounts to the same thing, Jo = 816

3

I (at - a.) (al - as) (a2 - as) I

(40)

Thus, for an arbitrary system of functions fk (z), for k = 1, 2, 3, that satisfy the condition fk (0) = ak , that are regular in the disk I z I < 1, and that map this disk univalently onto disjoint domains, we obtain the sharp inequality 3

fk (0) c

gl

Y 8 I (a1- a3) (at -- as) (as - as) I

(41)

Now, let ak, k = 1, 2, 3, be arbitrary. Equation (35) with C = 0 has the property that if we subject the unknown function w and the constants ak to the same fractional-linear transformation, that is, if we set W

cw

+d,

ak

cak + d ,

k =1, 2, 3,

(42)

ti then w as a function of z satisfies the same,differential equation a

k=l

(a1-a9) (al-as) I_ (a,- a1) (a3-a8) zew'4

a1-w

,

aq-w

(a]-al) (a8-a9) ay - tat

163


This is easy to verify directly by substituting the expressions (42) for w and the ak into equation (35). Using this property, we choose the transformation (42) in such a way that the points ak, k = 1, 2, 3, are mapped into the points ak = atk, where e = e 2n`13. Then, the equation for w reduces to the form (38). Consequently, ti ti its solutions are functions w = fk(z), with f k (0) = a k of the form (39) that map the disk I z I < 1 onto the three angles mentioned above. The functions w = fk (z), with Ak (0) = ak , that satisfy equation (35) with C = 0 are then of the form %

fk (z) =

k = 1, 2, 3,

afk (z) -F b

cfk(z)+d

(43)

and the domains Bk corresponding to them will possess the property (obvious but important for what follows) that the point w = o is a boundary point of at least two of the three domains Bk. Since we can verify directly that 3

3

fk (0) k=1

(a, - a.) (a, - al) (a. - a3).

-

H fk(0)

_

k=1

(a. - a2) (a, - a2) (a2 - a.)

Ip can be calculated in the present case from formula (40). Consequently, again in this case we have inequality (41). The case C = 0 does not always hold. For this case to hold, it is necessary and sufficient that the point w = -d/c corresponding to the point w = oo underti the transformation (42) lie on the boundary of one of the corresponding domains Bk , for k = 1, 2, 3, because it is only then that regularity in the disk I z I < I is ensured for all the functions fk (z) defined by formulas (43). However, inequality (41) always holds, that is, even when the ak are such that C 10. To show this, let us examine the question of the maximum of the quantity 3

I_

11 fk

(0)

k=1

(fl (0) -f2 (0)) (fi (0) -fa (0)) (f2 (0) -fa

(0))

over all possible systems of functions fk (z), for k = 1, 2, 3, that are regular and univalent in the disk I z I < 1 and that map this disk onto disjoint domains. Since I does not change when we replace the functions fk(z) with cj:k(z), where c is a constant, we can confine ourselves to functions fk(z) that satisfy the condition Ifk (0) I < 1. Now, just as above, let us show that I ff (0) I

4 Ifr (0) -fa (0) I,

Ifa (0) J C 4 Ifs (0) -f3 (0)

If9(0)1-4If3(0)-fr(0)I


164

From this we conclude that the family of functions fk (z) that we are considering is normal and that I < 83. Consequently, the least upper bound of the quantity I is finite and is attained by some system of functions f k (z), for k = 1, 2, 3. Let us set f k(0) = ak , for k = 1, 2, 3, and let us look at the question of the maximum of the quantity

J=

6 fk (0)

k=1

over all systems of functions fk (z), for k = 1, 2, 3, that satisfy the condition fk (0), that are regular in the disk I z < 1, and that map that disk univalently onto disjoint domains. According to what was said above, this maximum is attained by the functions f k(z). Therefore, on the basis of the exposition above, we conclude that the images Bk of the disk z I < 1 under the mappings w = f k(z) are such that the union Uk=lBk coincides with the entire w-plane. Therefore, there

exists a point a that lies on the boundary of at least two of the domains Bk. Let us define new functions k = 1, 2, 3.

These functions also map the disk I z < 1 onto disjoint domains, and the quantity I has the same value for them as for the functions f `(z) for k = 1, 2, 3 since I is invariant under a single fractional linear transformation performed on all the functions fk (z) for k = 1, 2, 3. This means that the functions f (z), for k = 1, 2, 3, are extremal under search for the maximum of the quantity 1. If we set rr f k (0) = 1/(ak - a) = ak , we conclude that the functions f k`(z) are extremal also for the maximum of the quantity

J'= Ik=1

I

with respect to all of the functions fk (z), k = 1, 2, 3, that satisfy the condition fk(0) = ak , that are regular in the disk Izj < 1, and that map this disk univalently onto disjoint domains. Since the point w = o lies on the boundary of at least two of the three domains onto which the disk I z I < 1 is mapped by w = f k* (z), it follows in accordance with the analysis made above that the functions w = f%`(z) are solutions of a differential equation of the type (35) with C = 0. But then, the maximum Jp of the quantity J is calculated, in accordance with (40), from the formula

4

64

81 Y g

1 (a; -- Q (al -

as) (a; - ay) I

§5. LIMITS OF CONVEXITY AND STARLIKENESS

165

This fact, in conjunction with the fact that the maximum of the quantity l is attained by the .functions f ` (z), leads us to the conclusion that this maximum is equal to 64/81 /. But then, for an arbitrary system of functions fk (z), k = 1, 2, 3, that are regular in the disk I z I < 1 and that map this disk univalently onto disjoint domains, we have I < 64/81 vl_, which is equivalent to inequality (41). We note, finally, that, since l is invariant under an arbitrary fractional-linear transformation of the functions fk (z) (but the same transformation for all three values of I), inequality (41) holds also when one of the functions fk (z) is meromorphic

in the disk IzI B9 bk bk

Consequently, Y,k=1akB2/bk < AB, from which we obtain (2). That equality holds in (2) only when the Sk themselves are rectangles completely filling the rectangle

of sides A and B is proved just as in Lemma 1. The possibility of conformal mapping of the strips asserted in Lemmas 1 and 2 follows from the fact that every simply connected region having more than a single boundary point can be mapped onto some rectangle in such a way that four given accessible points on the boundary are mapped into 01

z

8

the vertices of the rectangle. We prove this last Figure 11 assertion as follows: Let us first map our domain onto the upper half plane and let us denote by a 1i a2 , a 3 , and a4 the points on the real axis into which the four given boundary points are mapped. (We can arrange for all the ak to be finite.) We then map the upper halfplane onto the rectangle by using the function

-

C

dC

(C - al) (C - a1) (C - a1) (C -- a4)

Then, the points a I , a2, a 3, and a4 are mapped into the vertices of the rectangle.

174


As a consequence of Lemma 2, we obtain the result that the ratio of the sides of the rectangle is determined uniquely by the mapped domain and the points that are mapped into the vertices.

Theorem. 1) Let f(z) denote a member of S. Consider n rays issuing from the point w = 0 forming equal angles each with the next. On each of these n rays consider that point of intersection of the ray with the boundary C of the image of the disk z I < 1 under the mapping w = f(z) which is closest to the point w = 0. I

Then, at least one of these n points is at a distance no less than l/4 5/7-

from

w=0. Proof. Let us suppose first that the boundary C is an analytic curve and that the theorem is not valid. Then there exists a system of rays Lk, k = 1, , n, that issue from w = 0 at equal angles such that the boundary points on them that are closest to w = 0 are all at a distance not exceeding d < n 1/4 from w = 0. The function

w=fo(z)=dY4 -Z2

=az+...,

a=di/4 1, under the function w = fo W. Let us now partition the annulus Qr < I z I < 1 into a system of strips by drawing an arbitrary set of rays issuing from z = 0 in such a way that the following three conditions are satisfied: 1) The set includes all rays lk containing segments that the function w = fo (z) maps into segments OAk and all rays that are bisectors of the angles between adjacent lk (see Figure 12); 2) it includes all rays that the function w = fo(z) maps into curves tangent to C; 3) if this set includes a certain ray, it also includes the ray symmetric to it about the closest of the rays lk. The system of strips Sk constructed in this way can be partitioned into pairs of symmetric

v

1) Lavrent'ev and Sepelev[1930, 1937](Rengel [1933 ]).The proof presented here was given by the author [1939d] . The theorem can be generalized to the case of p-sheeted functions (see Goluzin [1948]).

§6. COVERING OF SEGMENTS AND AREAS

175

strips Sk, and Sk"" whose images Ik' and Ik " under the function w = fo (z) will have a common boundary segment on the rays Lk and will together constitute

Figure 12

a strip Ik ' U lk

which extends from so and returns to so . The entire w-plane exclusive of so and, consequently, the image of the annulus r < I z I < 1 under the function w = f (z) is partitioned into subsets contained in the strips Ik'U 1k" Also, by hypothesis, the number of such subsets of the image in question that lie in each strip 1k' U Ek" will not be less than two. Consider the pairs ak , and ak

of these subsets contained in the strips Ik'U Ik" and adjacent to s. Let

us denote by ak ' , bk ' and ak, bk " the sides of the rectangles into which they are mapped in accordance with Lemma 1 and let us denote by sk ' and sk their preimages under the mapping w = f(z) lying in r < z I < 1. Then, in accordance with Lemma 1, we have a'kr

bk,

ak

+ bkl

2a 1

(3)

1ogr Now, let us denote by Ak' and Bk' (Ak "" and Bk ") the lengths of the sides of the rectangles into which the strip Sk' (Sk ), and hence the strip 1A,

(ak 11) is mapped (as described in Lemma 1). Obviously, we may assume that AA, , = AV and B k ' = B k . Then, the strip I k ' U E k " is mapped into the rectangle Rk ' k with sides Ak ' and 2Bk ' such that corresponding to sides of length Ak ' are the parts of the boundary Ik ' U Yk lying on so. These last


176

mappings ak , and ak are mapped into strips contained in Rk' k and extending from one side of length 2Bk to the other. If we map these strips into rectangles as indicated in Lemma 2 and if we take for the continua on

the sides of length 2Bk, segments on £k' U IV' corresponding to the segments that, in turn, correspond under the mapping w = f (z) to the parts of the boundaries s k and s k N contained in Qr a_b, where a and b are both positive, we obtain

ak, + a,:,--,, 2

ak, a ,,

_

2

bk,

bk

V ak, akr, 4

2Ak,

Ak,

Ak

bk dkr + ak

Bk,

Bkr

B4r,

bk

But, in accordance with Lemma 1, we have Ak.)

(Ak,

Bk,+Bk"

_

2a

log Qr

Consequently, (g_L,

bk,

+

ak"

2S

log

1

Qr

When we compare this with (3), we obtain 2z'

log

1

2n 1

log r Qr But this contradicts the fact that Q > I. This proves the theorem for the case in which C is an analytic curve. In the general case, we need only apply this result to the function f(pz)/p C S, where p < 1, and then take the limit as p --' 1. By the same method we can

also prove the following theorem on the covering of the segments.


177

Theorem 2.1) Let f(z) denote a member of S. Then, in the image of the disk z I < 1 under this function there are n rectilinear segments issuing from w = 0 at equal angles (2rr/n) to each other the sum of the lengths of which can be made arbitrarily close to n.

Proof. Let us look at the image B of the disk z I < R, where R < 1, under the function w = f(z). We define the star of the domain B as the largest domain cdgttained in B that is starlike about w = 0, that is, that has the property that a straight line segment containing an arbitrary point in it with w = 0 is contained in it.', Corresponding to the annulus r < (z I < R in the w-plane is a doubly connected domain the inner boundary s of which will, for small r, lie outside the disk J w < r (1 - Sr) = p, where Sr -- 0 as r --p 0. Let us partition the part of the star lying outside s by drawing a finite system of rays issuing from w = 0 1) the system contains all tangents to the that have the following properties: boundary C of the domain B: 2) it is mapped into itself by a rotation about w = 0 through an angle 21rk/n for k = 1, -, n; and 3) the fluctuations in I w I on the portion of C between two adjacent rays (though not on the rays themselves) is less than a given t > 0. Since C is an analytic curve, such a system does exist. There are nm such rays, where m is an integer. Let us denote them in order by ll , 12, , l nm . Here, lm k+ q is obtained from lq by a rotation through an angle 2rrk/n (see Figure 14). Let us denote by pk (k = 1 , 2, , nm) the greatest distance between w = 0 and points of the part gk of the star lying between lk and lk+1 and outside s (for k = 1, 2, , nm, gnm+i = gl) By means of the function J= log (w/p), we transform this system of domains gk into a system of domains hk lying respectively in rectangles with sides of length log (pk/p) and ak . I

-

lK,F

Figure 14 1) Goluzin [1936a1. A generalization of this theorem was obtained by Bermant [1938, 19441.


178

Here, Gk"` , ak = 2n where amk+ 1 = a'1. On the horizontal sides of each of these rectangles is a boundary segment (see Figure 15).

I

dK

Figure 15

When the domain hk is mapped onto a rectangle with sides ak and bk such that the boundary segments mentioned above are mapped into the sides of length bk, we have, in accordance with Lemma 2,

ak : ak

bk J log Pk P

Therefore,

nm

Yk

Aftj bk k-l

m n-1

n-I

m

amk+1

at

J l-lk=olog Pmk+1 1=1 k=o log P

Pmkl PT

But the system of domains gk is the image of a system of strips contained in the annulus r < I z I < R under the mapping w = ((z) and the rectangles mentioned above with sides ak and bk are the rectangles mentioned in Lemma 1. Consequently, from what was said above, we obtain on the basis of Lemma 1 M

2n

log R

al 1=1

n-1 i1 k=o log

Pmk+1

Therefore, since 1'1=1 a1 = 27r/n, we have n-1 2

log

n-1

in

min L R ' k-o log-P 1=I r Pmk+1

If we now use twice the familiar inequality

2n U1

mI n

k=olog

Pmk+l P

(4)


n`

n n

I Ck k_1

n Ck, Ck

Y

179

0,

(5)

k

we obtain n-I nz

1

I k=0 log Pmk+i

n-1n

P

n-1

I1. log

k=o

log Pmk+l

k=0

P

Consequently, (4) yields n log

I

Pmk+i

P

n-I

R

log Pmk+i

max 1

P

k-0

or

(f n/ < max l

n-1 Pmk+l

k=0

Therefore, by taking the limit as r ---p 0, we obtain for some l1 in the closed interval 1 < l1 < m n-1

Rn c II Pmk+ii

(6)

k=0

or, again using (5),

n-1 nR c I Pmk+1ik=0

Let us now take an arbitrary system of n rays issuing from w = 0 at equal angles (2a/n) with each other and intersecting respectively gi 1 , 9m+11 , ' , 9(n-1)m 1 1 . The parts of these rays lying inside the star are of lengths X1

, A2 , .

. , an and hence are respectively greater than P11 - e, Pit - e,

' P(n-1)m+11

E.

,

Consequently, n

n-1

E 4 -- E Pmk+l, - nE ! n (R - E)

k=1

k=0

For R sufficiently close to 1 and a sufficiently small, this is arbitrarily close to n. This completes the proof of Theorem 2. As a special case (with n = 2) of this theorem, we obtain the result that there exists an interval of length arbitrarily close to 2 that passes through the point w = 0 and that lies entirely in the image of the disk I z I < 1 under the


180

mapping w = f (z) E S. Furthermore, when we rewrite inequality (6) in the n-1

log Pmk+1,

2n

form

log R

k=0

and take the limit as n

--1, 00,

we obtain the following inequality: 1) 1 2a

log R,

S

where w = pe`g' and the integral is over the boundary of the star of the image of the disk I z < R under the mapping w = f(z). On the other hand, if we raise (6) to the power 2/n and if we apply inequality (5) to the right side of the resulting inequality, then, when we take the limit as n = oe, we obtain the inequality 1

2

S p'd

where the integral is over the boundary of the same star. This integral is equal to the area of the star. By letting R approach 1, we obtain Theorem 3.2) if f (z) is a member of S, then the area of the star of the image of the disk I z I < 1 under the mapping w = f(z) is no less than rr. Starting with other considerations, we can obtain the stronger result:3)

Theorem 3 '. If f (z) is a member of S, then the product of the area of the star of the image of the disk I z I < 1 under the mapping w = f (z) and the area of the preimage of this star is not less than n2. Proof. Consider the star of the image of the disk I z I < R, where R < 1, under the mapping w = f(z) and let h denote its boundary and l the curve corresponding to it in I z < 1. This last curve is a closed Jordan curve. The function

F (x) =log f Z) = U-}- l V, F(0)=0i is regular in I z I < 1. Consequently, from Green's formula, we have, taking z = x + iy and using the Cauchy-Riemann conditions

UdV= S Uadx+Uydy a

1

_

SR

I

S,

1) Goluzin_[1917a . 2) Goluzin 3) Bermant[1938a].

/du av l az ay

OU dV`

Ty dz dz dy =

5 IF (x) I9 dx dy,

I

S,

s


181

d0

that is, by setting z = re`d` and f (z) = pe`g', log

r

or

rd +logpdy.

log pd + log x

A

1

(7)

1

On',the other hand, if we integrate by parts, we obtain

0= 3 (1S

= S log P- d'f +

ZZ) dz)

log I dp -

S log

`P) d log r

rd(

log p dy -

log r

i.e.

logpdp=S logrd+. But this last integral is equal to 2a log R because v' = const on the radial segments lying on A and log R on the remaining arcs. On the basis of this, inequality (7) takes the form

log pdq+log rdp:-:- 4nlogR.

(8)

Let us now consider the doubly connected domain B. bounded by the curve

A and a small circle y: w = e. From Green' s formula, we have, just as above log p

dt

- J=2

du dv log p

5

WI

(9)

P

I

with a double integral taken over the domain BE . Now, if we denote by SR the area of the domain B bounded by the curve A (that is, the area of the star) and note that the integrand in. the last integral in (9) is no less in the part of the disk w < SR/7r not belonging to the domain B than in the part (of equal area) of the domain B lying outside that disk, we conclude that du dv

SS B`

du dv

I'-

SS
0 and 0
0 (it obviously holds also with A < 0) we get the inequality (4)

SR'd(Dc5R'd(D.

C'

C

In particular, if C' is the circle w I = R0 , then (4) yields R'd(D _2itRo.

(5)

C

If we apply inequality (5) to the case in which C is C, and R0= M(r) + e, where c > 0, and then let a approach 0, we obtain from (2) the inequality 29

2aa

I f (re") I' dO

dr

M rr)l

,

integration of which with respect to r from zero to r leads to inequality (1). This completes the proof of the lemma. Lemma 2.1) Let f(z) E S. For 0 1/2. Then, for 0 < r < 1,

(14)

C(1-Iz1Y,

2n

2a

I f'(re'e) I drr< (1M' -r)a

(15)

where M 1 depends only on M and a. Proof. Since, by (14),

M(P)=ImIxPIf(z)I
2, µ are nonzero. composed of the numbers q = e2 , where v Proof. We set 7jj1

... ,

We denote by Av rV, the cofactor of the element

V 'q v'

of this determinant. In

accordance with the properties of determinants,

(0 for k=v',

L 71k

l 0 for k

=t

v'.

k= 1,

.,µ

if we multiply these equations respectively by rlkv, for k = 1, - , µ, and add the resulting equations, we get, the properties of the roots of unity, p.Av, = O,1.r

Since A L 0, it follows that A v v the proof of the lemma.

0 for v, v ' = 1,

, µ . This completes

Theorem 1.33 Let f(z) denote a function belonging to S. Let it and v denote a pair of natural numbers with 1 < v < µ. Suppose that the inequality 1) Goluzin [1948a]. Bazilevic [1951] and, independently, Milin (see Lebedev and Milin [19511) have given a sharp bound for Icnl in the class S. The bound given in the text was found by Milin (see Lebedev and Milin [1949] ). (In 1964, Milin showed that Icn I < 1.243 n(see §2 of the supplement).) For an improvement in the bounds for the coefficients of odd functions, see Alenicyn [19511. (In 1951, Gong Sheng (Kung Sheng = Kung Sun) showed that, for the coefficients of odd functions, Icn I < 2.55 (see §2 of the supplement).) 2) Schaeffer and Spencer [ 1943] and also Goluzin [1949c].

3) Goluzin [1948b].

§8. THE RELATIVE GROWTH OF COEFFICIENTS

191

I < cna, where a> 0, holds for all n = µk + v for k = 0, 1, . Then, the inequality Icn I < c ' na, where c ' depends only on c, a and ft, holds for all . For a< 0, this assertion does not hold for all p and v. n = 2, 3, Ic n

Proof. We define co

1al i- "

n=0

Then, J (x)

V, (z).

(1)

v!=l

Now, from the conditions of the theorem, we have in the disk I z I < 1 Co

00

I Cµn +. I rµn + v C cr ' (µn + V)a rµn

I fw. I (z) I C

n=0

n=0 `°O

ooo

7, (n+ 1)arµn 0, when we apply inequality (11) of §7 with in

=

µn/2a and x = r, we obtain (2a)a

a -1

µn

r2(1-r)a Consequently, (z)

cr (µa

jr W

2saQa .

- r)a

2`

n=1

)

C

Mlr (1 - r)'+a

Thus, in the disk- I z I < 1 we have M,r

(z)IC

Ifµ

-r)'+Q

(2)

where M 1 depends only on a, c, and It. Let us show now that satisfaction of inequality (2) in the disk satisfaction of the inequality (z)

with the same M2. We set 77,

e

2 277V

,

v' = 1,

z I < 1 implies

(3)

Mgr `'1

I

p.

For fixed z in the

192


disk z I< 1, let us look at the values of I f(r7,, z)I, v' = 1, , 1L. Suppose that the greatest of these absolute values is obtained for v ' = vI . Then, from Theorem 7 of §2 ( with n = 2) applied to I f(rit,I z) I and I f(riv, z) 1, where v,i i I

we obtain the .t - 1 inequalities Mr If (,q." Z) I
0. But M (P) C (l

P

p)e,

Vr)9 1, (13) reduces to the form Cr

r(1-r)4

log 1

1

r,

195

§8. THE RELATIVE GROWTH OF COEFFICIENTS

with I cT I < c, where c is an absolute constant. As a result, inequality (11) yields (n+1)Icn+IIr-nIcnIIrn+I (1c r)6/,log

l

1 r.

(14)

Finally, if we set r = n/(n + 1), we obtain inequality (9) from (14) for n sufficiently large and hence for all n = 2, 3, - - . This completes the proof of the theorem.

Theorem 2 '. For odd functions f (z) = En =I c zn_I z 2n-I belonging to the

class S, we have

Ilcn+tl-Icn_iIICcn 4 logn, n=2, 4, ...,

(15)

where c is an absolute constant.

Proof. Let us represent f(z) in the form f(z) = g(z2), where g(z) _ [f(z%)]2 E S. Suppose that the maximum value of Ig(z)I on I z I = r, 0 0. Then, by setting 00

>0,

1,2 (,t)dT=(v

(1)

we have 00

SX(t)dv

(v+l)a

(2)

0

Inequality (2) is sharp for each given v with equality holding only for the func-

tion a(r) = ±z(r), where p(r) = e-" for 0 < r< v and p(r) = e-r for r> v. [We note that a v satisfying condition (1) always exists because the lefthand member does not exceed f p e- Zrdr = %, and the right-hand member decreases

from % to 0 as v increases from 0 to oe .] Proof. That the function µ(r) belongs to the type of functions A(r) with given v is easily verified. Specifically, 00

CO

l19(T)dr- eAVdT+ a 9`dt=(v+ 2)e-1"_ JD

b)

00

X (T)dr.

(3)

r

Now, for an arbitrary function A(r) of the type in question, we have for r > 0

§9. SHARP BOUNDS ON THE COEFFICIENTS

197

(EL(T)-IA(r)I)(2e`"-p(t)-IA(r)1):0, because the left-hand memher is, for 0 < r( v, equal to e_v r> v, we have > µ(r) > I A(r)I. From this we obtain

(e_v

- IA(r)I)2 and, for

E.'('r)-A'(t) 1, we have the sharp inequality

ap=

IC e - P+1 +p 2

In particular, Ic ;2)1 < e-2 /3 + 2'1 = 1.013

(13)

.

Proof. The function f, (z) can be represented in the form f, (z) = n f(zp), where f(z) _ l,°, =1 c z" E S. We have

c2P+1=

P

(c3-PZPI c9

(14)

But, in accordance with Theorem 1, we have the sharp inequality

c3-P2P c9 1

2e

-2p-1 P+1+

This together with (14) yields (13). For certain special subclasses of functions of the class S, sharp bounds for the coefficients c" have been obtained. For this, we need Lemma 2. Suppose that a function f(z) = 1 + a1z + a2z2 + ... is regular in the

disk I z I < 1 and satisfies in that disk the condition l(f (z)) > 0. Then, Ian I < 2

for arbitrary n = 1, 2,

.

Proof. The condition 32 (f (z)) > 0 is equivalent to the inequality I 1 + f (z)I 11 - f(z)j. It follows that the function

z)f(z)-}-12 z+... B(f (Z)- 1

al

is regular in the disk Iz I < 1 and satisfies the condition I g(z)I < 1 in that disk. It then follows on the basis of the Schwarz lemma that Ia, I < 2. If we now denote by rlk, for k = 1, . . , n the nth roots of unity, we have 1

n

Consequently, the function n

n

f Ìkz0

k=t I

jf(1kzn)=1 k=1

t

az

200


also satisfies the conditions of the lemma. Therefore, Ian 2, . This completes the proof of the lemma.

2 for all n = 1,

Theorem 2.1) For functions f (z) £'n =1 cn Zn C S with real coefficients, we have the sharp inequalities

IC.I_n, n=1, 2, ....

(15)

Proof. By virtue of the univalence of f (z) in I z I < 1, we have, for arbitrary

z1 and z2 in IzI 1 and the domain B' is mapped into a doubly connected domain B" that does not include -o and that is bounded by this curve and by the circle Iz "I = 1. The composite of these two mappings constitutes a univalent mapping of the domain B onto the domain B ". Furthermore, on the basis of §3 of Chapter II, we conclude that this mapping sets up a one-to-one correspondence between the prime ends of the domain B and the boundary points of the domain B ". Here, just as in § 3 of Chapter II, by the prime end of the domain B that corresponds to the point z o on the boundary of the domain B ", we mean the set of all cluster points of all sequences of points of the

§ 1. MAPPING A DOUBLY CONNECTED DOMAIN ONTO AN ANNULUS

2 07

domain B corresponding in the domain B " to sequences that converge to z o . Let us now show that the domain B " can be mapped univalently onto a circular annulus 1 1 by means of the function t = log z onto the halfplane 3 ( t ) > 0. This mapping is not one-to-one because corresponding to each point z " in the domain Iz "I > 1 there correspond infinitely many points of the form t + 2kiri, for k = 0, + 1, + 2, . - . belonging to the halfplane $2(t) > 0. The domain B " is thus mapped into a vertical curvilinear strip in the t-plane bounded by the imaginary axis and an infinite analytic curve contained in the halfplane R (t) > 0. This strip is a simply connected domain B 1 with the property that, if a point t C B 1, then all points t + 2k ni, for k = 0, ± 1, + 2, - , also belong to B 1. Let us now map the domain B 1 univalently onto a vertical strip B 2 of the form 0 < R (w) < h in such a way that the boundary points t = - oo i, 0, and + oo i are mapped into the boundary points w = - oo i, 0, and +ooi. 1) Under this mapping, the point t = 21ri is mapped into a point wo on the positive half of the imaginary axis. By means of a similarity transformation in the w-plane if necessary, we can arrange to have w0 = 2ni. Then, if we denote by w = f (t) the mapping function and if we denote by t = q (w) its inverse, we conclude that these two functions satisfy the relations f (t+2ar1) = f (t) -}- 2it1, p (w+21r1) = p (w) -}- 2n1.

(1)

To see that this is the case, note that the functions f (t + 277i) and f (t) + 27ri both map the domain B 1 univalently onto the strip B 2 in such a way that the boundary points - oo i, 0, and + oa i are mapped in both cases into the points - oo i, 2iri, and + oa i respectively. Consequently, by virtue of the uniqueness, these functions coincide, so that we have the first of equations (1). The second of equations (1) is obtained in an analogous manner. Finally, let us map the strip B2 by means of the function C= ew onto the annulus 1 < et`. Again, the mapping is not one-to-one. If we now express : in terms of z, we obtain the function = ef(1°g z"), which maps the domain B " onto the annulus 1 < Ii < eh. This mapping is univalent because, by virtue of properties (1), the functions [;'= of (log Z") and z "= a 000g 0 are single-valued respectively in the domain B " and in the annulus 1 < I 1 < et`. Furthermore, it follows from what we said above that the function (;'= of ('Og is analytic in 1) Every strip has two infinitely distant boundary points, which we denote by ±ai.

208

V. UNIVALENT MAPPING OF MULTIPLY CONNECTED DOMAINS

the closed region B" and that its inverse is analytic in the annulus 1 < Ii < eh. Summarizing these results, we have Theorem 1. Every doubly connected domain B can be mapped univalently onto a circular annulus. This mapping defines a one-to-one correspondence between the prime ends of the domain B and the points of the boundary of the annulus. If the domain B is bounded by Jordan curves, the mapping is one-to-one and continuous exclusive of the boundary. On the other hand, if the domain B is bounded by analytic curves, then, in addition, the mapping function is analytic in B and the inverse function is analytic in the closed annulus. As a supplement to this existence theorem, let us prove a uniqueness theorem, one which reveals a significant difference between univalent mappings of simply and multiply connected domains. Theorem 2. The circular annulus onto which a given doubly connected domain B is mapped univalendy is unique up to a linear transformation: that is, if the domain B is mapped univalently onto two distinct circular annuli, then each of these annuli can be obtained from the other by means of a linear transformation. Consequently, the ratio of the radius of the larger boundary circle to that of the smaller is the same for the two annuli. Furthermore, the function f (z) that maps the domain B onto an annulus in such a way that a given point on the boundary of the annulus corresponds to a given point on the boundary of B is uniquely determined.

Proof. If a domain B is mapped univalently onto two distinct circular annuli, these annuli can be mapped univalently onto each other. Consequently, it will be sufficient for us to show that, if two circular annuli are mapped univalently onto each other, they can be obtained from each other by a linear transformation. And to do this, it is obvious that we need only show that, if two annuli r < Iz I < R and r < 14 < R' can be mapped univalently onto each other in such a way that IzI = r is mapped into W = r, 1) this last mapping is of the form C= e"z, where a is a real constant. This can be shown as follows: A function f (z) can be extended, by the symmetry principle, beyond the circle IzI = R into the annulus R < I z I < R 2/r and maps this annulus univalently into the annulus R '< I ci < R 121r. Then f(z) can be extended further into the annulus R2/r < Izi < R4/r3, etc. 1) We can always arrange for this by an inversion the radii of the boundary circles does not change.

under which the ratio of

§ 1. MAPPING A DOUBLY CONNECTED DOMAIN ONTO AN ANNULUS

209

Analogously, we can extend f (z) past the circle IzI = r into the annulus r2/R < IzI 0. Then, the function C= (C'(z) - a0)/a_1 provides the required mapping of the domain B. Let us turn to the general case. Consider the family 11 of all functions f (z) that map the domain B univalently and have the expansion f (z) = z + a1/z + in a neighborhood of z = oo. An example of such a function is f (z) = z. Let us pose the extremal problem: Out of all the functions of the family R, find the one that

maximizes the quantity R(e-2`Ba1) Let us first prove that this problem has a solution. Suppose that the entire boundary of the domain B is contained in the disk Izj < R. Then all functions of the family R are univalent in the domain jzj > R. Consequently, in accordance with Lemma 1, for these functions we have R (e-2` Ba1) _< R 2; that is, the numbers R(e-206a1) are uniformly bounded. Let A denote the least upper bound of these numbers. If this upper bound is not attained, there exists a sequence of functions fn(z) E Dl for which the sequence {R(e-2`Ba1)) converges to A. But, from Lemma 2, for functions of the family Dl we have if (z)/z1 < 2 in jzl > R. This shows that the condensation principle can be applied to the functions

ff(z)/z, that is, that the sequence { f,(z)/z} contains a subsequence (f,,k(z)/zI that converges uniformly in the interior of the domain Izj > R to a regular funcsuch that R(e-2`Bai0)) = A. But in accordance tion f0(z)/z = I + aJO)/z2 + with Lemma 2, we have l fnk(z)l < 2R' for functions fnk(z) on Iz = R', where R '> R. By virtue of the univalence of flk(z) in B, the same inequality holds also at points of the domain B that belong to the disk 1zI < R'. Consequently, the functions fnk(z) are uniformly bounded in the interior of the domain B0 obtained from B by removing the point z = oc. In accordance with Vitali's theorem, the sequence of these functions converges inside B0 to a regular function f 0(z), which, in accordance with Theorem 2 of § 1 of Chapter 1, is univalent in B0 and hence in B. Therefore, f0(z) E 2, which contradicts our assumption that the least upper bound A is not attained by the numbers R(e-2cea1). Thus,

§2. MAPPING ONTO A PLANE

213

the existence of a solution of the extremal problem posed is proved. Let us now show that the extremal function f0(z) provides the mapping referred to in the theorem. Let us suppose that the complement of the domain B with respect to its image B ' under the function C= fo(z) contains a continuum other than a straight line segment of inclination 9 with the real axis. Let B 1 denote whichever of the simply connected domains complementary to this continuum in the plane contains the point C= cc. Suppose that the function w = w(C) maps B 1 univalently onto the w-plane with rectilinear cut of inclination 0 with the real axis and that it has the expansion t;+ 01/C+ in a neighborhood of 4_ cc. Let us show that $2(e-2`61) > p. RiFemann's In accordance with theorem, the domain B 1 can be mapped univalently onto the domain It > R' in such a way that the function C_ C(t) inverse to the mapping function has the form C(t) = t + yo + y1/t + . The function w = w(C(t)) - yo = t + (131 + yl)/t + maps the domain It` > R ' univalently. onto the w-plane with rectilinear cut of inclination 0 with the real axis, and the function r= C(t) - yo = t + yl/t + . maps ItI > R ' onto a different domain. Therefore, on the basis of Lemma 1 we have at (e-0 71) < (e 9`B (N1 + Ti))

that is, R (e-z` 681) > 0. Now, let us look at the function w = w (f 0(z)). This function maps the domain B univalently and has the expansion z + (ai0> + O1)/z + in a neighborhood

of z =cc. Consequently, w(f 0(z )) E 9 and we have at (e-20 (4) + r1)) = R (e-20 a()) + at (e9te 1'1) > A,

which, however, is impossible. This shows that the function f0(z) meets all of the requirements of the theorem. At the same time, we have established an important extremal property of the mapping function, namely, the fact that it maximizes the quantity R(e-2`9a1). In the case of finitely connected domains, this theorem can be formulated as follows :

Theorem 1 '(Hilbert). Every n-connected domain B in the z-plane can be

mapped univalently onto the aplane with n parallel finite cuts of inclination a with the real axis in such a way that a given point z = a is then mapped into

214


C_ oc, and the expansion of the mapping function about z = a has the form

z-a -}-at(z-a)4-... 1

or

z4-i'-4-...,

according as a is finite or not. Some of the cuts referred to may consist of single points.

Let us now look at the question of the correspondence of boundaries under univalent conformal mapping of multiply connected domains. We shall confine ourselves here to finitely connected domains that have only accessible boundary points. We may assume that the point oc is in the interior of the domain. Let B denote such a domain and let K 1, K2, - - , Kn denote its boundary continua. Let us map the domain B univalently onto a domain bounded by closed analytic Jordan curves. This can be done as follows: we map the simply connected domain including z = ea and bounded by the continuum K 1 onto the interior of a circle. Under such a mapping, the domain B is mapped into a domain B' bounded by a circle Ki and the continua K21 , K,, . Then, we map the simply connected domain containing the domain B' and bounded by the continuum K2 onto the interior of a circle. Under this mapping, the domain B' is mapped into a domain B ° bounded by a circle KZ , an analytic curve K1 and the continua K3, , Continuing in this way, after n steps we arrive at an n-connected domain B(n) whose boundary consists of n closed analytic Jordan curves. The univalent mapping of the domain B onto the domain B(n) is a composite of successive mappings of univalent domains. Since these mappings define a one-to-one correspondence between the boundaries, the correspondence between the points of these boundaries defined by the mapping of B onto B( n) is also one-to-one. Furthermore, as in § 3 of Chapter II, we can show that the function inverse to the mapping function is continuous in B(") except at the point -a. If the domain B is bounded only by closed Jordan curves, we can show in addition that the mapping of B

onto B is also bicontinuous. If we now map the domain B(n) onto the plane with rectilinear cuts, then, just as in the proof of Theorem 5 of §3 of Chapter II, we can show that the mapping function is regular on the entire boundary of the domain B("). Combining all that we have said, we conclude that, in the present case, the mapping mentioned in Theorem 1' is one-to-one except for boundary points of the domain B. An important supplement to the existence theorem proved above is the following uniqueness theorem:

§ 2 MAPPING ONTO A PLANE

215

Theorem 2. There exists only one function performing the mapping mentioned

in Theorem I'. Proof. Let us suppose first that B is a bounded domain with boundary con, K,,. Let us suppose that there sisting, of closed analytic Jordan curves KI, C'= fl(z) and t;'"= f2(z) that map the domain B univalently are two functions onto the plane with parallel rectilinear cuts of inclination 0 and normalized as indicated in Theorem 1 '. Then both these functions are regular in B except at , n, they assume values that lie respectively the point z = a. On K, v = 1, c and Ne-` 9C°) = c v, for v= 1, , n; on certain straight lines -J(e-` c,, and -J(e-`Bf2(z)) = cv, where cv and is, on K we have that , n, are constants. It follows from this that the difference t; = c v = 1, F (z) = f 1(z) - f 2(z) is regular in B and assumes on K (for v = 1, , n) values lying respectively on certain straight lines d Consequently, if we take an arbitrary point Co that does not lie on any of the straight lines dv, we conclude that arg [F (z) - Col does not change as we move around any of the curves K. Therefore, on the basis of Cauchy's familiar theorem on the zeros of analytic functions, it follows that the function F (z) - Co has no zeros in B, that is, that F(z) does not assume the value Co in B. Thus, all values that F(z) assumes in B lie on the straight lines dv. However, this is possible only when F(z) = const in B. Since F(a) = p, it follows that F(z) a p, that is, f 1(z) = f2(z), so that the theorem is proved in this case. Now, let B denote an arbitrary finitely connected domain. Here, we may assume that it has no isolated boundary points. If there were two distinct functions performing the mapping described in Theorem 1 ', we could, by mapping the domain B univalently onto the bounded domain B0 with boundary consisting of closed analytic Jordan curves, find two distinct functions mapping the domain B0 univalently as indicated in Theorem 1 ', which, on the basis of what was said above, is impossible. This completes the proof of the theorem. In conclusion, we present an important relationship for the functions mentioned in Theorem 1' for various values of 0. Let B denote an n-connected domain bounded by closed analytic Jordan curves KI, , K and let C= jg(z, a) denote a function that maps the domain B univalently as indicated in Theorem 1 '. Let us show that the following relationship holds for arbitrary 0:

216


2

jo (z, a) = ei0 (cos 6 jo (z, a) -1 sin 6 j (z, a)).

(1)

The difference d(z) between the two sides of this asserted equation is a function that is regular in the domain B and that vanishes at z = a. Furthermore, all the values that d(z) assumes on any of the curves Ki, lie on a straight line 9 (e-' 64) = const.

Reasoning as in the proof of Theorem 2, let us show that d(z) = 0; that is, let us prove (1). This relation, which has been proved for a domain bounded by closed analytic Jordan curves is also valid for an arbitrary finitely connected domain B. For this, it is sufficient to map the domain B onto a domain B* bounded by closed analytic Jordan curves, apply(1) to B*, and then return to the domain B. This yields the relation (1) for B. Equation (1) yields jg(z, a) for arbitrary 0 as soon as we know j0(z, a) and l n/ 2 (z, a).

§3. Univalent mapping of a multiply connected domain onto a helical domain

In an analogous manner, we shall find the answer to the question of univalent mapping of multiply connected domains onto a plane with cuts along arcs of logarithmic spirals and, as limiting cases, onto the plane with radial cuts and with cuts along circular arcs of concentric circles. For constant 9 and c, the equation -9 (e ° Blog ') = c defines a logarithmic spiral in the plane with asymptotic point at the origin. This spiral has the property that it is intersected by an arbitrary ray issuing from the origin at an angle 0. This last follows, for example, from the fact that, if we shift to the plane t = log C, this logarithmic spiral is mapped into the straight line 9 (e-` 6t) = c with inclination 9 to the real axis, and the ray referred to is mapped into a straight line parallel to the real axis. For 9 = 0, the logarithmic spiral degenerates into a ray issuing from the origin. For 0= n/2, it degenerates to a circle with center at the origin. If we hold 0 constant and vary c, we obtain various curves constituting the family of logarithmic spirals of inclination 0. In all that follows, when we speak of logarithmic spirals of inclination 9, these are what we mean. Let us show that, for any simply connected domain B that has boundary points, it is possible to map B onto the plane with cut along an arc of a logarithmic spiral of inclination 0 in such a way that given points a and b of the domain B are mapped into 0 and oo and the expansion of the mapping function

§3. MAPPING ONTO A HELICAL DOMAIN

217

about z = b has the form z

b

+aa+at(z-b)-} ...

or

x+ao+ z

+-

(1)

according as b is finite or infinite. In the case in which the domain B has a single boundary point, this is obvious, and then the arc of the logarithmic spiral referred to degenerates to a point. On the other hand, if the boundary of the domain B is a continuum, let us first map B conformally onto the domain 1z 'I > 1 in such a way.that the point z = b is mapped into z '= 00. Then, the mapping function z '= q(z) has in a neighborhood of z = b, the expansion

za=tb +as+at(z-b)+...

or

a_tz+ao-}

y -{-...,

according as b is finite or infinite. This is possible by virtue of Riemann's theorem. Suppose that the point z = a is mapped into a point z'=a'. It now remains to establish the possibility of mapping the domain 1z 'I > 1 onto the plane with cut along a logarithmic spiral of inclination 0. But this possibility follows from the solution of the problem on the minimum of the quantity

R (e-2i 'log F'(a')) in the class 1. This problem was studied in § 3 of Chapter IV (the first application of Theorem 1 with a= 7r/2 - 0). It was shown there that this problem has a solution and that the extremal function C= F(z') normalized by the condition F (a') = 0 provides the required mapping. As a result, we see that the function C= cF(O(z)) = f (z) with f (0) = 0 and f'(') = oo with suitably chosen c provides a mapping of the domain B onto the plane with cut along a logarithmic spiral and has an expansion of the form (1). This result for simply connected domains can be generalized to multiply connected domains. For this generalization, we need the analogue of the minimal property proved in §3 of Chapter N, which we formulate as a Lemma. Of all functions J= F(z) = z + ao + al/z + that map the domain IzI > R univalently in such a way that a given finite point a and the point oo are mapped into the points 0 and oo respectively, the quantity 82 (e-Z` 6 log F'(a)) is minimized by the function F0(z) that maps the disk IzI > R onto the cplane with cut along an arc of a logarithmic spiral of inclination 0. Here log F'(z) means the branch that approaches 0 as z --. oo. We now have

Theorem 1.1) Every domain B in the z-plane can be mapped univalently onto 1) Grotzsch [1931].

218


a domain B' in the cplane that includes the points 0 and oo such that an arbitrary continuum of the complement of the domain B' with respect to the cplane is an are of a logarithmic spiral of given inclination 0. This mapping maps given points a and b of the domain B into 0 and oo and the expansion of the mapping or z + ao + function about z = b has the form (z - b) 1 + ao + a1(z - b) + according as b is finite or infinite. a1z-1 + Proof. We may assume that b = 00 because, if this is not the case, we can use the transformation z* = 1/(z - b) to switch from B to a new domain B. and we can then prove the theorem for this last domain, replacing a and b with 1/(a, b) and oo.

In the case of simply connected domains, the theorem was proved at the beginning of this section. To prove it in the case of a multiply connected domain B, let us consider the family 3R of all functions f (z) that map B univalently in such a way that a and oo are mapped respectively into 0 and oo and that have an in a neighborhood of z = 00. An example of expansion f (z) = z + ao + al/z + such a function is f (z) = z - a. We pose the extremal problem: Out of all functions of the family 9, find the one that minimizes the quantity R(e-2`Blog f'(a)), where log f'(z) denotes the single-valued branch in the domain B that approaches

0 as z Let us show that this problem has a solution. To do this let us choose a simply connected domain B' contained in B that includes a and oo. In this domain, all functions f (z) E 9 are univalent. Consequently, for these functions, the quantity R (e- 2` S log f '(a)) is bounded below by the same number as was calculated for a function that maps the domain B' univalently onto the plane with cut along an arc of a logarithmic spiral of inclination 0 in such a way that a and 00 are mapped into 0 and oc. Let us denote the greatest lower bound of the R(e-2c0log quantity f'(a)) for functions in 3R by A.

Let us show that this greatest lower bound is attained. We assume the opposite. Then, there exists a sequence of functions f"(z) E 9, for n = 1, 2, such that, as n -- 00, R (e-9`B log fn (a)) - A.

Suppose that the entire boundary of the domain B lies in the disk Izl < R. According to Lemma 2 of § 2, it follows that the entire boundary of the image of the domain Iz I > R under the function w = f"(z) = z + ao") + al" )/z + (where n = 1, 2, ) lies in the disk JC- ao")l < 2R and that 1f"(z) - ao")l < 2Izl in the


219

domain IzI > R. If Ial > R, this last inequality with z = a implies that Iao")I _< 21al. On the other hand, if lal < R, then the point Co = &(a) = 0 does not belong to the image of the domain IzI > R under the mapping C= f"(z} hence, it belongs to the disk IC- a("'I < 2R. Therefore, lao"Il < 2R. Thus, in all cases, lao"))l < 2max (lal, R) = M. But then, If" (x)/zl < 2 + ;11/R in IzI > R. Consequently, the condensation principle can be applied the sequence 1f"(z)/z{, so that it contains a subsequence {f"k(z)/z1 that converges uniformly in IzI > R to a regular function. The corresponding sequence tf"k(z)} converges uniformly in an arbitrary closed bounded subset of the domain IzI > R to a function f0(z) that is univalent in IzI > R. But since the values of C= f"k(z) corresponding to points of the domain B that lie in IzI < 2R belong to the disk a((0"'I < 4R and hence to the disk l l < 4R + M, the set of functions f"k(z) is ICuniformly bounded inside the domain B with the point z = oo excluded. Therefore the uniform convergence, mentioned above, to f0(z) holds also in the interior of the domain B, and f0(z) C M. The function f0(z) thus proves to be a solution of the extremal problem posed, so that this problem does have a solution. Let us show that the extremal function f0(z) provides the mapping indicated in the theorem. Let us suppose that the complement of the image of the domain B under the mapping C = f0(z) contains a continuum other than an arc of a logarithmic spiral of inclination 0. Let B1 denote whichever of the simply connected domains in the plane that are complementary to this continuum contains 0 and o-. Suppose that the function w = q (0 maps B 1 univalently onto the w-plane with cut along an arc of a logarithmic spiral of inclination 0 in such a way that 0 (0) = 0 and QS (oo) = 00 and suppose that 4 (0 has an expansion C + go + f31/C + in a neighborhood of C = -o. Let us show that

T(e l" log y'(0)) < 0. In accordance with Riemann's theorem, the domain B 1 can be mapped univalently onto a domain Itl > R in such a way that the inverse function has an expansion of the form J= C(t) = t + y0 + yl/t + . Suppose that the point t = a' corresponds to the point C = f 0(a). The function w = .0 (C(t)) maps the domain ItI > R univalently onto the w-plane with cut along an arc of a loga-

rithmic spiral of inclination 0 in such a way that the point t = a' is mapped into. w = 0. From the preceding lemma, we have sf (e-2:e (log gyp' (0) + log C (d))) < 51 (e-211 log C' (d)).


220

Therefore, R (e-2` Slog 0'(0)) < 0. Let us now look at the function w = q5(fp(z)) = O(z). This function maps the domain B univalently in such a way that a and oc are mapped into 0 and DC in a neighborhood of respectively and it has the expansion z + c 0 + c I/z +

_ . Consequently, it belongs to the family 9k and M (e2 ° log ' (a)) _ ,T (e-111 log ?'(0)) + R (e2d log fY (a)) < A,

which, however, contradicts the definition of A. This completes the proof of the theorem.

At the same time, we have established the following extremal property of the mapping function, which includes the distortion and rotation theorems for the family.of univalent functions that we are considering. 1) Of all univalent functions i F (z) defined on a domain B including w that have an expansion of the form

F(z)=z+ao+ z +..., in a neighborhood of z = o, the quantity R(e-208log F'(z)) with given z e B and given real 0 is minimized by the function defined in Theorem 1. In the case of multiply connected domains, the existence theorem that we have proved can be formulated in a different way:

Theorem 1'.2) Every n-connected domain B in the z-plane can be mapped univalently onto the t; plane with n cuts along arcs of logarithmic spirals of inclination 0 in such a way that given points a and b of the domain B are mapped into 0 and - respectively, and the expansion of the mapping function about z = b has the form z

- b+ao+a,(z-b)+... or z+ ao-}-i' +...,

according as b is finite or infinite. Some of these arcs of logarithmic spirals may degenerate into points. From Theorem 1', we obtain as special cases with 0 = 0 and 0 = 77/2 theorems on the existence of a univalent mapping of an n-connected domain onto the plane with radial cuts and onto the plane with cuts along arcs of concentric circles. 1) Gr&zsch [1931a], Goluzin [1937, 19471.

2) (Koebe [1918].


221

The correspondence of the boundaries under the mappings referred to in Theorem

is studied in the same way.as in §2. In particular, let us show that, if the domain B has only accessible boundary points, then the mapping is one-to-one in this case also except for points on the boundary of the domain B. On the other band, if the domain B is bounded by closed Jordan curves, the mapping function is continuous in B except at the point z = b. Finally, if the domain B is 1'

bounded by analytic Jordan curves, the mapping function is analytic on the boundary of the domain B. We also have a uniqueness theorem: Theorem 2. There exists only one function carrying out the mapping mentioned in Theorem 1'.

Proof. Let us assume first that the domain B is finite and bounded only by closed analytic Jordan curves K 1, , K,,. Let us suppose that there exist two functions C= f 1(z) and C= f 2(z) that carry out the mapping of the domain B mentioned in Theorem 1 '. These functions are regular on Ki, for v = 1, ,n and, on each of these curves, we have 3 (e

i0 logf1 (z)) = c,

and

(ete log ff (z));= C"'

where c v and c are constants. Therefore, it we set = F (z) = log (f I(z)/f 2(z)), then the function F(z) is regular in B and the values that it assumes on the curves K. for v = 1, , n, lie on a straight line segment. We then see, just as

in the proof of Theorem 2 of §2, that F(z) = const, that is, fl(z)/f2(z) const. But f1(z)/f2(z) - 1 as z b. Consequently, f 1(z) = f 2(z). Now, if B is a finitely connected domain, we can prove the theorem by reasoning analogous to that used in the proof of Theorem 2 of §2. We mention one relationship regarding the mapping functions which is analogous to (1) of § 2. Suppose that B is an n-connected domain bounded by closed analytic Jordan curves K1, ... , K,, and that C= jg(z, a, b) is a function that maps the domain B univalently as indicated in Theorem 1 '. Let us show that the following relationship holds in B for arbitrary 0: log j0 (z, a, b) = ei6 (cos 0 log j0 (z, a, b) - I sin 0 logj (z, a, b)) (2)

222


where suitable branches of the logarithms are chosen. The difference d(z) between the two sides of (2) is a regular function in B and the values that it assumes on , n, all lie on some straight line -J(e-Dgt) = const. the curves K,,, for v = 1, Reasoning as in the proof of Theorem 2 of §2, let us show that d(z) const = c, where c is equal to 0 for a suitably chosen branch of d(z). This yields equation (2), which is now proved for a particular form of the domain B. To see this, let us map the domain B onto a domain B. bounded by closed analytic Jordan curves and isolated points. We apply formula (2) to B. and then return to the domain B. As a result, we obtain (2) for the domain B. Some theorems on the existence of univalent mappings of multiply connected domains onto various other canonical domains will be given in §6. §4. Some relationships involving the mapping functions

Equation (2) of §3 enables us to find, for a given n-connected domain B a function je (z, a, b) with arbitrary 6 if we know the functions jo (z, a, b) and jn/2(z, a, b). But we can also establish relationships between these last functions and the functions jo(z, a) and jn/2(z, a) of §2 which correspond to the same domain B. To put these relations in the most symmetric form, we set

f' (z, a) = 2 (J x (z, a) - j o (z, a)), (1)

Q (z, a) = 2 (J x (z, a) + Jo (z, a)) and, analogously,

P (z, a, b) = 2 (log J,, (z, a, b) - log Jo (z, a, b)), 2 Q (z, a, b) =

(2)

(log j x (z, a, b) + log Jo (z, a, b)). 2

These relations then take the forms dz

d

P (z, a, b) = P (b, z) - P (a, z),

-dz-Q(z,

(3)

a, b)=Q(b, z)-Q(a, z).

On the basis of the same principles as in the preceding sections, we can

§4. RELATIONSHIPS INVOLVING THE MAPPING FUNCTIONS

assume that the domain B is finite and bounded by analytic curves KI, K21 K,,. Since (lo (z, a)) = const, 2 (11 w (z, a)) = const, T

on K. (v= 1,

223 ,

, n) for fixed a e B, that is, since jo (z, a) = jo (z, a) + const

j(z, a)_- j, (z, a)+const, 2

we also have on the basis of (1) P (z, a)

Q (z, a) + k, (a),

(4)

on Kv, where kv(a) is independent of z C K. Analogously, since 3 (log jo (z, a, b)) = const,

-3 (11og J. (z, a, b)) = const,

-

5-on Kv for fixed a, b C B, it follows on the basis of (2) that

P (z, a, b) = - Q (z, a, b) + k,, (a, b),

(5)

on K v, where k v (a, b) is independent of z C K v. Now, let us look at the integral 1' - tat } P (C, z) PP (C, a, b) dC,

taken over the entire boundary K = U'= I K, of the domain B in the positive direction. In accordance with (1) and (2), the functions F(j, z) and P (C, a, b) are, for arbitrary fixed z, a, b C B, regular in B. Therefore, in accordance with Cauchy's theorem, 11 = 0. On the other hand, applying (4) and (5), we have n r

1' - r=1 tat J

=

P (C, z) dP (C, a, b)

n

(-Q(C z)+k, (z))d(-Q(C, a, b) + k, (a, b))

I n

v=1

tat

Q (C, z) d Q (C, a, b)

Q (C, z) Qc (C, a, b) dC

224


(Here, we used the single-valuedness of the function Q (C, a, b) on each of the curves K u.) Since Q (C, z) and Q'(C, a, b) have simple poles in B at the point z and the points a and b respectively, by applying the residue theorem to the last integral we obtain

Q: (z, a, b) - Q (a, z)+Q(b, z). Remembering that I1 - 0, we then arrive at the second of formulas (3). Let us examine the integral 19 = 2nt - C Q (C, Z) Pi (C, a, b) dC

in an analogous fashion. On the it is obviously equal to PZ (z, a, b). On the other hand, we can show, just as above, that it is equal to P (b, z) P (a, z). Thus we obtain the first of formulas (3). One consequence of formulas (3) is that the derivative Qz (z, a, b) is an analytic function in B of the arguments a and b, whereas Pz (z) a, b) is an analytic function of a and W. Conversely, we conclude from these formulas that

the difference Q (z, a) - Q (z', a), where z and z ' belong to B, is an analytic function of a in B and that P (z, a) - P (z ', a) is an analytic function of a. If we let z' approach z, we see that analogous conclusions hold for the derivatives QZ (z, a) and PZ (z, a).

The last result can be derived in a different way, specifically, from the relations

Pp (z, a) = Pa (a, z), Q- (z, a) = Q. (a, z), which we can prove by examining the integrals 19

r = tat 1 P (C, Z) Qc (C, a) d`, la =tai } P (C, Z) P (C, a) dC

(6)

the same way as above. That the functions P (z, a) and Q (z, a) themselves are not necessarily analytic functions of a is shown by the example when B is the domain Jzl > 1. In this case, we easily see that

_

P(z, a)-1-1 1

z-a

a111-az,

Q(z, a)=1

1 -az

aJ'z-a


225

This method of deriving various relationships between functions that we do not know explicitly is called the method of contour integration.') Relations (4) and (5) on the boundary of the domain play the significant role in it. We now point out a curious analytic relationship with the preceding values of the function mapping the domain B onto a circular annulus with concentric circular cuts. Specifically, let us denote by J= f (z) a function that maps a domain B not containing - and bounded by analytic Jordan curves K,,, v = , n, onto the annulus q < ICI < 1 with cuts along arcs of concentric v = 1, circles (see Chapter V, §6, the case 0n/2 of Theorem 7). Suppose that the curves K1 and K2 are mapped onto the circles q and Just as before, we calculate each of the integrals I6

P (C, z, z') (logf (C))i a,

=tai

Ifi = 2x1

S P (C, z) (log f (C)) dC,

(7) (8)

h

where z, z 'E B, by two different methods. Keeping in mind this fact, together with (6) and (7), and the fact that log f (C) _ ]og f (C) +const on K, and

tats (1ogf (C))i dC= 1, tat J (logf (Q) dC=- 1, Kt

we can prove the following formulas:2) f (z) =f (z) eks (z, z') - ki (z,

f (z) = ej kt (z) dz - k2 (z) dz 1) Garabedian and Schiffer [19491 2) In the process, we need to evaluate the integral 2x1

Q G, z, z)(logf (C))i dC,

and, to do this, we need, by virtue of the multiple-valuedness of the function Q(C, z, z to make a cut in B connecting the points z and z ' and then replace the integral over K with an integral over a closed curve y encircling this cut. We then constrict the curve y around the cut. Integration by parts then leads to an integral that can be calculated in accordance with the residue theorem.


226

We note that z ' appears in (9) as an apparent parameter since f (z) is, in accordance with Theorem 7 of §6 of Chapter V, determined up to a constant factor. In conclusion, we derive an integral formula involving the function P(z, a, b). For any two functions f (z) and g (z) that are regular in B, we define their scalar product as

(f, g)= S

(11)

Sf'(z)g'(z)dxdy.

B

In accordance with Green's formula

Sf(z)g'(z)dz=sf(z)g' (z)dx -lf(z)g (z)dy K

K

SB S - t

d f (=a g' (z)

-

df(Za

g

Z)

(y

) dx dy = - 21 S S f' (z) g' (z) dx dy, B

J

where z = x + iy. For (f, g) we also have the contour representation g)

=

2i

S f (x)

(z) dz.

(12)

If we apply (12) to the case in which g(z) = P(z, a, b) and remember that f (z) P', (z, a, b) dz

= f (z) d 2 (log j, (z, a, b) -- log j0 (z, a, b)) f (z) d 2 (- log j x (z, a, b) - log jo (z, a, b)) V

_-Sf(z)dQ(z, a, b)=21cl(f(b)-f(a)), K

we obtain the desired formula (f (z), P (z, a, b)) = a (f (a)

- f (b))

We point out two applications of this formula. The first of these is the

(13)


227

Theorem (minimization of area). 1) Out of all functions f (z) that are regular in B and satisfy the conditions f (a) = q f (b) = 1, where a, b E B, the minimum value of the integral

(f, f)=SSIf'(z)I9do, B

which is equal to n/P(b, b, a), is attained only by the function (z, b, a)

fo (z) = P

P (b, b, a)

Proof. If we represent an arbitrary function f (z) of the type described in the theorem in the form f (z) = f0 (z) + (z), we conclude that the function 0(z) is regular in B and that 0 (a) _ o (b) = 0. For this function, we have in accordance

with formula (13) (gyp (z), P (z, b, a)) = 0.

Therefore,

(f, f) _ (fo, fo) + (p, fo) + (P, fo) + (y, F) _ (fo, fo) + (,P,

P)

It follows that

(f, f) with equality holding only when completes the proof of the theorem.

(fo, fo),

0, that is, only when f (z) - f o(z). This

As our second application of formula (13), consider a sequence of functions

k(C) such that kt,(b) = 0 for v = 1, 2,

,

that are regular in $ and satisfy

the orthogonality conditions 1

for v=v' v

l 0 for v

v' = 1

2 .. .

v'

and have the property that an arbitrary function f (z) with f (b) = 0 that is regular in B can be expanded in a convergent Fourier series in B: 1) (The problem of the minimum area of the image of a finitely connected domain has also been solved for the class of all functions that are regular in the domain that have at given points of it given initial segments of their Taylor expansions. See Alenicyn [19641).

228

V. UNIVALENT MAPPING OF MULTIPLY CONNECTED DOMAINS Co

f (z) = 2] cyky (z), c,

V-j

_ (f,

k0)

(It can be shown that such sequences of functions

exist.) Let us apply this to the function f (z) = P(z, a, b), where a and b belong to

B. Since, in accordance with (13), we have

cy = (P (z, a, b), k0 (z)) _ ik., (a), we obtain for P(z, a, b) the constructive formula 00

P (z, a, b) ='t }J k, (a) k, (z)

§5. Convergence theorems for univalent mapping of a sequence of domains Let us now consider the question of the convergence of a sequence of functions that are univalent in multiply connected domains in connection with suitable convergence of these domains and their images.

As a preliminary, we introduce the following concepts. Let {BJ, for n = 1, 2, , denote a sequence of domains in the z-plane that include the point z = on. We define the kernel of the sequence { Bn { as the largest domain B including z = on every closed subset of which is contained in each Bn from some n on. Obviously, for a kernel to exist, it is necessary and sufficient that all the domains Bn contain a fixed neighborhood of the point z = -a. We shall say that a sequence {Bn} of domains Bn converges to its kernel B (and we shall write Bn --+ B) if an arbitrary subsequence of these domains has the same kernel B. , then the sequence of the domains Bn has a For example, if B1 C B2 C B3 C kernel and converges to it. 'Theorem 1. Let {An) denote a sequence of domains An, n = 1, 2, . . , in the z-plane that include the point z = on. Suppose that this sequence converges to a kernel A. Let { fn(z)} denote a sequence of functions C= fn(z) such that, for each n = 1, 2, , the function fn(z) maps the domain An onto a domain Bn including the point C_ on in such a way that fn(o) = on and fn W = 1. Then, for the sequence .{fn(z)1 to converge uniformly in the interior of the domain1) A to a 1) The existence of a pole at z = co is not significant in the present case because uniform convergence of the sequence of the functions fn(z) is equivalent here to uniform convergence of the sequence of the functions fn(z) - z.

§5 CONVERGENCE THEOREMS

229

univalent function f(z), it is necessary and sufficient that the sequence IB"I have a kernel and converge to it, in which case the function C= f(z) maps A univalently onto B. We note that although the functions are defined at certain points z E A only from some n on, this does not destroy the sense of convergence at such points. Proof of the necessity. Suppose that the sequence i fn(z)y converges uniformly inside the domain A to a univalent function f W. Then f (z) is regular in A except at the point z = 00, where f (z) This maps the domain A onto a domain B' containing 00. has a kernel and that B' is contained Let us show that the sequence in that kernel. It will be sufficient to show that an arbitrary closed bounded domain B * contained in B' belongs to every Bn from some n on. Let 8 > 0 denote the distance from R* to the boundary of B'. Let us construct a grid of squares with sides of length 8/2. Consider the domain B. constituted by the union of all squares containing points in B. Then B` C B. and $. C B'. Let 81 > 0 denote the distance from $* to the boundary of B.. Corresponding to the domains B' and B. under the function = f (z) in A are domains we have A* and A.. Here, A* C A. and A. C A. But, for any point CO E If U) - Col > 81 on the boundary of the domain A.. On the other hand, it follows from the uniform convergence of the functions fn(z) in J. that there exists an n > 0 such that Ifn (z) - f (z)I < 81 on the boundary of the domain A. for n > N. Consequently, from the equation fn (z) - CO = V. (z) - f (z)) + V (z) - co)

we conclude on the basis of Rouche's theorem that the function fn(z) - Co has one zero in A., that is, that for n > N the image of the domain An under the mapping C= fn(z) contains an arbitrary point Co E B. This in turn shows that the sequence I Bn I has a kernel, which we denote by B, containing B' Now, let us consider the functions z = (An(C) inverse to the functions C- fn(z). For any closed subset of the domain B, these functions are defined on it from some n on. For sufficiently large positive r, the functions On(0 are all defined and univalent in the domain ICI > r and in that domain they have the expansions

230

V. UNIVALENT MAPPING OF MULTIPLY CONNECTED DOMAINS Vin)

/

J

Consequently, from Lemma 2 of §2, we have in ICI > I'Pn(C)-Q( n)I{2ICI, 0

so that Or. C2-F Ia)I

But the convergence of the sequence (&(z)} to a function f (z) to a function f (z) implies that the sequence {a0")} converges to a finite limit. Therefore, the functions On(O/C are regular and bounded in the domain ICI > r and the condensation principle can be applied to them, so that the sequence has a subsequence (0"k (0/0 that converges uniformly in the interior of the domain ICI > r to a limit function O(C)IC. On the other hand, the sequence 10"k (:)} converges uniformly to O(C) in an arbitrary bounded subset of the domain 11 > r. On the other hand, for all C E B" belonging to the disk ICI < 2r, we have, in accordance with .Lemma 2 of § 2, 1 yn (C) - m(on) I< 4r.

Consequently, the functions q"(t) are uniformly bounded in every closed subdomain

of the domain B that is contained in the disk ICI < 2r. Therefore, in accordance with Vitali's theorem, the sequence { 011k (0} converges uniformly to O (C) in the interior of the domain B exclusive of the point C = oo, and 0 (c) is univalent in B. Reasoning word for word as on p. 57, we can show that the function z = c¢(C) is the inverse of the function C= f W. It follows that 4(c) is independent of the sequence chosen, that is, that the sequence h itself converges in B. Thd function z = O(C) maps the domain B onto a domain A'. Reasoning as at the beginning of the proof but reversing the roles of the domains A" and we can show that A 'C A.

Now, it is easy to show that B '= B. If Co E B, then z o = O(to) C A and, consequently, Co = f (z o) E B '. This shows that B C B '. But we showed earlier that B 'C B. Consequently, B '= B. Thus, the function C= f (z) maps A univalently onto B. If we now take an arbitrary subsequence of the sequence 1Bn} and the corresponding subsequence of the sequence I f4 (z)}, which also converges to f (z), then by applying to these subsequences the conclusions derived above, we see

§ 5. CONVERGENCE THEOREMS

231

that the function C = f (z) maps A onto the kernel of this subsequence, which again is the domain B. Thus, Bn - B. This proves the necessity of the condition in the theorem. B. Let p denote a positive Proof of the sufficiency. Now suppose that Bn number such that the domain IzI > p is contained in each of the domains An and z + (n) + > r is contained in each Bn. If suppose that the domain ain)/z + in a neighborhood of z = 00 and if c does not belong to the domain Bn, then Ic - a(On'I _< 2p by virtue of Lemma 2 of § 2. On the other hand, I c I < p. Consequently, Iaon I _< 3p; that is, the coefficients a0(n) are uniformly bounded

with respect to n. Again in accordance with Lemma 2 of §2, we have 1fn(z) - ao")I < 2IzI in IzI > p, so that Ifn(z)I < 2IzI + 3p. In particular, Ifn(z)I < 5p on IzI = p. By virtue of the univalence of the fn(z), this last inequality remains valid at all points of An belonging to the disk IzI < p. Comparing the bounds on Ifn(z)I in the disk IzI p, we conclude that Ifn(z)I < 2IzI + 5p in every domain An. This shows that for every closed bounded subset of A, all the functions fn(z) from some n on are defined and uniformly bounded on it. Consequently, the condensation principle can be applied to an arbitrary sequence of these functions. Let us show now that the sequence { fn(z)} converges in A. Suppose that there exists a point z0 E A such that 1fn(z0)} diverges. Then, there exist two subsequences of functions that converge at z = z0 to distinct limits. Furthermore, these two subsequences themselves contain subsequences { fn,(z)} and { fnu(z)1 that converge uniformly in an arbitrary closed subset of A to functions f.(z) and f..(z), that are regular and univalent in A except at z = 00, and that have different values at z0. It follows from our proof of the necessity of the condition of the theorem as applied to the sequences {Bn,} and {Bnu} thai the functions f .(z) and f ..(z) map the domain A onto the kernels of these last sequences, that is, onto the same domain B. The function

z' =f* ' (f,

,

(z)) = + (x)

maps the domain A into itself and ifi(z)i z. Suppose that >fi(z) has, in a neighborhood of z = 00, the expansion

where a,,, for v > 0, is the first nonzero coefficient (which necessarily exists).

232


Consider the functions 02(z), 03(z), the expansions

defined in a neighborhood of z = oo by

W_3(z)_ (2(z))=z+3y+ .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

These functions also map the domain A univalently into itself. Since the domain IzI > p is contained in A, the functions t/i(z), for n = 2, 3, , are, in particular, univalent in IzI > p and the functions !(pz)/p = z + na,/p"+lzv are valent in IzI > 1. It then follows as a consequence of the area theorem that V+I I < 1, n = 2, 3, , which can be the case only if a = 0. This conIna ,,/p

tradiction shows that ,(z) = z, that is, that f.(z) = f..(z). This last fact in turn shows that the sequence 1 f(z)] converges in z0. Thus, the sequence 1f (n)1 converges everywhere in the domain A. The uniform convergence of this sequence inside the domain A now follows from Vitali's theorem. This completes the proof of the sufficiency of the conditions of the theorem and hence the theorem itself. One can also prove other theorems on the convergence of sequences of univalent functions on the basis of the manner in which they are normalized. We shall give one of these theorems. Just as above, we shall first introduce the following concepts: Let 1B,I , where n = 1, 2, , denote a sequence of domains contained in the z-plane: Suppose that a given point z0 belongs to each of the domains B. We define the kernel of this sequence with respect to zo as the largest domain B containing the point z = z0 and having the property that an arbitrary closed domain contained in it belongs to every B from some n on. Convergence to a kernel is defined just as before. Obviously, the kernel of the sequence of images of the domains B under a fractional-linear mapping of the z-plane is the image of the kernel B (of course, with respect to the image of the point z0). This does not violate convergence to the kernel. Furthermore, if we perform similarity transformations and then let p approach 1, the z '= pnZ on the domains B for n = 1, 2, kernel of the sequence 1B} remains unchanged and convergence to it is not violated. We shall now prove a convergence theorem. Theorem 2. Suppose that a point z = z0 belongs to each of a sequence of


233

domains An, for n = 1, 2, , in the z-plane. Suppose that the sequence }An} converges, with respect to z0, to the kernel A. Suppose that the functions C= fn(z) map the domains An univalently onto domains Bn in such a way that fn(zo) = Co and fR (zo) > 0 for n = 1, 2, . For the sequence {fn(z)} to converge uniformly inside the domain A to a univalent function, it is necessary and sufficient that the sequence (Bn} have a kernel with respect to 4 and converge to it. Then the limiting function f (z) maps A univalently onto B. We may assume that z 0 = Co = 0 because we can always arrange for this by making

translations of the z- and

planes. The proof of the first part of the

theorem is now based on Theorem 1.

Proof of the necessity. If fn(z) -' f (z) uniformly in the interior of A and f (z) is univalent in A, then f (0) f'(0) 0. Consequently, the sequence of the functions In (0)

Fn(z) =

1

In (z)l which are univalent in the domains An obtained from An by means of the mapping z '= 1/z, converges uniformly inside the corresponding kernel A'. Therefore, on the basis of Theorem 1, the sequence of the images Bn of the domains An under the mappings C= F,(z) converges to the kernel B'. Consequently, the sequence }BnI converges to a nondegenerate kernel B connected with B' by the relationship J'= The conclusion on the nature of the mapping C= f (z) follows. Proof of the sufficiency. Suppose that Bn --. B. Then, in the same notation, the sequence {Bn} has the kernel B' and it converges to that kernel. Consequently the sequence {Fn(z)} defined by

F.

(z)

=

fn (0)

fn(Z)

converges uniformly in the interior of the domain A ' to a univalent function F (z) such that F(oQ) _ o and F'(00) = 1. But then the sequence { fn(z)} defined by

fn z) =

fn (0)

Fn(z)


234

converges uniformly in the interior of the domain A to the univalent function

f(z)=

Ci

F(Z)

,

C/Q,

which maps A onto the kernel B of the sequence of the domains B. This completes the proof of the Theorem. §6. Univalent mapping of multiply connected domains onto circular domains. The continuity method

In §§2 and 3, when we were solving certain extremal problems, we proved theorems on the existence of a univalent mapping of multiply connected domains onto canonical domains. In the case of finitely connected domains, an incomparably stronger method of proof is the continuity method. The proofs that use this method are extremely similar to each other. They are based on the uniqueness of a univalent mapping of finitely connected domains onto the canonical domains in question and on a single topological theorem of Brouwer. We shall illustrate this method by proving an important theorem on univalent mapping onto circular domains, that is, domains bounded by a finite number of complete circles with no points in common (some of the circles may degenerate to points). We first prove a uniqueness theorem for circular domains. Theorem 1. There exists no more than one function that maps a given finitely connected domain B in the z-plane including - univalently onto a circular

domain in the -plane, also including -, and that has the expansion C= z + in a neighborhood of the point z = --. One can easily see that this theorem is equivalent to 't'heorem 1 '. Every function that maps a circular domain B (finitely connected) univalently onto another circular domain B' is a fractional-linear funcaiz-1

+

tion.

Since the two theorems are equivalent, we need only prove Theorem 1

,.

Proof. Suppose that the domain B is contained in the z-plane and is bounded by the circles K1, K21 , K,,, that the domain B' is contained in the t; plane and is bounded by circles K I', K2' , , K, and that the mapping in question maps Kv onto K v (v = 1, 2, - , n). Without loss of generality, we may assume that the domains B and B' contain the point w and that this point is mapped into itself.

§6. MAPPING ONTO CIRCULAR DOMAINS

235

Denoting the mapping function by 4 _ f (z), we then have f (oo) = -. The function f (z) can be extended across the circle K,, (v = 1, 2, , n) into the domain B,, obtained from B by inversion about K,,. It maps B univalently onto the domain B, obtained from B' by inversion about K. Consequently, the function , B unif (z) maps the circular domain consisting of the domains B, B1, valently onto the circular domain consisting of the domains B', B j , , B,. We reason analogously with these extended domains and continue the process indefinitely. As a result of such extensions in the z-plane, we obtain a limit domain G, which the function C= 1(z) maps univalently onto a limit G' in the 4 -plane. Let z '= S (z) denote a fractional linear function composed of an even number of inversions about the circles K1, K29 , K,, taken in any order. Then, the function f (S(z)) also maps the domain G onto the domain G' univalently since the function

S(z) maps the domain G into itself. Consequently, [(z) is invariant under the group of fractional-linear transformations z '= Sk(z), for k = 1, 2, . . , composed , K,,. The functions of a finite number of inversions about the circles K1, K21 z '= Sk(z), for k - 1, 2, .. , exclusive of the function z ' - z map the domain B , that have no common onto disjoint circular domains B(k), for k = 1, 2, , K,,. boundary points and lie each inside one of the circles K1, K21 Consider a disk Iz - al < r contained in B. Then, in accordance with Koebe's covering theorem, the domain B(k) (for k = 1, 2, - ) contains a disk of radius (r/4)ISk (a)I Consequently, the area of the domain B(k) is not less that Since the sum of the areas of all the domains B(k ), for k = n (rI Sk 1, 2, , is finite, we see that the series (a)1/4)2.

I Isk(a)I9

k=1

(1)

converges. If we now denote by Kk, for k = n + 1, n + 2, , all the circles that serve as boundaries for the domains B(k), for k = 1, 2, , and if we denote their radii by Rk for k = n + 1, n + 2, , we can show that the series

I Rk

(2)

k-n+ t converges. To see this, let us delete from the domain G the images of the point z = 00 under the mappings z '= Sk(z), k = 1, 2, . Then all the functions Sk(z) are regular and univalent in the remaining portion G* of the domain G. Therefore, from the general distortion theorem (see §4 of Chapter II), we have on K

236


(v= 1, 2,---,n) for all k= 1,

ISk(z)IM,ISk(a) where .tl depends on v but not on k or z. Therefore, we get the following expression for the circumference 2rrRk v of the circle Kk v onto which the function z '= Sk(z) maps Kv: 21rRk., =S Ky

K,

from which it follows that

Rky-MA IS,(a)I. If we square both sides of this inequality and then sum over all k and all v, recalling that each circle Kk is the image of a boundary circle of the domain B under exactly one of the functions z '= Sk(z), we obtain co

n

m

E R42 E M.Rv- Y Sk(a)I®

k=n+1

k=I

v=1

This proves the convergence of the series (2). Analogous conclusions hold for G'. Consequently, if we denote by Rk the radii of the circles into which the circles Kk, for k = n + 1, n + 2, , are mapped by the function C= f(z) defined on G, we conclude that the series 00

Y

Rk

k=n+I converges. It then follows, in particular, that the series co

E Al

k=n+l

(3)

converges, where Ak is the oscillation of the function f(z) on Kk, that is, Ak

max

If(z')-f(z,)I.

Let us now denote by G. a circular domain contained in G that contains B and that is bounded only by the circles Kk,, Kk,,, ... , Kk(,,,)

with indices k', k ", .. , k(-) sufficiently large that


237

v=I

where a is a given positive number. This is always possible by virtue of the convergence of the series (2) and (3). Now, for a sufficiently large circle K: lzl = R, we have, in accordance with Cauchy's formula, f (z')

1

1

z'-dz z - v=1 tai

f(z)=2ni

f (z')

z'-z dz' Kk(r)

(4)

for any point z belonging both to G. and the disk lzl < R. But f (z) dz' =

1

J z' - z

2ni

1

2tci

k(,)

C f (z') -f (z) dz' J

z' - z

k(v)

Now let d denote the distance from z to the boundary of the domain G.. It follows from what was just shown that m

m f (z

1

z' -z

I.d 2ni

v=1

Kk(v)

dz'

C

I "Al" d v=1

R,,(v)

m

M

Rk(r) `

okh) v-1

d'

v=1

But the left-hand member and the first term of the right-hand member of (4) are independent of the domain G. chosen and hence of e. Therefore,

f (z) =tai1 ` z'-z f (z) dz'. This shows that the function f(z) is regular in the disk lzl < R for arbitrarily large R, that is, that f(z) is an entire function. Since this function has a simple pole at z = -, we conclude that f(z) is a linear function. This completes the proof of Theorem I'. This existence theorem which we now wish to prove consists in the following:

Theorem 2.1) Every n-connected domain B in the z-plane can be mapped 1) This theorem was proved by Koebe. The continuity method was developed in his article [1918a] It has been simplified by the present author (see Goluzin [1938a]} Another proof has been given by Keldys [1939].

238


univalently onto a circular domain in the cplane. Among these mappings there is only one normalized mapping that maps a given point z = z E B into C= - and has an expansion in a neighborhood of the point z = a of the form z

a+al(z-a)+... or z+ z +...,

according as a is finite or infinite. Before proceeding with the proof of this theorem, let us recall a few concepts from n-dimensional geometry. Consider n-dimensional Euclidean space, in which a point is an n-tuple of finite real numbers (x1, x2, , xn). We denote this space by Rn and we denote a point in it by a single letter, for example, x or by an expression of the form x(xl, ... , xn) indicating in parentheses the specific real numbers defining it, these numbers being called the coordinates of the point x. Two points are thought of as coinciding if and only if their corresponding coordinate are equal. The , x(')) and x(Z)(x(Z), distance between two points x(1)(x(1), , x(2)) in the n n 1

1

-x(z))z]1/z An space Rn is defined as the number p(x(l)x«)) ' k1 (xCI) k k open ball of radius p with center at a is defined as the set of all points x E Rn whose distance from a is less than p. A neighborhood of a point a E Rn is defined as any open ball with center at a. A neighborhood is sufficiently small if the radius of the ball is sufficiently small. A set E C Rn is said to be bounded if it is contained in some ball. A point a E R is said to be a cluster point of a subset E of Rn if every neighborhood of the point a includes at least one point of E other than a. A sequence of points 5(m) E Rn, for m = 1, 2, , is said to converge to a point x(0) c R if p( (m), x(0)) --p 0 as in --. oc. A point a belonging to a set E C Rn is called an interior point of E if some sufficiently small neighborhood of a is contained in E. A set E C Rn is said to be closed if it contains all its cluster points. A set E C R is said to be open if it consists only of interior points. It is said to be connected if any two points in it can be connected by a continuous curve contained in E. By a continuous curve in Rn, we mean a set of points in the space R the coordinates of which can be repre, n, where each 01, (t) sented in the form xk = q,k (t) for a < t < b and k = 1, is a continuous function of t in the interval a < t < 6. All the properties analogous to the properties of point sets in a plane that were expounded in the geometrical introduction can be established for point sets in the space Rn. Now, consider a mapping of a subset E of Rn onto a subset E' of another


239

n-dimensional Euclidean space R,,. This means that to every point in the set E is assigned a point in the set E'. The mapping is said to be one-to-one or bijective if to every point of E there corresponds exactly one point of E' and vice versa. The mapping is said to be continuous if, for every sequence lx(')[ of points in E that converges to a point x(') in E, the sequence of images of these points converges in E' to the image x '(0) of the limit of the original sequence. It is easy to show that the inverse of a bijective continuous mapping of a closed set E C Rn onto a set E 'C Rn is also continuous. A one-to-one continuous mapping is also called a topological mapping. Obviously, the image of a closed set under a topological mapping is a closed set. The following theorem of Brouwer on topological mappings of arbitrary sets is known as the theorem on preservation of a domain. Theorem 3 (Brouwer). The image of an interior point of a set E C Rn under a

topological mapping is always an interior point of the image set E'; equivalently, the image E' of an open set E C Rn under a topological mapping is an open set. We postpone the proof of this important theorem to the following section. The proof of Theorem 2 by the continuity method is based on it. Proof of Theorem 2. By virtue of the results of § 2, any finitely connected domain can be mapped univalently onto the plane with parallel straight-line cuts at an arbitrary given inclination 0 and with suitable normalization. Therefore, it will be sufficient to prove Theorem 2 for domains constituted by the z-plane with rectilinear cuts parallel to the real axis and for a = .. Let us denote by 9 the family of all n-connected domains B consisting of the z-plane with n finite rectilinear cuts parallel to the real axis (numbered from 1 to n) and let us denote by 9R' the set of all n-connected circular domains B' in the cplane with boundary circles numbered from 1 to n. Two identical domains B and B' wi th boundaries numbered in a different manner are considered as distinct. Let us consider normalized (as indicated in the theorem) univalent mappings of domains B E 9R onto domains B 'C 91' such that identically indexed boundary curves of the domains B and B' are mapped onto each other. We consider domains in the families 911 and 9' as corresponding to each other if they can be mapped one onto the other as indicated. Then, from Theorem 1 of the present section and Theorem 2 of § 2, we conclude that


240

(a) to every domain B E 9 there corresponds no more than one domain B' E 9R',

(0) to every domain B' E 91' there corresponds a unique domain B E 9. Our problem is to show that to every domain B E 9R there corresponds exactly one domain WE W. The families 9 and 21 consist of domains defined by 3n parameters. For the parameters of the domain B we may take, for example, the coordinates of the origins of the cuts and their lengths. For the parameters of B' we may take the coordinates of the centers and the radii of the boundary circles. If we represent the domains B and B' by points in 3n-dimensional Euclidean spaces Ran and R3n and denote these points also by B and B', we conclude immediately from geometrical considerations that the point sets B and B' corresponding both to the domains B E 9 and to the domains B'E 9R' (and we shall denote these also by .9 and 9R') are connected open sets in Ran and Ran respectively. If we set up between the points of the sets 11 and 9' the same correspondence as exists between the domains of the families 3 and 9R', we conclude on the basis of (a) and 09) that to every point B 'E R' there corresponds exactly one point B E 2 and that to every point B E 2 there corresponds no more than a single point B' E W. Let us show that this correspondence is continuous, specifically that B1 --. B as Bi B' (that is, as the boundary circles of the domain B1' ap,proach coincidence with the boundary circles of the domain B '). Here, B1 corresponds to B11 and B corresponds to B'. If a sequence of the domains B' did not converge to a limit domain, we could choose from it. two subsequences }Bn 1}

and IBn2} that converge to the domains B and B0, where B I B0. On the basis of the convergence theorem, we conclude that the domain B' is mapped normwisc onto B and also onto B0, which, in accordance with (3) implies that B = B0. Here, we assumed that the boundaries of all the domains B1 are bounded, but this follows from Lemma 2 of §2. We mention also the following property of the limit domains: if B. E 9 and B. and the B,, if there exists a sequence of domains Bn E 91 such that Bn can be mapped normwise onto Bn E 9', then the domain B. can be mapped normwise onto B: E 99'. This is true because the sequence of the domains B converges to a domain B.' E 9R' since otherwise would there would be two subsequences }B,,'1} and }B'2} that converge respectively to domains B . and Bo,


241

where B: Bo. It then follows on the basis of the same convergence theorem that B; = Bo. But if B --> B. and B,,' B:, then we conclude, again on the basis of the convergence theorem, that the domain B. can be mapped normwise onto the domain B., which proves the assertion made. From this point, the proof of the existence theorem is fairly simple. We are

now dealing with a one-to-one continuous mapping of the open set Dl' of points in 3n-dimensional space R3- onto a subset (proper or otherwise) of the set Dl in another 3n-dimensional space R 3n- Consequently, Brouwer's theorem can be applied. According to this theorem, the image Do of the set 9R' is an open set in 3n-dimensional space. Let us show that Dto coincides with 9. Let B 1 denote a domain in the family Di that can be mapped onto the domain B' in the family V'. Such a domain can be obtained by a mapping of an arbitrary domain in 9' onto the domain in Dl corresponding to it. On the other hand, let B 2 denote an arbitrary domain in V for which we have not yet determined whether it can be mapped. Let us connect the point in the space Ran representing the domain B2 with the point representing the domain B1 by a continuous curve L contained in Dl . Let us proceed along this curve from the point B 1 to the point B2. If the domain B2 could not be mapped onto a domain in Dt', then, as we proceed along L from B1 to B21 we would encounter a point B. such that all the domains in DI corresponding to points of the curve L between B1 and B. could be mapped onto domains in D2' and the domain B. either could not be mapped onto domains in DI' or would be a cluster point of the set of points of L corresponding to domains that cannot be mapped onto domains in Df'. But this is impossible because, by virtue of the property of the limit domains mentioned above, the domain B. can be mapped onto a domain DI' and hence the point B. belongs to Dio, but then it cannot be a cluster point of domains that cannot be mapped onto domains in Dl' because go is an open set. Thus, all points of the curve L from B1 to B2 belong to the set D10. But then, again on the basis of the property of limit domains, B2 E DI0.

Thus, we have shown that the set go coincides with 9R and, consequently, for every domain in R there exists a domain in Dl' onto which the first can be mapped univalently and normwise. This completes the proof of Theorem 2. Since the proof that we have given used only the uniqueness theorem and a few trivial properties of circular domains, the same procedure can be used with

242


equal success in proving theorems on the existence of a univalent mapping of finitely connected domains onto various other canonical domains as soon as we have proved the corresponding uniqueness theorem for them. For example, we can prove in this way the following theorems of Koebe, 1) which generalize Theorems 1 and 1 ' of § 2 and Theorems 1 and 1 ' of §3:

Theorem 4. For a given point a E B, there exists a unique function mapping an arbitrary given n-connected domain B in the z-plane with boundary continua K1, , K univalently onto the cplane cut by n rectilinear cuts with given inclinations B1, , On to the real axis in such a way that the continua K,,, for , n, are mapped respectively into the cuts with inclinations 8,,, the v = 1, point is mapped into C= oo, and the expansion of the mapping function about z = a has the form z

1 a+a1(z-a)+...

or

z+ z +

according as a is finite or infinite. Theorem 5. Every n-connected domain B in the z-plane with boundary con, K can be mapped univalently onto the -plane with n cuts along tinua K1, arcs of logarithmic spirals of inclinations 01, , On respectively to the radial , n, are mapped directions in such a way that the continua K,,, for v = 1, respectively into arcs of inclination B,, given points a and b in B are mapped into 0 and - respectively, and the expansion of the mapping function about z = b has the form

Z-b

ao+at(z-b)+...

or

z+aro+ Z -...,

according as b is finite or infinite. The uniqueness of the mapping functions referred to in these theorems is proved in just the same way as Theorem 2 of §2 and Theorem 2 of §3 were proved.

By the continuity method, Koebe [1918a] proved, in particular, the following theorems: Theorem 6. Every n-connected domain B contained in the disk JzJ < 1 that 1) See Koebe [1918a] and Goluzin [1938a] Krylov [1938] has given a different proof of these theorems.


243

includes the point z = 0 and has the circle IzI = 1 as one of its boundary continua

can be mapped univalently onto the disk I < 1 with cuts along arcs of logarithmic spirals of given inclination B in such a way that zl = 1 is mapped into I = 1 and the mapping function C= f (z) is normalized by the conditions f (0) =0 and Oo) > 0. This mapping is unique. Theorem 7. Every n-connected domain B contained in the annulus q < IzI 2 must be the same for the two domains. In the second case, an analogous situation holds for one of the circular domains in question and for the domain obtained from the other by the transformation z '= l/z. Conversely, if these requirements are satisfied, the two domains can obviously be mapped univalently onto each other. Thus, for it to be possible to map two n-connected circular domains of a special form onto each other, it is necessary and sufficient that, in the case n = 2, one, and in the case n > 2, 3n - 6 suitable real quantities determining one of these domains coincide with real quantities determining the other. These quantities are called the moduli of the circular domains in question and of all the domains that are mapped univalently onto them.

If we partition all n-connected domains into classes, putting into a single class all domains that can be mapped univalently onto each other, then each such class will contain circular domains of a special form and, in accordance with what was said above, all classes of n-connected domains constitute a set depending in the case n = 2 on one and in the case n > 2 on 3n - 6 real parameters. §7. Proof of Brouwer's theorem I)

To prove Brouwer's theorem as stated in the preceding section, we need some additional information regarding the space Rn. , x satisfying a given linear A set of points x E R. with coordinates x 1, equation a,x, + a,x, +... + anxn = b,

where al, a2, .. , an are not all zero, is called an (n - 1)-dimensional plane or simply the plane Rn_1. The set of all points common to two (n - 1)-dimensional planes the left-hand members of the equations of which are linearly independent constitutes an (n - 2)-dimensional plane in Rn. In general, the set of all points common to k planes under the same condition constitutes an (n - k)-dimensional plane in Rn. An (n - k)-dimensional plane in Rn can be regarded as an (n - k)dimensional Euclidean space. Let x(k) (x ik ), 1) (Sperner [1928].)

,

xnk )), k = p, 1,

, n, denote n + 1 points in Rn that

§7. PROOF OF BROUWER'S THEOREM

do not lie in a single plane. The set of points X (X 1,

. ,

245

x,) whose coordinates

are determined from the formulas n

t =1,2,...,n,x1 o,

n

1j=1,

(1)

is called an n-dimensional simplex and the points x(k), 4=0, 1, , n, are called its vertices. 1) Let us denote this simplex by 2. The numbers Ai in (1) are single-valued continuous functions of x1, , xn. To see this, remember that, since the points x(k), k = 0, 1, , n, do not lie in a single plane, the determinant

cannot be equal to 0. This means that the system of n + 1 equations in (1) is a system of n + 1 linear equations in the Al that has a unique solution. Consequently, the Al are obtained as linear functions of the coordinates xi, for , n; hence, they are continuous functions of the xi. The set of points i = 1, of the simplex E that are obtained from (1) when one of the Aj is equal to 0 is called an (n - 1)-dimensional face or simply a face of the simplex E. In general, the set of points of the simplex E that are obtained from (1) if k of the numbers Ai are equal to 0 is called an (n - k)-dimensional face. An (n - k)-dimensional face of a simplex E is itself a simplex in (n - k)dimensional Euclidean space, specifically the (n - k)-dimensional plane passing through those vertices of the simplex E that lie on that face. The point x E obtained from (1) for Ao = Al = -' . = An = 1/(n + 1) is called the center of the , xn simplex E. By virtue of the continuity of the Ai as functions of x 1, (which we showed above), it follows that an arbitrary point x E E. obtained from (1) for all nonzero A is an interior point of the simplex E. Let us show that a simplex I can be partitioned into a finite number of simplexes the diameters of which (the diameter of a figure being the greatest distance between any two points of it) is less than an arbitrary number e > 0 in 1) In the case n = 1, the simplex is a line segment; for n = 2, it is a triangle; for n = 3, it is a tetrahedron.

246


such a way that, if two subsimplexes have points in common, then the set of all common points is a common face of some dimension or else a vertex of these simplexes. Such a partition can be made as follows, for example: let us look at all possible permutations jo, j 1, , jn of the numbers 0, 1, , n. Let us the set of points x E I that have the representation (1) denote by with ai0 < ai 1 < . < ain. _Let us show that Yio,i 1,...,in is a simplex. Let 6(k) (6 ik ), , n) denote the center of an (n - k)-dimensional , Cnk )) (k = 0, 1, face of the simplex E with vertices at x0k), xUk+1), En , x(in). Then,

Elk]

_ vE= kXV')

n-k+1

1= 1, ...

,

(2)

n.

From (2), we have XiUv1=(n

(n-v)E(v+11,

n-1'

Xi'n1 Consequently,

Xi

X

=(n-{-1)X10,

X, = (n -o -{- 1) (A/v - A1, 1), n

21 XV = v-0

' = 1, ...

,

n,

n

v=0

X,

v-0

It follows that a point x belongs to i o.i I i.. tin if and only if it belongs to the simplex with vertices 6(0), 6(1), . . . , 6(n). Therefore, the set 2io-i 1,...,i n coincides with this simplex. The simplex I is therefore partitioned into (n + 1)!

simplexes lio.il,...'in.

Two simplexes and lio.i{,...,in have in common only those points x for which there are equal numbers among the numbers A0, A1, ' ' ' , An, that is, for which certain of the All are equal to 0. These points constitute the common faces of different dimensions of the simplexes

Jn and

Tn particular, all these simplexes contain the


247

point 60). Let us now look at the question of the dimensions of the simplex keeping the notation given above for it. Let d denote the greatest ')I for i = 1, , n. We then have , n and of the numbers jxi') 0, n

(xi1 -xi1'))I_n+l'

n,

1=0

, n, must be equal to 0). (since one of the numbers x(j) - zip ' ), for j = 0, 1, , n, then , xn) E I and xi = In=oaixi' ), i = 1, Therefore, if x (x 1, EjO)

- xiI=

n

n

a/(Et)-xt/))

I

n+1

/=0

In particular, for points 6(k), k = 1, 2, I

Eio)

-

J=0

n+

, n, we have

Elk) I c n+ l.

Reasoning analogously with the (n - 1)-dimensional face of the simplex , 6("), we can show that with vertices at 6"),

IEII)-E,k)Icnn

n -}-1

nd do n+1

for the points 6(k), k = 2, 3, , n. We proceed in this way with faces of fewer dimensions. As a result, we see for do j, j'=0, that all the numbers not exceed nd/(n + 1). Consequently, the quantities corresponding to the quantity d for the simplexes Yio,i 1,...,i do not exceed nd/(n + D. If we now repeat this partitioning process for the subsimplexes with each of the simplexes Y'io,i 1'"''in , then with the simplexes obtained from these, etc., the simplex is partitioned after p stages into subsimplexes having in common only faces of various dimensions and having the property that the difference between corresponding coordinates of the vertices of each of them does not exceed (n/(n + i))"d. in absolute value. Let V denote any of these subsimplexes with vertices y(k)(y(k), , n. Then, if , ynk)) for k= 0, 1, n

y, y' E E', yi =

ajyii),

/=o

y;

_

'LV i

i=o


24 8

we have n

n

vc-y;l=J=0Oi-a;))'ill= J=0

Xi)(YV'-yi° n

(n+1)PdI (xj /

(n+1)Pd

i=0

/

and n

P (.Y, y')=

i=I

(yi-y;)9_

n28(,i+1)

eP

d'

2y

(n+1)P

n

d.

(3)

It follows that the diameters of all the subsimplexes after a sufficient number p of stages in this partitioning process will not exceed an arbitrary given number e > 0. This is the desired partition of the simplex 1. To prove Brouwer's theorem, we need first to prove two lemmas.

Lemma 1. Let I denote an n-dimensional simplex with vertices x(j) for j = 0, 1, , n. Suppose that the points of I are distributed among n + 1 closed sets Ei, for .j = 0, 1, , n, in such a way that the following three conditions are satisfied: (1) each point x E I belongs to at least one of the Ei; (2) x(') E Ei; (3) Ei does not contain a single point on the face si with vertices in x(k) for k # j. Then, the sets Ei, for j = 0, 1, , n, have at least one common point. Proof. Let us partition the simplex I into a finite number of subsimplexes Ql, a2P ' + UP in such a way that any two of these simplexes have in common only faces of some dimension or other. Let us show that among the ak there exists a simplex containing points of each set Ei for j = 0, 1, , n. Each vertex of the simplex ak belongs to one or more of the sets E1. If it belongs to the sets where v1 < vZ < < vy , let us assign to it the number vl as coordinate. To the simplex ak let us assign as coordinates the n + 1 numbers which are the coordinates of its vertices and to each face let us assign the n numbers that are the coordinates of the vertices of ak in it. Let us show that the number of simplexes ak of which all the coordinates are distinct is an odd number. This assertion is true in the case n = 1 because then I is a line segment which is divided into subsegments ak by a finite number of points.


249

The coordinates of the vertices of these subsegments are the numbers 0 and 1. If we proceed from the vertex of E with coordinate 0, the first of the ak that we encounter with different coordinates has coordinates (0, 1), the next has coordinates (1, 0), then (0, 1), etc. (Here, we are indicating the coordinates of points met successively as we move along I.) The last of these ak must necessarily have coordinates (0, 1) because the second end of I must have coordinate 1. This proves that the number of segments ak with distinct coordinates is odd. Let us now suppose that the assertion has been proved for an (n - 1)-dimensional simplex and let us prove it for an n-dimensional simplex. Let tk denote the , n - 1). If tk > 0, number of faces of the simplex ak with coordinates (0, 1, then the simplex ak has coordinates (0, 1, , n - 1, a), where 0 < a < n. Here, if a = n, we obviously have tk = 1. On the other hand, if a < n, then tk = 2 because the coordinates of all n + 1 faces of the simplex ak are obtained from , n - 1, a) by discarding a single number and we obtain the numbers (0, 1, , n - 1) only by deleting the number a when encountered twice. Let e (0, 1, , n) and let f denote the number of all simplexes ak with coordinates (0, 1, denote the number of all ak for which tk = 2. Then, from what was said above, P

E tk=e+2f.

k=1

(4)

On the other hand, if a face of a subsimplex ak lying inside I has coordinates (0, 1, . . , n - 1), it appears twice in the sum Y.k=1tk because each such face belongs to two of the simplexes ak Let g denote the number of such faces belonging to distinct ak. A face of a simplex ak lying on one of the faces of the , n - 1) only if simplex 1, on the other hand, will have coordinates (0, 1, this face lies on sn. But sn is an (n - 1)-dimensional simplex the points of which belong to the n intersections sn n E;, j = 0, 1, , n - 1, that satisfy conditions (1), (2), and (3) of the lemma. Our assertion is considered proved for the case of (n - 1)-dimensional simplexes. Consequently, Sn will have an odd number h of faces sk with coordinates (0, 1, , n - 1). Thus, we have P

2] tk = 2g+ h.

k=1

(5)

Comparing (4) and (5), we obtain e = 2 (g - f) + h; that is, e, the number of simplexes ak with coordinates (0, 1, , n), is an odd number. Consequently,


250

there exist simplexes with coordinates (0, 1,

-

points belonging to all the sets E1, for j = 0, 1,

,

n). Each such simplex contains , n, because its vertices

belong to distinct E1.

Now, suppose that we are given a sequence of positive numbers Ek, for k = 1, 2, , that approaches 0 as k --+ oo. Let us partition Y. into a finite number of simplexes having in common only faces of different dimensions with diameters less than E1. Suppose that among the subsimplexes al is a simplex containing points of all the sets E;, for j = 0, 1, , n. From what we have shown, such a simplex exists. Let us now partition into a finite number of simplexes of the

same kind but with diameters less than E2- And suppose that aZ is a simplex containing points of all the sets E. We continue the process. As a result, we obtain a sequence of simplexes all a'Z, . each of which is contained in I and the sequence of diameters approaches 0. In accordance with Weierstrass's theoren generalized to the space Rn, there exists in Y. a point a an arbitrary neighborhood of which contains infinitely many of the simplexes ak. Since the sets E;

are closed, the point a belongs to all the sets E1, for j = 0, 1,

, n. This com-

pletes the proof of the lemma.

Let us point out one way of partitioning the simplex I into sets E; as indicated in Lemma 1 such that the sets E; have a unique common point a, which is an interior point of 1. Suppose that Y. has the representation (1) and that corresponding to the point a are the values of the parameters A; equal to A;0), where 0, for j = 0, 1, , n. Let us form closed sets E1, for j = 0, 1, , n, by A1°) assigning to E; all points of the set E for which A; > in the representation lies in at least one of the sets E1 because, other(1). Each point of the set wise, we would have Ai < 0) for all j = 0, 1, - , n, which would contradict the equations

For the same reason, the sets E;, for j = 0, 1,

, n have the point a as their

unique point in common. Furthermore E; obviously contains the point x(j) and We shall use this fact in what follows. does not contain points of the face

Lemma 2. Let E denote a closed bounded subset of R. Suppose that E is covered by a finite number of closed sets Ek such that each point of the set E belongs to at least one of them. If E includes a unique point a belonging to no


251

fewer than n + 1 sets Ek and this point is a boundary point of E (that is, not an interior point of E), then, by changing the sets Ek in an arbitrarily small neighborhood of the point a, we can arrange to have every point of the set E belong to no more than n of the sets Ek. Proof. Suppose that E is a simplex of arbitrarily small diameter and that a is an interior point of L. We can get the simplex E by taking an arbitrary simplex, displacing it in R in such a way that some interior point of it occupies the position of a, and then performing a similarity transformation on it about a. Let S

denote the boundary of I (that is, the set of points on all of the faces of 1). Define D = S n E. The set D does not include any points common to more than n of the sets Ek. Consequently, every point x E D has a neighborhood whose intersection with E can be covered by no more than n of the Ek. In accordance with the Heine-Borel theorem, generalized to Rn, it then follows that there exists an c > 0 such that every ball of radius c with center at an arbitrary point x C D consists of points belonging to no more than n of the sets Ek. Let us now cover S with a finite number of closed sets Fi, for j = 1, 2, - , with diameter less than c such that each point of S will belong tp no more than n of the sets F. Let us show that such a covering exists. Let us partition S into (n - 1)-dimensional simplexes with diameters less than i2n and then let us represent each such simplex as the union of n closed sets with a unique point in common, each of which contains one of the vertices of the simplex. Let us take one of the vertices of the subsimplexes, for example b. Suppose that it belongs to the simplexes li, for j = 1, 2, - , h. The point b belongs to only one of the closed sets into which each of the simplexes Ej is divided. Consequently, b is contained in h such sets, let us say, Fill for j = 1, 2, , h, that lie respectively in Yi. Let us set -

h

Fk=U F,. We reason in the same way for all the vertices of the subsimplexes. Then, the set S is completely covered by the sets Fk, and each point x E S belongs to no more than n sets Fk because an interior point of a subsimplex belongs to no more than n of the sets Fk into which it is partitioned and, consequently, to no more than n of the sets Fk. On the other hand, a point on a face of a subsimplex can belong to only those Fk that contain the vertices of that face and hence to no more than n - 1 of the sets Fk. Furthermore, one can easily show that the

diameters of all the sets Fk is less than e. Thus, the sets Fk provide the

252


required covering for S. Let us now combine the sets Fk with the sets Ek as follows: 1) If Fk does not contain points of D, we combine it with any one of the sets Ek. 2) On the other hand, if Fk does contain points of D, we combine Fk with one of the sets Ek to which such points belong. Let us show that each point x E S still belongs to no more than n of the sets Ek. This is true because a point x E S that does not belong to D belongs to no more than a of the sets FA, and hence now belongs to no more than n of the sets El, On the other hand, if x E D, it belongs to no more than n of the original sets, that is, to E1, EZ, , E f, where f < n. On the other hand, x belongs to certain of the sets Fk, let us say to F1, F2, , Fg, where g < h. The sets F1, . . . , Fg are contained in the ball of radius E with center at x. But, from our construction, this ball contains points belonging to no more than n of the original sets Ek, let us say to the sets E1, E2, , Eh, where 1 < h < n. From 2), the sets F1, . . , Fg are combined with certain of the sets E1, , Eh, Consequently, a point x E S can belong to no more than n of the modified sets Ek, namely to certain of these sets that contain the original sets E1, , Eh. Thus, after we adjoin the sets FA, to the sets Ek, every point x E S belongs to no more than n of the sets Ek. Since a is a boundary point of the set E, the simplex I contains an interior point c that does not belong to E. Let us distribute the points of the simplex E and hence let us redistribute the points of the set K = E U I among the sets Ek as follows: If a point x E S belongs to the sets EvI, E 2 , - . . , Evp, where p < n, then the entire segment cx (one-dimensional simplex) belongs to all these sets and only to them. With this new covering of the set E U E, the point c is the only point that can belong to more than n of the sets Ek. But c does not belong to E. Consequently, E does not contain points belonging to more than n of the sets Ek. This completes the proof of the lemma. Brouwer's theorem. A topological mapping of a set E C R onto a set VC R maps interior points of E into interior points of E . Proof. Let a denote an interior point of the set E and let I denote an arbitrarily small simplex contained in E and including a as an interior point. Corresponding to the point a is a point a' in E ', and corresponding to the simplex is a set ' C V. Let us show that a ' is an interior point of the set V. Let us subdivide I into n + 1 closed subsets E;, j = 0, 1, - , n, in such a way that each E; contains only one vertex and no point of the opposite face and in such a way that a is the only point belonging to all the sets E;. The mapping of E


253

onto E' assigns to the sets E; closed sets E;' in V that have the unique comcommon point a'. If a' were a boundary point of E' and hence of 2', then, in accordance with Lemma 2, by modifying the sets Ej in a neighborhood of a', we , n, to have no point in common. With could arrange for the sets E;, j = 0, 1, such a covering, the corresponding modified closed sets E; have no common points and at the same time satisfy the conditions of the lemma, which, however, is impossible. The contradiction proves Brouwer's theorem.

CHAPTER VI MAPPING OF MULTIPLY CONNECTED DOMAINS ONTO A DISK

§1. Conformal mapping of a multiply connected domain onto a disk

In §1 of Chapter II, we showed that no regular function can map a multiply connected domain bijectively onto the disk JCI < 1. Let us now consider, in addition to regular functions defined on a multiply connected domain B, functions that can be extended into B along an arbitrary path. Here, by such a function we mean an initial element of a function and all possible analytic continuations of that element into B. As we shall now show, among these functions there is one with the property that all values that it assumes in B lie in the disk I C1 < 1 and that each value in the disk I C1 < 1 is assumed at exactly one point of the domain B. We shall call the correspondence that this function sets up between points of the domain B and points of the disk I C1 < 1 a conformal mapping of the domain B onto the disk ICI < 1. The conformality follows from the fact that all elements of the mapping function are univalent in the corresponding domains and, consequently, their first derivatives do not vanish anywhere (that is, at any finite point). The mapping function, although it is regular at every point of the domain B, is not univalent in B because, if it were, it would set up a one-to-one correspondence between the domain B and the disk I i1 < 1, and we have seen that this cannot be. In the case of a finite domain B, the inverse function can be extended into the disk I I < 1 along an arbitrary path and hence is a regular function in that disk. Before proceeding to a proof of the possibility of such a mapping of a multiply connected domain onto a disk, let us first mention the exceptional case when such a mapping does not exist. This is the case when the domain B B. constitutes the

entire z-plane with two exceptional points a and b. To see this, let us assume 254

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255

without loss of generality that a and b are finite and let us observe that, in this case, the function t = log ((z - a)/(z - b)) can be extended into B along an arbitrary path and it maps the domain B onto the t-plane with the point -0 excluded. The inverse function z = 0(t) is regular in that plane. Therefore, if there existed a function C= f(z) that maps the domain B conformally onto the disk ICI < 1, the function C = f (0 (t)) would be regular in t I < Do and it would not exceed unity in absolute value, and this cannot be the case. I

Let us turn now to the proof of the basic theorem. Theorem.1) Every domain B in the z-plane with more than two boundary points can be mapped con formally onto the disk I I < 1 in such a way that corresponding to a given point z0 E B and a given direction at that point there are (for a basic element of the mapping function) a point C= 0 and the positive direction of the real axis. This mapping is unique. Proof. We note first of all that the domain B can be mapped onto a domain B* contained in I CI < 1 in such a way that to each point of B* corresponds only

One point of B. To see this, let a, b, and c denote three boundary points of the domain B. By means of a fractional-linear mapping, let us map the z-plane onto the t-plane in such a way that these three points are mapped respectively into 0, 1, and o-. The domain B is then mapped into a domain B* that does not include the three points 0, 1, oo. Therefore the function C= A(t), inverse to the modular function t = g(z) (see §6 of Chapter II) can be extended into B* along an arbitrary path and the values that it assumes all lie in the disk I CI < 1 and occupy a domain B*. Under this mapping, to every point of the domain B* there corresponds as preimage only one point of the domain B* and hence only one point of B. By means of a fractional-linear transformation of the disk I CI < 1 into itself, we can always arrange for the point C= 0 and the positive direction of the real axis to correspond to the point z0 and a given direction at z0. ( Other points in the disk 1CI < 1 may also correspond to z0.) Proof of the theorem is now reduced to proving it for the case of a domain B contained in the disk z I < 1. The given point z o E B is the point z = 0 and the given direction at that point is the direction of the positive half of the real axis. I

1) This theorem was discovered by Poincare. Rado noted [1922-1923a] that the idea of the proof of Riemann's theorem that was given in §2 of Chapter II can be used to prove this theorem also.

256

VI. MAPPING ONTO A DISK

In this case, let us consider the set 9 = (f(z)} of all functions f(z) with the following properties: 1) they are defined by basic elements about z = 0 for which f (0) = 0 and f ' (0) > 0; 2) they can be extended into B along an arbitrary path; 3) the values that these functions assume in B as a result of analytic continuation all belong to the disk I CI < 1; 4) every value of 4 in the disk < 1 is assumed I

I

by every function ar no more than one point of the domain B.

An example of such -a function is the function f (z) = az, where 0 < a < 1. We pose the following extremal problem: Out of all functions f (z) E R, find the one maximizing f'(0), calculated for the basic element of the function. Since there exists a disk I z I < p, where p > 0, that lies entirely in B, the basic element of each of the functions f (z) C 9 is regular and less than unity in absolute value in this disk. Therefore, by the Schwarz lemma, we have f' (0) < 11p. This shows that the set of numbers f'(0) is bounded above. We denote the least upper

bound by a. Obviously, a> 1. There exists a sequence of functions fn (z) E R, n = 1 , 2, , such that if, (0)} converges to a. According to the condensation principle, the sequence of basic elements of the functions fn (z) has a subsequence that converges uniformly in the interior of the disk z I 0. If we now extend these elements into B along different paths, we see step by step in accordance with Vitali's theorem that the sequences of analytic continuations obtained converge to the corresponding analytic continuations of the function fo (z). Consequently, the function fo (z) can be extended into B along an arbitrary path and IA (z)I < 1 everywhere. Let us show that C= fo (z) assumes in B every value in the disk I CI < 1 at no more than one point. Let us suppose, to the contrary, that there exist in B two distinct points z I and z2 such that fo (z 1) = fo (z2) = c. Here, fo (z 1) and f0 (z2) are the values at the points z I and z2 of certain elements of the function f0 (z) that are obtained from the basic element by analytic continuation along certain paths 11 and 12 contained in B. Suppose that the disks I z - z I I < e and I z - z 21 < E have no common points, that they are contained in B, and that these elements assume the value c neither in the interiors of these disks, except at the center, nor on their boundaries. Suppose that Ifo(z) - c I > m > 0 on the circles I z - z I I = r and I z - z 21 = e. Let k denote an integer such that I fnk (z) - fo (z)I < m on these two circles. Then from I

fnk(Z)-C^ (Jnk (Z) - A

(z))+VO(z)-c)

§ 1. MAPPING OF A MULTIPLY CONNECTED DOMAIN

257

it follows, on the basis of Rouche's theorem, that the function 1 k (z) assumes the value c at two points of the domain B, which is impossible. This proves our assertion. Let us show now that the function f0 (z) provides the mapping mentioned in the theorem; specifically, that it assumes in B every value C in I CI < 1. Suppose, to the contrary, that there exists a number c such that I c I < 1 and f (z) c everywhere in B. Let us denote by t = q(C) the function used in §2 of Chapter II. We note that this function can be extended into I CI < 1 along an arbitrary path that does not pass through c and that 0 (0) = 0 and q ' (0) > 1. We see that the function F (z) _ 0 (fo (z)) C 3 and F' (0) = f6' (0) 0 (0) > a., which is impossible. It remains to prove the uniqueness of the mapping mentioned in the theorem. Suppose that C = fI (z) and C= f2 (z) are two functions providing this mapping. Then the function C' = f 1 (fZ I (C)) = OW and its inverse C= f2 (I1 (C )) are regular in the disks ICI < 1 and IC' I < 1, respectively. Therefore, the function 4' = q(c) maps the disk ICI < 1 bijectively onto itself in such a way that 9(0) = 0 and 0' (0) > 0. But then, 0(C) _ C; f 1 (z) = f2 (z). This completes the proof of 0

the theorem. Now let C= f(z) denote the function providing the mapping indicated in the preceding theorem. Let C= F W denote any other function mapping the domain B onto a disk but not satisfying the same normalization conditions as C = f (z). Then,

the function C' = F(f-1(C)) maps the disk 141 < 1 bijectively onto itself. Consequently, it must be a fractional-linear function. Thus, F(z) = (af(z) + 13)/(y f(z) + 8); that is, any other mapping function can be expressed as a fractional-linear function of f (z). In particular, if we take two distinct elements of the same multiplevalued mapping function f (z) at the point z0 and extend them along all possible paths contained in B, we obtain two functions, that is, two branches of f (Z), both of which map the domain B conformally onto the disk I C.I < 1. It follows from what has been said that these branches can be expressed each as a fractionallinear function of the other. Thus, all branches of a multiple-valued mapping func1tion can be represented as a fractional-linear function of just one of them and the corresponding fractional-linear functions map the disk ICI < 1 into itself. This conclusion enables us to study more deeply the nature of the multiplevaluedness of the mapping function f(z). Let f0(z) denote one of the branches of the function f (z). We denote by C' = S (C) = (a.C + 0)/(yC + 8) or more briefly, by S the fractional-linear function mapping the branch f0 (z) into the branch

258


f. (z); that is, f. (z) = S fo (z). Let us show that the set G of all "transformations" S corresponding to all possible branches of the function f (z) constitutes a group. Let us extend the branch S 1 f0 (z), S 1 E G, along a closed path such that extending the branch fo (z) along this path yields the branch S2 fo (z), S2 E C. We thus obtain a new branch S 1S2 f0 (z) of the function f (z). Therefore S 1S2 E C. Analogously, if we extend the branch fo (z) along a closed path the, extension of fo (z) along which yields a branch S f0, S E G, though in the opposite direction, we obtain a branch f. (z) = S f0 (z), where S. E G and we again obtain from Sf0(z) the branch fo (z) and at the same time the branch SS, f0 (z); that is, we have

f0 (z) = SS, fo (z), so that S. = S -1. Consequently, S -1 E G. Thus, G is a group. We note that the neutral element of this group is the identity transformation With regard to the function z = 0(0 inverse to C= f (z), when B does not < 1 along an arbitrary path and hence is contain -, it can be extended into regular in I I < 1. From this we conclude that, in the general case, its only singularities in the disk CI < 1 are poles; that is, that it is a meromorphic function in I CI < 1.

It follows from the relation z = 0(f(z)) that 0(Sfo(z)) = 0(f0(z)) or 0(SC) _ 95 (c), where S E G. This shows that the function 0(e) remains invariant under transformations belonging to the group G. Therefore, it is an automorphic function with group C. We noted above that all the transformations belonging to the group G leave the circle ICI = 1 unchanged. Let us show now that the fixed points of all these transformations lie on the circle ICI = 1. By fixed points of the transformation S, we mean points C satisfying the equation SC= C. A fractional-linear transformation other than the identity transformation has no more than two fixed points. Let us suppose that a given transformation S E G has a fixed point a such that

< 1. Then, 1/a will also be a fixed point of S. But the equation Sa= a implies that SC a in a sufficiently small domain 0 < ICI al < S. Suppose that I SC - al > m > 0 on the circle I C - al = S. Let 0 denote an arbitrary number in the domain 0 < IC - °'I < m. Since S(C) - f3 = (S (C) - a) + (a- 0), it follows from Rouche's theorem that the function S (C) - j3 has a zero y in the disk C - al 1. Therefore, all the fixed points lie on the circle I I = 1.

§1. MAPPING OF A MULTIPLY CONNECTED DOMAIN

259

If a transformation S E G other than the identity transformation has two fixed points on the circle ICI = 1, it is called a hyperbolic transformation. If it has only one, it is called a parabolic transformation. Let us show that, if S is either a parabolic or a hyperbolic transformation, the sequence of transformations S, S2, S3, . . does not include the identity transformation. Let us suppose first that S is a hyperbolic transformation with fixed points SI and 6. If we define the transformation t = T(C) = (C- CI)/(C- Cz), we see that J' = TST-1(C) is a fractional-linear transformation with fixed points 0 and oo. Therefore, C' - aC and, consequently, TS(e) = aT (C). This last shows that the transformation J' = S(C) is obtained from the equation

-s-a

Cs, a=const.

(1)

in (1) lies on the circle ICI = 1, then the point J must Now, if the point also lie on the circle ICS I = 1. Therefore, the arguments arg((t; ' - 0/'- 0) and arg ((C- CI) (C- Cz)) indicating the angles subtended by the segment blb2 at the points e' and C are equal. From this we conclude that a > 0. Furthermore, a 1 because we would then have C' = 4 which cannot be. Thus, the transformac f. Here, a tion S can be obtained from equation (1) by solving that equation for is positive and different from unity. The transformation J' = S"t; (where n = ± 1, ± 2, . ) can obviously be obtained from the equation

_

C'-C, C'-Cf-a

. C-C2

(2)

Therefore, it is always different from the identity transformation. Furthermore, it follows from (2) that the sequence of the functions PC and S-"J converge as n - + = to the constants SI and C2. This convergence is uniform on an arbitrary closed set not including the points z1 and z2. This last conclusion will be necessary for §2. We proceed analogously in the case in which S is a parabolic transformation with fixed point S1. By making the transformation t = T (C) = 1/(C- SI) we see that t' = TST-1(t) is a transformation with unique fixed point oo. Therefore, 7 , = C + a and, consequently, TS (C) = T (C) + a, where a = const. This shows that the transformation C' = SC is obtained from the equation C'-C,

C-C1a' +

(3)

260


where a X 0 because, in the opposite case, we would have ' = C, which is impossible. On the other hand, the transformation C' = S' C (for n = ±1, ±2,

)

is defined by the equation

_ +na --C1 1

C'

CI

(4)

Cr

Therefore, once again, it eannot be the identity transformation for any n. We note that the sequences of the functions S"C and S-' converge uniformly to 6 on an arbitrary closed set including the point CI as n +o. This property shows, in particular, that G is an infinite group, that is, that it contains an infinite number of distinct transformations and that f (z) is an infinitely-many-valued function in B. On the other hand, the function 0(0 assumes in ICI < 1 every value in B at an infinite number of points. All points in the disk ICI < 1 at which the function O(C) assumes the same value are called equivalent points. For every point a in the disk I a I < 1 there is an infinite set of points equivalent to a. These are all obtained from each other by transformations of the group G. Since these points are zeros of the function O(C) - 0(a), they do not have points of accumulation in I I < 1. It follows, in particular, that the group G is countable. Since the function f(z) is not single-valued in B, the question arises as to what conditions must be satisfied by the path 1 contained in B and issuing from and returning to a point z0 in such a way that as we proceed along it with some element of the function f (z) we return to the point z0 with the same element. Consider the case in which B does not include oo. If, as we proceed along 1 from zo with some element of f(z), we return to z0 with the same element, the corresponding point C= f(z) issuing from some point CO returns to Co after describing a closed path l' in the disk ICI < 1. The path l' can, by a continuous (for example. similarity) deformation be reduced to a. point that does not leave the disk ICI < 1. Obviously, under this deformation, the preimage 1 as it is continuously deformed also reduces to a point without getting outside B. Let us show that, conversely, if the path l can, by a continuous deformation, be reduced to a point without ever getting outside B, then, by proceeding along it with an arbitrary element of f(z), we always return to the original point with the same element of f (z). This is true because, otherwise, under such a deformation, the corresponding path 1', while being continuously deformed in I CI < 1, will go from the point Co to some other

§1. MAPPING OF A MULTIPLY CONNECTED DOMAIN

261

point equivalent to it and cannot be reduced to a point. Thus, for us to proceed in B along a closed path with some element of the function f (z) and return to the original point with the same element of f (z), it is necessary and sufficient that it be possible to reduce this path by a continuous deformation to a point without getting outside B.1) In conclusion, let us pause to construct a Riemann surface connected with the mapping that we have been considering. We confine ourselves to the case of a finitely connected domain B. If the domain B is n-connected, we can render it simply connected by means of n - 1 suitable transverse cuts (broken lines). We then take an infinite set of copies of the domain B so cut and consider them as lying one above another. We choose any one of them as the basic domain and denote it by Bo. We adjoin to Bo along each edge of each cut one of the remaining copies. We make this con-

nection-this gluing-along opposite edges of the cuts. As a result, we obtain a (2n - 1)-sheeted surface B I lying over the domain B and having (2n - 3) (2n - 2) free edges of cuts, that is, edges along which no gluing has been made. Then, we again adjoin along each of these edges one of the as yet unused copies of the cut domain B in such a way that we again perform the gluing along opposite edges of the cuts. We then obtain a new surface B2 with a finite number of sheets and some number of free edges of cuts. If we continue this way indefinitely, we finally obtain an infinitely-sheeted Riemann surface B.. If we consider the initial element fo (z) of the mapping function f(z) defined in the domain B0, the analytic continuation of this element into B is identical to its analytic continuation onto the surface B00 . As the result of all possible analytic continuations of f (z) onto the surface B., we obviously obtain a singlevalued function on that surface. Consequently, the function C = f (z) that we studied above also provides a one-to-one mapping of the surface B. onto the disk ICI < 1. The surface B. is called the universal covering surface of the domain B. The modular surface constructed earlier (see §6 of Chapter II) is, in this terminology, the universal covering surface of the domain consisting of the plane with the points 0, 1 and os excluded.

1) If B includes os, an analogous condition must be satisfied on the Riemann sphere.

262


§2. Correspondence of boundaries under a mapping of a multiply connected domain onto a disk

In § 1, we examined a function f (z) that maps a given multiply connected domain B conformally onto the disk CI < 1 only inside B. If we now let the point z approach the boundary of the domain B, we can easily show that the sequence of the corresponding points C approaches the circle ICI ="1 uniformly. In other words, for arbitrary e > 0, there exists a S > 0 such that, if the distance

from the point z E B to the boundary of B is less than S, the distance of the corresponding point C from the circle ICI = 1 is less than c. To see this, let E denote the set of all points z E B corresponding to points of the disk I Cl < 1 Then we need merely take for S the distance from the set E to the boundary of the domain B. When we make a more thorough study of the behavior of the function f (z) as z approaches the boundary of the domain B, we shall confine ourselves to the case of a finitely connected domain B. Such a domain can easily be mapped univalently onto a domain bounded by closed analytic Jordan curves and isolated points. The correspondence of the boundaries under such a mapping has already been studied. Therefore, for our purposes, it will be sufficient in the future to consider a domain B with a boundary consisting of closed analytic Jordan curves and isolated points.

Let a denote an isolated boundary point of the domain B, which we may assume to be finite. Suppose that the closed disk I z - a I < r contains no boundary points of the domain B other than a. As we go around the circle 1: I z - a I = r one time in the positive direction, the branch fo (z) of the function f (z) becomes the branch So fo (z), where So / 1 is one of the transformations belonging to the group G considered in §1. Furthermore, if we go n times around l not necessarily in the same direction each time), we end up at the original point with the element So" fo(z). Let us show that the transformation So is a parabolic transformation. Let us suppose, to the contrary, that So is a hyperbolic transformation with two distinct fixed points C1 and S2 on ICI = 1. If we choose a point on l and move around l an infinite number of times with branch fo (z), the corresponding point C describes a curve A in the disk ICI < 1 that approaches the circle I CI = 1 and, more specifically, approaches the points S1 and S2 If we continue fo (z) in the domain 0 < I z - al < r along an arbitrary path, we see that all values that the function assumes in the process lie in one of the domains, let us say B 1, into which the curve A partitions the disk CI < 1. If we let r


263

approach 0 continuously, the corresponding curves A, all passing through S1 and CZ, cover the domain B This shows that the inverse function 0(C), which is regular and bounded in B approaches a constant uniformly, on a sequence of arcs connecting the points 6 and S2 If we map the domain B 1 onto the disk It I < 1 and use Lemma 1 of §3 of Chapter II, we see that the O(C) = const, which, however, is impossible. This shows that S must be a parabolic permutation. If Co is its fixed point on ICI = 1, the curve A that we considered above must be a closed curve in Cl I< 1 whose only point in common with the circle ICI = 1 is Co. For values of z inside 1, the corresponding points C lie inside the curve X. But as r 0, the curve A is constricted to the point Co. Consequently, as we let z approach a, the branch fo (z) approaches Co uniformly. If we approach a with another branch S fo (z) of the function f (z), the corresponding point C = S fo (z) approaches the point SCo. Thus, as we let z approach a, the only limiting values of the function f(z) can be the points SCo, where ICo = 1 and S E G. When the entire boundary of the domain B consists of isolated points, it follows from what was said above that, as we let z approach the boundary of the domain B, the function f(z) can have as its limiting values only values belonging to a countable set of numbers, which are points on ICI = 1. It follows that, as C describes a path in the disk ICI < 1 that terminates on the circle ICI = 1 at some point different from these points, the corresponding point z = O(C) describes a path that cannot terminate at a definite point. Thus, in the present case, the function (k(C) does not have definite limits on ICI = 1 as r approaches any but a countable set of points on that circle from ICI < 1. Now, let K1 denote a closed curve constituting part of the boundary of B1. Let us encircle K1 with a Jordan curve l contained in B and defining, together with K1, a doubly connected domain B ' consisting only of points belonging to B. If we go around the curve l with branch fo (z) once in the positive direction, we arrive at the original point with a branch Silo (z). On the other hand, if we go around l several times (possibly some times in one direction and some times in the other), we arrive at the original point with branch S i ° fo (z). Here, S is a transformation of the group C other than the identity transformation. If we proceed around the curve infinitely many times with the branch fo (z), the corresponding point C describes in the disk ICI < 1 a Jordan curve A that terminates at the fixed points S1 and C2 of the transformation S (which, for all we know as yet, may coincide). If we extend f 0 (z) inside the domain B ', we obtain a multipleI

1

1

264


valued function in that domain. If, by means of a cut, we map the domain B ' into a simply connected domain, all the branches of the multiple-valued function will be single-valued in that domain and will map it bijectively onto domains adjacent to each other and filling up completely one of the domains, let us say B 1, into which the curve A partitions the disk ICI < 1. There is a continuous one-to-one correspondence of the boundaries under these mappings. Therefore, the arcs corresponding to the curve K 1 lie on the circle I CI = 1 and together they cover the arc s1C2 In addition to the arc S1S2 there are other arcs on ICI = 1 and these arise in the same way from other branches of the function f (z) than S i f o (z) and from other curves constituting parts of the boundary of the domain B (with one exception to be mentioned below). These arcs cannot overlap because, if they did, domains analogous to the domain B 1 would have common points whereas either domains analogous to the domain B' have no common points or the branches of the function f(z) considered in them do not coincide at a single point in them. It which does not reduce to a point, cannot cover comfollows that the arc pletely the circle I CI = 1; that is, S1 Cz. The only exception is the case when B is a doubly connected domain one boundary of which is an isolated point. In this case, the group G consists of the powers of one of its transformations which is a parabolic transformation. Thus, when we exclude this last one, we may assert that the transformation S 1 considered above is hyperbolic. If the point z C B approaches a point on K1 while describing a continuous curve, the corresponding point ' defined by an arbitrary branch of f(z) describes in ICI < 1 a curve that terminates at an interior point of one of the arcs obtained according to the type of the arc C1C2. On the other hand, as the point z approaches K along a curve that encircles K infinitely many times, the corresponding point C describes in < 1 a curve that terminates at one of the endpoints of these arcs. It follows from what has been said that, if the boundary of a multiply connected domain B does not consist entirely of isolated boundary points, then points on the circle I C1 = 1 fall into four classes: 1) interior points of all arcs defined according to the type of the arcs b1 C2 Corresponding to each such point is a unique nonisolated boundary point of the

domain B, and at each of these 0(4) is regular. 2) points of the type S1 and S2 are fixed points of hyperbolic transformations of the group G . There are countably many of these points. 3) points of the type of Cp, that is, fixed points of parabolic transformations


265

of the group G. There are also countably many of these.

4) all remaining points on the circle I4I = I. This classification was made from the point of view of the behavior of the function (k (C) in a neighborhood of the circle ICI = 1. If the function 0(C) approaches a definite limit as the point C on the circle IC1 = 1 approaches a point a on that circle, the point a belongs to class 1) or 3) because then the point z = O(C) approaches some point on the boundary of the domain B as Ca and, consequently, the point C corresponding to it approaches either an interior point of one of the arcs of the type 66 or one of the points of the type Co. Let us show that almost all points on the circle IC1 = 1 are points of class 1), that is, that the sum of the lengths of all arcs of the type Cl 2 is equal to 2rr. . Let . Let uk (c), for us arrange these arcs in a sequence {.(lk) y(k){, for k= 1, 2, k = 1, 2, , denote a harmonic function that is bounded in I CI < 1, that is equal = 1. and that is equal to 0 at all other points on to 1 on the arc S lk) This function can be constructed in accordance with Poisson's formula' I

1- r'

1

ttk(C)-2a

C==re'.

(1)

((k%(ak)

Consider the series

i

00

Z Uk (C).

(2)

y k=l For C= 0, this series obviously converges. Furthermore, if we denote by sn(S) the sum of its first n terms, we see that sn W < s n+1(J) in I C1 < I. It follows, in accordance with Harnack's theorem, that the series (2) converges uniformly < 1. inside the disk I tI < 1 and that its sum s (C) is a harmonic function in Furthermore, it follows from (1) and (2) that 2a

r 1-r' ed0=1 S(C)C2n of 1-2rcos(0-T)+r 1

< 1. If we now let C approach an interior point of the arc S 1k1S 2k 1, the 1 as 1 and, since sk(C) < s (C) < 1, it follows that s sequence sk(C) C approaches an interior point of that arc. Furthermore, if we map the disk I Cl < 1 (k) y ( sk) are into itself by means of transformations of the group G, the arcs ybvalues only on then mapped into each other (because O(C) has definite limiting

in

1) This formula will also be proved in Chapter IX.


266

these arcs). Consequently, the function s (C) is invariant under transformations of the group G. This last shows that, if we use the function z = O (C) to map the disk C1 < 1 onto the domain B, we obtain in B a single-valued bounded harmonic function s (f (z)) that is equal to unity at all nonisolated boundary points. Such a function is identically equal to unity in B because, otherwise, it either would attain a maximum or a minimum inside B or would approach an extreme value as z approaches isolated boundary points, which, in accordance with the maximum principle and the beginning of the present section, is impossible. Consequently, 1 in the disk 1. But

S (0)=

LEI 2a

S

k=1 t(k)t(k)

so that I' (length

de = 2n

length (Clk)C9kl),

k=1

27r. This completes the proof. §3. Dirichlet's problem and Green's function

A finite real function that, together with its first two derivatives,') is singlevalued and continuous in a domain B of the z-plane and satisfies Laplace's equation in B is said to be harmonic in B. Such a function can be regarded as the real part of some analytic function that can be extended in B along an arbitrary path. A function is said to be harmonic at a point if it is harmonic in some domain including that point. We note that, if u (z) is a harmonic function in a sufficiently small neighborhood of a point z = a with the point a itself deleted and if it is bounded below in that neighborhood, that is, if u (z) > m (or bounded above, that is, if u (z) < M), it will also be harmonic at z = a if its domain of definition

is extended to include the point z = a in a suitable manner. To see this, let v(z) denote the harmonic conjugate of U(z) and let t,) denote the increment in the function v (z) as z moves once around a sufficiently small circle I z - al = t (or a R in case a = ce). The function sufficiently large circle I z 2w

F(z)=e

(U + iv)

1) The derivatives here refer only to derivatives at finite points. In what follows, the harmonic functions mentioned may at times be multiple-valued, but in such cases an arbitrary branch of such a multiple-valued function will be single-valued in an arbitrary sufficiently small neighborhood of each point in the domain B.

§3. DIRICHLET'S PROBLEM AND GREEN'S FUNCTION

267

with either sign chosen is then regular and bounded in the neighborhood in question of the point z = a, and, with suitable choice of the value for F (a), regular at the point a itself. Obviously, F (a) is nonzero. Therefore, the function u + iv is also regular at z = a and the function u (z) is harmonic at z = a. It follows that a function u(z) that is harmonic and bounded in a domain B is harmonic at all isolated boundary points when it is extended in a suitable manner to these points. The maximum principle of harmonic functions has an important generalization: -Jf a function u(z) is harmonic and bounded above in a domain B (that is, its values in B do not exceed some finite constant) and if all its limiting values, as its argument approaches any except finitely many boundary points of B through values in B, are no greater than M, u (z) < M everywhere in B. An analogous generalization holds for functions bounded below.

Proof. We may assume that the domain B includes the point o. Let al, an denote the exceptional points referred to and let H denote the diameter of the boundary of the domain B. For arbitrary c > 0, consider the function a

v(z)=M-u(z)+e Iloglz Haktl' , an, which is harmonic in B. As z approaches boundary points other than a 1, this function approaches nonnegative limits since each term in the sum above has a nonnegative value. As z approaches one of the points ak, this sum approaches + oo, but u(z) is bounded above. Consequently, the limit of v(z) as z approaches any of these points is +Do. Since v(z) is obviously not a constant, it cannot have minima inside the domain B. Therefore, v (z) > 0 in B; that is, n

it(z)M+e

log

tl IZ -ak

k=l This holds for arbitrary c > 0. By letting c approach 0, we obtain u (z) < M, which completes the proof. Let us now pass on to Dirichlet's problem. Suppose that the domain B has a boundary K consisting of a finite number of closed Jordan curves without common points. The Dirichlet problem for such a domain consists in finding a func-

tion u(z) that is harmonic and bounded in B, that is continuous in B except possibly at countably many points on the boundary K; and that assumes prestated values at points of continuity on the boundary K. The boundary values are subject

268


only to the condition that they constitute a function on K that has a finite or countable set of points of discontinuity. Let us show that this problem always has a unique solution. Let us map the domain B onto the disk < 1, just as in §1. Consider the

set of points V on the circle I(' I = 1 belonging to the class (1) of §2. Let U W) denote a function whose values coincide with the values u (z ') at points z ' of the boundary K that correspond under this mapping to the points V on

I = 1. This real function U is defined, bounded, and continuous almost everywhere I) on the circle I C' 1 and satisfies almost everywhere on that circle the condition U (S (C )) = U('), where S is an arbitrary transformation belonging to the group G of §2. Thus, we arrive at the problem of finding a function U ((') that is harmonic and bounded in the disk ICI < 1 and that coincides with U(C') at all points of continuity of the function on ICI = 1. Consider the function I C'

UP=

2x

1- r'

ie

1

U ( e ) 1- 2r cos (0 - , f ) -}- r'

2a

tP dO C = re,

where the integral is in the sense of Riemann. By virtue of the known properties of a Poisson integral, this function is harmonic and bounded in I I < 1 and, under the shift to C' I = 1, it coincides with U (a') at all points of continuity of the function U(CH). On the other hand, any function U(C) possessing these properties is unique. To see this, suppose that there exists another such function U1(O. Then, for arbitrary C in 41 < 1, we have, in accordance with Poisson's formula I

2x

U1 (C)=2

e

U1(Ret)

RH-2RrR9

H

ret', cos(er 'p) -}-r' de' C=

where r < R < 1. If we now take the limit as R -' 1, remembering that we can take the limit under the integral sign (Lebesgue's theorem), we see that U1(C) U(C) in ICI < 1. It follows from the uniqueness property that U (C) remains invariant under transformations of the group G. In fact, the functions U(S(C)) and U(C) are both solutions of the problem that we have just examined and hence they are 1) In the present case, with the possible exception of a countable set of points on ICI= 1.


269

identical in CI < 1. If we now return to the z-plane, the function U(5) becomes the function u(z), which is single-valued and bounded in B and which satisfies the Dirichlet problem posed above. It also follows from what was said above that this problem has a unique solution. We note that if certain of the curves constituting the boundary K degenerated into points, the Dirichlet problem for the domain obtained would not always have a solution. To see this, note that the function u(z) which provides a solution to the Dirichlet problem for such a domain is, by virtue of what was said above, a harmonic function at all isolated boundary points. *Therefore, it is uniquely determined by its values on the remaining boundary curves. This shows that the values of the function u(z) at isolated boundary points cannot be chosen arbitrarily. Thus, we have proved the existence and uniqueness of the solution to the Dirichlet problem for an arbitrary finitely connected domain B bounded by Jordan curves and for arbitrary given values on the boundary K that constitute either a continuous function or a function with a finite or countable set of points of disContinuity. To find the solution of an arbitrary Dirichlet problem for a given domain B, it is sufficient to know the solution of any one particular Dirichlet problem. Here, it is a question of finding the Green's function. The Green's function for a domain B is defined as a real function g (z, C) satisfying the following three conditions: 1) for every C E B, it is a harmonic function of z in the domain B except at the point z = J; 2) g(z, C) approaches +oo as z - C in such a way that the difference g (z, C) - log 111z remains bounded if is finite and in such a way that the difference g (z, 00) - log I z is bounded if C = o.; 3) as (z, C) approaches the boundary K, the function g(z, C) approaches 0. In accordance with the maximum principle, g(z, C) must be positive everywhere in B. If a Green' s function g(z, C) exists, then, for finite S the difference g(z, C) log (14 z -el) when assigned the appropriate value at z = J, at the point z = is harmonic in B and continuous in B and assumes on the boundary K the values -log(1/jz - CI); that is, this difference is a solution of a particular Dirichlet problem for the domain B. Conversely, if u(z) is a solution of this Dirichlet problem, then the function log (1/I z - CI),+ u(z) is obviously a Green's function for the domain B. An analogous situation holds for C= 00. Under a singlesheeted mapping of the domain B by means of a function z* = f(z) onto another ,domain B* bounded by Jordan curves, a Green's function g(z, C) is transformed I

into the function g(f-1(z*), fwhich, obviously, is a Green's function for the domain B*. This shows that, if we know a Green's function for one domain,

270


we can, by means of univalent mappings, obtain Green's functions for various other domains of the same order of connectedness. In the case of a simply connected domain B, there is no Green's function, as the function -log I f(z)I indicates, where t = f (z) is a function that maps a domain B univalently onto the disk It I < 1 in such a way that the point z = C is mapped into t = 0. For multiply connected domains, there is no such simple relationship with conformal mapping.

In the case of a simply connected domain B bounded by a closed Jordan curve, an analytic function P (z, C) the real part of which is a Green's function g(z, C) for a domain B is obviously continuous on B except at the point z = . If, in addition, the boundary of the domain B is an analytic curve, then p (z, is regular on that curve. It turns out that these properties also hold in the case of finitely connected domains. Let us consider first a finitely connected domain B bounded by arbitrary closed Jordan curves. Let g(z, C) denote a Green's function of B and let h(z, C) denote its harmonic conjugate. As we shall see, the last function is non-single-valued in B. Consider the function F(z) = g - ih. Let us take an arbitrary points z0 on one of the boundary curves of the domain B and let us describe a circle y about it sufficiently small that it intersects` the curve and there are no other boundary curves either inside the circle y or on it. The part of the domain B lying inside y consists of simply connected domains. We denote by B0 whichever one of these domains has the point z0 on its boundary. Let us map B0 onto the halfplane `fi(t) > 0 in such a way that z = z0 is mapped into t = 0. If we denote the mapping function by z = 0(t), the function F(OW) is regular in the domain (t) > 0 and its real part u (t) is also continuous in (t) > 0 and is nonnegative. Noting that u (t) = 0 at points on the real axis sufficiently close to t = 0, we describe a circle t I = p sufficiently small that u(t) = 0 on the diameter lying on the real axis. Let us now define a function NO on t = p as follows: U (t) coincides with the function u (t) on the upper semicircle and it assumes at points in the lower semicircle the same values as the function u (t) assumes at the conjugate points in the upper halfplane with with opposite sign. The function U(t) is obviously continuous on the circle I = P. Consider the integral I

U(t)2i

2+c U(Peie)p'-2prps-rs

It defines a function U (t) that is harmonic in

cos(B-cp)+r'dA,

I

t I < p and continuous on I t I < p .


271

It is obviously equal to 0 at points on the real axis and it coincides with u (t) on the upper boundary semicircle. Consequently, U (t) coincides with it (t) on the 'entire boundary of the upper half-disk and therefore throughout this entire halfdisk. If t(t) is an analytic function in It 1 : p and its real part is the function U(t), then the difference $(t) - F(O(t)) has real part equal to zero in the upper half-disk. Therefore, this difference must be a constant in t I < p. This last fact shows that the function F(0(t)) can be extended in a neighborhood of t = 0 beyond the real axis and, in particular, it is a continuous function at the point ,it = 0. Returning to the z-plane, we conclude that the function F(z), considered in 9, is continuous at the point z0. Since z0 is an arbitrary point on the boundary of the domain B, this means that F(z) is continuous on B except, of course, at the point C. Furthermore, since t ' (0) is obviously nonzero, it follows that f' (z) = $ (t)/ (t) 0 in a part of a sufficiently small neighborhood of the point z0 contained in B. This means that the zeros of the function F' (z) cannot cluster on the boundary of the domain B. Consequently, F' (z) and hence pa (z, have only finitely many zeros. Finally, since the function w = fi(t) maps the part of the neighborhood of the point t = 0 lying above the real axis into the part of the neighborhood of the point w = CO) that lies to the right of a vertical line passing through the point w = (N0), it follows that, as we move along the boundary of B in a neighborhood of z0, the function h(z, C) always changes in the same direction. More precisely, as z describes the boundary of the domain B in ,the positive direction, h(z, C) increases. Now, if z0 is an interior point of an analytic arc on the boundary of B, then the function 0 (t) is regular at the point t = 0. But then, the function F (z) and hence the function p (z, C) is regular at I

the point z 0.

By analogous reasoning, we can obtain a generalized theorem on harmonic functions: Let u(z) denote a function that is harmonic in a domain B bounded by closed

analytic Jordan curves. Suppose that u(z) is continuous in B and that its values .on the boundary of B constitute an analytic function of a parameter determining points on the boundary (that is, a function that can be represented in a neighborhood of each boundary point in the form of a series of powers of the parameter in question). Then an analytic function F(z) the real part of which is u(z) is regular in B.

Proof. Just as above, we shift to the t-plane. Then, the values of u (z) on

272

V1. MAPPING ONTO A DISK

the boundary of the domain B in a neighborhood of a point z0 can be represented by a Maclaurin series of powers of the real. parameter t. In the complex t-plane, this series defines a function that is regular at t = 0 and real on the'real axis (in a neighborhood of t = 0). When we return to the z-plane, this fast function becomes a function T W that is regular at the point z o and equal to u (z) on the boundary of the domain B. in a neighborhood of the point z o . The difference F(z) - 'Y (z) is regular in the portion of the neighborhood of the point z0 lying in B, and its real part is continuous, including boundary points of the domain B, at which points its value is 0. Resting on this last remark, we can again show that this difference is regular at the point z0. But then, the function F(z) itself is regular at the point z o. Let us now show how, once we know a Green's function for the domain B, we can find the solution of an arbitrary Dirichlet problem for that domain. Let us first consider the case of a domain B with boundary K consisting of closed analytic curves. Let u(z) denote a solution of a Dirichlet problem for the domain B with given boundary values which define on K an analytic function of a real parameter determining points on K. Green's formula 1) S S (utul-u1Ltu)dxdy=- S lud- uldn)dsBi

is applicable to the functions u(z) and u1(z) = g(z, 0 and to the domain B. obtained from B by subtracting the disk I Z - CI < E.2) Here, the integral on the left is over the domain B. i that on the right is over the boundary KE of the domain Be in the positive direction; ds denotes an element of'arc on KE; and d/dn denotes the derivative with respect to the inner normal. Since Au = Au 1 = 0 in BE and since u 1 = 0 on K, the preceding formula reduces to u gdRds=O. dgds(1) I zHere, the first integral is independent of c. Remembering that the function g (z, C) is of the order of magnitude of log c on the circle I z - CI = E, we easily u

dgds-+ -

tf

1) This formula is also valid for a domain B including z = o., as we can see if we apply it first to the domain Be obtained from B by subtracting the domain IzI > 1/E, let C approach 0, and then proceed as described in the text. 2) For finite C. In the case t = a., we take the domain Izt > 1/E.


273

conclude that the last integral approaches 0 as a --b 0 . With regard to the middle integral, we have d log

lim

.-.o

u dg ds = lim

It

1

ds

z

do

ds`

u

lim C

Iz

27cu (C). i

Thus, if we take the limit in ( 1) as a -+ 0, we obtain u (C)

2a

u (z) do ds.

(2)

This formula is known as "Green's formula". It expresses the values of the function u(z) inside the domain B in terms of its values on the boundary of B. Since the function p (z, C) = g - ih is regular on K and hence dg/dn = dh/ds (one of the Cauchy-Riemann conditions), we can put the preceding formula in a different form:

u (C) = 21-

it (z) dh (z, Q.

(3)

Let us show that formula (3) holds for a finitely connected domain B bounded by arbitrary closed Jordan curves and for an arbitrary given function u (z) on the boundary K, where the integral on the right should be understood as a Stieltjes integral. Let g(z,. C) denote a Green's function for the domain B and let 8 > 0 denote a number less than the distance from any of the zeros of the function p'(z, t;) (where p = g - ih) or the point C to the boundary K and also less than 'half the distance between different boundary curves constituting K. On the set consisting of all points of the domain B whose distance from the boundary K is 8 or greater, the function g (z, C) has a positive minimum A0. Consider the set KA of points of the domain B defined by the equation g(z, C) = A, where 0 < A < Ao . This set consists of analytic curves. To see this, let zo denote a member of KA. The points of KA in a neighborhood of zo have a parametric representation z = z (t) that can be obtained by solving the equation p (z, t;) = A + it. Since pi (z, 0 for z = z0, this parametric representation defines an analytic arc in a neighborhood of zo. Furthermore, the subset K,, of the set KA consisting of points at a distance no greater than 8 from a given boundary curve L of the domain B consists of a single closed analytic Jordan curve surrounding L because,

274


otherwise, B would contain a domain in which g (z, t;) would be a harmonic function and on the boundary of which g (z, would be equal to A, which is impos-

sible since we would then have g(z, ) = A in B. It follows that the inequality g(z, t;) > A defines in B a domain Bk including the point z = J with boundary KA consisting of closed analytic Jordan curves approximating the boundary of the domain B. Now, suppose that u (z) is a function that is harmonic in B and continuous on B. Let us apply formula (3) to the domain BA. Since g(z, t;) - A is obviously a Green's function for the domain BA and since the function a = h(z, t;") increases as z moves around KA in the positive direction, so that we may take a as parameter determining points on KA, it follows that it (C) = 2 5 it (z) dh (z, () = 2

it (za (a)) da.

(4)

ltx

But u(z) is continuous on B. Therefore, if we also express the points on K in terms of a = h (z, t;), we conclude that, for every e > 0, the values of the function u(z) at points on K and KA corresponding to the same value of a will, for sufficiently small positive A, differ from each other by an amount less than e. Consequently, the integral in the right-hand member of (4) approaches the integral

over K as A -. 0. By taking the limit, we obtain it (C) =

(z,, (a)) da, 2a 1c u

that is,

u ) - 2>< 5 it (z) dh (z, t)

(5)

Finally, formula (5) remains valid when the function u (z) has a finite number of points of discontinuity on the boundary K but remains bounded (Iu(z)I < M) in B. To see this, let us include the points of discontinuity on K in the interiors of small arcs (z (ak ), z (ak )), such that I I ak - ak I < t/2M and let us denote by K. the remainder of the boundary K when these arcs are removed. Then, l

(u (za (a)) - it (zo (a))) do I

2

+

I it (z). (a)) - u (zo (a)) I do.

But off these arcs, u (zA(a)) -- u (zo (a)) uniformly. Consequently, for sufficiently small A, the last integral is also less than a/2. This proves formula (5) in the present case.

§4. APPLICATION TO A UNIVALENT MAPPING

275

We note that univalent mappings of the domain B transform formulas (3) and (5) into the corresponding formulas for the image. In conclusion, let us prove the symmetry of a Green' s function; that is, let us prove that g (z, C) = g (C, A. Let us apply formula (5) to the function u (z) _ g(z, C) + log 1z - Cj, which is harmonic in B and continuous on B. Since g (z, C) = 0 on K, we obtain

g(z,C)+ logI z-CI=Z 1ogI z'-CIdh(z, z). It Follows from this formula that g (z, C) is a harmonic function of C in B except at the point C= z, in a neighborhood of which it also has a singularity of the same type as the function g (C, z). Therefore, the difference d(z, C) = g (z, C) g(J, z) is a harmonic function of each argument throughout the entire domain B. g(C, z) < 0; that is, But, as we let z approach K, we have lim d(z, C) d(z, C) < 0 for every in B. On the other hand, as we let C approach K, we have lim d (z, J) = lim g (z, ) > 0; that is d (z, C) > 0 in B. Consequently,

d (z, C) = 0, that is g (z, C) = g (t z) for every z and C in B. §4. Application to a univalent mapping of multiply connected domains We can use the solution of the Dirichlet problem to give new proofs of theorems on the existence of a univalent mapping of finitely connected domains onto 'Certain canonical domains, for example, onto the plane with rectilinear parallel cuts or onto the plane with cuts along arcs of logarithmic spirals of equal inclination. Let us stop here to prove Theorem 1 ' of §2 of Chapter V on the mapping of an n-connected domain B in the z-plane onto the c-plane with finite parallel straight-line cuts of inclination 0 to the real axis. As we know, it will be sufficient to consider the case when the domain B does not include z = oo and is bounded by closed analytic Jordan curves K 1, K2, , K,,, where K is assumed to be the external boundary. Consider n - 1 functions Uk (z), for k = 1, , n - 1, that are harmonic in B. Suppose that each uk (z) is equal to 1 on Kk and equal to zero on all Kk, for k' k. For each it, let vk (z) denote the harmonic conjugate of uk (z). The functions vk(z) are in general multiple-valued in B but, in accordance with §3, they are continuous on B. Let us denote by r'k( the increment in the function vk(z)

276


as z moves around KI in the positive direction (with respect to the domain B). One can easily show that the determinant jck,1I of the numbers a k.t is nonzero. Let us suppose, to the contrary, that this determinant is zero. Then there exist real constants kk , for k = 1, . , n - 1, not all equal to zero, such that Ik=1Akcok,1 = 0, 1= 1, ... , n- 1. The function

n-1

f(z)= YjXk(uk+1vk), k_l which can be extended into B along an arbitrary path, returns to its original value as z moves around each of the curves K, encircling K. But then, this function is single-valued in B and hence is regular in B. The real part of the function f (z) is constant on each of the curves Kk, for k = 1, , n. Then, reasoning as in the proof of Theorem 2 of §2 of Chapter V, we can show that f (z) = const in B. But this is impossible since the real part of the function f (z) vanishes on the curve K and is nonzero on at least one of the curves Kk for k = 1, , n - 1. This contradiction proves that Icuk.,I 0. Now, suppose that u0 is a harmonic function in B that is equal to -3`(e'`B/(z - a)) on the boundary K = Uk=1Kk and that v0 is its harmonic conjugate. Let us choose real constants Ak, k = 1', , n - 1, such that the function n-1

ve +

Xkvk

will be single-valued in B. This is possible since 1cok.,i / 0. Let us show that the function

f(z)=z

1

a +el01

-L

n-I

(110+1VO+ Z `k(uk+lvk)) k=1

is univalent in B. This function is regular in B except at the point z = a and assumes values on Kt (for l = 1, - , n) that lie on some straight line d, at an inclination 0 with the real axis. This shows that, if Co does not lie on the straight lines dl, l = 1, - - , n, the increase in the function arg (f(z) - CO) as z moves around the boundary of the domain B in the positive direction is equal to zero. On the other hand, this increase must be equal to 2n times the difference between the number of zeros and the number of poles of the function f(z) - Co in the domain B. Since f (z) - Co has a single simple pole in B, namely, at z = a,

it follows that f (z) assumes the value 4 at exactly one point in B. If the function f (z) assumed any value C1 at more than one point in B, this would also be true of points sufficiently close to S1, which, by virtue of what was shown above,

§5. MAPPING OF AN n-CONNECTED DOMAIN

277

cannot be the case. Thus, the univalence of f (z) in B is proved and so is the fact that the image of the domain B under the function C = f (z) is a plane with parallel rectilinear cuts of inclination 0 with the real axis. The function f (z) + c yields the same mapping, and for some c we have, in a neighborhood of z = a, the expansion I

z-a -{--al (z----a)+..., if a is finite and the expansion z + a1z-t +- - ' if a = oe. By analogous reasoning, we can prove Theorem of §3 of Chapter V regarding the mapping of a finitely connected domain B onto a plane with cuts along arcs of logarithmic spirals of equal inclination. §5. Mapping of an n-connected domain onto an n-sheeted disk

In conclusion, let us look at the possibility of mapping an n-connected onesheeted domain B onto an n-sheeted disk C1 < 1. By an n-sheeted disk, we mean here a Riemann surface lying entirely over the disk CI < 1 and having over each point of that disk exactly n points (a branch point of order p is considered as p points). This mapping is assumed to be effected by a bijective regular function. If we do not wish to resort to the concept of a Riemann surface, the mapping should be understood in the sense that, if 4 = f (z) is the function performing the mapping, then the function f (z) - Co has exactly n zeros in B for arbitrary Co such that I C0I < 1 (with a zero of order m counted m times) and has no zero at all in B for any Co such that I Co I > 1. Theorem. Every n-connected domain B in the z-plane that has no isolated boundary points can be put in one-to-one correspondence with the n-sheeted disk ICI < 1 by means of a regular function. If B is bounded by closed Jordan curves K1, , K,,, then the function defining this correspondence is continuous in B and it can be normalized in such a way that, at n given points al, , any where , n, it assumes a single given value a such that I a.I = 1 ak a Kk, for k = 1, and, at two given points b and c of the curve K that are distinct from the an and that are distributed along with the an in a definite order on K., assumes given values f3 and y distinct from a and distributed, along with a, in the same order on the circle I CI 1. Under this normalization the mapping is unique. (Here, "order" refers to the direction of motion around the boundary of the domain B and the boundary of the n-sheeted disk.)

278


Proof. As always, it will be sufficient to consider a domain B that does not include - and that is bounded by closed analytic Jordan curves Kk, k = 1, - , n, where K,, is the outer boundary. Suppose that the curve Kk (where k = 1, -- , n) is defined by the equation z = zk (t), for 0 < t < 1. Suppose also that zk (0) = ak and that the point zk (t) moves along Kk in the positive direction as t increases. Instead of the disk I < 1 and the n-sheeted disk mentioned in the theorem, we can obviously choose an arbitrary simply connected domain G that can be mapped onto the disk and the n-sheeted surface over it respectively. For such a domain lying over the (w = a + iv)-plane, we take the strip G : 0 < R W < 1 and seek a function w = f(z) = u + iv that is regular in the domain B and that maps B onto the n-sheeted strip lying over G and having the property that w --, + - i as z approaches points ak through points in B. To construct the function w = f(z), let us choose on each of the curves Kk, for k = 1, - , n, a point bk = zk (tk), where 0 < tk < I. These points, together with the points ak partition the curves Kk into arcs Kk : z = zk (t) for 0 < t < tk and Kk : z = zk (t) for tk < t < 1. Suppose that u (z) is a bounded harmonic func-

tion in B that is equal to zero on the arcs Kk and equal to 1 on the arcs Kk . Let v(z) denote the harmonic conjugate of u(z) (defined up to a constant term). In general, the function v(z) is not single-valued in B. Let wk, for k = 1, -, n, denote the increase in v(z) as z moves around Kk in the positive direction. Here, in a neighborhood of the points ak and bk, the motion is along small arcs contained in B. The numbers wk depend on the choice of the points bk. If we succeed in choosing the points bk in such a way that all the numbers wk are equal to zero, then the function f (z) = u + iv will be single-valued and hence regular in B. Let us show that, in such a case, the function f W serves as the required mapping. To do this, let us investigate first the behavior of f (z) in a neighborhood of the points ak and bk. Let us map the simply connected domain bounded only by the curve Kk and containing the domain B onto the halfplane J) > 0 in such a way that the points ak and bk are mapped respectively into The mapping function z = qS(t;) is regular at ; = 0 and has an expan0 and 0, in a neighborhood of that point. , where cl sion z = 0 (C) = ak + c 1t; + This transformation maps a(z) into a function that is harmonic and bounded in a domain of the c-plane that contains the real axis on its boundary and that has the values 0 and 1 on the positive and negative halves of the real axis respectively. But the function (1/rr) arg C has this same property. Therefore, the difference f(0(C)) - (1/77i) log C can, in accordance with the symmetry principle, be continued

§5. MAPPING OF AN n-CONNECTED DOMAIN

.

279

past the real axis and it is a function that is regular on that axis except at the point 1= 0. It follows that the function f(z) = (1/7ri) log (z -ak) is regular in a neighborhood of the point z = ak. In the same way we can prove that the function f(z) + (1/7ri) log (z - bk) is regular in a neighborhood of the point z = bk. It follows from what was said above that the function v(z) approaches +00 uniformly, at the same speed as the function -n-1log I z - akl, as z approaches ak through values in B, and that v(z) approaches -- uniformly at the same speed as the function -n-'log (111z - bkl), as z approaches bk through values in B. This .property enables us to show that the function w = f (z) maps the domain B onto the' n-sheeted strip 0 < $2 (w) < 1. Let w0 denote any point in the strip 0 < Y (w) < 1. Since the function f(z) does not assume the value w0 in sufficiently small neighborhoods of the points ak and bk for k = 1, , n, the number of .zeros of the function f(z) - w0 in B is equal to (21r)-1 times the increment in arg [f(z) - w0] as z moves around the boundary K = Uk=IKk of the domain B in the positive direction, where, in neighborhoods of the points ak and bk, the path is along small arcs contained in B. But as z moves around the curve Kk (for k = 1, , n) in the positive direction, beginning at some point zo a Kk, the corresponding point w = f(z) first moves downward along the imaginary axis and then moves along a sufficiently distant curve to the line S% (w) = 1 and then moves upward along that line. It then moves along a sufficiently distant curve back to the imaginary axis and returns along it to the initial point z0. This shows that the

increment in arg [f(z) - w0] as z moves around the curve Kk in the positive direction is equal to 2n. Then, the increment in arg [f(z) - w0] as z moves around the entire boundary of K in the positive direction is equal to 27rn. Consequently, the number of zeros of the function f (z) - w0 in B is equal to n no matter what the value of w0; that is, the function w = f (z) does indeed map the domain B bijectively onto the n-sheeted strip 0 < R(w) < 1. We assume that the function f (z) is single-valued in B. , G) n Let us show that values t 1, , t,, for which all the numbers rv 1. are equal to zero do exist. Choosing tR arbitrarily in the interval 0 < t < 1, let us consider the different tk such that 0 < tk < 1 for k = 1, 2, , n - 1. , n, are functions of t 1, Then u (z), v (z), and &)k, k = 1, , t,-I- We set

tt(z)=tt(z, t1, ..., t,-t), v(z)=v(z, ti, ..., to-), Dk=(lk/4, .. ., to - r), k = 1, ..., n.

280


Let us show that there exist numbers t1,

,

t,,-, such that

k=1, 2, ..., n.

Wk(t1, .. .,

(1)

To do this, we mention first a few properties of the functions W k (t 1, '

to-1): are continuous functions of each of the

1) The functions CJk(t1, ,

variables tv for 0 < tY < 1, where v = 1, accordance with §3,

, n - 1. This is true because, in

a

tt (z, t1, ... , to -1) = 2a

A

S dh (C, z) = 2a

k=1Kk

(h (bk, z) - h (ak, z)), k=1

,

and hence u(z, t1i au/ax, and au/dy are continuous functions of z, , as z varies in B and each tv varies in the interval 0 < tv < 1 for ti,

v=1, ,n-1. But wk (t 1,

,

1) is determined from the formula

dv- -aydx+azdy

wk

k

where the integral is over an arbitrary closed analytic curve Ck contained in B and encircling only the curve Kk. Consequently, each wk (t1, , t,,_1) is a continuous function of tv in.the interval 0 < tv < 1, for v = n - 1. 2) Each wk (t 1, , is an increasing function of the parameter tv for v k. To see this, note that, for tv > lv, the difference A", k = wk 01, ... , t", ... , to-1) - wk 01, ... , t", ..., to-1) is the increment in the function vv(z) as z moves around the contour :Kk. This function is the conjugate of the function uv(z), which is harmonic in B and vanishes on K except on the arc of the curve K,, between the points zv(ty) and zv(tv), on which it is equal to 1. Since uv > 0 in B, we have auv/an > 0 on Kk. This is a strict inequality because vanishing of the above derivative, together with vanishing of the derivative auv/as, would mean that the derivative of the function fv(z) = uv + iv,,, vanishes at some point on Kk and hence the image of the domain B under the function w = fv(z) would get outside the strip 0< 92(w) 0 on Kk and, on the basis of equations (3), we have a) k < 0.

Now, by using properties 1)-5), let us show that the system (1) has exactly one solution. It follows from properties 1) and 5) that, for every system of values , t_), 0 < t l < 1, such , 1, there exists a unique value t 1 = r1(t2, 92; is continuous with respect to 0 with rl (t2i , , that w 1 (rl, t2, -

-

-

t2i

. , In- 1 Since, in accordance with property 2),

w 1 (rl, t2,

,

In-,) increases

with increase in t2, , In- 1 and, in accordance with property 4), decreases with increasing rl, it follows from the identity w1 [T1 (t9, ... , to-1), {y, ..., tn_r] = 0

that rl (t2i -, tn_0 increases with increasing ti, for v / 1. Let us define the functions wk(t9, ..., to-1)=wk(r1(t9, ...,

tn_1), t9,

..., tn_1), k=2, ...,

lt.

282


These functions are continuous with respect to t2, , tn_ 1. For each k, the obviously increases with increasing tv for v k. function wk (t2, - , Furthermore, it follows from (2) that, for all t2, , tr_1,

wk (t2, ..., to-1)=0, k-2

,

n - 1) increases with It then follows that wk (t2, or k = 2, 1) is posiincreasing tk. Finally, it follows from property 5) that (Jk (t2, , tive for tk = 0 and negative for tk = 1. Consequently, the functions wk(t2, , satisfy conditions analogous to conditions 1)-5), with the number of functions and the number of variables decreased by one. We reason in the same way with these functions: there exists a unique value t2 = r2 1) n the interval , 2 (t3, 0 < t2 < 1 that satisfies the equation w2'(t2, , 1) = 0. Here, r2 (t;, , tn_1)

is continuous and increases with increasing t for v>2. Now let us consider the functions Wk (t3, ... , to-1) - ok ('t8 (t3, ... , trt_1), t3i ... , tn_1), k = 3, ... , h, and continue in the same way. After n - 1 steps, we arrive at the uniquely defined system of values tn_1 = 'tn_I = const, tn-9 = 'cn-2 (tn_1),

, n - 1. But then, in accordance with equation (2), we also have wn(r1, ..., rii_1) = 0. This proves that there exists a system of values t1i satisfying equations (1) and hence that there exists a function w = f(z) mapping the domain B bijectively onto the n-sheeted strip such that wk (r1,

, rn_ 1) = 0 for k = 1,

,

0 < Pw < 1 in such a way that the given points ak E Kk, k = 1, 2, . , n, are mapped into w = + i-. Let us now prove the uniqueness of the mapping. In the treatment above, the only arbitrariness was in the choice of the value of t in the interval 0 < t < 1 and the constant term up to which the single-valued function f (z) was defined. Suppose now that, in addition to the point an, two arbitrary points b and c are defined on the contour Kn and suppose, for definiteness, that the points an , b, and c occur in that order as one moves in the positive direction around Kn. If c = z (tn), for 0 < to < 1, we fix that value t = to as indicated above and from it we can find the system of values t1, , tn_1 which determines up to a constant term the single-valued function v (z). We choose this term in such a way that the point b C Kn is mapped into a given point 8 on Rw = 0. Then, the points an, b,

§6. SOME IDENTITIES

283

and c on K are mapped respectively into + ioo, S, and - ioo and the function w = f (z) is uniquely determined. This completes the proof of the theorem. The proof that we have given is due to Grunsky.1) The question arises whether the idea of the proofs of numerous existence theorems can be used in the present case to establish the existence of functions mapping an n-connected domain B onto an n-sheeted disk by using some extremal property of these functions. It turns out that this is possible and that in this connection the problem centers around the maximum value of If' (a) 1, where a is a finite member of B, out of all the functions f (z) that satisfy the condition f (a) = 0, that are regular *in B, and that do not exceed unity in absolute value. The proof is given in §3 of Chapter XI. It uses a number of boundary properties of analytic functions which will be explained in Chapters IX and X. §6. Some identities connecting a univalent conformal mapping and the Dirichlet problem

In §4 of Chapter V, by using the method of contour integration, we established certain relationships connecting various functions that map a given finitely connected domain univalently onto canonical domains. In the present section, we shall continue the application of this method2) by introducing into our study appropriate harmonic functions defined as the solutions of certain Dirichlet problems. Specifically, suppose that B is an n-connected domain that does not include oo and that is bounded by n closed analytic Jordan curves K. for v = 1,--. , n. Suppose that a)v(z) (for v = 1, . , n) is a function that is harmonic in B, equal to 1 on Kv, and equal to zero on all Kµ, for µ v. We note that the functions ojv(z), for v = 1, - , n, satisfy the relation n

a Ya

(z) =1,

(1)

because the sum on the left is a harmonic function in B and it is equal to 1 everywhere on the boundary K = U v=I K. The harmonic function iv(z) conjugate to wv(z) (v = 1, , n) is not in general single-valued in B. Let (Z). (z))K, as usual denote the increment in the function tvv(z) as z moves around the curve Kµ in the positive direction (that is, positive with respect to the domain). Sup-

1) Grunsky [19371.

2) See Garabedian and Schiffer [19491.


284

pose that

( (z))K = 2aPl,

µ, v=1, ..., n.

If we now set wv(z) = cov(z) + iwv(z) (for v = 1, . , n), then the analytic funcsatisfies the conditions

tion

1) W.(z)_-W.(z)+c,,,, on Kµ, cµ

const,

(2)

2) (W, (z))K = 2it1Pt,.

(3)

3) WY(Z) is regular in B. Turning to the contour-integration method, we shall use the functions P(z, u), Q (z, u), P (z, u, v), and Q (z, uy v) introduced in §4 of Chapter V. With respect to these functions, we recall that they satisfy on K. (v = 1, . , n) the conditions

P (z, u) = - Q (z, it) + k. (u),

(4)

P(z, u, v)_- Q(z, u, v)+k,(u, v),

(5)

where kv(u) and kt,(u, v) are independent of z in K. We note that the functions kt,(u) and kv(u, v) will appear explicitly in the relationships that we derive below.

Now consider the integrals

t = tai 5 P (C, u) WY (C) dC,

Is =tat

P (` u, v) w, (C) dC.

(6)

5

In accordance with Cauchy's theorem, each of these integrals is equal to 0. If we use formulas (2) and (4) to transform the first of these integrals, we obtain n

II =

J P (C,

u) dw, (Q (C, u) -+ kµ (u)) dw., (C) c

c KK

IL

KKµ

n

l

(- Q (C, u) + k, (u)) dW, (C)

W=

_-

I,

a

l

271

Q (G u) Wy (C) dC + YI kl, (u) Pµ. µ=1 n

W (u) + Y, kµ (u) PI,. µ-1 As a result, we arrive at the formulas (for u E B)

W;(u)_ I k,,(u)PIL. M

µe1

(7)


285

Transforming the second of the integrals (6) in the same way, we obtain

+ 1'

n

Q (C. U, v) %'(Q A -

= - tat

kµ (u, v) P1, W=1

KKK

To evaluate the remaining integral, we make a cut y in B from u to v. Since the integrand in the domain so obtained is single-valued, the integration over K can be replaced with integration over an arbitrary closed analytic Jordan curve C contained in B and containing the cut y but no point of the boundary of B. Integration by parts then leads to the following: n 19

k (a, v) Pµ,

(C, u, v) w. (C) dC + µ=1

n

w (u) - w, (v) +

kN, (u v) Pµ. V.

µ=1

As a result, we obtain the formulas n

w (v) - w, (u) _ E kµ (u, v) Pµ,

v = 1, ... ,

n.

(6)

µ=1 Formulas (7) and (B) are interesting in that they provide expressions for the functions wv(u) and wv(v) - wv(u) (defined as solutions of Dirichlet problems)

in terms of the functions kAL (u) and kIL (u, v) (defined in terms of univalent mappings). With an eye to finding inverses of these formulas, let us show that the rank of the matrix I1PIL,vII is equal to n - 1 and let us at the same time clarify

the nature of the indeterminateness of the inverse problem. To do this, let us consider the system of linear equations n

E Pµ .kµ=0, µ-l If this system has nonzero solutions AL, for R = 1,

(9)

, n, then the function

n

I) wµ (z) j

µ=1

is single-valued and consequently regular in B. Also, it assumes on each curve K. (for v = 1, , n) values lying on some rectilinear segment. By the same reasoning as was used in the proof of Theorem 2 of §2 of Chapter V, we can show that this function must be constant in B. But then the function µ=1 " A (D (z) µk will also be a constant. Since this constant assumes the value av on K. (for

286


v= 1,

, n), it follows that Al= A2 = .. = An. This shows that the only possible system of solutions of equations (9) is the system Al = A2 = ... = An .

Now, on the basis of (1), we have n

µ=1

wµ (z) = const in B,

(10)

so that the increment in the sum on the left as z moves around any of the curves in question is equal to 0; that is n

Ei Pµ,0, µ=1, ..., n.

(100)

µ=I

From this we conclude that the numbers Al = A2 = .. = An do indeed satisfy the system (9). This shows that the rank of the matrix lI Pµ, vll is equal torn - 1. Furthermore, it follows from what we have shown that the functions kµ(u), for µ = 1, . , n, can be determined from formulas (7) and that the functions kµ(u, v), for p = 1, , n can be determined from formulas (8), in both cases up to a com-

mon term. On the otter hand, the differences kµ(u) - kµ, (u) and kµ(u, v) kµ, (u, v), for p, p' = 1, , n, are uniquely determined by these formulas. This last fact is significant, for example, for formulas (9) and (10) of §4 of Chapter V. Let us now turn to the Green's function g(z, C) for the domain B. We denote the corresponding analytic function of z by p(z, t), so that g(z, 0 = 2 (p (z, Obviously, the function p (z, C) has a logarithmic singularity at the point z = t and is not single-valued in B. The multiple-valuedness is easily shown: (p (z,

C))K _ ` dp (z, C) _ t 12`

V

V

a' (P (z, Q) as

ds

-

ag (z, C) ds = - 21tlw (C).

t AS

Thus, the function p (z, t) possesses the properties C10) 1) p (z, C) = - p (z, C) + const on KY; (10,11) 2) (p (z, C))KV = - 21tw (C); 3) pE (z, C) is regular in B except at the simple pole z = t with residue -1. Consider the integrals 18

=tat

P (C, u) pz (C, z0) dC and 1 = tat S P (C, u, 'U) pz (C, z(,) 4


287

with u, v E B. If we evaluate these integrals first by using the residue theorem directly and then by using (4), (5), and (10 as above, we arrive at the following formulas: A

pz (u, zo) = Q (zo, u) + P (z0, u) -

kµ (u) wµ (zo), µ=1

P (v, zo) - P (u, zo) = Q (zo, u, v)

(11)

+ P (zo, u, v) -

kµ (u, v) wµ (zo). IL-1

we separate the real parts in (11), we obtain the curious formula n

g (v, zo) - g (rt, zo) =1 og j , (zo, u, v)

og pµ (u, v) . wµ (zo)

(12)

Here pµ(u, v), for p = 1, . , n, are the radii of circles on which lie respectively , n, under the mapping w = jn/Z(z, u, v). the images of the curves K., for p = 1, By using the symmetry of the Green's function and replacing zo with z E B, we can also rewrite (12) in the form g (z, v) - g (z, u) _ R (log j n (z, u, v)) -

a

log Pµ (u, v) . wµ (z)),. µ=1

Then, together with (11) we have the formula

p(z, v)-p(z, n)=logj (z, it, v)- Elog pµ(u, v) w,(Z)j-

µ=1

The purely imaginary constant term on the right is not written because we have not normalized the imaginary parts of the functions p (z, C) and w. (C) in any way. The actual result is obtained as a relationship of the analytic representation of functions that are meromorphic in the domain B and real on the boundary K. These functions are known as Schottky functions. Let f(z) denote such a func-

tion that has in B only simple poles ak, for k = 1, , N, with residues rk. Furthermore, suppose that f(z) is regular on K. Consider the integral Ia

=2

P (C, u, v)f (C) A.

From the residue theorem we have N

15 = }J rkPZ (ak, u, v). k=1

On the other hand, by using (5) and the relationship

288


-

f(z)=f(z) we obtain n

16 = I r=1

2aI

t

on K, (v=1, ..., n),

(- Q (C, it, v) + kr (u, v)) df (C) r

N rk Q: (ak, tt,

Q (C, u, v) f' (C) dC' =

= tat

v) +f (u) -f (v)

k=1

(The last integral is calculated by use of the cut y connecting u and v, just as before.) As a result, we obtain the relationship N

f (it) -f (u) = E (rkQ; (ak, it, v) - rkPZ (ak, u, v)).

(13)

A-1

By using formulas (3) of §4 of Chapter V, we transform (13) to the form N

f (u) - Z (rk Q (u, ak) - rkP (it, ak) k=1

N

=f (v) - E (rkQ (v, ak) - rkP (v, ak)) k=1

Since the left-hand member depends only on u and the right-hand member only on v, we conclude that the function f (u) has a representation in the domain B of the form

f (u) _

(rkQ (u, ak) - rkP (u, ak)) + A,

(14)

k=1

where A is a constant. , N, cannot be arbiIt turns out that the numbers ak and rk, for k = 1, trary. It is easy to establish certain relationships between them. For z E Kv, we obtain from (14) on the basis of (4) f (u)

_

N

}J (rkQ (u, ak) + rkQ (u, ak) - rkkr (ak)) + A k=l N

N

=2Z'T(rkQ(it,ak))- Z rkkr(ak)+A k=1

k-1

(15)

Since f (z) is real on K., we have N

Erkkr (ak) -A)=O, v=1, ..., n. k=1

(16)

289


If we multiply (16) by Psum with respect to µ = 1,

, n, and use relation-

ships (7) and (10), we obtain the conditions k=1

rkw"(ak))=0, =1, ..., n.

(17)

It is a remarkable fact that conditions (17) on the numbers ak and rk are not only necessary but also sufficient for the function (14) constructed from them to be, for suitable A, a Schottky function. To prove that conditions (17) are sufficient, suppose that they are satisfied. Then, by using (7), we can transform them to the form 0

y

nn

E PP. .3

P=1

k=1

rkkµ (ak)) = 0, y ^ 1, ... ,

n.

(18)

If we consider (18) as a system of equations of the type (9), we conclude on the basis of what was proved above that this system has only equal solutions, and this leads to n relations (16) with suitable A. But then, it follows from the identity (15) as applied to z C K. that the function f(u) defined in accordance with formula (14) is real on K,, for all v = 1, - - , n, so that f (z) is a Schottky function. , n, Let us stop for an application of this result. Let ak, for k = 1, denote arbitrary points in a domain B. From the identity n

w,(z)0, zEB,

v=1

we conclude that the system of n linear homogeneous equations n

(etewY

(ak)) = 0, k = 1,

. . .

has, for arbitrary 0, the nonzero system of solutions A 1 = .. = An = 1. Consequently, the rank of the matrix IIg (e`Bwv(ak))II is less than n. But then the system of equations n

rk (e;eW

(ak)) = 0, Y = 1, ... ,

n,

also has nonzero real solutions in the rk, for k = 1 , - , n. Consequently, conditions (17) are satisfied for the numbers rke `9 and ak , for k = 1 , , n. Therefore, there exists a constant A such that the corresponding function f(u) defined in accordance with formula (14), which, in the present case, is the function n

t

! (z) = E rk k=1

n

(e l1 Q

(z, ak) -

e-'OP

(z, ak)) + A =

k=1

rke1B1-e (z, ak) + A

290


[this last on the basis of formula (1) of §2 of Chapter V], is a Schottky function. The only poles that this function can have in B are included among the ak. If the number of these poles is m, where m < n, then, by evaluating the increment (arg [f(z) - c]) K, we easily conclude that f(z) assumes every value not lying on suitable intervals of the real axis at exactly m points of the domain B. Consequently, the Riemann surface consisting of m sheets of the w-plane with suitable cuts lying over the real-axis is the single-valued image of the domain B under the bijective function w = f(z). In conclusion, let us look at functions that are regular in the region B with modulus equal to unity everywhere on the boundary K. It follows easily from Cauchy's theorem on the number of zeros that if such a function F(z) has exactly N zeros in B, it maps the domain B onto the N-sheeted disk I C1 < 1. Suppose now that the function F(z) has in B only simple zeros ak, for k = N, and that lim s-.alt z

Then, the functions

F(z) ak

-

= , k=l,..., N. rk 1

F (z) + F (z) and I (F (z)

(19)

-F

(z))

are obviously Schottky functions in the domain B. If we apply formula (14) to them, keeping (19) in mind, we obtain the following formulas: N

F(z)+F(z) = Y, (rkQ(z, ak)-rkP(z, ak))+A, k=1

I

(20)

N F (Z) - (z)Z) "? (rkQ (z, ak) + rkP (z, ak)) + B,

where A and B are constants. Furthermore, from (20) we obtain N

F(z)=- Zrkp(z, ak)+C, k=I

N F (z) _

L rkQ (z, ak) _r 'jh-1

In the present case, conditions (17) take the forms N

N

2(Erkw,(ak))=O, 9(Zrkwy(ak))=0, v=1, ..., n, k-I

k-1

(21)


291

and are equivalent to the complex conditions N

I rkw (ak)=0, v=1, ..., n.

(22)

k=1

The question regarding the sufficiency of conditions (22) for the possibility of construction of the corresponding function F(z) by means of formulas (21) is not answered here. Remark. The functions that we nave been considering include, for example, the functions of §5 that map the domain B onto an n-sheeted disk. However it is possible to exhibit examples of simpler functions of this kind. Specifically, &ch of the two functions Pi (z, u) Ps (z, u, o) { J 1 (z) = Qs (z, u) , A (z) = Qz (Z' 1, v) ,

(23)

where u, v E B, is such a function and each of them maps the domain B onto the 2n-sheeted disk CI < 1. We confine ourselves to proving this assertion only for the first of the functions (23), assuming, as above, that the domain B is bounded by closed analytic , n. Jordan curves K,,, v = 1, In the first place, it follows from formula (1) of §2 of Chapter V that, for z, u C B.

je (z,_ u) = ei0 (cos 6j; (z, u) - I sin 9j' (z, u)) # 0. Therefore,

T Jo (z, u)

J'. (z, u) # I tan'9,

(24)

0

for all real 0. This means that, for all z, u C B, the ratio on the left side of (24) cannot assume purely imaginary values. And since this ratio approaches 1 as z --* u, we conclude that it assumes only values in the right halfplane. But then, we have in B

Qz (z, u) = 2 (jo (z, u) +j'a, (z, u)) # 02

Thus, the function Qz (z, u) has in B a unique double pole z = u and has no zeros at all. In the second place, if z = zv(s) is the equation of the curve K. and s is the length of arc on K. measured in the positive direction with respect to the domain B, then, when we differentiate (4) with respect to s , we obtain


292

P (Zr (S), U) zv (z) _ - Qz (z,, (S), U) z (S).

(25)

It follows that on Kv, Pi (z, u)

If, (z)

(26)

Qi (z, U)

On this basis, we conclude that the functions PZ (z, u) and QZ (z, U) have no zeros on K. To see this, suppose, for example, that QZ (a, u) = 0, where a E K. Then, in accordance with (26), we have Pz (a, u) = 0 and hence jo (a, u) _ .0 (a, u) = 0. Therefore, in accordance with formula (1) of §2 of Chapter V, we have 1B (a, u) = 0 for all 0. This means that, for all 0, the mapping £= 1e(z, u) maps the point a into the endpoint of the boundary cut of the image of the domain B. On the other hand, if in a neighborhood of z = a we set

J (z, u)=cl(z-a)+..., f,a(z, u)=dl(z-a)+... we conclude from the geometrical interpretation of the argument of the derivative of an analytic function that the ratio c 1/d1 is purely imaginary. Therefore, since je (z, u) = e18 (c1 cos 9 - Id1 sin 9) (z - a) + ... ,

it follows that the derivative 1e (z, u) has at the point z = a a zero of no less than second order for some real 0, and this contradicts the univalence of the function je (z, u) in the domain B. This proves the assertion. Thirdly, from (25) we have(( arg

Q'(z,

)/K-(argz'(s))K-2(argz'(S))K=4a (n-2).

Since (arg Q. ,(z, u))K = - 4n, it follows from what we have proved that (arg Pz (z, u))K = 4rr (n - 1).

This last equation shows that the function PZ (z, u), which is regular in the region B, has exactly 2n - 2 zeros in B. But then the function f1 (z) has exactly 2n zeros in B. This information, Cauchy's theorem on the number of zeros, and property (25) enable us to conclude that the function 4 = f 1(z) assumes in the domain B every value in the disk I CI < 1 at exactly 2n points and it does not assume any other values at all; that is, it maps the domain B onto a 2nsheeted disk. This completes the proof of the assertion.?) 1) In a similar way, Garabedian and Schiffer [19501 have constructed functions that map the domain B onto an n-sheeted disk.

CHAPTER VII THE MEASURE PROPERTIES OF CLOSED SETS IN THE PLANE

§1. The transfinite diameter and tebygev's constant In contemporary questions in the theory of functions of a complex variable, a significant role is played by certain specific ways of measuring closed sets in the complex plane. We begin with one of these ways, which was proposed by Fekete. 1) Let E denote a closed bounded infinite set of points in the z-plane. For n points z 1, z 2, .. , z. C E, we define n

V (z1' zy, ... ,

za) = rl (zk - zl),

n ,:- 2-

(1)

k, 141

k e y". This shows that

e-7n - max

zEEn

(z) I,

v = 1, 2, .. .

where E v(z) is the Cebysev polynomial for E. This means that, for every n, there exists a point zn on the boundary of the domain B(n> such that an < Itv (z,,)I. All the cluster points of the sequence (zn , for n = 1, 2, -, lie on E. Let z0 denote one of these points. Then, this last inequality implies a-y < v v Itv (z0)I and consequently, a-y R by means of a normalized

314

VII. MEASURE PROPERTIES OF CLOSED SETS

function J= F(z) that satisfies the conditions F(oo) = oo and F'(-) = 1. Thus, we have Theorem 3. 1) The capacity, and hence the trans finite diameter, of a bounded continuum E are equal to the conformal radius of the domain complementary to E and containing oo. In conclusion, we present one more theorem connecting Green's function with iebyit'ev polynomials.

Theorem 4. Suppose that a domain B with boundary K contains oo and has Green's function g(z, oo). Let tn(z) denote the l,ebysev polynomials for K with zeros on K. Then n

y

m i in 00

I tnmz) I

_ eg (z

mE

I

K

t n (x) I ,

this limit being uniform in the interior of B. Proof. Consider again the functions v,, (z), for n = 1, 2, . , defined by formula (4). Since these functions are harmonic and nonnegative in B, the condensation principle can be applied to them. According to this principle, the sequence {vn(z)I itself and any subsequence of it contain subsequences that converge uniformly in the interior of B to harmonic functions. Here, the limit functions are always nonnegative in B and, in accordance with Theorem 1 they vanish at so. Consequently, these functions vanish identically in B. This means that the sequence Ivn(z)j itself converges uniformly in the interior of B to 0. This is equivalent to the assertion of the theorem. §4. Harmonic measure of closed bounded sets Together with the methods considered above for measuring closed bounded subsets of a plane, R. Nevalinna 2) introduced another measure, which has proved very important in the theory of functions.

Let a denote a closed bounded set of points in the z-plane and let B denote any of the domains complementary to a. Let f3 denote a closed Jordan curve in the domain B that has no points in common with a. Let us remove 0 and its interior from B and denote the remaining portion of B by B18. Let B(n), for 1) Fekete [19231.

2)R. Nevanlinna [1936a]. All the material of the present section is taken from that article.

§4. HARMONIC MEASURE OF CLOSED BOUNDED SETS

315

n = 1, 2, , denote a sequence of domains contained in B and containing (3. Suppose that their boundaries an consist of a finite number of closed Jordan curves and that the sequence exhausts the domain B. Finally, let B('), for n = 1, 2, , denote the domain obtained from B (n) by removing the curve S and the domain bounded by it. We denote by to (z, an, B( )) a function that is harmonic in B I , equal to unity on a., and equal to zero on P. Obviously, this function is nonnegative and less than unity in B(n). Furthermore, the difference to(z, a,n B()`, 18 to (z, an+1' Bn+1) since it is nonnegative on the boundary of the domain BP(n) 18 is positive in the interior of BQ I . This means that for fixed z E B'6, the quantity to(z, a BQ I) is defined, beginning with some n and decreases with increasing n. We can now apply Harnack's theorem, concluding from it that (n)) = !o (z, a, BQ) exists everywhere in B18 and is a harmonic limn oo o. (z, an, BP function in BR. Obviously, 0 < to (z, a, BR) < 1 in B'8. We note that to (z, a, BR) does not depend on the choice of sequence B1°I because, if there are two sequences B(n' ) and B(n" of this kind, then, for arbitrary n', there exists an n" such that B(R ) C B I and conversely. Consequently, in the first case we have the inequality to(z, a ,, B') < (o (z, a ,,, and, in the second case, we have the opposite inequality. The quantity &j (z, a, BP) is called the harmonic measure of the set a with respect to the domain B, the curve (3, and the point z E BR. We note that, as we let z approach points of the set a through values in BR, the harmonic measure to (z, a, BR) does not always approach unity. For example, it will not approach unity as z approaches isolated points of the set a because at each of these points to(z, a, BR) is a harmonic function. However, we can show that, if to (z, a, BR) j 0, then, as we let z approach points of the set a through values in BQ, the limiting values of the harmonic measure oo (z, a, BQ) are bounded below by a positive number. To see this, let go denote a closed Jordan curve in B that surrounds the curve A and does not contain any point of the set a. On P0, the function &)(z, a, BR) has a positive minimum 9 such that 0 < B.< 1. Let zQ denote an arbitrary point of the domain BR that lies outside

go and let n be sufficiently great that 00 and zo lie in B(nI. Since in B 1, it follows that a) (z, a., BQ I) > 9 on 00 Furthermore, to(z, an, B I) = I on an. Consequently, in accordance with the

(0 (Z, a, BQ I) > a) (z, a, B

maximum principle, we obtain ao (z0, an, B18 (n)) > B for all n sufficiently great. As we let n approach oc, we see that &o(z0) a, BQ) > B > 0. It follows that all

the limiting values of &j (z, a, BR) as z approaches a exceed 0.

316


The question arises as to ways of telling when the harmonic measure of a given set a is zero or positive. The vanishing or not of a harmonic measure co(z, a, BA) holds simultaneously for all points of a domain BR because, if it vanishes at a single point in Bp, it must vanish everywhere in BA by virtue of the maximum principle. However, the same phenomenon holds with regard to the curves S chosen in B. To see this, suppose that for some curve S, we have denote another curve of the same kind. Let y and y' co (z, a, BR) = 0 and let such that y' denote closed Jordan curves lying in B and enclosing S and

lies inside y and there are no points of the set a between y and The curves y and S' together form the boundary of a domain B0 contained in B. Set 0 = max Z E y, w (z, y, B o), 0 < 0 < 1. Now, let a denote any positive number. For

n sufficiently large and for z E y, we have w(z, an, BA (4) < e because the convergence of the sequence (w(z, an, B ))E to zero in BR implies its uniform conA vergence to 0 inside BA. Let us denote by A and A' the maxima of the function Co (z, an , Bp;) on y and y'. Since the difference w (z, a n, B $ ) ) - W (z, an, B 4 >) is a harmonic function in the portion Bin> of the domain B(n) that is bounded by the curves an and

y' and since it is equal to 0 on an and does not exceed A' on y', it follows that A < A' + e. Furthermore, since the difference w (z, an, B(n) - Aw (z, y, B 0) is a.

harmonic function in Bo that vanishes on S` and is less.than 0 on y, it is less than zero in Bo and, in particular, on y', where the maximum of the function w(z, an, B 4' ) is equal to V. Consequently, on y' we have A' 0 for any domain B complementary to a and for any curve S C B. Just as in the proof of Riemann's theorem in §2 of Chapter II, we can, by a suitable transformation, arrange for the domain B to have exterior points. Let B0 denote the domain bounded by the curve S and some other closed Jordan curve lying outside B. Then, B( n) C B0. Consequently, just as above, we can show that, for z E BA(n) w (z, a,

By letting n approach oo, we see that

B 0) > 0.

§4. HARMONIC MEASURE OF CLOSED BOUIpED SETS

317

co(z, a, BQ)>a)(z, a0, B0)>0. It follows from this last assertion that, if the harmonic measure of the set a with respect to some domain complementary to a is equal to zero, then a does not contain a continuum and the complement to a is the unique domain including oo. We now point out the following simple properties of sets of zero harmonic measure.

Every closed subset of a closed bounded set of harmonic measure zero is also a set of harmonic measure zero. This follows immediately from examination of the differences between corresponding functions of the type W (z, an, B( )). 18

The sum of two closed bounded sets of zero harmonic measure is a set of harmonic measure zero.

To show this, denote the two sets by a1 and a2 and let S denote a curve in the complement to a = a1 + a2. Let {B(1)} and {BZ"I} denote two sequences that exhaust the complements to a1 and a2 respectively and that contain S. Then, for a given point z in the domain B complementary to a and containing S, we have (0 (z, at n, Bln)) < 7t,

w (z, as, .,

B(2n))

\2'

for sufficiently large n. Here a 1,n and a 2, n are the boundaries of the domains B(1 and B(z) . Let B(n) denote that one of the domains in the intersection of B in and B (z) that contains S . For suitable choice of the domains B (n) and Ben) (they can, for example, be constructed from squares as at the beginning of §2), the boundary of the domain B(n) consists, for every n = 1, 2, , , of a finite number of closed Jordan curves. Therefore, the domains B(n), for n = 1, 2, exhaust the domain B. The function

,

0 - co( z a

w (z,r an, y B(n))

r

1-

' 1, n B`'n)

w

(z, as, n a9B(n') 9 r

which is harmonic in B(n) is equal to zero on S and does not exceed zero on an . 18 Consequently, in Bp we have w (z, a, Bp) < w (z, at, B1) + w (z, as, B9),

that is, w (z, a, BR) = 0. We shall now prove the theorem of R. Nevanlinna, which is fundamental in the theory of harmonic measure.

318


Theorem I. A closed bounded set has zero harmonic measure if and only if its capacity is equal to zero. Proof. As a preliminary, we shall make a necessary observation concerning Green's function. Let B denote a domain containing oo and bounded by a finite number of closed Jordan curves. Let_g(z, 00) denote its Green's function and let h(z, 00) denote the conjugate function. Let y denote Robin's constant. Formula (1) of §3 is then applicable and from it we obtain the inequalities

logd,(z)cg(z, oo)-7clog dg (z),

(1)

where d1(z) and d2(z) are the minimum and maximum distances from the point z to the set of boundary points of the domain B. We shall very shortly have occasion to use these inequalities. Turning to the proof of the theorem, we may assume that the closed bounded set referred to in the statement of the theorem is contained in the disk I z I < 1 because, otherwise, we could arrange for this by means of a similarity transformation without affecting the property to be proved. Let us denote this set by a and let us denote by B the domain complementary to a that includes *a. Let B' denote a domain including 00 that is bounded by a finite number of closed Jordan curves contained in I z I < I and satisfying the condition B' C B. Let us denote by g(z, 00) and g'(z, 00) the Green's functions for B and B'. Let us denote by co P (z, a', B'P) the harmonic measure of the boundary a' of the domain B' with respect to the subset BP of the domain B contained in I z I < p, where p > 1. (1) If a has positive capacity, then the constant y = limz.,00(g(z, -)-log Iz1) is finite. Let p denote a fixed number > 2 and let MP and mP denote the maximum and minimum values of the function g' (z, oo) on I z I = p. The difference g, - (1 - w p)mP is a harmonic function in BP and it is nonnegative on the boundB'P. ary of Consequently, this difference is also nonnegative in B that is, g, > (1 - a) P)mP

(2)

in B P . If z 1 is a point on the circle I z I at which g' (z, oo) has a minimum, which we denote by m1, then inequality (2) yields

mt

(1 - (OP (z1, a', BP)) m

But m1=min jz1=1(g'-logI z1) log (p - 1) + y'. Consequently, we obtain from (3) T > (1 - WP (z1, a', B'P)) (log (P - 1) + 7'),

so that

log (P -1)

7

wP (z, a', BP):1 - T +log(9-1)

7+1og(9-1)

> 0,

since y' < y. This shows that if the capacity of the set a is positive, the harmonic measure of a is also positive. (2) Now suppose that the capacity of the set a is equal to zero. Since the difference (1 - a)P)MP- g' is a harmonic function in BP and since it is nonnegative on the boundary of B P, it follows that this difference is also nonnegative in B o and this yields

g' < (1 - 0P)MP.

(4)

For a point z 2

on I zj = 1 at which g' attains a maximum, which we denote by M 1, inequality (4) yields

Mi C (1 - WP (z9, a', BP)) MP. But

MI -max(g'-log IzI):7, Izl=1

and, in accordance with inequality (1),

MP` log (P+1)+T. Consequently,

so that

T c(I

wP(z9i

min w c w (Zs, IzI=I1, P

a, BP))(log (P+1)+T'),

a', B') c 1P

T"

7'+1og(P+1)

If we fix p > 1 and let B' (a sequence of domains exhausting B) approach B, then y' approaches oc and consequently min,,,=161 (z, a', BP) approaches 10 zero. Since the function o (z, a', BP) on I zj = I then converges uniformly to (a (Z' a, BP) where BP is the portion of the domain B lying in the disk I zj < p, it follows that t) (z, a, B ") - 0 because the function o U (z, a', B'P) would otherwise have a positive bound on I z = 1. Therefore, the harmonic measure of

320


a is zero. This completes the proof of the theorem. As simple applications of the theory of harmonic measure, we shall now prove two theorems, one of which deals with removable singularities of harmonic functions and the other with the maximum principle for harmonic functions.

Theorem 2. Let a denote a closed bounded set of points in the z-plane that is contained in a domain B. If the harmonic measure of a is zero, then every function u(z) that is harmonic and bounded in B except at points of the set a is harmonic in the entire domain B if we define it in a suitable way on a. On the other hand, if the harmonic measure of a is positive, this property does not always hold.

Proof. Suppose that the harmonic measure of a is zero. Then we can assume without loss of generality that B is a bounded domain with boundary consisting of a single closed Jordan curve 0 and that u(z) is a harmonic function on 8 because otherwise we could consider not B but a suitable domain of the type described contained in B and containing a. Let u0(z) denote a function that is bounded and harmonic in the domain B and coinciding with u(z) on the boundary

of B. Let us show that u0(z) a u(z) in B. For a point z0 C B and given c> 0, there exists'a domain Bp I, contained in B including z0 but not containing a, the boundary f3 + an of which consists of a finite number of Jordan curves and which has the property that w(z0, an, B (n)) <E. If (u(z)I M in B - a, then Iu0(z) I < M in B. Consequently, for the difference v(z) = u(z) - u0(z) in B - a, , we have Iv(z) I < 2M. Now, let us consider the function v(z) + 2M6)(z, an, B18(n) ).. This function is harmonic in B( ), equal to zero on S, and nonnegative on a n .

Consequently, v(z) + Ma (z, an, B( )) > 0 in B( )

Therefore, v(z0) >

- 2Mc, > - 2Me. Since c is an arbitrary positive number, v(z0) > 0. Treating the function v(z) - 2Mco (z, an, BP (n)) , we can show that v(z0) < 0. Consequently, v(z0) = 0; that is, u0(z) = u (z) in B - a. It follows that if we define u (z) on a equal to u0(z), then u (z) . becomes harmonic in B. On the other hand, if a has positive harmonic measure, then, for example, the domain of definition of w(z, a, B), which is harmonic and bounded in B - a cannot be extended to a in such a way as to make w (z, a, B) a harmonic function in B because this function, since it is equal to zero on S, would be identically equal to zero in B - a and then the set a would be of zero harmonic measure. Theorem 3. Let u(z) denote a function that is harmonic and bounded above in a domain B. If lim:. Z u(z) < m for all points Z* on the boundary except for a set of harmonic measure zero, then u(z) < m everywhere in B.

§5. APPLICATION TO MEROMORPHIC FUNCTIONS

321

Proof. Suppose that u (z) < M in B. Let us suppose that there exists a point z0 in B at which u(z) > in. It follows from the hypotheses of the theorem that there exists a point zl E B such that u(z1) < u(zo). Let us describe a small disk S around z 1 in which u (z) < m 1, where m < m 1 < u (z 0). From the hypothesis of the theorem, the set a can be contained in the interior of a finite number of Jordan curves lying in the complementary domain containing B. Let us denote the union of these Jordan curves by an, so that, for the domain B( n) with boundary an U 8, we have w(z 0 , a n , B (1)) m 1. This completes the proof of the theorem. §5. An application to meromorphic functions of bounded form A function f(z) is called a meromorphic function of bounded form in Izl < 1 if it can be represented in that disk as the ratio of two bounded regular functions . f (z) _ T(z) (z)

(1)

We may assume that I,(z)I < 1 and IVi(z)I < 1 in Iz I < 1. A test of whether f(z) is a meromorphic function of bounded form is given by Theorem 1. 1) Let f(z) denote a function that is meromorphic in the disk Iz I < 1 such that f(0) no. For f(z) to be a function of bounded form, it is necessary and sufficient that the quantity T (r) = 2n \ log If (re") I dO + 1 log kkr

if

1)R. Nevanlinna [1922].

I

e

bk

322


be bounded above for 0 < r < 1. Here, the bk are the poles of the function f(z) other than the point z = 0. These zeros are indexed in such a way that a pole of multiplicity n is counted n times. The summation is over all poles bk in the disk I z I < r. The symbol log a is defined by

(logs if a>1

+

log a=j

if0 e and to the function -s I W. According to that theorem, - s I (w) < - mE in B I; that is, s 1(w) > mE in B I. But mE + oc as e - 0. Consequently, s 1(w) _ +

in B, which contradicts the finiteness, proved above, of the function s W. ibis contradiction proves the theorem.

In connection with other results dealing with the application of the theorem on harmonic measure to the study of analytic functions, we refer the reader to the monographs of R. Nevanlinna [1936a] and Privalov [1950].

CHAPTER VIII MAJORIZATION PRINCIPLES AND THEIR APPLICATIONS

§1. An invariant form of the Schwarz lemma

In §2 of Chapter II, we established a simple but extremely important property of bounded functions, known as the Schwarz lemma. A very simple and widely used form of this lemma is as follows:

Theorem ('the Schwarz lemma). Let f W denote a function that is regular in the disk I z I < 1. Suppose that f (0) = 0 and that if (z)I < 1 in I z I < 1. Then z) < I z I throughout the disk I z < 1 and f' (0)j < 1. Also, in the inequality I f (z)j -}z 1, the equality holds, for points z 0, only if f (z) = rz with I e If we now discard the requirement that f (0) = 0, that is, if we consider an arbitrary function f(z) that is regular in the disk z < 1 and satisfies the ine-

quality IfWj < 1 in that disk, then, if we set

f I

C+

)

z

(C1+2C) =c1C+ I+zC+Z) ...,

-f(z)f

C1T I-If(Z)I, for fixed z in the disk z < 1, we see that the function g(C) satisfies the conditions of the Schwarz lemma. Therefore, the inequalities mentioned in the conclusion of that lemma hold. From this we obtain a further generalization of the I-IZ18f(z),

I

Schwarz lemma.

Theorem 2. If a function f (z) is regular in the disk I z I < 1 and if l1 (z) I < 1 in that disk, then, for arbitrary C and z in the disk I z I < 1,

f (C) -f (Z) Ic C- Z 1 -f (Z)1(C) 329

1 - zC

f

(1)

330

VIII. MAJORIZATION PRINCIPLES AND APPLICATIONS

and, for arbitrary z in the disk I z I < 1,

If,(x)1=11If(Z)I

(2)

with equality only for f(z) = E (z + a)/(1 + a z), where I E (= 1 and a I < 1. We note that, if we set z = 0 in 1, we obtain the result that, in ICI < 1,

f(C)-f(0)

(3)

I -f (0)f (C)

After some simple manipulations, as in §4 of Chapter IV, we can reduce this inequality to the form

f(C)-f(o)1-Ifo)CIIaICI' 'IÌI-If(o)II'ItI''

(4)

From this in turn we get the inequality I + if If(t)I I+C, IfOIICI 1

)

(5)

I

Here, for C 0, equality holds only when f() = E (C+ a)/(1 + a t;')., where

I E I = 1,

Ial 0 and that a is a segment of the real axis. Then, as one can easily verify, we have (in the case of a halfplane, the harmonic measure on the boundary is defined as above)

§4. HARMONIC MEASURE. SIMPLEST APPLICAWONS

345

w(z,a,B)=1, it

(8)

where (k is the angle subtended by the segment a at the point z. 2. Suppose that B is the half-disk I z I < 1, a (z) > 0 and that a is the boundary semicircle. Then, one can again verify easily that w (z,

(9)

B)

where 0, and continuous except on the positive half without the point z = 0

plane

of the real axis. Suppose that f (z) , a as z approaches the origin along this 0 in the angle half axis. Then, f (z) --> a uniformly for arbitrary 8 > 0 as z

0<argz 0. Let us choose arbitrary c in (0, 1) and let us choose r > 0 such that If (z) < e on the interval a, of the real axis from 0 to r. If we apply inequality (2) to the halfdisk B,.:

z I < r, 3 (z) > 0, we obtain log I f(z)I ---- w(z,a,.,B,)loge

(10)

in B,. Let\us find a lower bound for w (r, a,, B,). To do this, let us find an explicit expression for this quantity. The function C= rz/(r - z)2 maps the halfdisk B, into the halfplane (C) > 0 in such a way that the point z = 0 is mapped into the point C= 0 and the interval a, is mapped into the positive half of the real axis. We denote this last by a. If z and C are corresponding points, then Also, by virtue of (8),

w (z, a B,) = w (C, a, B)-

_71-11R

, 0<argC 8/2tr= 8


346

in the domain I z I < r' , 0 _< arg z < n - 8. Consequently [cf. (10)], I f (z) _< E S. Since S does not depend here on r or c, this means that f(z) --' 0 uniformly as z -+ 0 in the angle 0 < arg z < n - 6. This completes the proof of the theorem.

Theorem 4. Suppose that a function f(z) is regular and bounded in the half plane (z) > 0 and 1 is a Jordan curve lying in that halfplane that terminates at z = 0. Suppose also that f(z) --, a as z --+ 0 along 1. Then, f(z) --, a uniformly as z --, 0 in the angle 0< arg z < n - 6 for arbitrary 8> 0. Proof. We can again assume that a = 0 and I f (z)I < 1 in (z) > 0. Let c denote a number in the interval 0 < e < 1. Let us choose r sufficiently small that If (01 .5 e on the part of the curve 1 lying in the half-disk B,: I z I < r, (z) > 0. Let us denote by a, the part of 1 extending from z = 0 to the first point at which 1 intersects the circle I z I = r. The arc a, partitions the half-disk B, into'two domains B, and BT. Suppose that B, is adjacent to the negative half of the real axis. By applying inequality (2) to B,. , we obtain log E.

Here again, it remains only to find a lower bound for -(z, ar, Br ). Let us denote by 0, the portion of the boundary of the domain Br complementary to a,, and let us denote by ar the portion of the boundary of the domain B r complementary to f3, . Finally, let us denote by a" the interval (0, r) of the real axis. Then, from the extension principle, we have w (z, ar, B,.) . w (z, a Br) : w (z, ar, Br) But, as follows from the proof of Theorem 3, the quantity a) (z, 0-,", Br) has a positive lower bound S1 in the sector I z I < r', 0 < arg z < n -6 with sufficiently

small r' in (0, r). ,

e

Thus I f (z) I < Esl in the part of B , lying in this sector. Treating B,s in

the same way, we can show that there exists a S2 > 0 such that I f (z)I < E 2 in that part of B,. lying in the sector I z I < r" , 0< arg z < n with sufficiently small r" in (0, r). Suppose now that 0 < $ < n/2. If a is the greater of the numbers S1, 82 and if r1 is the smaller of the two numbers r' , r" , then If(z)I <eS for I z I < r1 and h < arg z < n -$ . This proves the uniform convergence of f (z) to 0 as z - 0 in the angled < arg z < n - $. This completes the proof of the theorem.

Theorem 5. Suppose that a function f (z) is regular and bounded in the half (z) > 0 and is continuous except at finite points of the real axis. Suppose

plane

§4. HARMONIC MEASURE. SIMPLEST APPLICATIONS

347

that f(z) approaches limiting values a and b as z approaches - and +along the real axis. Then a = b and f(z) -+ a uniformly as z

cc in the half

plane Z(z) > 0.

Proof. If we apply Theorem 3 to the function f (-1/z) - b, we conclude that this function approaches zero uniformly as z -+ 0 in the angle 0 < arg z < n - $, where $ > 0. Furthermore, we can show, in a manner analogous to the proof of Theorem 3, that the function f (- I/z) - a approaches zero as z --+ 0 in the angle $< arg z < n. For $< n/2, we see from this that the function f (z) approaches both a and b as z. -+ cc in the angle $ < arg z < n - $. Consequently, a = b and f(z) , a uniformly as z -+ oo through values in the halfplane (z) > 0. This completes the proof of the theorem. By means of a conformal mapping, we obtain from Theorem 3 the following two analogues of this theorem.

Suppose that a function f(z) is regular and bounded in the angle $1 < arg z < $Z and that it is continuous except at finite boundary points of that angle. Suppose that it has definite limits a and b as z approaches o along both rays of that angle. Then, a = b and f(z) -+ a uniformly as z -+ oc in that angle.. Theorem 5". Suppose that a function f(z) is regular and bounded in a simply connec ed domain B bounded by a Jordan curve, that it is continuous on B except at a point z = on the boundary K of the domain B, and that it has finite limits b as' z approaches c along K from the two sides. Then, a = b and f(z) is continuous on B if we define f(c) = a. Theorems 3-5 are due to Lindelof1) and they apply to bounded functions. We shall now give another theorem on bounded functions, this one due to Milloux.2) It is of a somewhat different nature from the preceding ones. Theorem

.

Theorem 6. Suppose that a function f(z) is regular in a simply connected domain B that is contained in the disk I z I < 1 and has boundary K consisting of an arc a of the circle I z I = 1 and of a continuum f3 belonging to the disk z I < 1 and containing z = 0. Suppose that I f(z) I _< 1 in B and that none of the limiting values of If(z)I exceed e, where 0 < e < 1, as z approaches points on JS through values in B. Then, in B, If(z)I<e11 1) Lindelof [19151. 2) Milloux [19241.

areslnl+1t1/EIC (1-IZII. (11)

348


For any zo in the disk Iz I < 1, there exists a domain B containing zo and in it the function f(z) is such that equality holds in the first of inequalities (11) at

z=zo. Proof." When we apply inequality (2) to the function f(z) for z E B, we obtain log If (x) I < w (x, P, B) log e.

(12)

Here, the question consists in finding a bound for the harmonic measure cw(z, 8, B).

We shall first need to prove an inequality for univalent functions. Suppose that a function g(C) is regular and univalent in the disk ICI < 1 and that it does not assume the value zero in that disk. Then, the function

g` h(Q)

C)-g(C.) 0

g'(CO)(1-ICoj)

-c+...

will, for arbitrary Co such that ICo I < 1, be regular and univalent in IKI < 1 and will not assume in that disk the value -g(to)/g'(t;') (1 - I Co12). By Koebe's theorem it then follows that I g(Co)/g '(co) (1 - K0!2)! > 4; that is, 4 g(Ca)

1-ICoI

(13)

Equality holds in (13) in the case of the function g(') = [(1 + 77.0/(1 - 770] 2 where 1771 = 1. If we set Co = rei9 in (13) and integrate with respect to r from 0 to r, we obtain, for arbitrary C in the disk I CI < 1 the inequality llog

with equality holding under the same condition. Finally, if g(C) is such that g(0) = 1, then for z = g(t;) we have in I z I < 1

-ylzl

1±ylzl'

(14)

This is the desired inequality. Let us now see about finding a bound for w (z, 0, B). We map the domain B onto the half-disk B `: I CI < 1, (C) > 0 in such a way that the arc a is mapped into the boundary diameter of the half-disk. Suppose that the point zo E B is 1) The proof given here with the sharp inequality was given by E. Schmidt [1932].

*4. HARMONIC MEASURE. SIMPLEST APPLICATIONS

349

mapped into a point Co, Idol < 1. By means of a supplementary mapping of the half-disk B ' into itself, we can arrange to have the point Co lie on the imaginary axis. Specifically, let us draw through Co a circle orthogonal to the circle ICI = I and to the real axis. Denote by a the point of intersection of this circle with the real axis. When we map the upper half of the C-plane into itself in such a way

that the points - 1, a, and + 1 are mapped into the points -1, 0, and + 1 respectively, the half-disk B' is mapped into itself and the orthogonal circle mentioned above is mapped into the imaginary axis. In particular, the point Co is mapped into a point on the imaginary axis. Thus we may assume that Co = ito for to > 0. We denote by z = g(C) the inverse function providing the mapping mentioned. We have zo = g(ito). The function g(C) can, according to the symmetry principle, be extended beyond the real axis, and it maps the complete disk ICI < 1 univalently onto a domain that does not contain the points 0 and oo. Inequality (14) is therefore applicable to it. If we denote by j9 ' the boundary semicircle in B , we now have by using (9) 4 4 .w (zo, B) = to (Co, P', B') = arc tan to = arc tan Co R which, by (14), yields (zo, P, B}

arc tan

tan = i1-

+

IzoI

IzoI'

4

arc tan

I --

zo I

TT

zo I

(15)

,+,-,l z° l Zo-,

Sin

2tan 4 _i-IzoI =f -{-lane

i --1-1 zo I

and, consequently, I zo I -2I arc sin r-1+1zol*

Therefore (15) can be written in the form (o (z

> a arc sin i

zoI

(16)

where zo is an arbitrary point of the domain B. Inequalities (12) and (16) also lead to the first of inequalities (11). Equality holds in it only when B is a disk with cut along a suitable radius and the function f(z) is such that, as z assumes values on the half-disk B', the function log I f (z) I is a harmonic function that is equal to 2 (1 - 96/n) loge on the boundary, where 96 is the same as in formula (9). From the inequality

350


arc sin

1-Izl' 1-IzI> 1-IzI I+IzI

1+Iz1

2

we get the second of inequalities (11). This completes the proof of the theorem.

Now, we shall finally give an application of the theory of harmonic measure to a single question associated with the principle of the maximum modulus. We recall that, if a function f (z) is regular in a domain B and if none of the limiting values of its modulus exceed M as z approaches any of the boundary points of the domain B, then if (z) I < M throughout the domain B. The situation is quite different when the requirement of boundedness of the limiting values of I f(z)I is violated even at a single point of the boundary. Then, the modulus principle no longer holds. For example, the function f (z) = e-" is regular in the upper halfplane and its modulus is equal to 1 at finite points of the real axis. Furthermore,

I f(z)I > 1 in the halfplane (z) > 0. The reason for this is the fact that the limiting values of the function are not all finite as z approaches the point z = o. in (z) > 0. Now, we have

Theorem 7.1) Suppose that a function f(z) is regular in the halfplane (z) > 0 and that all the limiting values of its modulus are bounded above by 1 as z approaches any finite point on the real axis. Then, either (1) If(z)I < 1 in (z) > 0 or (2) there exist positive numbers A and R such that

M(r)>eAr

(17)

where M(r) denotes the maximum value of I f(z)I on the semicircle Izi = r,

(z)>0 for r > R. Proof. Let us denote by B the half-disk I z I < r, (z) > 0 and let us denote by a. its boundary semicircle. From inequality (2), we have 1

log I f (z) I C w (z, a, B) log M (r).

(18)

But by (9), if z = x + iy,

w(z,a,B)= 22

-azc tan r y Xt. ac tan r+X 11- 2

1)Phragmen and Lindelof [19081.

(arc tan

x

arc tan

§5. ON THE NUMBER OF ASYMPTOTIC VALC"ES

351

Consequently, for fixed z and sufficiently large r, cu

(z,a,

B ti 2 (l r-yx + ry--}- x ) = 2r. re2yr - x° 7c

4y

.

nr '

(19)

Suppose now that the inequality I f(z)I < 1 does not hold everywhere in B. Let zo = x0 + iyo denote a point in B such that I f(zo)I > 1. Then, from (18) and (19), we have log If (zo) I < 11M log M (r)

r which shows,that, for suitable A > 0 and sufficiently large r, inequality (17) holds. This completes the proof of the theorem. By performing a conformal mapping, we obtain from Theorem 7 a generalizing theorem that will be used in the following section. 4yo

r _, co

Theorem 7'. Suppose that a function f (z) is regular in an angle of magnitude no with vertex at z = 0. Suppose that the limiting values of I f(z)I do not exceed 1 at finite boundary points. Then either (1) If WI < 1 in the entire angle or (2) there exist positive numbers A and R such that

M (r) > e''110

(20)

where M (r) is the maximum value of I f(z)I on an arc of the circle I z I = r lying in th angle in question, where r > R.

§5. On the number of asymptotic values of entire functions of finite order Let us give an application of the theory of harmonic measure to a more complex question, one dealing with entire functions. Suppose that f (z) is an entire function and that l is a continuous curve in the z-plane that extends from some finite point to oo . If, as we let z move along

1 towards 00, the function f(z) approaches a definite limit a, then a is called the asymptotic value of the entire function f(z) defined by the curve 1. Suppose that two curves ll and 12 of the type described define finite asymptotic values a1 and a2. If a1 a2, these values are considered distinct; if a1 = a2, then let us agree to consider the curves ll and 12 as defining distinct asymptotic values only in the case when there is some R > 0 such that the curves 11 and 12 cannot be connected by a continuous curve lying in the domain I z I > R and having

352


the property that the oscillation of the function f (z) on it is arbitrarily small. In the theory of entire functions, this convention identifies the asymptotic values of the entire function f(z) with nonalgebraic (transcendental) singularities of the inverse function. If an entire function f (z) is such that, for some finite k > 0 and sufficiently large r, we have M (r) < e rk , where M (r) = maxe Zl =r I f (z)j then f (z) is called an entire function of finite order. The greatest lower bound p of the numbers k that possess this property is called the order of the entire function f (z). It was assumed that the number of asymptotic values of an entire function of finite order is finite and is bounded by a quantity depending only on the order p. The complete answer to the question was given by Ahlfors.1) We shall give his solution, using the method of strips, which we have already used in §6 of Chapter IV.2) Lemma. Suppose that there are a finite number n of nonoverlapping radial strips Sk, for k = 1, 2, , n, in the annulus ro < I z I < r that extend from the circle I z I = ro to the circle I z I = r. Let ak (rk) denote the arc of the circle z = ro (I z I = r) that serves as part of the boundary of the strip Sk. Suppose that none of these arcs degenerate to a point. If we map these strips conformally onto nonoverlapping domains F,k: ro
1 and is independent of k, then the grouped series W

f (z)=panr (z)+ E pan+l, k=1

Panr (z) = Coza° +

k

Xn

k

(3)

(z),

... + CnrzXnr,

Zn + 1

pank+I,knk+1(z)=Ck 1 z

b+I an

--... +Cnk+Iz k+1, k=1, 2,

.. ,,

converges in a sufficiently small neighborhood of an arbitrary point on the circle z = 1 at which f (z) is regular. 1) Proof. Suppose that the function f (z) is regular at the point z = 1 (if it is regular at a point z = e` °', we could consider the function f (e-taz)). Let us set

z = a(t;) = Q' (1 + C)/2, where p is a positive integer. Since u(1) = 1 and la (C) I < 1 for I C1 < 1 but C 1, the function F ([;) = f (a (O) is regular on CI < 1. From (3) we now obtain the transformed series OD

F(C)=f(a(C))=panr (a(C))+

(4)

k_1

k

Here, the two quantities constituting the subscript to Q are the lowest and highof C that appear in the corresponding polynomial. If we take p > est 1/(q -J 1), so that (p + 1)/p < q, we have (p + 1)Ank < plink+I In this case, the series (4) is a grouped series with respect to the power series representation of < p, where F(C) about 0. Since this last series converges in some disk p > 1, 't also converges on the set of points C into which a sufficiently small neighborhood of the point z = 1 is mapped. But then, if we shift back to the variable z in (4), we conclude that the series (3) converges throughout a sufficiently small neighborhood of the point z = 1. This completes the proof of the theorem. In a sense, the following theorem is a converse to Theorem 1. Theorem 2. Suppose that a power series f (z) =

F+n=ocnzn

has radius of con-

1) In particular, if the inequality hn+1 > qhn, where q > 1, is satisfied for all n = 1, , then the power series itself can be regarded as grouped. In this case, pi,n, Xn+1(z) = enzxn+ 1. In accordance with Theorem 1, it will therefore converge in a sufficiently small neighborhood of every regular point on I z I = 1. But this last is impossible because a power series cannot converge outside its circle of convergence. This means that, in the present case, the function f(z) cannot be continued beyond the circle of convergence. This is a familiar theorem of Hadamard.

2,

358


vergence 1 and that corresponding to it is a grouped series that converges uniformly in a sufficiently small neighborhood of some point z0 such that I z0I > 1. Then, f(z) can be represented as the sum of a lacunary series and a series with radius of convergence exceeding 1. Proof, It follows from the conditions of the theorem that there exists a neighborhood y contained in I z I > 1 on which the grouped series converges uniformly. This means that there exists a sequence of partial sums snk (z) = L.nk0 CnZn, k= 1, 2, , of the original power series that converges uniformly on A. We may assume that nk+1 > 2nk for k = 1, 2, . Consider the function -

uk(z)=

klogls"k(z)I-log Izl

(k = 1, 2, ...),.

which is harmonic in the z-plane except at the zeros of the polynomial snk (z). Obviously, there exists a positive number Ka such that, for k > K1, the function uk (z) < - q on y, where q is a positive number independent of z and It. Now, let r denote a positive number. Define S = e'"Z , so that S < 1. Since the maximum value of Isnk (z)/z"kI in z I > S is attained for Iz I = S, if we set mS.k = max I-S ISnk (z)I, then, in j z > S and, in particular, in I Z I > 1, we have

Uk(Z)criklogma,k-logs=riklogma,k+ 2 Since

lim ma k = 0

k -.co

it follows that there exists a positive number K2 such that uk (z) < E in I zI > for k > K2. Inequality (2) of §4 can be applied to the domain bounded by the circles I z I = 1 and y and to the closed set E representing the circle Iz I = R, where R > 1, lying inside this domain. (By means of a conformal mapping, inequality (2) of §4 can obviously be carried over to the present case.) As a result, we see that, for k > K = max (K1, K2), we have uk (z) < - A1q + A2E on z I = R, where Al and A2 are positive numbers depending only on R (note that K depends on e). If we choose a sufficiently small that -A1q + A2E = -rl K, we have on the circle z what amounts to the same thing, I

I Snk (z) I

(Re --'1)4k.

By using this inequality, we obtain from the formulas

§6. THE HYPERCONVERGENCE OF POWER SERIES

cm

Snk (z)

('

1

ai

359

dz, m ` nk,

zm

IzI - R

bounds for the coefficients:

IC.I
p(w) and p' (wo) = p(wp). Therefore, in Gro we have A' (w) <X(w) and A* (wo) = A(wo) though only on condition that the function It (A - t) increases in the interval 0 < t < B f. Consequently, when this last condition is satisfied, the metric (5) is a support metric for the metric (4) and it satisfies condition (1). With regard to points at which f '(z) = 0, as was shown above, at such points A(z) = 0 or else the metric ds = A(z)I dz I is regular and hence can, itself, serve as support metric. What has been said leads to the conclusion that if A is such that the function f (A - t) increases in the -interval 0 < t 3Bf, we have in I z I < 1 the inequality ds < da, that is,

V If'(z)I

1

Iz

2

(6)

Since O (A - t) increases in 0 < t < Bf, if we replace p (w) with Bf in (6), we obtain

V_A If'(z)I 0. Let zo denote a point in IzI < 1. Let Azo denote the domain contained in the disk I z I < I zo I, containing the disk I Z I < I zo 12, and bounded by an arc of the circle I Z I = IzoI2 and the arcs of two curves passing through zo and tangent to the circle I Z I = I z012. Then wo = /(zo) belongs to Azo. Proof. In accordance with inequality (3) of §1 as applied to the function f(z) = z-IO(z), the point wo = ,(zo) lies in the disk with boundary defined by 1) This inequality can also be proved if we replace the assumption that the image of the disk j Z I < 1 under the mapping w = F(z) is starlike with the assumption that it is symmetric about the real axis. For more information on this, see Littlewood [19441 .

374


the equation I(w - a1zo)/(zo - alw)I = I zo1, where a1 = 0'(0) and 0 < a1 < 1. If we now take a= al in the interval 0 < a< 1, let us find the envelope of all the disks obtained. These disks are obviously contained in the disk Iw I < I zo I and their centers lie on the segment (0, zp). The equation of the envelope is determined from the equations

F (w, a)=., (log z - aw )= log I zo 1, 0 z°

Fa(w,

C

w - azo

-

w

zo - atm

' =0.

(14)

If we replace the first fraction in the second of these equations with its complex conjugate, the equation takes the form OR

r

1

\ (i-V-ago) (za-aw))

=0 or

tm - azp zo-arw0.

This equation together with the first of equations (14) leads to the conclusion that (w - azo)/(zo - aw) = ± i Izo 1, i.e. w = [(a ±ilzo 1)/(1 ±ialzo I)]zo on the envelope. As a varies in the interval 0 < a < 1, this determines the arcs of the circles passing in one case through the points zo, i Izo Izo, and -zo (corresponding to the values a= 1, 0, and -1) and the other passing through the points

zo, -iIzo Izo, and - zo. It follows that the circles I(w - alzo)/(zo - alw)I = 0 < a 1 < 1, cover the domain ; tzo mentioned in the lemma. This completes the proof of the lemma.

Izo 1,

Theorem 6.1) Let f (z) and F (z) denote two functions defined and regular in the disk I z I < 1 and suppose that F(z) is univalent. Suppose that f(0) =

F(0) = 0, that arg f'(0) = arg F '(0) [this condition is dropped if f'(0) = 0], and that f(z) -< F(z). Finally, suppose that f(z) / F(z). Then,

If(z)I 0, we take 8 in the interval 0 < S < n such that f(t) - f(00)I < E/2 whenever I t - 001 < S. Suppose that 10 - 00I < 8/2. Then, since 2n

c P(r, t-O)dt=1,

(3)

2x a'

we have 2n

u (r, 6) --f (00) = - (f (t) -f &)) P (r, t - e) dt. I

Let us partition the integral on the right into two integrals I1 and I2' the first over he arc (00 - S, 00 + 8) and the second over the remaining arc A of the circleIC I = 1. Noting that P(r, 0) > 0 in the disk ICI < 1, we obtain Ap+b

I1tI=4n S

2n

P(r, t-6)dt_

P(r, t-0)dt= 2.

00-a

Since I t - 01 > 8/2 on A and hence, P(r, t - 0) < P(r, 8/2), we have

1121-

1-ra 1-- 2r cos

2

+ r'

2n S

If(t)I dt.

This shows that I2 0 as r -s 1. Consequently, there exists an rt > 0 such that 1121 < e/2 whenever r > 1 - ri. Consequently, for r > 1 - n and 10 - 001 < 8/2, we have I u (r, 0) - f (00)1 < c, which completes the proof of the theorem. As a consequence of Theorem 1, we obtain the familiar theorem that, if the function f(t) is continuous for all t, then the function u(r, 0) defined on the circle ICI = 1 as the limit of u(r, 0) when the approach is through points inside the disk I CI < 1 is continuous on I C I < 1 and u (l, 0) = f(0). On the other hand, if f (0) is continuous on some arc of the circle I C I = 1, then the function u (r, 0) defined on that arc as the limit of u (r, 0) when the approach is through points inside the disk I C I < 1 will be continuous in the union of the open disk I < 1 and the I

set of interior points of that arc. Theorem 2. If a function f(0) has a finite derivative at 0 = 00, then

au(r, 0)/a0 - f'(00) as t;'= re`' approaches the point e`00 through points in

382

IX. BOUNDARY PROBLEMS FOR ANALYTIC FUNCTIONS

the open disk JCI < 1 along any nontangential path (that is, a path that lies in ie0 some angle with vertex at e that, sufficiently close to eie0, lies entirely inside the disk ICI < I). Proof. Let us first prove the theorem for the case of a function f(0) that is linear in the interval (a, a + 27r): f(0) = A0 + B and that is extended as a periodic function outside that interval. Then, from (1) we obtain

:du(, 0)= 1

2f(t)dp(r, t--0)dt=-

a a

0)

Sexf(t)dP(r,

t

2 a

dt

a

(f (a -}- 21r - 0) -f (a -{- 0)) P (r, a - 0) a+2x

f'(t)P(r, t-0)dO=-AP(r, a.-O)+A.

} 2n a

It follows that, for a < 00 < a + 2n, the derivative au (r, 0)1x0 --. A as rei0 e`00 along any path leading to the point e`60 from inside Icl < 1. Let us turn now to the general case. It follows from (1) that 2x

du (r, 0) d9

1

= 2a

sin (t - 0) f (t) P (r, t - 0) I - 2r2rcos (t - 0)

r'

dt.

(4)

Let us denote by uI(r, 0) the function obtained from (1) if we replace f(t) with the linear function f(00) + (t - 00)f'(00 ) in the interval (a, a + 277), where eai e` e 0 and extended as a periodic function outside that interval. From what we have shown, au, (r, 0)100 - f'(00) as re` a -__, Now, from (4) and the same relationship applied to u 1(r, 0) we obtain, ree`0o.

membering the periodicity of f (o), du (r, 0)

de -

du, (r, 0) de

-

a+ 2x 1

T7C

S

r f (t) -f (00) l t - 0o

- f' (00))

a

(t - 00) sin (t -- 0) X P (r, t - 6) 1-2r2r (5) cos (t - 0) -f- r' d t Let us show that the last factor in the integrand in this equation is bounded for t E (a, a + 277) if rece is sufficiently close to e`0b and lies in an angle with vertex at a `90 and sides making angles `0.0, where 0 < a 0 < n/2, with the tangent to the circle Cl I= 1 at the point a We write this factor as follows:

§1. LIMITING VALUES OF POISSON'S INTEGRAL sin

Q= 2r (t

383

(t

et 00). ret

(6)

0)

°

But j," - re`BI = Ie'('-'9) - rl > Isin(t - 0)I (this last is obtained by replacing the absolute value of the expression as a whole with the absolute value of the imaginary part). Consequently,

2rIt-00I[

F9_'1__ Well

Furthermore, if we assume for the moment that I t - 001 < a.0 and note that x

reie=etB we have

2

se

0, a0 f(r) + (v + 1) e, that is, f(r) v --+ oo. But this is impossible because of the uniform boundedness of the func-

tions f(z). By giving e a sequence of values ev that approach zero and choosing all points for which (2) holds with different ev , we see that fk (x) converges everywhere in (a, b) except possibly at a countable set E of points. The limiting function f (x) is defined and monotonic on the set (a, b) - E. Consequently, at points of the set E it has left- and right-hand limits. If we define the value of the function f(x) at points x in E as the arithmetic mean of these limits, we obtain a function f (x) that is defined and monotonic throughout the interval (a, b). Also, fn (x) -+ f(x) in (a, b) except possibly at countably many points. This completes the proof of the theorem.

Regarding the representability of harmonic functions by means of a PoissonStieltjes integral, we have Theorem 2. ') For a function u(r, 0) that is harmonic in the disk JCJ < 1 to 1)This is true because every function of bounded variation can be represented as the difference of two nondecreasing functions and the total variation of each of these functions does not exceed the total variation of the given function. 2)Plessner [1923].

§2. REPRESENTATION OF HARMONIC FUNCTIONS

387

be representable in that disk by means of the Poisson-Stieltjes integral (7) of §1, where a(t) is a function of bounded variation, it is necessary and sufficient that u(r, 0) belong,to h1. For u(r, 0) to have such a representation in the disk 0 in I[; I < 1. Proof. 1°. If equation (7) holds for u(r, 0) in ICI < 1, by representing the function a(t) as a difference of two nondecreasing functions, we also represent the function u(r, 0) in that disk as a difference of two nonnegative harmonic functions. -Let us denote these by u1(r, 0) and u2(r, 0). Then, for arbitrary r in the interval 0 < r < 1, we have tic

2'

2%

I u (r, 0) I A 0

S u1(r, 0) de -F- S u2 (r, 0) de = 2-n (ut (0) + U2 (0)),.

(3)

0

or

Hers and in what follows, we shall write u(r, 0)I=U = u(0). Then it follows that

u(r,Eh1.

2b. Conversely, let us suppose that u(r, 0) E h1. Consider the function a, (0)401 u(r, t) dt which depends on the parameter r, where 0 < r < 1. Since

Ia,(e) S u(r, t)IdtCM 0

and

k=1

Ok+1

n

n

ar (0k) - Qtr (0k+1) I c

1

h=1

S

tt (r, t) dt 1 c 2ivM,

ek

0-01 K, 2x

u (r, 0)

Zn

Pk''

u (pk, t) pk9 - 2p kr cos (t- 0) -}- r9 A

388


If we integrate by parts, we obtain 27ru (r,

Pk' - r'

e) = aPk (27t) Pks - 2pkr cos 9 -}- r° 2x

q

Pk - l

aPk (t)

!

at pill -26 r cos (t - 9) dt.

0

If we now take the limit as k -+ o and remember that aP,k (2n) = 277u(0) = a(2n), we have 2x

2nu (r, 0) = a (2a) P (r, 0) - a (t) aP (rdt - e) dt. Inverse integration by parts now yields formula (7) of §1. This completes the proof of the first part of the theorem. The second part is proved analogously.

Corollary 1. For a function u(r, 0) to belong to the class hi, it is necessary and sufficient that it have a representation in ICI < 1 in the form of the difference of two nonnegative harmonic functions.

Proof. Suppose that u (r, h) E h I Then, u (r, h) can be represented in ICI < 1 in accordance with formula (7) of §1. Consequently, as was shown above, it can be represented in the form of the difference required. The converse follows from (3). Corollary 2.I If a function u(r, 0) belongs to the class h I, it has definite' limiting values constituting a boundary function u(9) almost everywhere on the circle ICI = 1. This follows from the theorem just proved and Theorem 4 of §1. To establish conditions for representability of harmonic functions by means of Poisson's integral and for many other questions that will concern us later.on, we need to have some auxiliary inequalities, which we shall now prove. We shall use the generally accepted notation. Specifically, we shall write Ax) E L" on E (where p > 0 and E is a linear space) to indicate that I f(x)I" is summable on E. For p = 1, we shall write simply L instead of L I.

Holder's inequality. If g (x) E L" and h (x) E L 9 on E, where p > 1, q > 1,

and 1/p,+ 1/q = 1, then g(x)h(x) CL on E and

Sg(x)h(x)dxI _(SIg(x)IPd.c)h/P(SIh(x)Igdx) '1Q.

(4)

389


Proof. We note that, if a, b, a and ig are all positive and a + 8 = 1, then aa'b,8 < as + 13 b.

(5)

To see this, note that the function 0 (x) = b,8xa - ax - 3b increases in the interval 0 < x o.

Since S6(b) = 0, it follows that O(x) < 0 for 0 < x < b; this proves (5) for a > b.

By reversing the roles of a and b, we can prove (5) in an analogous manner for

a < b. Let us replace a, 8, a and b in (5) with l/p, l/q, ap and bg. Then, for a, b, p and q all positive with 1/p + l/q = 1, we have the inequality

p ap + bq

.ab

(6)

Now, since the functions I g (x)I 1 and I h (x)I q are summable on E, it follows from 'o) that g(x) h(x) is summable. Furthermore, from (6) we have 11

I g (x) I

g(x)IPdx)'IP

I h (x) I

((Ih(x)Igdx)'lq

Y

dx

.+ 1

i g (x) IP

I h (x) I g

dx,

SIg(x)IPdx

E

that is, we obtain inequality (4). It is easy to show that equality holds in (4) only when I g (x) I P = A I h (x)I ' everywhere on E for some constant A.

Minkowski's inequality. If g(x) E LP and h (x) E LP on E, where p > 1, we have E

I g(x)+h(x)I"dx)

(7)

"n

E

E

Proof. By using (4), we have 51 9(x) +h (x) IPdx- S I g(x) I E

I g(x)+h (x)

+Sih(x)I

IP-t dx

g(x)+h(x)IP-'dx

(SIg(x)IPdx)'

(SIg(x)+h(x)Iq(p-1)dx) '1q

+(S I h (x) Ipdx)''P (S i g(x)+h (x) Iq(p-t) dx)11q. E

E


390

Since q(p - 1) = p, this leads after some simplification to inequality (7). In accordance with the remark concerning inequality (4), it follows that equality holds in (7) only when g (x) = kh (x) on E for some constant X. Using Holder's and Minkowski's inequalities, let us establish a test for the possibility of taking the limit under the Lebesgue integral sign. Specifically, If a sequence of functions fn (X)' for n = 1, 2, , defined on E approaches a limit function f(x) almost everywhere on E and if

SIf, (x)1 dxcM, F,

where p > 1 and M is finite and independent of n, then Jim S fn (x) dx = S f (x) dx. n-oo E E

(8)

Proof. In accordance with a familiar lemma of Fatou, we have the inequality

SIf(x)V'dxc lim SIfn.(x)IPdx, n-+00 E

E

from which it follows that f(x) C LP on E. Furthermore, by applying the inequalities that we have proved, we see that, for an arbitrary set e C E, S (f (x) - f (x)) dx l C S I fn (x) - f (x) I dx e

(S I fn

(x) -f (x) IP dx)"P (mes e)li4

(mes e)'/a [(S I fn (x) I P dx)' IP 0

+ (S I f (x) I P dx)'/p11

(mes e) 11q . 2M 1/P,

By making mes e sufficiently small, we can make the right-hand member of this inequality arbitrarily small independently of n. And this, in accordance with a familiar test for the possibility of taking the limit under the Lebesgue integral sign, yields (8). Let us now turn to the representation of functions that are harmonic in 4 < 1 by means of Poisson's integral (1) of §1. We note first of all that not every function u(r, 0) that can be represented in the form (7) of §1 can also be represented in the form (1) of §1. For example, the function


391

u(r, 0)= (1+1) which is harmonic and nonnegative in ICI < 1, by Theorem 2 can be represented in

the form (7) of §1. But its limiting values on the circle It; l = 1 are equal to zero everywhere except at the point C= 1. Therefore, if it could be represented in the form (1) of §1, we would have f(t) = 0 in (0, '2n) in accordance with Theorem 1 of §1 and hence we would have u(r, 0) = 0 in ICI < 1, which is impossible. We do have 1)

Theorem 3. For a function u(r, 0) that is harmonic in the disk ICI < 1 to have a representation in the disk ICI < 1 by means of a Poisson integral (1) of §1, where f(t) E L°, where p > 1, in (0, 2n), it is necessary and sufficient that

u(r,0)Elhp. Proof of the sufficiency. If u (r, 0) E hp, where p > 1, then u(r, 0) E h and, in accord ce with Corollary 2 to Theorem 1, we conclude that u (r, 0) has almost everywhere on the circle ICI = 1 limiting values along all nontangential paths, and in particular along the radii. Furthermore, these limiting values define a function u (0) C L° cause, by Fatou's lemma, 2x

2x S

I u (6) Ip dA - lim S I u (r, 0) Ip d0.

r-1 0

0

Consider Poisson's formula

'-r'p'-2prcos(t-e)--r'dt, r1. §3. The limiting values of analytic functions Using the preceding results regarding harmonic functions, we can easily derive a very simple result regarding the existence of limiting values of analytic functions. Specifically, suppose that a function f(C) is regular and bounded in I C I < 1. Then, its real and imaginary parts represent bounded harmonic functions in I C I < 1 which belong, in particular, to the class h1. In accordance with §2, these functions have definite limiting values almost everywhere on ICI = 1 as C approaches this circle along nontangential paths. Therefore, f(C) also has definite limiting values along nontangential paths almost everywhere on that circle. These limiting values define a function on ICJ = 1, which we shall denote by f(C) or by

f(e`0) where J= e`0. This very simple result can now be generalized to the class N (introduced by R. Nevanlinna) of functions f(C) that are regular in the disk ICI < 1 and that can be represented in that disk in the form of the ratio of two bounded functions:

f(C)=`f ) We can always assume that unity in the disk < I. I

(1)

and fi(C) are bounded in absolute value by

I

Another way of characterizing the class N is on the basis of Nevanlinna's theorem:

394


Theorem 1. For a function f(c) / 0 to belong to the class N, it is necessary and sufficient that the integral 2x

+

log If (rei) I dO

(2)

0

be bounded by some constant M for 0 < r < I.

Proof. 1) If the function f(C) / 0 belongs to the class N, that is, if it can be represented in the form of the ratio (1) with Io(t;)I < 1 and I(C)I < 1 in the disk ICI < 1, then, since If(C)I < 1/Ii/,(C)I in that disk, we have 2x

+

2x

log If (rei°) I A

log 14' (re'°) I d6.

(3)

0

0

Cm+1 + ...,where m> 0, then in accordance with Jensen's formula as applied to the function ,/i(C)/C'", we have, for 0 < r < 1, Now, if ,/i (C) = c.C' + cm

+1

2

2

r logICkl+mlogr.

S log(retB)Id6-log Ic,,, +

0 0 such that I bn (C)I > 1 - e for It; I > 1 - rl, which it is possible for us to do. For 1 - 77 < r < 1, we have 2x

re b(((ee

fP )j

M

dOc(1-r)P

(4)

But since this integral is a nondecreasing function of r in the interval 0 < r < 1, inequality (4) also holds for 0 < r < 1 - 77, that is, throughout the entire interval 0 < r < 1. If we fix r and let n approach oc, we obtain, remembering the arbitrariness of e > 0, the result that, in 0 < r < 1,

404

IX. BOUNDARY PROBLEMS FOR ANALYTIC FUNCTIONS 2x

S Ih(rera)IPdO<M.

0

This proves that h (C) C Hp and, in fact, that the least upper bounds of the integrals (1) for f(C) and h(C) in the interval 0 < r < 1 are equal. This completes the proof of the theorem.

Theorem 2. 2) If f(C) E Hp, where p > 0, we have 2a

2x

im S If(red) Ipd9=S If(e10)19d9

(5)

and 2+c

f(e'e) IP d0 = 0. lim S I f (rere) r_1 0

(6)

Proof. It will be sufficient to assume that f (C) j 0. If we represent f (C) in accordance with formula (2), we conclude from Theorem 1 that the function [h(C)]P"2 C H2. Consequently, by virtue of Theorem (2) of §2, the real and imaginary parts of this function can be represented in ICI < 1 -in terms of their limiting values by means of Poisson's integral. But then, the function [h (J)]P / 2 itself will'have such a representation in ICI < I. 2a

hp/2 (Tet°) = 2n

hP12 (eu) P (r, t -

9) dt.

(7)

In accordance with the Bunjakovskii-Schwarz inequality, this yields 2cc

I h (rele) IP C [S I V12 (e") I P'/2 (r, t - 0).pl/2 (r, t - 0) dt]2 0 2

CSI h(err)I'P(r, t-0)dt, 0

so that 21; S

0

21t

I h(re") I"dO<S I h(e")IPdt. 0

But since If(C)I < Ih(C)I in ICI < 1 and I f(O = I h(C)I almost everywhere on ICI = 1, we obtain from (8), for 0 < r < 1, 1) F. Riesz [1923].

(8)

§4. BOUNDARY PROPERTIES OF FUNCTIONS IN CLASS HP 2x

405

.2%

S If(1eia)I"dO

S

0

0

I f(e')IPdo.

(9)

Therefore, 2w

2w

limy s if (re") I P dA C S If (e") IP dA. 0

(10)

0

[The existence of the limit on the left follows from the monotonicity of the integral (1).]

On the other hand, by applying Fatou's lemma to the integral on the left in (8), we obtain the opposite inequality. Therefore, equation (5) holds. To pr\ ove (6), let c denote any positive number. Then, let us choose 8 > 0, such that, or any set e with mes e < S we have fe I f(eee)I P dO < e in the interval (0, 2a). Sine I f(reee)I -- I f(eie)I almost everywhere in (0, 2a) as r--+ 1, by taking an arbi y sequence of numbers r n < 1 that approaches 1 as n c, we conclude on the basis of Egorov's theorem that there exists in (0, 20 a set el such that mes eI < 8 and I f(rn eoe)I -' If(ei9)I uniformly outside e1 as n Therefore, if we denote the complement to e 1 with respect to (0, 20 by E, we have

3Jim SIf(rneie)IPdA=SIf(eie)IPdA.

(11)

It follows from (5) and (11) that lim S I f n-oo e,

(rnece) Ia

dA= S I f (ete) I P dA. el

(12)

But the right-hand member of (12) is less than E. Therefore, there exists a positive number N such that fe 1 I f (rnefe)I P dO < e for n > N. Now, let us choose N sufficiently great that we also have the inequality I f(rne`0) - f(eie)I P <e on E whenever n > N. Then, since

Ia-bl"C(Ia(+IbI)P=(max(2Iai, 2Ib1))P =2amax(1aIP, IbIP)C2P(IaIP+IbIP), for arbitrary complex numbers a and b, we have 2%

S If (r.ere) -f (e") I P dO c S I f (rnete)

-

f(eie)1P dO

E

+S2°(If(rnere)IP+I.f(ei) I")dO--- 217E+2p2a=(27-, +2P+I) e. el

(12)


406

Since c is an arbitrary positive number, this means that 2%

lim S

n-mp

I f (rnete) -f (e") la d9 = 0.

Also, because of the arbitrariness of the sequence (rn1, equation (6) follows. This completes the proof of the theorem. Theorem 3. 1) For a function f(C) that is regular in ICI < 1 to have a representation in It; I < 1 in terms of Poisson's integral, it is necessary and sufficient that f(t;) E H1. If this condition holds in ICI < 1, we have Poisson's formula 2+i

f(C)=2 f(e")P(r, t-0)dt.

(13)-

Proof. If f (C) can be represented in the disk ICJ < 1 by the formula 2x

f(C)=2 p (t)P(r, t-0)dt, where 9(t) E L for t E (0, 27r), then it easily follows that, for 0 < r < 1, 2x

2x

S If(reta)Ide<S I p(t)I dt, 0

0

that is f(t) E H1 Conversely, if f (C) E H I, then, for. 0 < r p, then f(C) E H Proof. If f(C) C HP, where p > 0, then, in the representation (2), we have h(C) C HP. Consequently, Poisson's formula (13) 4.

2s

h9

h"(e")P(r, t-0)dt

T1-

S 0

is applicable in I I < 1 to the function [h()]" C H I. Since I f (0I < I h (C)I in ICI < 1 and I f(e`0)I = I h(e`0)I almost everywhere on ICI = 1, this formula leads to the inequality 2n

If(C)IP2- If(e")IPP(r, t-0)dt. Therefore, if If(e`0)I < M almost everywhere on ICI = 1, it follows that < 1. On the other hand, if f (e `B) C L Q, where q > p, then, by I f (C) I < M in using Holder's inequality, we have 2x

9-P

If(C)IP 0,

s' (T) dT,

and since h

S (T + h) - 3 (T) dc =--

1

t

s (T) dT -

s (T) dT

a

=h [

t

A

a h

t+ h

a

t

s(T)dTs(r)dT, mes G (k), for k 1, 2, define G t= U k 1 GC so that mes G C = mes E,. Consequently, the set N = E - G t is a set of measure zero on ICI = 1, and we have therefore represented E as the union of the set N of measure 0 and the closed sets G (k), for k = 1, 2, But corresponding to the set N t is the set NZ of measure zero on C and, obviously, corresponding to the sets Gk) are the closed sets G=k) on C. Consequently, the set E. corresponding on C to the set E, is the set NZ U Uk 1Gik), which is a measurable set. Its measure is positive because otherwise, E t would, from what we have C

proved, be of measure zero. In an analogous way, we can show that corresponding

to a set of positive measure on C is a measurable set of positive measure on ICI = 1. This completes the proof of the theorem. Remark. It follows from part 2 of the proof of Theorem 2 that a function _ ).(z) that maps the domain B onto the disk JC1 < 1 is absolutely continuous on

422

X. FUNCTIONS ANALYTIC INSIDE A RECTIFIABLE CONTOUR

C if we regard it as a function of arclength on C because the inequality Elsk - s kI < c implies I I X (z (Si)) - X (z (S;)) I C i7 (e). 11

Theorem 3. The mapping z = 0)(C) in Theorem 1 is conformal almo t everywhere on ICI = 1 or, respectively, conformal almost everywhere on C.

Proof. Obviously, we can consider the conformality of the mapping only at points on ICS = 1 in the one case and on C in the other. Since the derivative dw(e`t)/dt exists and is equal to zero almost everywhere on the circle CI = 1, there exists almost everywhere on C a unique tangent such that in a neighborhood of a point of tangency, the curve C lies on both sides of the normal to C at that point. Let us show that angles are preserved at all points on ICI = 1 that are mapped into points of the curve C with this property. Let 4U denote one of these points on ICI = 1 and let zo denote the point on C corresponding to it. Then, every branch of the, function %

u (C,

CO) = arg

M - (to) C-Co

is continuous in the closed disk ICI < 1 except at the point Co. But since the curve C has a tangent with the property described at the point zo, the function u (C, CO) approaches a definite limit a as C approaches Co along the circle ICI = 1 (this value being the same for approach from the two sides). We take this limit as the value of is (C, C.) at the point Co. Obviously, the oscillation of the function is (C, CO) in the portion of the disk ICI < 1 contained in a sufficiently small neighborhood of the point Co is less than 3n. This shows that the function u(C, CO) is a bounded harmonic function in ICI < 1 and it can be represented in that disk in terms of its values on the circle ICI = 1 in accordance with Poisson's formula. These values define a continuous function on ICI = 1. Consequently, the function u (C, CO) is continuous on the closed disk ICI < 1 including the point C., where it is equal to a. Now, consider an arbitrary curve A lying in the disk ICI < 1 with endpoint Co, and let l denote the corresponding curve in the domain B. Since u (C, CO) -- a as C, t through values belonging to the disk ICI < 1, if the curve A has a definite tangent at the point Co, the curve l will have a definite tangent at the


423

point z0 and vice versa. Also, if these curves have tangents at the points indicated, a will be the angle of rotation of the tangent in the transition from the curve A to the curve 1. Since this angle of rotation is the same for all pairs of curves A and 1, this is equivalent to conservation of angles at the point Co in the shift from the disk ICI < 1 to the closed domain 8 or the other way around. This completes the proof of the theorem. Let us now look at the question of the correspondence of boundaries under conformal mapping of domains bounded by smooth Jordan curves. By a smooth Jordan curve C: z = z(t), where a < t < b, we mean a curve possessing at every

point a tangent that rotates continuously as z moves along C, that is, a curve for which the angle 0(z) of inclination of its tangent to the real axis is a continuous function of the point of tangency. In the case of a closed curve, we must also have 0(z(b)) = 0(z(a)) + 21r. Obviously, a smooth curve is a rectifiable curve. Therefore, 0(z) can also be regarded as a function of the arclength s on C: 0 = 0(s). We now have

Theorem 4.1) Let z = a) (4) denote a function that is regular in the disk < 1 and that maps that disk univalently onto a domain B bounded by a smooth closed Jordan curve C. Let L'= a. (z) denote its inverse. Then arg &)%) is a continuous function in the closed disk ICI < 1 and arg a.'(z) is a continuous function in the closed domain B. Furthermore, on the circle ICI = 1,

argni (()=9(Q-argC- 2.

(4)

Proof. As a preliminary, we point out that, by virtue of the smoothness of the curve C, for any point a E C and any number t > 0, there exists an arc on C containing the point a as an interior point such that the angle of inclination of the chord connecting any two points of that arc to the real axis differs from the angle of inclination of the tangent at the point a by an amount less than t. This follows from the fact that the angle of inclination of the chord is equal to the angle of inclination of the tangent at some interior point of the arc intercepted by that chord.

Now let us assume that the condition of the theorem is satisfied and let us consider the function 1) Lindelof [1916].


424

u (G i) = arg

Celt

-C

(C) ,

which is, for arbitrary fixed r> 0, a harmonic function in the closed disk Icl < 1. We assert that there exists a finite constant K such that lu (C, r)I < K for all C in the closed disk ICI < 1 and all r in the interval 0 < r < n. If this were not the case, there would be a sequence for n = 1, 2, , such that Mn = as n -, oc. Obviously, r, -4 0 as n --, oc. Furthermore, max 1 CIS1 lu (C, we can, by shifting to a subsequence of the arcs r, if necessary, arrange to have m converge to a the sequence 1 ,1 of points on i C 1= 1 at which lu (t,,, point Co on the circle ICI = 1. But because of the smoothness of C, the sequence converges to the angle formed by the tangent to C at the of the values is ((, point zo = w(Co) and the tangent to the circle I = 1 at the point Co. This contradiction proves the assertion made.

Now let a= e4 denote an arbitrary point on the circle I = 1 and let 0 denote the angle between the tangent to C at the point a = w(a) and the tangent to the circle I C I = 1 at the point a. Then, for given e > 0, there exist q > 0 and S > 0 such that, for Irl < 8, we have

lu(C T)-?I< a on an arc of length q containing points on each side of a. Let us show that this inequality also holds in that portion A of some neighborhood of the point a that lies in the disk C1 < I. If we set U r 0 and I k I< 1 in accordance with Poisson's r) formula 2n

U (G

r) = 2L V

= 2n

U (e "t, r) P (r,

t - B) dt

a

U(e", 2) P (r, t - 9) dt + 2n

U(e", r) P (r, t - 6) dt,

where C= reie and I is the portion of the circle ICI = 1 complementary to the arc (>/i - q , t/i + t?). Therefore, U (C,

r) l

P (r, t - O) dt +

47C

4-17


+X+

TI

S P(r, t-6)dt- 0 such that the inequality IC- aI < rll, where ICI < 1, implies I NC, r)l < t, that is Iu (c., r) - 01 < t for all r such that Irl < S. Then, for CC A, where A is the portion of a neighborhood of the point a mentioned above, we have the limit relation u (C, r) ..._, arg to '(c) as

r -- 0. Therefore in A,

Iargw (C)-pI<e. This shows that arg a)' (C) is continuous in the closed disk ICI < 1 if we define this function at the points a such that IaI = 1 as being equal to the angle that we have been considering.

The continuity of arg X'(z) in 9 now follows from the fact that X'(z) _ 1/w'(c). Since 0 = B(s) - Vi - n/2, formula (4) also follows from what we have shown. This completes the proof of the theorem. Theorem 5. Under the conditions of Theorem 4, we have log w%) C H1 and

w'(0 C Hp for all p > 0. Proof. From the inequality between the arithmetic and geometric means, we

have, for 0 < r < 1, 2%

I n

log I w' (re1t) I dt - lim t 2n 1 b n.oo n

zni

log l w' (re n k) I 1

n

lim log n-ao

n

)I

k- 1 1

imo log

d k

I w (re n

n

n

2at 1

k

I w' (Ten

2cn

log

2a } I w (re`t) I dt.

(4 ')

Consequently, in accordance with Theorem 1, the first interval is bounded in the interval 0 < r < 1. But since the integral 2n S

0

I arg w' (re`°) I d6


426

is, by Theorem 4, bounded in the interval 0 < r < 1, it follows that log co' (i) E H I. The second assertion of the theorem follows from Theorem 7 of § 5, Chapter IX asapplied to the function u(p, 0) = arg co'(reè), which, in accordance with Theorem This completes the proof of the 4, is continuous in the closed disk theorem.

Theorem 6 (Kellogg). Let z = to(G) denote a function that is regular in the disk JAI < 1 and maps that disk univalently onto a domain B bounded by a smooth closed Jordan curve C. Suppose that the angle of inclination 0(s) of the tangent to C as a function of the arclength s on C satisfies the Lipschitz condition Ja,

O 1. Then cj"(C) E Hp. Proof. From the condition of the theorem, we find, by using H61der's inequality s

le(s)-0(s)I=S0'(s)ds 0, we can choose rk such that

If (z'(so))-rkI< a . For a sufficiently small, e

a

If (z' (so + o)) -f (z' (so)) I da o

I f (z' (so + o)) - rk I do.

o

Pf

+c

If(z'(so))-rkIds< 2 + 2 =s.

But this means that for so outside E,

0 as a-.0.

(a) G

Under the assumption that we have made, let us find a bound for each of the integrals in (6). We have (a) do

Rr=l J

a

(1+m)a-eiet

I

'4 (a)

do

C SeI (I+ m) a-ciel0'('

or, replacing the modulus in the denominator with the imaginary part,

Rte

cos+o -1

a

Scos4+o

_e

2 mx

()I.

I 0, the class of all functions f (z) J 0 that are regular in B and have the property that the integral

S If (z) I°ds cr

E

(1)

is bounded for 0 < r < 1 (V. I. Smirnov). By changing to the variable C in this

integral, ;e asil. see that the function f (z) belongs to the class EP if and only '(t;) E Hp. Thus, by § 4 of Chapter IX and § 1 of the present chapter, if the function f (z) belongs to EP, then f (z) has definite limiting values f (z ') almost everywhere on C, over all nontangential paths; If (z ')I is summable on C; and if f (co

limy S I}(z)Ipds` If(z')IPds. Cr

(2)

C

The class EP for p > p, can also be defined in this way, but it can also be defined independently of conformal mapping, namely, as the class of functions f (z) that are regular in B and have the property that, for each of them, there exists in B a sequence of rectifiable closed Jordan curves Cn, for n = 1, 2, , which converge to C and for which the integrals

S If(z)Ids C.

are bounded by a finite quantity independent of n. 1) Obviously, every function f (z) E p satisfies this condition. Let us prove the converse. Suppose that, for

a function f (z) that is regular in B, there exists a sequence of curves Cn with the property described. Let z = con(C )denote a function that maps the disk ICI < 1 conformally onto a finite domain bounded by the curve Cn in such a way arg w' (0). Then, by the theorem on the conthat con (p) = w (0) and arg vergence of univalent functions, the sequence of the functions con(C), n = 1, 29 , 1) Keldys and Lavrent rev [1937].

§5. CLASSES OF FUNCTIONS. CAUCHY'S FORMULA

439

converges uniformly in the interior of the disk ICI < 1 to the function w (C). Let

us set ,P (C) =f (w (C))p w' (C), T. (0 = f (w (C)) ° W. (C).

Since, for every n, the function f (wn(C)) is bounded in ICI < 1, it follows that On(:) E HP. Therefore, for 0 8/2, we have

If(z)I c e If(z')Ids and, consequently,

9n (z) Ip C xb 5 I q,, (z') 1-0 ds

0, there exists a polynomial P(z) such that 5

I F (z) - P (z) Is ds < e.

C

Theorem 3. For the family of all polynomials in z to be complete in a domain

B, it is necessary and sufficient that the curve C satisfy condition (S). Proof of the sufficiency. Suppose that condition (S) is satisfied. Then, in accordance with Theorem 2, we have equation (3). Now,' if F(z) E E2, then E H2 and, consequently, in accordance with Theorem 2 of f (C) = F (W (C)) W § 4 of Chapter IX, 2x

iii or

5

I f (ete) - f (rete) J2 d6 = 0

r lim S r-1 C IF (z) -f (rX (z))

I s ds -- 0.

This means that, for given e > 0, there exists an r in 0 < r < 1 such that

5IF(z)-f(rX(z)) C

X'(,&)I9ds<e.


450

But the function f(r)((z)) is continuous on B. Consequently, in accordance with Walsh's theorem, l) there exists a polynomial Q(z) such that, in B, If (rx (z)) X' (0) - Q (z) I < e. Let us form the polynomials Pn(z)= Q(Z)P". 2(Z)' We have

I F(z)-Pn(z)I$dsCs(IF(z)-f(rX(z))Y X ()I _C

C

p ..

+ If (rX (z)) (Y X (z)

, , 9 (z) "

iC (0)) I

+ I Pn. 2 (z) (f (rX (z)) Yx' (0) - Q (z)) I )gds. Of

Consequently, by applying Minkowski's inequality, we obtain

(IF(z)_Pn(z)I2ds )1/9 _(SIF(z)-f(rX(z)) VX7 (z) I'ds)

1/2

C

+ max If (rX (z)) I zFB

(S IF2(z)-Pn.g(z)I'ds)1/9 C

+ e \ CS I Pn, g (z) 11 ds)19 Here, the right-hand member is, for sufficiently. large n, less that 2f. Since e is an arbitrary positive number, this proves the sufficiency of the condition of the theorem.

Proof of the necessity. Suppose that the system of polynomials is complete for the domain B. Then, for the function f7(z) E E2, there exists a sequence of polynomials Pn(z) such that lim

(z) - Pn (z) I2 ds = O.

JI

n-00 C

(4)

If we now apply Minkowski's inequality twice, we have 5 I Pn (z) I' ds)1/' = (S I P. (z) C

(5 I x '(z) I ds)1I9

Ig d$)1'9 + (S I x '(Z) I ds)1'9 ,

C

C

X' (z) - Pn (z)12 ds

(5 I

(S I P,, (z) I2

which, in accordance with (4), yields lim 1) Cf. Walsh [1960], p. 47.

IPn(z)Igds=2Tr

C

(5)

§7. APPROXIMATION IN MEAN

451

On the other hand, we obtain from (4), by use of Cauchy's formula, the result that

litn P.(z)=

n-.oo

in B. 1) In particular, limn-+cPn(0)=

Therefore, if we set pn(z) = Pn(z)/Pn(0), pn(0) = 1, we have, in

accordance

with (5), n

i oo

S I P,, (z) I9 ds = 21rw (0).

In accordance with §6, this yields d(o) = &)'(0); that is, condition (S) is satisfied. This completes the proof of the theorem. The results that we have obtained are closely connected with the theory of orthogonal polynomials for a domain B. By a system of orthogonal polynomials, we mean here a sequence of polynomials for n = 0, 1, - , such that, for each n, K,,(z) is of degree n and

SKm(z)Kn(z)ds=

1,

if

01

if

m=n,

(6)

min.

Obviously, an arbitrary polynomial in z can be represented as a linear combination of the polynomials Now, if F(z) E E1, then the numbers

cn=SF(z)K,,(z)ds, n=0, 1, ...,

(7)

are called the Fourier coefficients of the functions F(z). Theorem 4. For the family of all polynomials in z to be complete in a domain B, it is necessary and sufficient that an arbitrary function F(z) in E2 satisfy the closedness condition

F(z)11ds= C

k=1

Ck

where the ck are the Fourier coefficients for F(z). Proof. Let p(z) denote an arbitrary nth-degree polynomial and let 1) See proof of Theorem 6.

(8)

use

452


represent it in the form P (z) _

k=0

r)

c Kk (z).

(8

For f(z)EE2,

SIF(x)-P(x)I9ds= SIF()12ds+SIP(z)I'ds C

- 2,9R

F (x) P (x) ds) = S I F (z) l i ds + C

- 2(Ek-0

n

CkCk)=SlF(z)I9ds- E ICkI9+

k=0

C

I ck I9

k=0

C

E

ICk-C;12.

k-0

It follows that the minimum for the integral (9)

I F (x) -P (x) I e dt C

yields a polynomial p(z) determined in accordance with formula (8) with ck = e k. Here, this minimum is equal to

JF(z)12ds-

IckI'.

k=0

(10)

For the system of all polynomials in z to be complete, it is obviously necessary and sufficient that the difference (10) approach 0 as n oo for an arbitrary function F(z) E E2, that is, it. is necessary and sufficient that equation (8) hold. This completes the proof of the theorem. By combining Theorem 4 with Theorem 3, we immediately obtain Theorem 5. For an arbitrary (unction f (z) E E2 to satisfy the closedness formula, it is necessary and sufficient that C satisfy condition (S). We now have

Theorem 6. When condition (S) is satisfied, an arbitrary function F(z) E E2 can be expanded in B in a Fourier series 00

F (x)

ckKk (z),

k=0

ck = S F (z) Kk (z) ds, C

§7 APPROXIMATION IN MEAN

453

that converges uniformly inside B.

Proof. Since the minimum of the integral (9) out of all polynomials in z yields a polynomial defined in accordance with formula (8) with ck = ck, for k = 0, 1, , n, it follows that, for these polynomials pn(z), lim

IF(z)-Pn(z)I9ds=0,

(12)

if. condition (S) is satisfied. Now, by Cauchy's formula, we have in B I

F (x) -Pn (z) = 2z1

F (z') -Pn (z') dz .

z' - z

If E is a closed set in the domain B and if S is the distance from E to C, then, for

I F (z) -Pn (z) I

I F (z') -Pn (z) I ds 2sg

F(z)-'Pn(z)I

C

From this.we conclude in accordance with (12) that pn(z) - F(z) uniformly on E. Since pn(z) is a segment of the Fourier series (11), this proves the theorem. In particular, if we apply Theorem 6 to the function f (z) = X '(z) and keep in mind the fact that then 2x

c,=S YT X (z)Kn(z) ds=SK.(w(ete))

dO

2rrKn (0) ylw (0),

we obtain

'Theorem 7. For curves C satisfying condition (S), we have in B the formula LJ Wk (0) Kk (z))

X'(z)=2a

k

(13)

2' 1K, (0) 1.2

k=0

where the convergence of the series in the numerator is uniform inside B.

CHAPTER XI SOME SUPPLEMENTARY INFORMATION

§ 1. Gluing theorems Gluing theorems establish the existence of analytic functions that obey certain relationships on the boundaries of certain domains. The nature of these

relationships is clear from the statements of the theorems we now present. Theorem 1. Suppose that a function x'= g(x) sets up a continuous one-to-one mapping of the interval 1: - 1 < x < 1 into itself, mapping each endpoint onto itself. Suppose also that, as a function of a complex argument, g (z) is regular and univalent in a sufficiently narrow circular tune with vertices at the points ± 1 and containing the interval 1. Under these conditions, there exist two functions w = f 1(z) and w = f2(z) that map the semidisks B 1: IzI < 1, (z) > 0, and B2: Izl < 1, Nz) < 0, respectively onto two disjoint domains G1 and G2 obtained from the disk JwJ < 1 by making it a smooth cut A that is analytic at all interior points. 1) Furthermore, the mappings are such that, on 1,

A (x) =fs (g (x)), that is, such that any two points x, g(x) E I that lie on the boundaries of the domains 81 and B2 are mapped into the same point X. Proof. It follows from the conditions of the theorem that the function z g(z) is regular and univalent in a sufficiently narrow segment 8(1) bounded by the interval l and the circular arc contained in ` (z) > 0 and that, on this segment, `(G(z)) > 0. We can take the segment B(1) in such a way that the values of the interior angles at the vertices are equal to n/2n, where n is an integer. Let B(2) denote the segment symmetric to B(1) about the x-axis (see Figure 17). In B11) we consider the function w = g1(z) = g(z), and in B12> the function w = 92W = z. For arbitrary x E 1, these functions map the domains B") and 1) That is, such that any interior arc of it is an analytic curve. 454

§ 1. GLUING THEOREMS

455

B(2) in such a way that the points x and g(x) in the interval l are mapped into the same point w of that interval (see Figure 18). Let us now map the univalent domain made up of the images G(1) and C(2) of the domains B(1) and B(2) under these mappings and of the interval l univalently onto the lune Bc' U B(2) U 1 in the w '-plane in such a way that the points ± 1 are mapped into themselves. Under this mapping, the domains G(1) and G(2) are mapped into domains adjacent to some smooth curve Al that is analytic at all interior points (see Figure 19). If a funcFigure 17 tion w'= f (w) provides this mapping, then the function w '= f (gk(z)) = gk +2(z), k = 1, 2, will provide mappings of the domains B(1) and 8(2) onto these same adjacent domains. In accordance with the symmetry principle, the functions w'= gk+2(z), for k = 1, 2, also map the

Figure 18

Figure 19

"double segments" B(3) and BW bounded by the interval I and the circular arcs inclined to l at the angle rr/2"-1 in an analogous manner. Let G'3) and G(4) denote the images under these mappings and then let us map the domain C(3) U G4 )U kl univalently onto the lune B(3) u B4 )U l in the w "-plane in such a way that the points ± 1 are mapped into themselves. Under this mapping, the domains G(3) and G(4) are mapped into domains that are tangent along some smooth curve A2 that is analytic at all interior points. By applying the symmetry principle n - 1 times as indicated, we obtain the mappings mentioned in the

XI. SOME SUPPLEMENTARY INFORMATION

456

theorem. This completes the proof of the theorem. Theorem 1 is a modification of a gluing theorem of Lavrent 'ev. 1) By using

Lavrent'ev's theory of quasiconformal mappings, Volkovyski%Z) succeeded in obtaining several other gluing theorems that play an important role in the theory of Riemann surfaces. In these theorems, the requirements imposed on functions of the type g(x) are weaker. Another procedure for obtaining gluing theorems is based on the use of the boundary properties of analytic functions that were obtained in Chapters IX and X. Specifically, let us prove Theorem 2.3) Let a and b denote two distinct points on the circle jzj and let yl denote one of the arcs connecting them. Let g(z) denote a function defined on yl that possesses the following properties: 1) g(z) is regular and has nonzero derivative at all interior points of the arc yl; 2) the function z '= g(z) maps the arc yl bijectively onto the complementary arc Y2 on the circle Izl = 1 and maps the endpoints into themselves. Under these conditions, there exists a function

W=F(z)= z +alz+ ...,

(1)

that is regular in the closed disk jzj < 1 except at the points 0, a, and b and that satisfies the equation4)

F(z)=F (g (z))

(2)

at all interior points of the arc yl. Proof. Consider the family IR of functions F(z) of the form

i

F(z)= +z+f(z) where f (z) E H2 and f (0) = 0 (with regard to the class H21 cf. § 4 of Chapter IX) that satisfy almost everywhere on yl the relationship f (F(g (z))) = at (F (z)).

(3)

1) Lavrent'ev [1935] 2) VolkovyskiT [1946] 3) Schaeffer and Spencer [1947] 4) In a monograph by the same authors [1950] the existence of a function F(z) that is univalent in Iz) < 1 is proved.

§ 1. GLUING THEOREMS

457

An example of such a function is the function

F(z)= _1_Z. We pose the extremal problem: Out of all functions F(z) E 9, find the one minimizing the integral (4)

IF =S iZ S If (z) I9 ds. 1

Let m denote the greatest lower bound of the integrals (4) in the family P. Let us show that, for some function F(z) E 9 , we have IF = m. Let us suppose the opposite. Then, there exists a sequence of functions F,,(z) = z-1 + z + for n = 1, 2, , such that IF. m as n - - . Let us define CD

fn (z) _ E Ckn1,Zk. k=1

Then, 00 (nl 9

(") 9

l-.1k=1

r 9k

n, v=1, 2, ...

Therefore, the functions fn(z) are uniformly bounded inside the disk Izl < 1 and

the condensation principle is applicable to them. Let } f,,k(z)} denote a subsequence of functions that converges uniformly inside the disk Izl < 1 to the function fo(z). Obviously, fo(z) E Hz. Furthermore, in accordance with Minkowski's inequality, we have, for

0 0 and B2: Iz I < 1, 'Az) < 0 under suitable meromorphic functions w = g1(z) (see Figures 20 and 21) G

Figure 20 Figure 21 Figure 22 and w = c2(z) in such a way that both mappings map the arc 1 common boundary diameter. Obviously, the function g(x) = 024-1 1W), satisfies the conditions of Theorem 1 of § 1. On the basis of this theorem, there exist functions z '= Vi 1(z) and z '= 02(z) that map B 1 and B2 univalently onto two complemen-

tary domains G1 and G2 of the disk Iz 'I < 1 (see Figure 22) in such a way that the points x and g(x) (for - 1 < x < 1) on the boundaries of B 1 and B 2 are mapped into the same point of the disk Iz 'I < 1. But then the functions


464

w = 01(0 i 1(z)) and w = map the domains G1 and G2 onto Fn and Ak respectively in such a way that the common portions of the boundaries of these functions coincide. In accordance with Riemann's theorem on analytic continua-

tion, we conclude that both these functions define a meromorphic function w = F(z) on the disk IzI < 1 that maps that disk bijectively onto the surface F + 1. By a suitable normalization, we easily obtain the function w + 1(z), fn + 1(0) wo, f + 1(0) = 1, which yields such a mapping of the disk IzI < r + 1, where rn+1 < oo on Fn+1.

2. Let us now consider the case when the surface to Fn, of two triangles Ak and

F

Ak' contained in Fn. If a and b are the endpoints of the arc 1 and a lies in Fn + 1' we remove from the surface Fn a sufficiently small one-sheeted or multiple-sheeted disk K: Iw - al < E constituting a neighborhood of the point a on Fn + 1 but cut along 1. The surface Fn with points belonging to K deleted from it and the disk K, with a sufficiently small arc VC 1 with endpoint at a adjoined can obviously be regarded as the single-valued images of the semidisks B1: IzI < 1, %z) > 0 and B2: IzI < 1, %z) < 0 (see Figure 23) under meromorphic

Figure 23 functions. Here, both functions map the common boundary diameter onto the portions of the boundaries lying over the circle Iw - al = E. Therefore, reasoning

as in case 1, we can show that the surface Fn U 1' is the image of a finite disk under a suitable meromorphic function. When we note that the surface Fn U 1' is obtained from Fn by shortening of the cut 1, we continue this shortening process indefinitely so that as a result we exhaust the entire cut 1. If we now use the

§2. MAPPING OF SIMPLY CONNECTED RIEMANN SURFACES

convergence of the corresponding normalized functions w =

465

where

which was proved above for the surfaces F , we conclude that the surface F,, + 1 is the image themselves, where n = 1, 2, of the disk Izi n - 1), then the functions 61 = F1(z i , , for 1= 0, , n - 1, would map a neighborhood of the point (x1, , xvm) in 2m-dimensional Euclidean space onto an open set in n-dimensional space containing the point (0, 0, , 0). The intersection of this set with the S0-axis (61 = 62 = . '= 6n-1 = 0) is an open segment containing the point (0, 0, , 0). Consequently, the function F0(z 1 , ... , z,) would not be minimized when z k = zk, k = 1, , m. From what we have shown, we conclude that there exist real numbers A0, X1, , an-1, not all equal to 0, such that the 2m equations hold:

as =0, k=1, ..., 2m. kt

(5)

470


Keeping (2) and (3) in mind, we conclude from (5) that, for the values Ak mentioned above, the partial derivatives with respect to x and y (where z = x + iy) of the function

- Agg(z, a)

vanish at z = zk, for k = r,

ill (z) T ... -1-

(z)

, m. This function is obviously not a constant

since the Ak are not all zero. Let us define analytic functions p(z, C) and wk(z), k = 1,

, n, by

8 (z, C), = at (P (z, C)), wk (z) _ M (Wk (z))

(see §6, Chapter VI) and let us define n-1 w (.z) _ - Aop (z, a) + 1=1

A1w1(z).

From the latter equation, we conclude that the derivative w '(z) is not identically zero but that it has zeros at all the points z 1, , Z M. Since the real part of the function w(z) is constant on each boundary curve K1, for l = 1, , n, in the case in which w'(z) has no zeros on K, we obtain the result n

(arg

(z))K

t=1

(arg

(z))K,

n

[(arg

)K1- (arg

W

--

_

2a (n - 2),

where ds is the element of arc length on the boundary K. We note that the.function w' (z) is regular at a when A0 = 0 and we note that w '(z) has a simple pole z = a in B when A0 0. Therefore, we conclude from Cauchy's theorem on the number of zeros and poles that the number of zeros of the function w'(z) in B distinct from z = a does not exceed n - 2 if A0 = 0 and is exactly equal to n - 1 if A0 0. This result will also hold when w ' (z) has zeros on K, as follows from the same theorem of Cauchy but this time with consideration of the zeros on

§3. EXTREMUM FOR BOUNDED FUNCTIONS

471

the boundary. Here, the zeros on K are counted with half multiplicity. I) This result leads to the first important conclusion regarding the extremal function fp(z). The zeros of the function fo(z) in the domain B are also zeros of the function w ' W. The number of these zeros does not exceed n - 1. If there are exactly n - I of them, then A0 i 0.2) , zm, m < n - 1, constitute all the zeros of the Now suppose that z 1, function fo(z) inside B. Then, the function

U(z)=log Ifo (z) I +g(z, a) +

k-l

g(z, Zk)

(6)

is harmonic in B and U(z) < 0 in B. Since this last inequality is satisfied in the part of the domain B contained in a small neighborhood of an arbitrary point of the boundary K, if we shift to the disk by means.of a univalent mapping and use the boundary properties of nonnegative harmonic functions in a disk (see §2 of Chapter IX), we conclude that U(z) has definite limiting values almost everywhere on K along nontangential paths and these values define a summable function on K. Now, let f2(z) denote the varied function obtained from the relationship log I fs (z) I =log I fo (z) I + u (z),

(7)

where u(z) is a harmonic function in B. To make f2(z) single-valued in B, we require that the conjugate v(z) to u(z) be single-valued in B. For If2(z)I not to exceed 1 in B, it will be sufficient to require that u(z) be bounded in B and that U(z) + u(z) be nonpositive almost everywhere on K. This is true because, in the first place, by virtue of (6), and (7), m

If2(z)I=e U(z)+u (z) e

-B (z, a) - E 8 (z. Zk) k=1

I.fo (z) I

(8)

in B. Furthermore, under the assumptions made, u(z) also has definite limiting 1) For a direct derivation of this theorem for the function w '(z), consider neighborhoods not only of its zeros lying in B but also of its zeros lying on K. Then Cauchy's theorem can be applied to the logarithmic derivative of w'(z). If we then let the neighborhoods of the zeros shrink to a point, the difference between the number of zeros and the number of poles of the function w '(z) in B will be equal to (argw'(z))K. This last expression means the sum of the increments in argw '(z) over the set of arcs constituting K and not encircling the zeros of w '(z). 2) (A more complete study of the zeros of the function fo(z) has been made by Garabedian [19491)

472


values almost everywhere on K. From Green's formula

0(z)+u(z)-2a J

z)ds,

where K. is the set of curves g(z ', z) = e (where a is a sufficiently small positive number), we obtain Green's formula for the domain B 2a J

U(z)+u(z)-

(U(z) 'u(z))

dg (z', z) ds

(9)

by taking the limit as c --p 0. (As always, the normal n is directed inside the domain B.) It follows that U(z) = u(z) < 0 in B. Therefore, in accordance with (8), Ifz(z)I < I in B. Thus, f2(z) E IR. If we rewrite (7) in the form log ff (z)

z-a

I

= log

fo (z)

I z-a

I + u (z),

and let z approach a, we obtain log'I f9 (a) I = log I fo (a) I + u (a).

Then, by virtue of the extremaliry of the function f o(z), we have u(a) < 0. By using Green's formula (as derived above), we see that this leads to the inequality

u(a)= 2n c u(z)

dg (z', a

(10)

Furthermore, if we denote by KE.1,1 a curve in B that is close to Kt and defined by the equation i1(z) = 1 - e, where a is a sufficiently small positive number, and if we denote by KE,I,k, where k 1, a curve in B close to Kk and defined by the equation &)I(z) = e with the same e, then the single-valuedness of v(z) in B can be expressed in the form of n - 1 equations (for sufficiently small e): (v (Z)) K,.

dv (z) _ - S

1, 1= S K., 1,1

k

K 1.l (m1(z)-e)

s

I

Kr11, k

do

do

ds = -

ds+2a

Td u

S

(w1 (z) + e)

K+ 1,1

S

duds=

n

ds

§3. EXTREMUM FOR BOUNDED FUNCTIONS-

u

S

n

(z) !L do z) ds + 2e

do

s

473

ds = 0,

K,0 1, 1

UK,

1, k

kal

1=1, ..., n-1. Here, Fo is a fixed small positive number. This is equivalent to satisfaction of the n - 1 limiting (as E -- 0) equations

u(z)d!-

ds=0, I=i, ..., n-'1.

Turning now to the function (6), let us suppose that, for some q > 0, there exists a subset E of K of positive measure such that U(z) < - q for z E E. Then, for u(z) we may -take the function defined by the boundary values

U (z)

-

(0 on K outside E n

(

I

s ll- °

dg (z, a)

d'k`

do + k=1 k^ do J

on E,

where the Ak, for k = 1, ... , n - 1, are constants and F is a constant of either sign but sufficiently small absolute value. (To construct u(z), we need only shift from the domain B to the disk ICI < 1, as was done in § 3 of Chapter VI, then use Poisson's formula to construct a harmonic function on that disk from the new boundary values, and finally return to the domain B with it.) For this function u(z), conditions (11) take the forms n-1 dg dm1

do do ds +

kk

k=1

c duk 1

do . do ds = 0'

(12)

CCC

1=1,...,n-1, and, by virtue of the choice of the Ak, for k = 0, 1, , n - 1, they must be satisfied. But conditions (12) constitute a system of n - 1 homogeneous linear equations with n unknowns Ak, k = 0, 1, , n - 1. Therefore, this system has nonzero solutions. Let us now denote by Ak, k 0,...,n - 1, any nonzero system of solutions and let us construct from them the function u(z) in (10). We then obtain an inequality in which the left-hand member includes the factor c, which maybe of either sign. This leads to the condition

474


'Xo)

n-1

(dgdn

a))2ds+

I

dwk do

a

k=l

dg (z a) do

ds = 0.

(13)

If we multiply equation (13) by -A0 and equations (12) by Al (for each 1 = , n - 1) and then add all the resulting equations, we obtain 1,

(_X6 11, a)

+X1

do

d&)

+...+kn_t

do

dwn_y

do

) ds=0.

It follows that, almost everywhere on E, nom)-11

- o do + Lj Aff

Xk

k-1

do = 0.

Since, in addition, everywhere on K we have

-0+

n-I

d

dw

k_1

0,

kk dSk

we conclude that the function n

w'(z)

AoP1 (z,

a) + 2] Akwk (z), k-1

which is regular on K, vanishes on E. But then w '(z) __ 0 in B and hence the function

- kog (z, a) +

Akwk (z)

k-l

is constant in B. However, this cannot be the case because of the choice of the , n. This contradiction shows that U(z) = 0 almost everyAk for k = 0, 1, where on K. But then, if we apply Privalov's uniqueness theorem (cf. §2 of Chapter X) to a suitable subset of the domain B, we conclude that U(z) - 0 in B; that is, in B we have m

IogI/o(z)I =-g (z, a)- Z g(z, zk) k-1

and

m

I p (z, z ) ffe (z) = e

k

I

§3. EXTREMUM FOR BOUNDED FUNCTIONS

475

Now, the conditions under which f0(z) is single-valued in B take the form 2a

(argfo (z))K, = wt (a) + I wl (zk) = V1,

l =1, ... ,

(14)

n,

k=1

where the vi are integers. Obviously, vl > 1, for 1 = 1, over l = 1, . . . , n, we obtain

m+1=

n

r= l

, n. If we sum (14)

vl:n.

But m 0. Let us denote the extremal functions by f (z, p), thus pointing out their dependence on p. Then, we can easily see that between suitable extremal functions we have the relationship f (z, pq) = zf (z, p). This last means that the case of arbitrary p can be reduced to the case in which q < p < 1. We shall now consider this case. We note that the case p = 1 is trivial because we then have f (z, p) _- 1.

Lemma. If the function f (z) is regular in the annulus q < Izl < 1 except possibly at a single simple pole on the segment - 1 < z 0. Let us define

F (z) =

2

(f (z)+f (z))>

This function also satisfies the conditions of the lemma and it assumes real values for real values of z. If F(z) is regular in the open annulus q < Iz I < 1 and F(z) coast, then obviously F(z0) < 1. Let us now consider the case in which F(z) has a simple pole on - 1 < z < - q. Since IF(- 1)I < 1 and IF(-q)I < 1, it follows that F(z) assumes on the interval -1 < z < - q every real value of absolute value greater than 1. On the other hand, it easily follows from Cauchy's theorem on the number of zeros and poles that the function w = F(z) assumes every value w such that Iw I > 1 at exactly one point of the annulus q < Iz I < 1. Consequently, IF(z)I must not exceed 1 anywhere on the interval q < z < 1 and, in particular, F(z o) < 1. If equality holds here, then F'(z 0) = 0 and a neighborhood of the point z 0 would be mapped by w = F(z) into a multiple-sheeted neighborhood of the point w = 1 or of the point w = - 1. But then, F(z) would assume certain values w such that Iwi > 1 at certain points of the annulus q < IzI < 1. Thus, we have shown that F(z0) < 1 with equality holding only when F(z) = const. But F(z 0) = f (z 0) and, consequently, f (z 0) < 1. Now, if f (z 0) = 1, we have F(z 0) = 1 and then F(z) - 1. On q < z < 1 and - 1 < z < - q, this yields R (f (z)) ° 1, Consequently, the image of the interval - 1 < z < - q will be the


478

entire line R (f (z)) = 1, and the image of the interval q < z 1. On the other hand, it follows just as above that the function f (z) assumes every value w exceeding I

in modulus at exactly one point of the open annulus q < zl < 1. This con-

tradiction proves that f (zo) must be less than 1. This completes the proof of the lemma.

Theorem. Out of all functions f (z) that are regular in the closed annulus q< lzl < 1 such that l f (z)l < 1 on the circle lz l = 1 and i f (z)l
r

f(P(z))=-27vi

J C p(z)

A,

(7)

RCCCin

the counterclockwise direcwhere the integral is over the circle C: tion. Let us expand the fraction 1/(p(z) - 4) as a function of z in a neighborhood of z = = in a Laurent series I

P(z) - C

-I _

m

bk (C)

(8)

k=1

which converges uniformly on C with respect to C if IzI > r. From this result we obtain the equations

482


b, (C) = bs (C) _ ...

= bn-1(C) _ 0,

bn (C) =1,

bn+1 (C) + alb. (C) _

(9)

0,

(C) + aibn+1 (C) + asbn (C) = 0,

(10)

bon-1 (C) + aib2n-s (C) + ... + art-lbn (C) _ 0;

bm (C) + albm-1 (C) +. . . + ambm-n (C) = Cbm-n (C),

m > n.

(11)

The fact that (11) holds also for n < m < 2n - 1 follows from (9) and (10). When we substitute the expansion (8) into (7) and keep (2) in mind, we obtain by equating coefficients of like powers

ak- tai S f (C) bk (C) dC,

k ^ 1, 2, .. .

(12)

C

Let us now return to the determinant A. Let us replace the coefficients ak in it with their integral representations (12) denoting the variable of integration in the with row (for v = 1, 2, nants,

,

m) by

,,. Then, by the properties of determi-

Am = (tai )m S S ... 1 f (C1) ... f (Cm) CC

C

X

b1 (C1)

bs (C1)

bs (Cs)

bs (Cs)

... ...

bm (Cm) bm+1(Cm) ...

b,,, (C1)

bm+1(Ca)

dC1 ... dCm.

(13)

bsm-1(Cm)

For m n, let us set m = np'+ q, where 0 < q < n, and let us transform the determinant in the right-hand side of formula (13). Specifically, let us add to each column from the last to the (n + 1)st the n preceding columns multiplied respectively by a,,, an_1, . , al and let us use (11). We then obtain a determinant of which the with row (for v = 1, 2, has the form b, (C,), ..., b,+n-1 (Q,

C,,b, (C.),

, rn)

C,b,+1(C,), ..., C,bm+,-n-1 (C,).

Then (for m > 2n), let us add to each column from the last to the (2n + 1)st the n preceding columns, multiplied by a,,, 1, .. , al respectively. Then, again using (11), we obtain a determinant whose with row has the form

§5 TRANSFORMATION OF ANALYTIC FUNCTIONS

483

by (CY), ..., by+n-1 (C. ), C,b, (C'), ..., Cvbv+n(Cv) C b (Cu), ..., C9bm+y 9n-1(Cv)

After p such operations, we obtain a determinant in which the with row has the form

by (Cj, ..., Cyb

Cubv (C'), ... ,

(C') ... , Cpbp (Cv), ..., C'bv+q-1(C,)

Let us substitute into (13) the determinant thus transformed and let us put the integral signs back into the rows. We now change the symbols for the variables of specifically, we denote the variable of integration in all elements integration of the with column by ,,. Then, putting the integral signs back in front again, we obtain 1

r

...

b, (CI) m

S f (C1) ... f (Cm)

`4'n - \ 2nt)

b9 (C1)

C

.

.

Cn+1b1(C,,+1) ...

bn (C,1)

bn+1(Cn)

.

.

.

.

.

.

Cn+1b9 (Cn+1) ... .

.

.

.

.

.

bm (C1) ... bm+n (C,) Cn+1b,n (Cn+1)

... C9nbn (C9n)

Cs9n-f-1b1(C2, 1)

... C9nbn+1(C9n) Q. 4-1b9 (C9n+1)

..

.

.

.

.

.

.

.

.

.

.

...

... Cmbq (Cm) ... Cmbq+1 (Cm) .

.

.

.

.

.

.

dC1 ... dCm

(14)

C9nbm+n (C9n) CCn+Ibm (C9n+1) . . . Cmbm+q-i (Cm)

Let us now perform the same operations in the determinant in this equation, but this time on the rows instead of the columns. After p steps, we arrive at the determinant (see p. 484) composed of p2 n x n matrices, 2p rectangular matrices, and one

q x q matrix. It is convenient to regard the last 2p + 1 of these matrices as truncations of n x n matrices, of which we shall have occasion to speak below. On the basis of (9), in every square matrix of the type indicated, all the elements to the left of the secondary diagonal are equal to zero. Furthermore, if in each matrix we add the first k rows (k = 1, 2, , n - 1) multiplied respectively by ak, , a 1 to the (k + 1) st row and keep (9) in mind, we arrive at a matrix in which only the elements of the secondary diagonal are nonzero. Since these operations are performed

simultaneously on the complete rows of the determinant (15), it is transformed into a determinant in which only the elements containing b remain nonzero. But , a,,. b = 1. Consequently, the determinant (15) and hence A. do not depend on a1,

CA+tb9 (Cn+1) ... C9nbn+1(C'en)

Cn+ibn (C.+1) ... Cen b9n-1 (C9n)

b9 (C1) ... bn+t (CA)

bn (Ct} ... 69n t ltin)

+1b9 (.np}1) ... Cmbq+1(Cm)

Cnp+tbn (Cnp+i) ... Cmbn+1-q (Cm)

C

C4+1b1 (Cnp+1) ... Cmbq (Cm)

Ci bq (Ci)

...

Cib1(Cl) ...

Cn bn+q-i (Cn)

Cibn (Cn)

...........................................

.................................................................................

.................................. ......................... :..................................................................... ...............

Cn+tbn (Cm+i) ... Cgnb9n-1(C9A

(Cn+1) ... Csnbn (+dn)

Cibn (Cl) ... Cnb9n-1(Cn)

Cn+1b1

Crt+ib9 (Cn+1) ... C9nbn+l (C9n)

Crtbrt (Cn)

C1b9 (Ci) ... Cnbn+1 `C,.)

C1b1 (Cl) ...

Cmbq (Cn)

.......

Cnp+lbq (Cnp+1)' ... Cmpb9q-1 (Cm)

C p+1b9 (Cnp+1) ... Cmbq+1 (Cm)

Cnp+ 1b1 (Cnp+1) ...

................................................................

........

.......................................................... :..............................................................................:........ ...................... ........ .................... ..........................

Cn+ibi (Cn+1) ... C9nbn (C9n)

bl (C1) ... bn (Cn)

(15)

00

0.

§5. TRANSFORMATION OF ANALYTIC FUNCTIONS

485

On the basis of what has been shown, to calculate A.*, we can now set Then, ak (where v = 1, 2, ) and ak = 0 for all of k. Consequently, A, is of the form n

I

at

at

as

79

ap+t ...............................................................................:.......................... a3

ap+t

a3

as

(16)

........................................ --........ .......................... ,.........................

a p+t

........................................................................................

ap+i

m=np+q,,

where the unfilled-in spaces are occupied mostly by zeros. We now put the columns numbered n, 2n, , 2np into the 1st, 2nd, , pth positions (without changing the order of the remaining columns) and we put the rows numbered n + 1, 2n + 1,

,

n(p - 1) + 1 into 2nd, 3rd,

, pth positions. Then we arrive at a determinant

that is equal to the determinant AP multiplied by a determinant in which, for every nonzero q, the qth row from the bottom consists only of zeros, so that the determinant is equal to zero. On the other hand, if q = 0, this last determinant will have the same form as (16) with p replaced by p - 1. Repeating the same operation, we show after p steps that the determinant (16) is equal to + AP. This completes the proof of (4). 'Therefore, n

D* - lim

m-00

my

Am I = lim

P-00

Anp =11m

P- W

12,p2

I _AP la = ,

which is equivalent to (6). This completes the proof of the theorem. We turn now to some applications of this theorem. Theorem 2. For functions that are regular in an infinite domain B with


486

boundary K consisting of a finite number of closed Jordan curves and having the the inequality D < d(K) expansion f(z) = Ek=1ak/zk, in a neighborhood of z given by Theorem 3 of §2, Chapter VII, is sharp. Here, D is defined as in Theorem 1 and d(K) is the transfinite diameter of the set K. Proof. We shall go through the proof only for the case in which the curves mentioned in the theorem are all analytic. Then, if we denote by g(z, ,) the Green's function for the domain B, we conclude that this function is harmonic on K. Therefore, for sufficiently small S > 0, the equation g(z, -) = - S defines a set of closed analytic Jordan curves approximating the curves in K and defining domains B 8 containing B. If g S (z, ,) is the Green's function for the domain B g, if t n, g (z), for n = 0, 1, are the Chebysev polynomials for the boundary K 8 of the domain B g which have all their zeros on K g, if mn, g = ti maxZEK8Itn,8(z)I, and if d(K3) is the transfinite diameter of Kg, then one can easily see that g g (z, 00) = g(z, -) + S and, consequently, n

Itn,a(z)

d(Ka)=d(K)e--8 lim

n- 00

e8(z.-)+a

mn, a

(§3, Chapter VII). The convergence in this last formula is uniform in every closed subset of the domain B g, and in particular on -ff. From this last it follows that there exists a positive number N such that, for n > N, we have It n, S (z)I > Mn. S ti on B. This means that, for n > N, the set E* = E(I t n, 8(Z)1 < Mn, 8) lies entirely outside the domain B. Now consider the function 00

f (Z) =

z2k

(17)

k=1

By setting up the determinants Ak for this function, we easily see that for n = 1, 2, . Therefore, D > 1 for it. On the other hand, since it is regular in Iz I > 1, it follows from Theorem 3 of § 2, Chapter VII, that D < 1. Consequently, D = 1. From the function (17), we now form the function f. (z) _ f (t n, g (z)/mn, 8) = 1k ak/zk. This function is regular outside the set E * and, mn, g . But in particular, in the domain B. By Theorem 1, D* = nV/ n , 8 D S = d (K g) = d We- 8. Consequently, for sufficiently large n we have D * > d(K)e- 8 - S. Since 6 can be chosen arbitrarily small, we can make D' A 2n

+ 1 = 1,

1

§6. ON p-VALENT FUNCTIONS

487

arbitrarily close to d(K). This proves the sharpness of the inequality mentioned in the theorem. This completes the proof of the theorem. Theorem 3. For any complex numbers a and b, an arbitrary positive integer n, and a continuum E such that d(E) < l a - b I/4, an arbitrary polynomial Pn(z) = z" + c 1zn-1 + - + cn assumes either the value a or the value b somewhere in the complement of E with respect to the z-plane. The number " a - b /4 cannot be replaced by a larger number without additional restrictions on E. Proof. It was shown in §2 of Chapter VII that D = limn+ccnI= 1/4 for the function f(z) = - log (1 - 11z) = 11z + 1/2z2 + . If we now apply Theorem 1 to this function and to the polynomial p(z) = (pn(z)b)/(a - b), we see that D* _ n a - bl/4 for the function f* (z) = log Pn (Z) - b

ak

vn(z) - azk k=1

If all the zeros of the polynomials pn(z) - a and pn(z) - b lie in the continuum E, the function f.(z) is regular in the supplementary domain B containing -. 4< Consequently, by virtue of Theorem 3 of §2, Chapter VII, D* = n 1a d(E). But this contradicts the conditions of the theorem. Consequently, the polynomial pn(z) assumes in B at least one of the values a, b. That the inequality for d (E) cannot be improved is shown by the example of the polynomial pn(z) _ zn + a' and the set E = E(zn E e), where e is the segment connecting the points 0 and b - a. In this case, we have, in accordance with Theorem 2 of §1, n Chapter VII, d(E) = ja - bl/4 and this polynomial pn(z) assumes the values a and b respectively at the point z = 0 and at the points Jb - a . e 21tki/n, where , and all these points lie on E. This completes the proof of the k = 1, 2, theorem.

§6. On p-valent functions 1) A function w = f(z), which is regular or meromorphic in a domain B in the complex plane is said to be p-valent (for p = 1, 2, ) in that domain if no com-

plex value is assumed by f(z) at more than p points in B, that is, if the 1) Goluzin [19401.

488


Riemann surface onto which the function w = f(z) maps B bijectively covers each point in the w-plane with no more than p sheets. In the present section, we shall consider the following classes of p-valent functions: S, is the class of functions of the form

w=f(z)=z"(1'+atz'+avz2-}-...),

(1)

that are regular and p-valent in Izl < 1. 1p is the class of functions of the form

that are p-valent and regular in > 1 except for a pole at J= . p is the class of functions in lp that do not assume the value zero in the domain 16 > I. For these classes of p-valent functions and also for certain of their subclasses, we now give a number of sharp inequalities regarding the initial coefficients and we shall establish the analogues of the area, covering, and distortion theorems, which are well known for univalent functions. 1°. Lemma. If F(4) E Y-p (where p > 1), then, for 1 > 0 and p > 1, a

IF(pere)Jad9:0.

do

(3)

Proof. The function w = F(C) maps the domain 1 onto a Riemann surface RP bounded by an analytic curve L P and having the maximum number p of sheets in a neighborhood of w = -. Let R and ID denote polar coordinates in the w-plane. Consider the integral

S R' dO, 1>0, P

(4)

over LP in the direction that puts LP on the right. This integral is the limit of the Riemann sum constructed as follows: From the point w = 0, we draw a suffici ently large number n of rays G 1, G 2, ... , G. that are not tangent to L P and do not pass through multiple points of L P. We denote by A01, A 2, ... , Otpn the magnitudes (nonnegative) of the angles between pairwise-adjacent rays. There


489

are finitely many points of the curve L p that lie over Gk (for k = 1, 2, , n). Let us index these points A1,k+ A2,k, ..., Ank,k We denote their distances from , Rnk,k. Let us suppose that they are indexed the point w = 0 by Ri,k. R2,k, in such a way that Rl.k >R2.1, > ...> Rnk,k. We assign to the point Al,k (for l = 1, 2, , nk) the number el,k, which is defined to be + 1 if the coordinate $ increases as we move around LP in a neighborhood of Al,k and equal to - 1 if I' decreases. Then, the integral (4) is the limit of the Riemann sum n

a

1

[e1,kR1+

e9,kRa.k--

a + ... +enk.kRRk

(5)

k]D'Pk

But if we move along Gk (for k = 1, 2, , n) from w = - to w = 0, it follows from geometric considerations that the numbers of sheets RP lying above segments of the ray Gk between AI,k and Al+l,k (for 1 = 1, 2, - ) are respectively equal to p - V,1 Es .k , and these numbers do not exceed p. Therefore, 1

1=1, 2,...,nk, k=1, 2,..., n.

s=1

Thus, if we set a1, k =

es, As

s=1

all the numbers 01,k will be nonnegative. Furtherniore,

a1.kel,k,

Q1.k-01-1,k-el,k,

1=2, 3, ...

,

nk.

Consequently, we can rewrite the sum (5) in the form n

r px

LJ [Ql. k t l', i,

x

a

a

l9. k) + a,. k lR9. k - \9. k + ... + aRb. k Rna. k] 'Ok

k=1

It follows that this sum is always nonnegative. Therefore, the integral (4) is also nonnegative. On the other hand, since the curve L p has the parametric representation w = F(pe`B), where 0 < 0 < 2n, we have

490


R'd(

R'de d8.

S

0

But if we set

Re", where

peie, we have

a_p aR de-"Rdp Consequently, 2,,

Rd=p a

2s

aRd g R 1 OP =

dP

Rk A.

(6)

dl

0

P

On the basis of what was said above, we then get (3). This completes the proof of the lemma.

20. Beginning with this lemma, let us prove the following theorems: Theorem 1. If the function (2) belongs to the class Ir (where p > 1), then 00

nIap+n

(7)

Proof. From (3) with A = 2, we have .a

0 ! p9 (p-n)

I an I9

> 0, P > 1,

n=1 or Co

21 (p

1

n)lanl9P9>0.

n=1

By letting p approach 1, we get (7), which completes the proof of the theorem. For p = 1, Theorem 1 coincides with the theorem on areas for univalent functions. Theorem 2. 1) If a function (2) belongs to the class Yp (where p > 1) and if

a1=a2=...=ak_1=0 (where k>1), then, for n=k, k+ 1,---,2k- 1, we have

IanIC2p with equality holding only for a function

(8)

2p

F(Q `Cp(1 +n)n, I(9) 1) For p = 1, Theorem 2 was proved by the author [1938, where it was erroneously

formulated for the class 11 instead of the class 2i.


-

491

belonging to the class Ip. Proof. )From the lemma as applied to the function (2), we have, for A > 0 and p > 1, 2x 1F(pe'°)I9Xd6

aP `

0.

(10)

3'

For

1, we set [F (C)]'' = C"p (1+ ct C-1+ ...).

Then, (10) yields OD

(n - Ap) P 2 (n_).pl < 0, co =1. A=O

If we let p approach 1, we obtain

I (n - )a)I C. 11

n=1

Xp

But the expansion [F(t)]" in I C I > 1 begins as follows:

Consequently, cn = Aa,,, where n = k, k + 1, , 2k - 1. Keeping this in mind, we obtain from (11) the following inequality for A < k and n = k, k +

2k-1: (n-Xp)X,Ia.11<Xp, that is, 9

p

Ianl <X(n_Ap) If we now set k = n/2p, we obtain (12)

IanlcLPn Equality can hold in (12) only if c

0 for v n, that is, only if 2,0

F(t)=Cp(1 +jn)n, ICI=1. The function (13) belongs to the class lp because the function

(13)


492

is univalent in ICI > I. This completes the proof of the theorem. Theorem 3. If the function (2) belongs to the class EP (where p > 1), then

IayIc2p,

(14)

with equality holding only for a function of the form

InI=1.

F(C)=Cp(1 +C!-),Pp Furthermore, for n = 1, 2, 3,

.

(15)

,

Ia.ICAn.pp

(16)

where A,,,p is a finite quantity depending only on n and p.

Proof. The first of inequalities (14) follows from Theorem 2 for k = 1. To prove the remaining assertions of the theorem, we set

F(C)=V1/1 +C'+c'+ C

for

I

...1p

C2

I > 1. Then, from the lemma, with A = 2/p, we obtain, by letting p approach

1 from above, co

21 (n-1)IcAI9 1/2, that is, Ic 12! 1/pP + 4°. Let us find another bound for the absolute value of the derivative of functions belonging to the classes 1P and Xp . Theorem 7. If the Function (2) belongs to the class YEP (where p > 1), then, in the disk ICI > 1, we have

as, ..., aP-t)ItICIl where c(a1, a2, ... , aP_1) is a finite quantity depending only on a1, a2, ... , ap_1. If the function (2) belongs the class lp (where p > 1), then, in the disk IcI > 1, IcIP

IF'(C)Isc(p)

where c(p) is a finite quantity depending only on p. Proof. From (2), we have, in the disk

IF'(Ql 1,

+...

00

nll lnnl

IaP 11+1 n=1 co

=plCln-t+(p-1)1aiIICIP-9+... +1 a9-tl+y n-1

IC

nl'

ap'l.

cPICIP-1+(P-1)la1IIC19 +...+lap-tI 00

+

n=1

s

nlaP+n l'

(Istn-u n=1

from which Theorem 7 follows by virtue of Theorems 1 and 3.

We note that the order of the inequalities given by Theorem 7 is sharp since

F'(p)= psoo. for the function

F(C)=ppC-1

1

§7. ON THE CARATHEODORY-FEJER PROBLEM

497

which belongs to the class Y_P' for arbitrary. P> I.

§7. Some remarks on the Caratheodory-Fejer problem and on an analogous problem 1)

In subsection 1° of the present section, we shall give a simple derivation of the results of Carathe'odory and Fejer2) dealing with the problem of the extendability of a polynomial in z (by adding terms of higher powers of z) to a power series representing a rational fraction with constant modulus on the circle IzI = 1 and on the minimal property of the maximum of the modulus of such a fraction in the open disk Izl < 1 among all functions that are regular in that disk. This derivation is based on simple results dealing with the solution of the problem of coefficients in the case of bounded functions. In subsection 2°, we consider the extendability of a polynomial in a similar fashion, starting with results dealing with the problem of the coefficients in the case of functions with bounded mean values of the modulus on concentric circles. Here, the minimization of the maximum value of the modulus is accordingly replaced with minimization of the maximum of the mean value of the modulus. A particular case of the extendability of a polynomial that is considered in subsection 2° has more than once found application (by Landau, Fejer, Szasz, and others) in inequalities relating to bounded functions. However, the problem of sharp inequalities has remained unsolved if the particular extendability referred to does not exist. As will be shown in subsection 2°, our second method of extension enables us to indicate a path for obtaining sharp inequalities in all cases, thus removing this gap at least theoretically. We note further that in subsection 1° we shall, by beginning with the solution of the Caratheodory-Fejer problem, give a simple derivation of the final form of the solution given by I. Schur to the problem of coefficients for bounded functions in the case of interior points of the domain of the coefficients. We shall use the following notation:3) We denote by B the class of functions f (z) = c 0 + c 1z + that are regular in the open disk Iz I < 1 and that satisfy in that disk the condition If (z)I < 1. We denote by H 1 the class of functions f (z) = c 0 + c 1 z + . that are regular in the disk Iz I < I and that satisfy the condition 1) Goluzin [1946a]. 2) Caratheodory and Fejer [1911].

3) (Here, the notation used by the author [1946a] is kept.)

498

XI. SOME SUPPLEMENTARY INFORMATION 2%

2rz

If (rel) I de

l

for 0 0. Beginning with the conIrk - ckI < E, where k = 0, 1, cept of a neighborhood, we define cluster points, interior and boundary points of a set, open and closed sets, and convex sets in the usual way. The space Rn can, in case of necessity, be regarded as a 2n-dimensional Euclidean space with Cartesian coordinates corresponding to the real and imaginary parts of the numbers

CO, C1,...,cn-1 , cn_1) E Rn such that the We denote by B(n) the set of points (co, c1, numbers c 0, c 1, , cn_1 are the first n coefficients of some function in the

class B. We denote by HP) the analogous set for H1. These are bounded sets in 3n because, for all k = 0, 1, , we have Ic k I < 1 for both the functions in B and the functions in H1, by.virtue of the integral representation of the ck. Since the principle of compactness holds.for functions in B and H1 and since the limit. of a sequence of functions belonging to either of these classes also belongs to the same class, it follows that B(') and H ln) are closed sets. Furthermore, since for functions f 1(z) and f2 (z) belonging to one of these classes the 2(z) for . 0 < A < 1 also belong to the same class, it functions A f 1(z) + (1 - W 2(z) follows that B(') and H ln) are convex sets. It is also obvious that the origin , 0) is an interior point of these sets since the polynomials c 0 = (0, 0, C 1z +

- + cn_lzn-1 belong, for small ck, both to B and to H1. 1°. Theorem 1 (Schur). To points (c0, c1, ... , cn_1) on the boundary of B(n) there correspond in B only fractions of the form an-1 + a0 + a1Z T ... +

a0Zn-1 an_1zn-1

(1)

Proof. 1) Let us suppose that a function f (z) E B other than a fraction of the 1) The basic idea of the proof is borrowed from Schur. See Schur [1917] and also Bieberbach [1927].

§7. ON THE CARATHEODORY-FEJER PROBLEM

499

, form (1) corresponds to a point (c0, c 1, on the boundary of Br" ). Following Schuur, we de{fine a{sequence (fk(z)I inductively by

J o (z) =J (z)r J k ('z) = Z fk-1 (Z) -fk-1 (0)

1 -fk-1 (0) fk-1 (Z)

k = 1,

`2,

...

(2)

Let us show that I fk(0) < 1 for k = 0, 1, , n - 1. Since the fk(z) belong to B whenever they are defined, we have Ifk(0)I < 1. Let denote the first E, where JEI = 1. function in the sequence such that Ifv(0)l = 1. 'Then, , f0(z) from the formulas inverse 'then, by calculating the functions f y_1(z), to (2), v

() = E, / k-! (z) = fk-1 (0) '+' Zfk (Z)

(3)

1+fk-1(0)Zfk (Z)'

we can show by induction that l v-k (Z) = Ek kNo+'F Y12 ++

}k

OZk ,

I gk l = 1, k = 1, `2, ... ,

.

Here we can assume the Ek equal to 1 because otherwise we could arrange for this by changing the arguments of f3p, , Ok. In particular, f0(z) = f (z) is also a fraction that reduces to the form (1) when we multiply its numerator and denominator by 1 + z + + z"- "-1 and this, by the hypothesis of the theorem, is impossible. Since Jfk(0)l < 1 for k = 0, 1, , n - 1, the functions (2)are defined for k = 0, 1, , n - 1 (that is they have denominators not identically equal to 0). For these functions, the inverse formulas {k-1 (Z) = fk-1 (0) "+' Zfk (Z) 1 +fk-1 (0) Zfk (Z)

I

J

k = n - 1, ... , 1, fo (z) =J (z)

(4)

are also meaningful. Now keeping the values fk(0) for k = 0, 1, , n - 1, let us calculate the functions f k(z), for k = n - 1, , 0, from the formulas

f,*-1 (z)=f"-1(0), fI-1(Z)=

fk-1(0)'+' zfk(z)

1 +fk-1 (O)zfz (z)

k=n- 1,..., 1.

All the f k(z) belong to B. From (4) and (5), we have

fk-1(Z) -fk-t (z)

_

(1 - fk -1(0) 1') z (fk(Z) -fk (z)) (1 +fk-1(0) zfk(z))(1 +fk-t(0)zfk (z))

_-z k (z)

(Z)

(5)

500


where the t4k(z) are, for k = n - 1,

, 1, regular ink{I zI < 1. Therefore

f (z) - {J0 ('z* W = Zn-1p1 (z)... pn_i W V n-I W -fn-1 (0)).

Since the point z = 0 is a zero of the right-hand member of multiplicity no less than n, the first n coefficients in the expansions of f (z) and f o(z) about z = 0 are identical; specifically, they are c 0, , cn_1. On the other hand, since fk(0), for k = 0, 1, , n - 1, are rational functions of C 0, , Cn-1, C 0, , Cn-1 and since c 0, c 1, , cn_1 in our notation satisfy the inequalities Ifk(0)I < 1 for k = 0, 1, , n - 1, these same inequalities are satisfied for all systems of numbers (c 00*, , cn-1) sufficiently close to the preceding ones. The functions , cn_1 belong (by virtue f:(z) calculated from fk(0) corresponding to c0, c1, of formulas (5)) to the class B. On the other hand, if we take for fo(z) the polynomial c00 + + Cn_1zn1 and then calculate the fk(z), for k = 1, 2, , n - 1, from formulas (2), these fk(z) again satisfy.(4) Just as above, we can show from (4) and (5) that the function f (z) - f o(z) now has a zero of multiplicity at least n at z = 0. Consequently, the first n coefficients of the function f p(z) are c 0, , c n-1; that is, this function corresponds to the point (c 0, , c _ 1), which therefore belongs to B (n ). This.shows that (c 0, , c,,_ 1) is an interior point of B(n ) in contradiction to the assumption. Therefore, only fractions of the form (1) can correspond to boundary points of B(n). This completes the proof of the theorem.

Theorem 2 (Caratheodory and Feje'r). For any polynomial c 0 + c 1z + Cn_1z"-1 J 0, there exists a unique') fraction of the form R (z)

-

as + .+.. -+

-±,hiZn-t

,

A

0,

+

(6)

that is regular in Iz I < 1 and that has as its first n coefficients in its expansion about z = o the numbers c 0, c 1, , cn_ 1. Out of all functions f(z) = c o + that are regular in I zI < 1 and have the same first n coefficients c 0, c Iz + c 1,

, cn_1, this function and only it minimizes the quantity Mf = maxlz 10, k=1,...,n,

(11)

... 0

............... Ck_t

Lrk_y

...

ED

0

0

...

1

be satisfied.

Proof. We note that the determinant D"(,A) (cf. (7)) is an even function of A because, if we divide each of the first n rows and the first n columns by - 1, we obtain D"(A). Therefore, the determinant D"(1) is of the form (11).

We first prove the necessity of the conditions of the theorem. If the point , , qc"_1) lies in B(" lies inside B(n). then the point(gco, for some q > 1. Consequently, there exists a function f (z) E B whose expansion . On the other hand, the function f 1(z) = begins qc o + - + qC"_ 1z"-1 + z"-1 f (z)/q = c 0 + + Cn_1 + satisfies in I z I 0 for k = 1, 2, , n - 1. This proves the necessity of conditions (11). We now prove the sufficiency of the conditions of the theorem. If conditions, (11) are satisfied, then the equations Dk (,A) = 0, for k = it - , n, cannot have roots in the interval 1 < A 1 is a multiple root of DE,,(h) = 0, it follows from (13) that all the Dk,k(A5) are, for k = 1, 2, - - , 2µ, different from 0 and have the same sign. It then follows from (12) that Dk,k(A5) = 0, for k = 1, 2, - , 2;L, since Dµ (A5) = 0. In particular, -

D1, 1(A*) = - A*Dµ_I (A*) = 0,

which cannot be the case. This shows that the equation D,,,(A) = 0 has no multiple roots in 1 < A p for I Z I < 1; 3) the class BG of functions f (z) that are regular in the disk I z I < 1 and assume in that disk only values belonging to a given closed convex region G. Our purpose is, first of all, to establish sharp inequalities regarding functions belonging to the class B in which only functions of the form f (z) = ezm, where 1c l = 1, can be extremal. Such inequalities dealing with the derivative were examined by the author earlier. The basic tool was then the invariant form of the Schwarz lemma. 5) Here, we shall use an essentially different method, one 1) See, for example, Landau [1929a]. 2) Ibid.

3) Szasz [1920]. 4) Goluzin [1950]. 5) Goluzin [1945]. We note that by the same procedure Dieudonne had already

obtained the following results [1931a]: If f (z)= 10=lcnzn E B, then If'(z)I < 1 in the closed disk IzI < XF2 - 1 but not always in a larger disk (see also Caratheodory [1936]).

§B SOME INEQUALITIES FOR BOUNDED FUNCTIONS

515

that will enable us to study questions in an extremely general and exhaustive form. This same method can be applied at the same time to the establishment of inequalities regarding functions in the classes R and BG. 1°. We shall establish some general theorems. 'Theorem 1. Let m denote a nonnegative integer and let Yn (n = m, m + 1, denote complex numbers such that y,,, 0 and Y-n _m lyn I < -. Then

1) If m = 0, a necessary and sufficient condition for the inequality C6

clToi

21 T.C.

n-0

(1)

to hold for all functions f (z) = Em=ocnz" E B is that 0 ` To

2

I TnCn)l

n-o

on

ICI=1.

(2)

Equality holds in (1) only for f (z) = c o, where Ic o I = 1;

2) If m > 0, then a necessary and sufficient condition for the inequality

m-I

00

Tm

_ Ttm-ncn

1

1

Z Tncn + TM

n-m

n-0

to hold for all functions f (z) _ IW=ocnzn E B is that 00

I n=m

TnCn-M)

2

on ICI = 1.

(4)

Equality holds in (3) only for f (z) = c,nz", where Icm I = 1.

(We note that, by virtue of the assumptions of the theorem, the series in (2) and (4) converge uniformly on the circle It; I = 1 and the series in (1) and (3) converge for all f (z) E B since we then have Icn I < 1 (n = 0, 1, Proof. -Since the functions f (z) = Y,W=oc nzn E B have limiting values along radial paths almost everywhere on the circle Iz I = 1, if we denote these limiting values on ICI = 1 by f (z) and take the limit as r approaches 1 from below. in the formulas cn

2n!

Cn+,

I C =r

ICI=r

!

' = o (n=0' It

...

516


which hold for 0 < r < 1, we obtain the formulas

cn = 1 f C)Cn+tdC tai

f (C) Cn dC =O (n=0, 1, ...),

where C is the circle 141 = 1 and the integrals are Lebesgue. On the basis of this and the uniform convergence of the series lm=mynCF on the circle we have

1

1

7m

f (C)

i

7ncn = 2a1

Cm+t

n=m

\

d

7n 7mCn-m

Co

f (C) I

1

2n!

7m

19-M

2a!

-C

7nCm-11

m+i l

C(C) 251

\ 7m

C

n=m

m-n

7n7m

n=m

I 7-Cm-n)

nm

A

-

Co

I

f (C)

2a!

Cm+i

- tai

C(C r

n

n-m

'V 7m nm

7m

d C

Cn m 2M

L (m+)i [2D1

27!

(7m001

a-m

7nCm nl

-1A-

1

_1 TnC9m-ro 7in n-m+l

that is, m - l

Co

7m

Tncn +

7m n-0

a=m

T m-ncn Co

' -L C(Cr 12`9 2n!

y 7nCm-n) -1] dC.

(7m

(5)

n-m

For m = 0, the second summation on the left disappears. If condition (4) is satisfied on the circle JCJ = 1, we obtain from equation (5)

m-l

cc

7m

I Tncn + -1

n=m

7m

L T9m-ncn

n-0

00

G

1

2a

[281

1

m

l TnCn-m)

n-m

-

I&P =

1,

where C = e`0; that is, we obtain inequality (3) and, by setting m = 0, inequality (1). Since the function in the square brackets in equation (5) is not identically equal to zero on the circle JCJ = 1, equality holds in the relation (3) or in (1)

§8. SOME INEQUALITIES FOR BOUNDED FUNCTIONS

517

only when If(C)I = 1 and arg ([(C)/[,m) = coast on a set of points of positive measure on the circle ICI = 1. But then [(C)/Cm = const on the set mentioned, and this can be the case only if f (z) = e is zm in the disk Iz I < 1. Then equality does indeed hold in (3) or (I). Thus, the sufficiency of the conditions of the theorem is proved. Let us suppose now, conversely, that inequality (3, or inequality (1) if m = 0, holds for all ((z) E B. If we apply this inequality to a function Co f(z)=zn

1-az

-zm(-a+ A=1 (1 -IaI2)an 1Znl E B,

for which

CO=C1=...=Cm_1-0, Cm=-Oy Cn=(I -ICE

I2)Zn-m-1

(n=m+1, ...), IaIIz(+...+zm-1Cm-1I

I

on ICI=1,

that is,

/I

on

ICI=1.

(24)

The inequality

I 1 -Cmzml -IzI I 1

zm-1Cm-1

=1 -Izim-IzI(1+

Izim-1)=1-IzI-2Izim,

where ICI = 1, and its sharpness for even m (equality holds when _ - e-àrgz) imply that inequality (24) holds for all z for which 1 - IzI - 2 Izim > 0, that is, for IzI 2rm$i+2rm+i-1=0 and 2rm+rm-1=0. , we have rm < q and, consequently, 2qm + q - 1 > 0. By letting m approach - in this last inequality, we see that q > 1. Thus q = 1. This completes the proof of the theorem. If limm,W rm = q < 1, then, for all m = 1, 2,

1) In the case of odd m, the corresponding greatest number is obviously equal to the

greatest r in the interval 0 1, let rm denote the unique positive root of the equation 4rm + r - 1 = 0 and set rl = 1/3. For all functions f (z) C B, we have

Ism(z)I+Irm(z)I--- 1,

(25)

in the disk I z I < rm with equality holding only for f (z) = c0, where Icol = 1. For even m, the number rm cannot be replaced with a larger number. We note that

r, I, the maximum of either of the functionals Ut (0 (log F (C)))

I T (log F (C))

(12)

that are regular in the in the class V of functions F(C) = C+ ao + a1/C+ domain JCI > 1 excluding the pole C= - and that map the domain ICJ > 1 onto domains with complement that is starlike about C= 0 is attained only by a function of the form F(C)=C(1et_)a°,

alt, A910, A1+A9=2.

(13)

The case when

t'(IogP(C))=0 at an extremum is excluded from consideration.

We note that, for 0 < Al < 2, the function (13) maps the domain > I onto the w-plane with two rectilinear cuts issuing from w = 0 at an angle other than it. In a number of special cases of the type that we are considering, the extremal functions (9) and (13) degenerate into functions with k2 = 0. For example, this is the case in questions of maximizing or minimizing W`(z)I WI in the class S* or

536


of minimizing IF%)I in the class V. However, this degeneration does not always occur. For example, in the question of maximizing IF'(C)i in the class V for the function F(C) = C+ eiQ'C1 + const, we have I F, (C) 1==1 + It'll

,

whereas, for the function (9) with 0 < AI < 2, we see that obviously F'(a) as C approaches any point on ICI = 1 that corresponds to boundary points on

w=0. An analogous situation occurs in the problem of maximizing IF%)I in the subclass 12 of odd functions F2(C) belonging to the class V. These functions have. the representation

F9 (C) = F (0, where F(C) E V. It follows from Theorem 3 for k = 3/2 that the only extremal function here is a function of the form X1 / e1P X2 F2(C)=C 1 - eta) C2 l\1 - C,

where Al and A2 are nonnegative constants-such that Al + A2 = 1 and where a and 0 are constants. Here we conclude just as above that, for ICI close to 1, we have 0 1+l.

This inequality leads to inequalities (14) and (15). This completes the proof of the theorem.

It is still an open question whether inequality (15) is valid in the entire domain ICI > 1 under the conditions of Theorem 5. The corresponding question with regard to a bound for larg f'(z)I in the class S* has an affirmative answer. This will be shown below. 4°. Let us establish a theorem for the inverse of a starlike function. Although, as we have mentioned, the application of the variational method set forth here to the particular question of solving this problem is not new, we shall nonetheless give its solution, treating it as an application of the general 'Theorem 2 of subsection 30 Theorem. For given z in the disk IzI < 1, the function argf'(z) in the class S* attains its maximum and minimum only in the case of functions of the form 1) See the author's article [1945] and also Theorem 5 of §8, Chapter XL


538

.f(z)=

(1-etaz)2

(1)

where a is a real number. Proof. Without loss of generality, we may assume that z = r > 0. The problem that we are considering is a special case of the problem of Theorem 2 of subsection 30 when 1(w) = ± iw. It follows from the proof of that theorem that the function a(t), corresponding to an extremal function f(z), has in the interval - n < t < n no more than two points of discontinuity, which must be roots of the equation

F(t)

ur )_ f(r) 1 C (1-ee ttr)l )-0' C= rf'(r)

t(1ê ;tr +

(2)

(this is equation (10) of subsection 3° with I(w) = + iw). If there are two points

of discontinuity, the function

(t)=t (l log (1-e ttr)-C1

tre(3)

(cf. (10') of subsection 3° with 0(w) = iw) has the same value at the two points. Let us suppose that there are two such points of discontinuity t 1 and t 2, indexed so that t 1 < t 2. Then, the function F(t) = tp'(t) has one zero inside each of the intervals (t 1, t 2) and (t 2, t 1 + 2n). By ridding equation (2) of denominators, we reduce it to the form

at (e" (1

-

e-tt

r) (1 - eit r)9 -4-- Ce tt (1 - ett r)9) = 0

and we can then reduce this to the form

at (e-9" r - e` t (l +C+2ra)+2r+ra+2Cr -ett (r9+Cr9))=0. If we now replace the first two terms with their complex conjugates, which does not change the real part, we obtain the .equation

at (e91t r - e't (1 + 3ra + C + Cr') + 2r + ra + 2Cr) = 0.

(4)

Then, by setting C= e`t, we write this equation in the form of a fourth-degree algebraic equation

rC`-(1+3r3 -C -Cr9)C3+a9C9+a3C+a4;0. This.equation must be satisfied by the numbers bk = eItk, for k = 1, 2, 3, 4,

(5)

§9. A METHOD OF VARIATIONS

539

corresponding to the numbers tk mentioned above. We now have from (5) 4

4

L

Ck) I = 1

COS tk = + 3r +1 +

rn

OR (C).

k=1

k=1

On the other hand, since R(((z)/z('(z) > 0 in Izl < 1, we have

(C)at ( rf'f(r)(r) J

+r'

Consequently, 4

costk-::k.1

r

+3r+1-}-r° 1-r _3+r+ (1-r)9(2+r)>3+r. r

r(1+r)

1-}-r

This shows that

cos tk > r, k -1, 2, 3, 4. It follows, in particular, that cost > r for t 1 < t < t 2. But then, since dt

(l log (1 -e 't r))=-

e-11

(1

f tr)1r ecoitr I'

we conclude that the first term in (3) decreases in the interval (t1, t2), so that 81 (1 log (1 - e-tt, r)) > 81 (1 log (1 - e- its r)).

Furthermore, remembering that the extremal function ((z) is of the form z f(z)= (1-e-ttlz)zz(1-e tt2z)s t-a)

0< A 1,

we obtain

C= f (r)

1

rf' (r) -

1+e 111 r

1-e tt,r

(

1-] 1 )

`tsr e-tear e

Therefore, by setting

-ttk 1-e ttkrr

Ck= 1 we have

-4-e

k= 1,

2,

-.

(6)


540

C

le itlr 1-e -e-"'r

_

rC

(ic(-e- t11r

le e1'r

1-a-11'r )

_

-

e-1t'r

1-e111r

1

))_T

1-e-1t'r

Cl-Cs

DI (i 1

xC1+ (1-A)C.

2IS-1 ((C2-C1)(ACi+(1 -A)Ci)) C

2I

(1 - )

= I C 1' 2

(1-{-- a+1t1 r

1 -{-

a-1t, r)

1-a+1tir 1-et1Yr

so that C

lei11r

_ jC I' 2

C

(C1C2)

2r (1- r) (sin t1- sin t,)

I1-e't'rj2.I1-ehi2rI9 (t Z), which contradicts what was said above, namely, that the function Tli(t) has equal values at the two points of discontinuity t 1 and t2- This contradiction shows that a(t) cannot have two points of discontinuity in the interval - n < t < n. Consequently, the function f (z) must be of the form (1). This completes the proof of the theorem.

5°. The same method of variations can be applied to tht solution of extremal problems in the class of typically real functions. A function

f (.Z)=z+csz9+... ,

(1)

that is regular in the disk IzI < 1 is said to be typically real in that disk if it is real on the diameter - 1 < z 0 for 2(z)>O 0,

A1-f- A9 =1,

where t1 and t2 are numbers in the interval - 1 1 onto the class of functions w = f a domain whose complement is starlike about the point w = 0; the class of functions in I that map the domain ICI > 1 onto a domain with convex complement; SM (M > 1) the lass of bounded functions f (z) E S: I f (z)I < M for IzI < 1;

1m (m > 0) the class of functions F(C) E I such that IF(C)I > m for S(1) the class of functions f (z) = az + a 2 z 2 + , where a > 0, that are regular and univalent in the disk IzI < 1 and satisfy the inequality

1;

I1(z) I < 1 for IzI < 1. R C

T

R

L

the class of functions f (z) that are regular in the disk I z I < 1; the class of functions f (z) that are regular in the disk I z I < 1 and satisfy the inequality R(f (z)) > 0 for IzI < 1; that are regular in the disk the class of functions f (z) = z + c2 z2 + IzI < 1 and satisfy the inequality %f W)Nz) > 0 for IzI < 1; the class of functions f (z) that are regular in the disk I z I < 1 and that satisfy the inequality f (z 1) f (z 2) 1 for arbitrary z 1 and z 2 in the disk IzI < 1; the class of functions f (z) that are regular in the disk I z I < 1 and that satisfy the inequality f (z 1) f (z 2) -1 for arbitrary z 1 and z2 in that disk; L the subclasses of univalent functions in the classes R and L respectively.

The classes of functions that are subclasses of several of these classes will be denoted by the corresponding indexes. For example, S( ) is the class of

§1. BASIC METHODS

565

functions f (z) E S(k) f SM, and SR) is the class of functions f(z) E S(I) l SR. Other notation will be explained when introduced.

§1. Basic methods of the geometric theory of functions of a complex variable

10. The principle of areas. The principle of areas (area is nonnegative) was used in its simplest form by Gronwall (cf. the area theorem in §4 of Chapter II). By using the theorem on areas, Bieberbach proved the familiar theorems in the `class S and I (cf. §4 of Chapter II). The principle of areas was used in a finer fashion by Prawitz [1927] and Grunsky [1939]. Later, Goluzin (see Chapter XI, §6) proved the theorem on areas for p-valent functions. In more general cases, the use ,of the principle of areas leads to the proof of a "generalized theorem on areas". Furthermore, such a theorem is the starting part for obtaining inequalities of various kinds. One of the first theorems of this nature was obtained by Lebedev and Milin [1951]. Lebedev [1961] gave a generalization and strengthening of this result, which we shall now briefly expound. Let (oo, aI, , an) denote the class of all systems {fk(z)Jn of functions w = fk(z), for k = 0, 1, , n, that map the disk Iz < 1 conformally and univalently onto pairwise disjoint domains such that fo(0) = oo and fk(0) = ak, for n, where aI, aZ, k= 1, , an are fixed points. We denote by Bk(r), for It = 0, 1, , n, the image of the disk I z I < r, where 0 < r < 1, under the function w = fk(k). We denote by B (r) the complement of the set U k = 0 Bk(r) with respect to the extended w-plane. Suppose that a function Q(w) has a regular and singlevalued derivative in the domain B (r0), where 0 < ro < 1. The problem is to calculate the area a (r), where ro < r < 1, of the image of the domain B (r) under the function 6 = Q (w):

o(r)=

I Q'(w)dw,

(1)

B(r)

where dri is the element of area. Consider some single-valued branch of the function Q (w). Then, in the annulus r0 < I z I < 1 with appropriate cut, we have the expansion 00

Q (f4 (z)) =

= P(k)zZ q +

q-l

00

b(k)z'F + P(k) log Z,

q-0

(2)

in which only the coefficients 0" o and the branch of logz depend on the choice

SUPPLEMENT

566

of branch of the function Q W. If we compute o (r) and let r approach unity, we obtain in the limit the following inequality (generalizing the theorem on areas): n

n

CO

k=0q1

q I bgk' I9

n

00

< E Z q IP9k' 19 - 2 t k=0q=I

Z p'k'(bok) k=0

k

art

J=0

p

(3)

,

in which the form of the summation with respect to j is associated with the particular choice of the branch of Q W. It is easy to give various necessary and sufficient conditions for equality to hold in this conditional inequality. For n = 0, we obtain the generalized theorem on areas presented by Lebedev and Milin [1951]. If F (C) belongs to E and if the function Q (w) is regular in the complement of p0, where po> 1, under the function w = F(C), the image of the domain then, by setting CO

CO

Q(F(C))= J, bn Cn + b0+ Y Pnr.n' 1 < n=1

we have

0 for IwI < 1, 0 < t < m. Then, the solution w = f (z, t), f (z, 0) = z of the differential equation

W -wp(w, t), 0ct 0. Then, for sufficiently small real A, the solution w = f (z, t, A), f (z, 0, A) = z of the equation dw dt

1 + k (t) ea"19 U) w

--(1 + 1`'q1 (t)) w 1- k (t) eiA

(1) w,

0

t < 00,

is, for every t in the interval 0 < t < oc, a regular and univalent function of z in the disk Izl < 1 and satisfies the conditions t

f(0, t)=0,

fZ(0, t)=e

-1-AJ*li(z)di °

Relying on this fact, we obtain the varied function f*(z) E S: CO

f* (z) =f (z) + A 5 [L (z, t) lit (t) - IN(z, t) 71s (t)] dt -+ X2R (z, t),

(1)

0

where L (z,

t)1+ k (t) f (z, t)

t) =f (z) - f (z) f,f (z, (z, t) 1- k (t) f (z, t)

N(z, t)=f'(z) f ((ztt) ( 1

2k

k((t) ( (z, tt))'

and R (z, t) is a function that is uniformly bounded with respect to t inside the disk Izl < 1. By a suitable choice of T11(t) and rt2(t) in (1), we obtain the varied function f*(z) E S: X

f* (z) =f (z) + X

k=l

m

Ak P (x, zk) - k E AkQI (z, zk, t) k=1

M

- Jl E A kQs (z, Zk, t) + X9R* (z, k-1

t),

(2)

where the Ak are arbitrary complex numbers, the zk are points in the disk Izl < 1, R*(z, t) is a function that is uniformly bounded with respect to t inside the disk Izl < 1, and

§ 1. BASIC METHODS

P (z, zk) = Q1 (z, zk, t)

571

I

zkf (?k)

f (zk) )

f (z)2f-f(Z)s (zk) '

Z (zk, t\ = ( zff (zk, t)

Qs (z, zk, t) = (Zf Z (zk, t) f (zk, t)

f

( (z)

(z)

(f (z) -.f' (z)

f (z, 0 fz (z, t)

f (zk, t-f (z, t f (zk, t) --f (z, t)

f (z, t) +f (zk, t) f (z, t) l fz (z, t) 1 -f (zk, t) f (z, t) Jf

If we set t = 0 in formula (2), we obtain a formula that differs only insignificantly from Goluzin's formula (24), of §3, Chapter III. If we apply it to a function f (z) E S that is extremal in some problem, we obtain a differential equation for f (z) that contains the parameter t. We also obtain differential equations for f (z, t) and g (z, t), for the boundary of the image of the disk under the function w = f (z), and for the function k (t) in Loewner's equation. By this procedure, Lebedev [1951, 1951a] obtained only certain qualitative results for extremal functions in the problem of coefficients for functions in the class S. With greater success, Kufarev [1951, 1954, 1956b] (see also [1963]) combined Goluzin's variational method and Loewner's method of parametric representation. The variational-parametric method constructed by Kufarev for solution of extremal problems also leads to a differential equation for the function g (z, t) (and consequently for f (z) and f (z, t)). Furthermore, by using this equation and Loewner's equation, we reduce the problem of finding the function k (t) in Loewner's equation to that of solving a boundary problem for a system of differential equations. If from this system we succeed in finding the function k (t), then integration of Loewner's equation reduces to determining the extremal function and the solution of the extremal problem. By means of Kufarev's variational-parametric method, mathematicians of the Tomsk school have solved many difficult extremal problems in the geometric theory of functions (for some of these see §2 of this supplement). Goluzin's variational method has been extended to multiply connected domains (see Kufarev and Semuhina [1956], Aleksandrov [1963a], Gel'fer [19621) and to multivalent functions (see Gel'fer [1954, 1956], and Goodman [1958, 1958a, 1958b, 1958c]). It is easy to obtain variational formulas of the type of Goluzin's formulas for classes of functions that can be represented by means of a Stieltjes integral (convex, starlike, typically real, etc.). However, it seems more convenient to us in such cases to use other methods, in particular, other variational formulas. We shall speak of such methods in subsection 8°.

572

SUPPLEMENT

5°. The method of extremal metrics. These methods rest on inequalities between the lengths of curves belonging to a family and the areas of the domains filled by them. Grotzsch [19281 was the first to use this method as a method in the theory of univalent functions and he called it the "method of strips". His method is expounded in Chapter IV, §6. By means of it, Grotzsch solved numerous problems for both simply connected and multiply connected domains. Many of these can now be solved more easily by other methods, in particular, by the method of contour integration. The method of strips was perfected by Ahlfors [19301. He proved a very important inequality, known as the principle of length and area. We shall give a formulation of this result. Let f (z) denote a function that is regular in a domain B and let n (w) denote the number of roots of the equation f(z) = w that lie in B. Define ZR

p(P)= 2w J 0

n(pe10)dg,

p>0.

Let l (p) denote the total length of the level curves if (z)I = p in B and let a denote the area of B. Then 00

(P)' dP b

P P (P)

< 27ta.

The reader can find a proof of this result, certain of its applications, and a suitable bibliography in the book by Hayman [19581. In 1946, Ahlfors and Beurling gave a new formulation of the method of extremal metrics and this formulation later enabled Jenkins [19581 to give a further development of this method. The name "method of extremal metrics" is connected with

the fact that special metrics are introduced in certain of the studies in a domain filled by a family of curves. 6°. The method of quadratic differentials. The significant role of quadratic differentials in the solution of extremal problems was already disclosed in the familiar investigations of Grotzsch (see subsection 5°). In particular, in his theorems on the existence of conformal mappings of multiply connected domains onto canonical domains, the latter are determined in a number of cases by the trajectories of a quadratic differential. The role of quadratic differentials also showed up in the solution of extremal problems by the variational method. As we know, the method of interior variations usually leads to a differential equation for extremal functions and for the boundaries of the corresponding extremal domains.

§1. BASIC DOMAINS

573

Here, these equations are expressed by means of suitable quadratic differentials, and, to investigate the problems of integrating these equations, we need in a number of cases to have a qualitative picture of the trajectories corresponding to these quadratic differentials. This aroused interest in the behavior, local and global, of the trajectories of a quadratic differential. The first systematic investigations in this direction were the works of Schaeffer and Spencer [1950], Jenkins [1954a], and Jenkins and Spencer [1951]. Furthermore, the theory of the trajectories of quadratic differentials on a finite oriented Riemann surface was developed by Jenkins [1958, 1960c]. In particular, he obtained a very general "fundamental structural theorem " [1958, Chapter III].

The fundamental result of the method of quadratic differentials is Jenkins's "general theorem on coefficients" [1958, 1960a, 1963a]. This theorem deals with a set of univalent functions f1(z) that are regular or meromorphic in disjoint subdomains Al of a Riemann surface R and that map these domains into disjoint subdomains of R. It is assumed that the domains Al are bounded by the trajectories of the quadratic differential Q (z) dz 2 on R. 1) Under these conditions, one can establish an inequality containing the coefficients in the expansions of the functions f 1 (z) and Q (z) in neighborhoods of each pole of Q (z) of order m > 2. If Q (z) has in a neighborhood of z = o- (we use a local parameter that maps the pole into the point z = ea) and expansion Q(z) = a/z2 + terms of higher degree in z-1 (that is, a second-order pole of the quadratic differential), then an admissible function f1(z) must have an expansion f1(z) = az + ao + a1/z + terms of higher degree in z-1 . If Q (z) has in a neighborhood of z = oc an expansion Q(z) = az n 4 + terms of lower degree in z, where m > 3, an admissible function f1(z) must satisfy the condition f1(z) = z + az-k + terms of higher degree in z-1 , where k > %m - 2 (Jenkins [1960a]). Jenkins's theorem on coefficients was supplemented by Tamrazov [1965c] for the case in which the differential Q(z) dz2 does not have poles of order higher than 1. This very general theorem enables us to obtain systematically many familiar results in the geometric theory of functions: with it, one can easily prove the distortion theorems for functions that are univalent in the interior or exterior of the

1) With regard to the definitions in this subsection, see the monograph by Jenkins [1958].

SUPPLEMENT

574

unit circle, one can study sets of values of these functions and their derivatives, one can obtain a number of results regarding univalent functions without common values, and one can prove existence theorems for mappings of multiply connected domains onto canonical domains. With the aid of the "general theorem on coefficients", Jenkins posed and solved a number of difficult extremal problems. (With regard to some of these results, see §2 of this supplement.) 7°. The method of symmetrization. At the present time, we know several methods of symmetrization. We pause briefly for circular symmetrization proposed by P6lya. For a simply connected domain B containing the point z = 0, one can easily prove that there exists a unique simply connected domain B. with the following property: if a circle I z I = r, where 0 < r < no, intersects the domain B along arcs whose lengths add up to l (r), where 0 < l (r) < 277 r, then this circle in-

tersects the domain B,, along a single arc of length 1(r) with center at the point z = -r. The domain B. is symmetric about the real axis. If w = f(z) and w = f.(z), where f (0) = f,(0) = 0, are respectively functions that map B and B* univalently and conformally onto the disk Iwi < 1, then If (Z)

I

1-If(Z)I'

I f* (Z)

I

1-If*(Z)I

s

z E B.

There are other inequalities connecting certain quantities for B and B. The method of symmetrization rests on an inequality of this sort. It is expounded rather completely in the books by Hayman [1958] and Jenkins [1958].

We note that this method, especially in combination with the method of extremal metrics, made it possible to solve a number of extremal problems that had not been solvable by other methods (see §2 of this supplement). A generalization of the method of symmetrization to multiply connected domains has been given by Mitjuk [1964a] (see §3, subsection 2° of the supplement). 8°. The method of integral representations. The solution of extremal problems is simplified in the classes of functions that have an integral parametric representation. We shall consider classes A of functions that have a representation by means of a Stieltjes integral b

f (z) = S g (z, t) dR (t),

(1)

a

where [a, b] is a finite interval, where g (z, t) is a fixed function that is regular with respect to z in some domain B for a < t < b and continuous with respect to

575

§1. BASIC DOMAINS

in that interval for every z E B (the kernel of the class), and where 1t(t) is a function that is nondecreasing on the interval [a, b] and that satisfies the condition 1t(b) - p(a) = 1 (the parameter of the class). In what follows, we shall denote the set of such functions p (t) by M [a, b]. For example, the class C of functions f (z) such that f (o) = 1 is represented t

by the formula it

f(z) = S

1

z dp (t),

1, (t)

E M [- it,

it].

(2)

The class of typically real functions f (z) (the class T) is represented by the formula (see Chapter XI, §9) cx

f (z)

1- 2z cos t

zs d[., (t),

R (t) E M [- it, it].

(3)

In the solution of extremal problems and problems on finding the domains of the values of the functionals and systems of functionals in the class of functions that can be represented by means of a Stieltjes integral, we sometimes find the following theorems) useful. Suppose that the set 2 of points (x1, .. , x n ) in n-dimensional Euclidean space Rn is represented by the formulas b

xk= gk(t)dt',(t), k-1, 2,..., n,

(4)

a

where the gk(t) are fixed functions that are continuous on the interval [a, b] and iz(t) is a function in the class M [a, b] (these functions being, in general, different for different points of R). Then R is the closed convex envelope of the set of points

xk=gk(t), k=1, 2,..., n, ar tcb, and every point in R can be represented by formulas (4), where p(t) is a piecewise-constant function in M [a, b] with no more than n points of discontinuity. This theorem and special cases of it have been repeatedly used by various authors (sometimes, unfortunately, without reference to Caratheodory).

I) This theorem follows easily from the results of Carathe'odory [1911] (see also F. Riesz [1911]).

SUPPLEMENT

576

Caratheodory's theorem cannot by any means be applied to all extremal problems in the class of functions that is represented by formulas (1). In such cases, two variational formulas proposed by Goluzin (formulas (7) and (8) on p. 528) are very useful. It is possible to give other analogous variational formulas in the class A (see, for example, Zmorovic [1952]), and from one of these formulas we can obtain variational formulas of the type of Goluzin's formulas in the class S (see Chapter III, §3). Goluzin's formulas and Zmorovic's formulas usually lead immediately to the fact that the extremal function is of the form m

M

f (Z) = }J Akgk (z, t), k=1

Ak ,:- 0, E Ak = 1,

(5)

where the tk are points in [a, b], and they indicate the value of m. We give yet another variational formula in the class A. Let f (z) denote a member of A. Since g (z, r) E A for every r in a < r < b, the function f,x (z) = (1 - A) f (z) + Ag (z,

0 < A < 1,

r),

also belongs to this class and we have the following variational formula in the

class A: b

f* (z) =f (z) + A (g (z, t) - f (z) =f (z) + X S [ga (z, ti) - g (z, t)] dµ (t), O1 but it can hold for the second only when C 1 = C 2' By the same method, Bazilevic [1951] obtained, for all S I and C2 in the domain ICI > 1, a bound for the functional

log F(C1)-F(C9) F (C1) -f- F (C2)

C1- C2 C 7R C2

in the class £m21 . From it, we obtain the sharp inequalities

11

l+ IF(Cj)I

IC j_

F (Cl)+F(Cs) C1-C_ F(C1)-F(C9) C1H-Ca

1

IF(Cj)I l+ICjl 1JI F (Cj) I m

1+ IF( j)

1

I Cj l 1

1

IC-l

SUPPLEMENT

578

By using the generalized area theorem, Milin (see Lebedev and Milin [1951]) established the following sharp inequalities for F (C) E y( 2), for arbitrary p and 1/r> 1: q and for arbitrary C1 and C2 on the circle (1-r2)2(IP+qI+IP-ql)

_ F(C1)-F(C9)IP IC,-C2 P

(1 +rp) a (IP+qI-LP-q1)

X

I C1+C2 Iq

F(CC)+F(Ca) Iq

2 (IP+qI-IP-qI) C (1+r') I

r2)

_2

(IP+qI+IP-ql)

In this particular case these results have a simple geometrical interpretation and they have led to a number of covering theorems for the classes S and S. (see §2, subsection 2° of this supplement). General inequalities of the type of the distortion theorems, which constitute an application of formulas (2) on p. 118, have been obtained by Kolbina [1952a]. In particular, a generalization has been obtained of Theorems 3 and 4 of § 2, Chapter IV.

Nehari [1953] was the first to use systematically in the theory of univalent functions a method based on the classical minimal property of the Dirichlet integral for a harmonic function obtained by finding its singularities. His technique has led to a number of new results in this category of questions. We present one of his theorems. Let f(z) denote a function that is regular and univalent in the disk Izj < 1

and satisfies the inequality If W1 < 1 in that disk. Let z1, z2,..., zn denote points in zI < 1 and let all a2,..., an denote complex constants such that Then n

n

avaµ logf µ .v=1

(zz)

-f

(zµ)

µ

1 -< 1

µ.v=1

log 1 if (za)Z (z.

(2)

v

Using this inequality, Nehari obtained, for the coefficients in the expansion of a function f (z) = a0 + a1z + a2 z2 + that is regular in a neighborhood of the point, a necessary and sufficient condition for f (2) to be regular, univalent and bounded in modulus by unity in the disk IZ < 1. Together with the class of functions that we have considered above, let us also consider the class S. By letting M approach oo, we obtain the corresponding

579

§2. UNIVALENT FUNCTIONS IN DISK AND ANNULUS

results for univalent functions without the boundedness condition. In particular, we obtain a necessary and sufficient condition for a function to belong to the class S, one that is equivalent to Grunsky's univalence condition 119391 for meromorphic functions. By means of Loewner's method of parametric representation, Slionskii [19591 obtained in 1956 results for bounded univalent functions that are directly connected with a number of theorems in §2 of Chapter IV. Thus, Slionskii obtained a necessary and sufficient condition for a function to belong to the class Im. These results also include a strengthening of Goluzin's Theorem 1 of §2, Chapter IV for the case of the class Y,m. In particular, inequality (1) follows. Suppose also log f(X)_y (y)

(D (x, Y)

f(X)f(y)'

f (X) f (y)

1

-2

(x,y)=log

Ixl 2, Alenicyn [19561 obtained the following result in this direction by extending the corresponding results of Nehari: If the functions Q z), for v = 1, 2, , n, map the disk I zI < 1 univalently onto disjoint domains and if z = 0 is a regular point of these functions, then, for arbitrary real constants av 0 such that Iv=1 av = 0 we have the inequality

SUPPLEMENT

584

11 I f v (0)

v=1

Ia" < [

fk (0) -J v (0) I-9akav

(5)

1z.Ei k 1 onto a domain B2 contained in B. The domains B1 and B2 are disjoint. Let B (all a denote the set of points M (x I. x), 2 where xk = 11'k (0)I if ak 1' z and x2 = if' 2(.)' if a2 = . Lebedev [1955b] obtained the result that, in the case in which B is the entire plane (including w = Dc), the set &B (0, oo) is the domain 0 < x1 X 2 < 1, x1 > 0, with x1x2 = 1 only when f 1(c) = tc', for ICI < 1, and (2(C) = c/t, for ICI > 1 and 0< t < oa, and the set &B (a1, a2) is the domain 0 < x1x2 < Ia1-a2I2,

x1 > 0 with x1x2 = Ia1 - a2I2 only when f1(C) = (a1k1 - a2c)/(k1 - C), (2(C) _ (a1k2 - a2c)/(k2 - c), and Ik1k21 = 1. Furthermore, by applying Goluzin's variational method, Lebedev determined the region &B (a1, a2) for cases when B is the plane with the point oo excluded and when B is the disk Iwi < R. As a consequence of these theorems, one can again obtain the results of Lavrent'ev [1934] and Kolbina [1952] (here B is the unextended plane) and those of Kufarev and Fales [1951] (here, B is a disk).

585

§2. UNIVALENT FUNCTIONS IN DISK ANNULUS

Let Ro denote the set of all functions f (z) that are meromorphic and univalent in the disk I zI < 1 and satisfy the condition 1(0) = 0. Let R.. denote the set of all functions F(C) that are meromorphic and univalent in the domain ICI > 1 and satisfy the condition F (oc) = cc. Let 2 denote the set of ordered pairs

(A(z), F(c )) such that f(z) / F(c) for every z in Izl < 1 and every c in ICI > 1. For the class 9, Alenicyn [1956a] used Nehari's method to obtain the following inequality for arbitrary z in I z I < 1, arbitrary c in J CJ > 1, and arbitrary con-

stants a1 and a2: {al log Z'f (!) f (0) + 2a1; log (1 - f (Z) + log F' (C)

f (z)

1

F (t)/

F' (oo)l

Iall2log(1-Izl')- I%1glog(1

-Itll,).

(The values of the logarithms on the left-hand side are taken on suitable branches.) This result leads to inequalities involving various functionals both in the class of functions mentioned and for univalent mappings. In particular, Alenicyn obtained [1958] the following inequalities, which do not involve the derivatives of the functions in question:

Let f1(z) and f2(z) denote two functions that are meromorphic and have no common values in the disk I z I < 1. Suppose that fl(0) = 0 and (2(0) _ c. Then, for arbitrary points zI, z2 in the disk I z I < 1,

log(1

A(ZO) '-2log(1-IztI)(1--Iz9I9),

H(a - I ztz2 I/l"(1- I zj I') (l - I z2 I') These inequalities are sharp for arbitrary z1 and z2 such that Iz1l = lz21 = 1 with equality holding only in the case of univalent functions. These results lead immediately to a number of sharp inequalities for functions in the classes R and L and functions associated with them (see Alenicyn [1956a, 1958]). For other results in this direction, in particular their extension to a circular annulus, see §3, subsection 20 of this supplement. The same functional 11(zl)/f2(z2) in the class of functions indicated was studied by Lebedev [1957] by means of the variational method: for the class R, he found in explicit form the boundary of the range of the functional 6= f (zo)/F (Co) or, equivalently, of the functional 6= f (r)/F (p), where r = Iz0I and p = I Col. The pairs of functions corresponding to given boundary points of this range can be determined from the differential equations that we have obtained for

SUPPLEMENT

586

them. As a consequence, sharp bounds have been obtained for If (r)/F (p)I and I log (1 - f (r)/F (p)) 1, the radii of disks that prove to be the ranges of values of f'(0)/F(p) and f (r)/F'(oo), the range of values of the system (I f (r)I, 1/IF(p)I) in

the class I, and also the range of values of the function f W in the class R. The range of values of the system (If "(°)/f '(0)I, I f '(0)/F'(00)I) in the class R was determined by Ulina [1960). This led to the solution of the problem of maximizing the functional 'I = I f ,1(0)/f ,(0)I a I f '(0)/F'(°)I Q , a > 0, )9> 0 in

that class. Lebedev [1961) has obtained a number of results of extremely general nature in problems dealing with disjoint domains by beginning with the area principle. He considered the class 2 (-, a1, , an) of all systems {fk(zW, of functions fk(z), for k = 0, 1, , n, that map the disk I z I < 1 conformally and univalently onto disjoint domains in such a way that fo(0) = oo and fk(0) = ak, k = 1, , n, where a1, - , an are fixed points. By using the generalized area theorem for functions in this class (see §1, subsection 1° of this supplement), Lebedev obtained a number of distortion theorems in the class Dl (m, a1, an), in particular, inequalities of the same type as the distortion theorems of Goluzin in the class 1. The following results are among the applications of these general theorems. If {fo(z), f1(z){ E$!f(-, 0), then 2a

2n 1

If, (ele) Is d9

T7-;

d

I fo (eye) Is

d9 IbI, f1(z)=(I al - = const, 171I =1.

I b.z9)lz

a

From this and the familiar result of Rogosinski (Theorem 1, §8, Chapter VIII), it follows that, for functions f (z) in R (or L), tic 1 2a

S b

I f (e!e) I2 d9

1

with equality holding only for

f(z)

=

z P±jTP-li1z

,

Pal, I iI =1.

(7)


587

Inequality (7) strengthens the previous result of Lebedev and Milin [1951]: if f (z) E R (or L), then 2n 1

2x

C

I f(ere) I d6

5

with equality holding if and only if f (z) = 77z, where 1771 = 1.

In the problem of conformal radii of disjoint domains, Lebedev [1961] obtained the following result: , n, denote fixed numbers such that Let yk, for k = 1, 0 yk = 0. Then, , an), we have for { fk(z)In E 9l (oo, al, n

11

I

ak - at

2s {Tk1I}, (8)

where f0'(0) = limZ_0 1/zf0(z). The cases in which equality holds in (8) were all listed. Inequality (8) generalizes to the case of complex yk the inequality known earlier for real yk. A recent paper of Jenkins [1965] is closely related to this class of questions. In that paper, Jenkins started with the area principle and established a more general inequality of the distortion-theorem type for the class fit, defined above, of pairs of functions If (z), F (C)1 that depend on a large number of parameters. From that inequality, one can obtain a number of particular inequalities for the functions f (z) and F (C) and for functions in the classes R and L. Furthermore, by means of the extremal metric, Jenkins obtained a number of results for the functions f (Z) and F (C) in question, though omitting the requirement that they be univalent, and

for the classes R and L. A number of theorems of the distortion theorem type have been established for more general classes of functions, in particular, for functions that are p-valent in mean (in various senses of the word). A function w = f(z) that is regular in the disk I z I < 1 is said to be p-valent in mean with respect to circumference in IzI < 1 if it maps the disk IzI < 1 onto a Riemann surface R lying above the w-plane such that the total angular measure of open arcs lying on R over an arbitrary circle Iwi = p, where p > 0, does not exceed 217p. The function w = f (z) is said to be p-valent in mean with respect to area in the disk IzI < 1 if it maps the disk IzI < 1 onto a Riemann surface such

SUPPLEMENT

588

that the part of its area that lies over an arbitrary disk IwI 0, the equation f (z) = w either has exactly p roots in I zI < 1 for every w on the circle IwI = p

or (2) has fewer than p roots in IzI < 1 for some w on the circle IwI = p. If p is a positive integer, the property of being weakly p-valent is a less stringent condition than the property of being p-valent in mean with respect to circumference, but being p-valent in mean with respect to area does not imply weak p-valence or vice versa. We denote by FO (respectively F° or FI ), where p is a positive integer, the class of functions of the form C p+9Z7+9 +...' f (Z) disk= za + C p+ tzP+1 +

that are regular in the IzI < 1 and p-valent in mean with respect to circumference (respectively p-valent in mean with respect to area, weakly p-valent) in that disk. A powerful tool in the theory of conformal mapping is the following symmetrization principle. Suppose that a function w = f (z) = wo + c + , where c 0 and p > 1 , is regular in the disk IzI < 1. Let B = Bf denote its range in the disk IzI < 1. Suppose that the domain B* obtained from B by symmetrization about a straight line or ray passing through the point wo lies in a simply connected domain B0. Suppose that a function w = 4(z) = w0 + c' z + is regular and univalent in the disk I z I < 1 and maps it onto the domain Bo. Then ,z'

1

ICpI 1, the theorem was proved by Kobori and Abe [1959].

This theorem has led to a number of distortion and covering theorems for regu-

lar functions, in particular, for functions in the classes FO and F, (see, for example, Hayman [19581).

A generalization, also obtained by Hayman [1951a], of Fekete's familiar


589

theorem on the transfinite diameter (Theorem 3, Chapter VII, §3) for the case of nonunivalent mappings1) has also found applications in this class of equations. For weakly p-valent functions, Hayman [1951a, 19581 has obtained the following generalization of Bieberbach's classical bounds on the modulus of a function and its derivative in the class S (see Chapter II, §4). denote a function belonging to FP'. Then, Let f (z) = z' + cp +1 zp +1 +

cp+tI -2p

(9)

Also, for IzI = r, where 0 < r < 1, we have the sharp inequalities rp

(1+r)2

rp f(z)I (1-r)'p

if, (z) I -p (1 -r) If (z) I

=p(1-1 r)9p r) .

Furthermore, w = f (z) assumes every value in the disk IwI < 4-P exactly p times in the disk IzI < 1. Equality holds in all these inequalities only for the function

f (z) _

'P ap,

161=1.

(10)

By combining the method of extremal metrics that he developed and the results of the symmetrization method and by generalizing his own results for the class S [1953a], Jenkins obtained [1955, 19581, for f(z) E F?, the least upper bound of I f (r2) I, where 0 1. Furthermore, a is finite if and only if £ is a normal family, and a = 1 if and only if all functions in the family 2 are convex.) In particular, for a = 2, we obtain the classical distortion theorems for univalent functions (see Chapter II, §4). We now stop for certain results of the distortion-theorem type for doubly connected domains. In analogy with Loewner's parametric representation of univalent functions in a disk, Komatu [1943] gave a parametric representation of functions w = f (z) that are regular and univalent in the annulus I 1 with cut along a Jordan curve. Here, the circle IzI = 1 is mapped onto the circle IwI = 1 and f (1) = 1. Goluzin [1951e] gave a simpler variation of the solution of this problem in a different form. Let K (m/M) denote the class of functions w = f (z) that are regular and univalent in the annulus m 0, that do not vanish in that annulus, and that, for m < r1 < r2 <M, map the circle IzI =r into the interior 1 of the image of the circle I z I = r2. Let K (m/M, 1/1) denote the class of functions f (z) E K (m/M) for which f (1) = 1 and m < 1 < M. Following the method indicated in the article by Goluzin cited above, Li En-pir [1953] considered the problem of

SUPPLEMENT

592

generalizing Loewner's parametric representation for functions in the class K (m/M, 1/1) that map the annulus in < I z I < M onto the domain obtained from the w-plane by deleting two Jordan curves that terminate respectively at the points 0 and oc. Lebedev [1955c] gave the complete solution to this problem. The parametric representation that he obtained makes it possible to obtain a number of in-

equalities in the class K(m/M) and its subclasses with a complete clarification of the question of extremal functions. As an example, we give the following result. Let w = fl(z) and w = f2(z) denote functions in the class K (m/M, 1/1) that map the annulus m < I zl < M onto the entire w-plane with cuts along a segment of the real axis with endpoint at w = 0 and along the ray lying on the real axis extending in the opposite direction from the coordinate origin. Suppose that for f 1(z) the finite cut lies on the positive half of the real axis and for f2(z) on the negative half. For the functions f1(z) and f2(z), we have explicit expressions (see §2, subsection 4° of this supplement). We have the following theorem (Lebedev [1955c, 1955d]):

For f (z) E K (m/M, 1/1) and I z I= r, where mco, the classes IC are empty and the class I0 consists only of functions of the form f(z) = cfo(z), where I c I = 1. For c E (0, 1], the only extremal functions are functions of the form f (z) _ ESC(-z) in the case of the first problem and of the form f(z) = Egc(z) in the second, where I E I = 1 and w = IL (z) = c z + maps the disk I z I < 1 univalently onto the disk Iwi < 1 with cut along a segment of positive radius. For c = 1, this cut is constricted to the point w = 1. For c E (1, c0), the qualitative nature of the extremal mappings is different. The solution of the problem in this case depends on extensive use of the theory of quadratic differentials, the general theorem on Jenkins's coefficients, and Tamrazov's supplement [1965c] to that theorem. For arbitrary fixed c E (1, c0), the maximum pf in the class I is attained by all the functions (and only by these functions) f (z) = c f B(z), Ic I = 1, where the functions w = fe(z) = cecez + , - $(c) < 9 < $(c), constitute a one-parameter family of mappings one of which (namely, w = fo(z)) is symmetric about the real axis and for two of which the family is degenerate along one of the three arcs constituting the inner boundary of the image. The problem of minimizing of in the classes l (1 < c < c0) has an analogous solution. By the same procedure, we can make a complete analysis of the form of the extremal mappings and the question of their uniqueness in the problem of Duren mentioned above (see Tamrazov [1965c]). In the investigation of questions associated in an essential way with classes

594

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of mappings of a variable doubly connected domain that are not compact in themselves, the basic method is clarification of the existence of finite bounds for certain functionals in these classes. A number of extremal problems of this type for bounded mappings are solved in the papers by Tamrazov [1962, 1962a, 1962b, 1963, 1965b]. In these articles bounds have been found for the values of the functionals in question that depend only on the modulus of the doubly connected domain. We present the following results dealing with unbounded mappings of an annulus. Let g denote the class of regular univalent mappings w = f (z) of the annulus B: r < Izj < 1 for which the bounded component of the complement of the domain Bf with respect to the entire plane includes the points w = 0, 1 and contains the image of the circle IzI = r. Let pf denote the distance between the images of the circles Izl = r and Izj = 1 under the mapping in question. Whereas the problem of finding inf Z =1 If (z) I in the class Zg can be solved in a simple manner by the method of circular symmetrization (the solution of this problem also follows from the familiar results of Teichmnller [19381), solution of the problem of minimizing p1 in the class 9 required the application of variational methods (Tamrazov [1965c]). Specifically, we have the following theorem: Suppose that a function w = g(z) maps B univalently onto the entire w-plane with cuts -oo < w < _t (where t = t (r) > 0) and 0 < w < 1. In the class 21, we have pf > t with equality holding if and only if f (z) = g (e z) or f (z) = 1 - g (e z), where Ie I = 1.

2°. Geometric properties of a univalent conformal mapping. Among the results obtained under this heading a significant place is occupied by covering theorems. The distortion theorems mentioned at the beginning of subsection 1° of §2 of this supplement have led to a number of such results. Specifically, Bazilevic [1951] obtained the following sharp bound on the linear measure of a covering of the circle jwl = p by the image B (r) of a disk of the form lzi < r < 1 under functions

in the class S: If a function f (z) E S, then, for arbitrary r in the interval 0 < r < 1 and arbibitrary x _> e"I er, the intersection of the circle Iwl = p with the domain B (r) has linear measure not exceeding the intersection of that circle with the domain B*(r) corresponding to the function f *(z) = z/(1 - ez)2, where I kI = 1. This result was obtained independently, with the method of areas, by Lebedev

and Milin [1951] for p2 > (2/f)r/(1 - r2).


595

The extremal property of Koebe's function that we have mentioned leads to bounds on the area a(r) of the image of the disk I z I < r, where 0 < r < 1, under mapping by functions in the class S and its subclasses S(k) , to bounds on the mean modulus of the function, and to other bounds (cf. the articles mentioned above). For example, in the class S(2) Bazilevic obtained the inequality c (r)

rrl (l ±r4)Y + c (r).

This inequality deviates from the sharp inequality by no more than c (r), which approaches zero as r -- 1. For the mean modulus of a function f (z) E S, he proved the inequality 2%

2a

S If (re"')I dp
r in such a way that the circle 1z] = r is mapped into the circle I wj = r. Goluzin showed (see p. 144, Theorem 2) that the maximum value of the third transfinite diameter of the complement of the image of the domain ICI > 1 under a function in the class I is equal to the transfinite diameter of the continuum consisting of three segments connecting the coordinate origin with the points representing the cube roots of 4 (see also articles by Garabedian and Schiffer [1955], Reich and Schiffer [1964]. However, the four intervals connecting the origin with the points representing the four fourth roots of 4 no longer form an extremal continuum maximizing the fourth transfinite diameter in the family of all continua whose outer conformal radius is equal to 1 (Garabedian and Schiffer [19551). In this connection, we give the following result: by assuming symmetrization of a special form, Szego [1955] showed in a very simple way that the outer conformal radius of a system formed by n segments the product of whose lengths is given and which issue from the coordinate origin at angles each equal to the next is maximized when all these segments have the same length. It follows from this and from Lindelof's principle that, if the function w = f (z) E S*, then, for n arbitrary points al, a2, , an lying on n rays issuing from the point w = 0 at equal angles and belonging to the boundary of the image of the disk I z I < 1 under the mapping w = f (z), we have ata9 ... an

with equality holding for a function of the form f (Z) = z/(1 + e z")2"" , where jcl =1


597

n

Associated with the familiar theorem on 1/4 solving Szego's problem on the covering of n segments for the class S (cf. p. 174) is the following result of Xia Dao-xing [1958]: let KP (where p is a fixed number > 1) denote a family of convex domains B in the w-plane that have the following properties: each domain B includes the point w = 0, and its interior conformal radius about w = 0 is equal to 1; for every boundary point of the domain B, there exists a circle of radius p n, that passes through that point and encircles the domain B. Let wk, for k= denote the boundary points of the domain B C KP that lie respectively on n arbitrary rays issuing from w = 0 at equal angles: T,a (p) = min min max I Wk I B E K, (wi..... w,) k

By use of the extremal metric, one can show that this minimum is attained by a .domain bounded by a regular n-sided circular polygon whose sides are arcs of a circle of radius p. This enables us easily to determine the quantity Tn(p) If we let p approach oo, we get the solution of Szego's problem for the entire class S. By the same method, Xia Dao-xing [1956] obtained certain theorems on the covering of intervals under a univalent conformal mapping of an annulus. Tamrazov [1965] obtained some very general theorems on the covering of curves under a univalent conformal mapping that generalize the familiar results on the covering of segments and provide us with more precise information. For example, he obtained a precision of Rengel's theorem [1933] on the covering of n segments in the case of functions that are meromorphic and univalent in a disk, and the analogous result for an annulus. Closely associated with questions on the covering of segments is the wellknown problem of Lavrent'ev (see Chapter IV, §4) on the determination of the region maximizing the conformal radius in the family of all simply connected regions containing the coordinate origin but not containing - or a given system of points aI, , am (for m > 1). This problem was solved by Gel'fer [1958, 1960] for the case of a family of fundamental domains of groups of fractional-linear transformations associated with elliptic or automorphic functions. Koebe's theorem on the covering of the disk Iwi < % by the image of the disk I z I < 1 under a function w = f (z) C S has been sharpened by Jenkins [1960] to the case of the class SR. Let Bf denote the image of the disk IzI < 1 under the mapping w = f (z) and define D= fl fESR Bf. Applying the phrase "general theorem on coefficients", Jenkins found the set D determining the boundary

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functions of that set in terms of the theory of quadratic differentials. On the basis of Lavrent'ev's results in the above-mentioned problem of maximizing the conformal radius in their simple particular case, Kuz'mina [1962] determined in explicit form the set D 1 = fl fE S [B B f], where B 1. is the domain symmetric to fu Bf about the real axis. It follows from the form of the boundary functions of the set D 1 (they have real coefficients) that the sets D 1 and D coincide. A number of generalizations of Koebe's theorem on the sharp bound for the modulus of the function f (z) E S in the disk I z I < r, where 0 < r < 1, have been obtained by the Romanian mathematician Mocanu [1957, 1958]. Fel'dman [1963] obtained more general results in the same direction. For example, for a domain b contained in the disk Izl < I and bounded by a closed Jordan curve defined by parametric equations in polar coordinates, he found in certain particular cases the domains U fE S b f and fl fE s b f, where b f denotes the image of the domain b under the mapping w = f (z), and he also solved the analogous problems for the class S. Cernikov [1962] found the set Ufssbf in the case in which b is a disk I z l< r, where 0 < r< 1. We give one result dealing with problems of this nature for an annulus. Let {B} denote the family of doubly connected domains in the z-plane that contain the point z = 0, have the circle I z I = 1 as one of their boundary components, and satisfy the following condition: under a univalent mapping of the domain B onto the disk Iwl < I with cut along a segment of positive radius that maps the unit circle. and the coordinate origin into themselves, the point z = 1 is mapped into the point w = 1 . Tamrazov [1962b] determined explicitly the majorant domain containing an interior boundary of an arbitrary domain in the family of domains in I B I with given fixed Riemannian modulus and pointed out a number of properties of these domains. The behavior of the level curves (the images of concentric circles Izl = r < 1) also shows clearly the degree of distortion under a univalent conformal mapping of a disk. This question has been well studied for sufficiently small values of r. For example, limits of convexity and starlikeness have been found (see Chapter IV, §5). It is also natural to study the behavior of functions in the class S on circles with a displaced center. The well-known results of Nevanlinna and Grunsky asserting that an arbitrary function in the class S maps a disk with center at the point z = 0 of radius not


599

exceeding R. = 2 - V3 onto a convex domain and that such a function maps a disk with the same center but of radius no greater than RS = tanh (7r14) onto a starlike domain (Chapter IV, §5) have been supplemented by Aleksandrov [1958a, 1959, 1960] as follows: 612 with center at the point %', where Every disk of radius p < 2 - v/3 161 < 1, is mapped by an arbitrary function in the class S onto a convex domain. The bound on p is sharp. Every non-Euclidean disk with non-Euclidean center at a point 6, where 161 < 1, the non-Euclidean radius of which does not exceed rr/2 is mapped by an arbitrary function in the class S onto a domain that is starlike about f (6). This result is sharp. Goluzin (see Chapter IV, §5) introduced the concept of "generalized starlikeness". The following theorem of Aleksandrov [1960] deals with his results on the limits of generalized starlikeness in the class S: Every non-Euclidean disk with non-Euclidean center at a point 6, where 161 < 1, the non-Euclidean radius of which does not exceed nn/2 is mapped by an arbitrary function in the class S onto a domain of the form Dn[f (6)]. This result is sharp. In particular, every disk of radius not exceeding Rns = tanh (nn/4) with center at the point z = 0 is mapped by every function in the class S onto the domain Dn(0).

Of special interest is the problem of the change in the curvature of the level curves under a univalent conformal mapping. We present the basic results that have been obtained in this direction. Miros`nicenko [1951] first showed that the following inequality holds, giving the greatest lower bound of the curvature K. of a level curve Lr (the image of

the disk Iz = r) in the class S for 2 -

< r < 1:

(1-+19 r K'l 1-4r+r' ll -r He also showed that the inequality

K,

1- 2 (k+1) rk + rek (1 + rk)2/k

r

(I - r)o

gives the greatest lower bound in the class S(k), where k > 2, for

SUPPLEMENT

600

k + 1 - k2 + 2k < rk < 1. These bounds are attained by the functions f (z) _ z/(1 + z)2 and f (z) = z/(1 + zk)2/k, respectively. Beginning with Loewner's parametric representation, Korickii [19601 showed that these inequalities are valid in the entire unit disk. Up to the present time, the least upper bound of the curvature of the level curves in the class S has not been found. By using Theorem 4-of §3 of Chapter IV, Korickii [1957, 19601 found the following least upper bound for the curvature KP (the image of the circle = p > 1) for functions in the class 11k1, where k > 1:

K_ P[P°k+2(k-1)pk+1] p < (P -1) oft+ 1)2/k

which is attained by the function F(C) = (ck + 1)2/k/C. In certain subclasses of S and 1, the problem has been solved completely. For example, ZmoroviZ [19521 obtained greatest lower and least upper bounds for the curvature of the level curves in the classes S(k), where k > 1, and in the class I by using the integral representations of these classes. .

In 1952, Bazilevic first found in the class S* the greatest lower bound of the curvature of the level curves by the method of embedding of the class S* in a larger special class of functions that are regular in the disk IzI < 1 and in the class S(k)* , for k _> 2; Korickil [1955) did this by the same method. Korickii [19551 also obtained the greatest lower and least upper bounds of KP in the class 1(k), for k > 1. Aleksandrov and Cernikov [19631 first found the least upper bound of the curvature of the level curves in the class. S(k)*. For k = 1, this least upper bound is attained for all 0 < r < 1 by the function f (Z) = z/(1 + z)2 . For k > 2, this bound has a different form for different values of r: for 0 < r < r0, it is attained by the function f (z) = z/(1 + zk)2/k ; for rp < r < 1, it is attained by the function

f (z)-

51412/h

[(1 + -",?I G -Z where ji1, p.2, and ro depend only on k and r (with 0 < p.l, N.2 < 1, µl + µ2 = 1).

The investigation of the behavior of the level curves as we approach the boundary of the unit disk is a difficult problem because the behavior of the level curves as r --+ 1 may prove to be very complicated. For example, it is natural to assume that the level curve Lr1 embedded in the curve Lr2 (where rl < r2) is


601

"more regular" in the geometrical sense than the curve Lr2. However, Bazilevic and Korickii [19531 showed that, under a mapping of the disk IzI < 1 by a function in the class S, the number of points of inflection of the level curve Lr and the number of points of destruction of its starlikeness (points on the level curve at which the direction of rotation of the radius vector changes as the point z moves around the circle IzI = r in a specified direction) can change nonmonotonically with increasing r; that is, if r1 < r2, it may happen that the level curve Lr1 has more points of inflection or more points of destruction of starlikeness than the level curve Lr2 does. Furthermore, this paradoxical behavior of the level curves can occur even for bounded functions in the class S. Nonetheless, for sufficiently rapid growth in the absolute value of a function, there will necessarily be a certain amount of regularity in the behavior of its level curves as r ---' 1. Specifically, Bazilevic and Korickii [19621 proved the existence

of absolute constants ak and as such that: If an arc of a level curve Lr of a function f (z) E S is contained in the annulus

akR(r) 1, and that map the circle jzj = 1 onto the circle lwl = 1. Suppose that a function w = f *(z, R), where f `(1, R) = 1, maps the annulus B univalently onto the domain Iwl > 1 with cut w > P.

If f(z) E t(B) and if df is the shortest distance from the origin to the inner boundary of the domain B f, then df > P with equality holding if and only if f (z) _ c f *(riz, R), where IE I = 1711 = 1.

Kubo obtained an analogous inequality for bounded functions in R (B): l f (z) l <M, where M> R, for z E B. For univalent functions, this last result had already been shown by Grotzsch [1928] and Komatu [1943]. The method of symmetrization proposed by Szego [1955] for starlike domains has been generalized by Marcus [1964] for arbitrary sets. Let SZ denote an open subset of the z-plane that does not include the point oc and let z0 denote a point in 11. For an arbitrary disk z - zo i < p, where p > 0, contained in El and for arbi-

trary 0 in the interval 0 < q < 27r, let us consider the set E of points z E U such that 1Z - z01 = r > p and arg (z - z0) = 0. Let us define L,

5

dr ,

R ('P) = P eXP LP

.Furthermore, for arbitrary n = 2, 3, .. , we define

LpLn n-I

2ak

k=0

n-I RI In (T

R(n)

+2nk)

= P exp LP(n) (p).

k=0

Obviously, R(") (0) is independent of p. We shall refer to the transformation of the set 1 into the set of points z such that z - z0 = re`s, where 0 < r < R"")(0) and 0 < 0 < 27r, as the S" transformation of the set SZ with center at the point z0. Under the S"-transformation, an arbitrary domain is mapped either onto the plane or onto a domain that is starlike about the point z0 and possesses for n > 2,

604

SUPPLEMENT

n-fold symmetry of rotation about that point. As Marcus has shown, such symmetrization does not increase the interior radius of the domain. By using this last result, Marcus obtained the following theorem: is a member of R. Suppose that the Se,Suppose that f (z) = z + c2 Z2 transformation with center at the point w = 0 is applied to the domain Bf. Then,

V-4

,

0 1, then, in the notation used in the theorem of subsection 1°,

IcPI-Icfl 11141

Pk,

k?1

are the nonzero roots of the equation f (z) - w0 = 0 in where z 1, z2, , zk, the disk I z I < 1 and p 1, P2' ' ' p k' are their multiplicities. Mitjuk [1965d] also obtained an analogous strengthening of Hayman's theorem on % that we have given and of Marcus's theorem. In connection with the above-mentioned result of Kubo [1958], we mention that Mitjuk [1965b] obtained a more general covering theorem for functions that are regular in an annulus. Specifically, he considered a broader class of functions and studied the multiplicity of the covering produced by them. We mention also that Mitjuk [1965c, d] extended Hayman's theorem of subsection 1° to the case of functions that are weakly p-valent in an annulus. For functions of this last class, he obtained a theorem on the linear measure of the covering of arcs of a circle, thus generalizing the theorem of Kubo [1954] mentioned above.


605

As a supplement to the results of Goluzin and Bermant regarding a covering performed by regular functions in a disk (see Chapter IV, §6), we present the following theorem (Jenkins [19511).

If f (z) C R, then in the image B f of the disk fz1 < 1 under the mapping w = f (z) there exist n straight line segments issuing from w = 0 at equal angles the sum of the lengths of which is arbitrarily close to n/r*, where p* = rrr* 2 is the area of the preimage of the star of the domain B f. In connection with these results, we have the following theorem of Antonjuk [1958] for functions that are regular in an annulus. Suppose that the function w = f (z) is regular in the annulus B: 1 1, where C is a contour in B that is not homologous to 0. Let B f denote the star of a finite doubly connected Riemann surface B f onto which the annulus B is mapped by the function w = f (z) with respect to a system of rays issuing from the point w = 0 (the existence of such a star is proved). We have the following inequality connecting the area P* of the star Bf with the area p* of its preimage:

(p* +'t) (P* +'r)

2Rb

with equality holding only for a function of the form f (z) = e z, where lc I = 1. Hazalija [1958] had already obtained in 1951 the inequality p * > 77 (R2 - 1). A number of results dealing with the problem of finding a lower bound for the area of the image of a disk and an annulus and also of the star of that image have been obtained by Mitjuk [1961b, 1965a-d]. 3°. Bounds on the coefficients of univalent functions. The problem of the

coefficients in the expansions of univalent functions has played a significant role in recent years in the general turn taken by the geometric theory of functions. As we know, for functions f (z) = z + c 2z2 + in the class S, it consists in determining for every n > 2 the ranges of values of the system of coefficients Ic2, , c, of functions in that class. A particular case of this problem is the problem of finding sharp bounds for the coefficients. Success in this search would lead to proof or refutation of a well-known conjecture of Bieberbach. In recent years, a number of problems associated with this class of questions have been solved. Schaeffer and Spencer [1950] used their own form of the variational method to determine explicitly the range of values V3 of the system of coefficients Jc2, c3}

606

SUPPLEMENT

in the class S. Although V3 is a body in 4-dimensional real Euclidean space, the property of symmetry with respect to rotation admits a simple description of this set if we determine its three-dimensional cross-sections for which, for example, b = 0 or b 3 = 0 (c k = ak + ib k, where ak and b k are real). The boundary of the 2 range of values of each of the systems {a2, a3, b3), {a2, b2, a3} is expressed in explicit form in terms of elementary functions. + , Garabedian and In the class I of functions F(C) = C + ao + Schiffer [1955] obtained by the method of variations the sharp inequality Ia3I < + e -6 . This proves that the bound found by Goluzin [1949c] for the coefficient of C-3 for odd functions in the class I is sharp throughout that class. In 1955, Garabedian and Schiffer [1955a] first proved the validity of Bieberbach's conjecture for the fourth coefficient. This proof uses both the method of variations and Loewner's parametric method. The ranges of values of the systems of initial coefficients in classes of bounded functions in the classes S(k) and J(k) were investigated by Bazilevic [1957] by means of Loewner's parametric representation method. Thus, in the class SM) of functions f(z) = z + Ck+lzk+1 + CZk+lz2k+1 +---,where cvk+l = a vk +1 + ib vk +1 for k = 1, 2, , Bazilevic determined in explicit form the range of values of the system {jck+1l, IC2k+111 and, under the assumption that bk+l = b2k+1 = 0, the range of values of the system {ak+1, a2k+l1 and he exhibited all their boundary functions. Bazilevic posed and solved the same problems for the class 1mk) and also for the classes of functions inverse to the functions in SM and J(k) , respectively. Charzynski and Janowski [1959] determined the range of values of the system {c 2, c 31 in the class SM by a method analogous to that used by Schaeffer and a-1C-1

Spencer [1950].

Interesting bounds on the coefficients of univalent functions, bounds which from a new standpoint explain how the vanishing of some of the initial coefficients affects the growth of the remaining ones, were obtained by Jenkins [1906b], by extending his general theorem on coefficients to the case of quadratic differentials of a suitable form [1960a]. We present some of his results.

Let n denote a positive integer and let F (C) denote the function F (C) _ C + 2°°_ (a . /C') E Y,, where a . = 0 for j < (n - 1)/2. Then,

607


IanI -

nl +

(1)

Let M denote the class of functions f(z) meromorphic and univalent in the disk Izi 2, if the function f(z) = z + Y._ j-2c . z1 E M, and if c . = 0 for j < (n + 1)/2, then >

c,1j n 2

1

(2)

.

Equality holds in (1) and (2) only for the functions n + 1, a) and n, a) = (1 + e`naC n)2/n . n - 1, a) respectively, where

F-1 (z-1;

Inequalities (1) and (2) were obtained by Goluzin for the classes 20 and S respectively as special cases of results that are valid for p-valent functions (see Chapter XI, §6). If n > 0 and 00

Cl C

F(C)=C+ -n

then, for arbitrary real V j, we have the sharp inequality

{- a "1an+t + 8e Flan

- 2 ne

6 8(n+1) 89- 4(n+1)

S91og 4 -}- n1 1 ,

4.

(3)

For n = 0, inequality (3) remains valid for all F(C) E 1 when 0 < S < 4. All the extremal functions are indicated. A consequence of inequality (3) for functions of the type indicated is the inequality a3n+1 I C n

l{-

l

[1 + 2 exp

1- 2 n n 2/]'

(4)

From inequality (3) we get an analogous result for functions that are univalent in a disk. In particular, we have the following sharp inequality: Let n denote an integer greater than 2 and let f(z) denote the function f(z) = z+Y.I-n C. z1EM. Then C2.-1

n

1 l [1 + 2 exp 1- 2

n n

(5)

SUPPLEMENT

608

Inequalities (4) and (5) had been obtained earlier by Goluzin [1949c] for functions in the classes l(n +1) and S(" ^1) respectively.

In particular, inequality (4) shows that the function /

tn+i)

2/(n+ 1)

=C+ n+ 1 L_n-+-...,

which maximizes Ia1I and Ia2I in the class E and satisfies for all n the differential equations for the function maximizing Ia"I in that class is not extremal for a2"+1, n > 1. In connection with this, we note that all odd-subscript coefficients Clunie [1959a] constructed along the lines of Littlewood's well-known construction a function in the class I for which I a" I > n 1 +8 , where 8 is a positive constant, for an infinite sequence of values of n. This function maps the domain ICI > 1 onto a domain bounded by a Jordan curve. At the same time, the inequalities 2 IanIC n+ l, n=0, 1, 2, ...,

(6)

are valid in the class I*. Specifically, Clunie [1959] proved inequalities (6) in a simple manner for n > 1 and for functions F(J) = C+ a1/C + a2/C2 +. _ C 2. The only extremal function is the function c-1 F "(eiJ), where IE I = 1. By using the same idea of proof, Pommerenke [1962a] obtained inequalities (6) for all functions

F(C)=C+a0+a1/C+... EY.* for n=0, 1, 2,..

.

In 1960, Charzynski and Schiffer [1960] gave a much simpler proof of the inequality I c 41 < 4, which shows that this inequality is actually more elementary than Loewner's result Ic31 < 3. These authors also obtained [1960a] a geometrical

proof of the inequality Ic4I < 4. In both cases, the proof is based on an inequality for suitably chosen combination of the coefficients c 2, c 3, c 4, of a function in the class S. From this inequality, the inequality Ic4I < 4 is obtained by means of simple inequalities that follow from the area theorem. We shall present the first of these proofs of the inequality Ic4I < 4 in a somewhat modified form. The proof rests on Grunsky's familiar condition [1939] for univalence (see also Chapter IV, § 2). We present this result in the special case of functions that are regular in the disk I Z I < 1:1)

1) The article by Garabedian, Ross and Schiffer [1965], gives an elementary proof of this theorem that is based on the area method.


609

A necessary and sufficient condition for a function f (z) = z + c2 z 2 + that is regular in the disk Izl < 1 to be univalent in jzj < 1 is that the inequality amnxmxn

(7)

I xn I9

f

m,n-1 n=1 be satisfied for every sequence xn} such that the right-hand member of this inequality converges, where the coefficients amn are determined by the expansion Co

In f (z) -f (C)

z-C

m,n=0

amnxMV,Ixl 0. Then, from (8) we have

1) Inequality (8) can also be obtained directly from inequality (12) of Chapter IV, §2, with n = 3 and the same values for x1, x2, and x3.

SUPPLEMENT

610

3 +ffl {2(cq-0(cg- 4 c.) -+- 2c,-l2c'}, and, in view of (9), this leads to the inequality

cs-l 4-I

2c$-2q2'

If we now set c9 = 2xe"P,

l = 2xe

'

0 C X C 1,

a cos 2 p,

we obtain, using the notation y = Isin(3q5/2)I, acs-1I=2xy,{12c'-19c$}=- 3x3+ 3 xsy9

and, consequently, cd

? + 8x9 - 14x3 3 3

- (8x9

4 3

x8)y9 + /

8

j-3

x vi - x9y.

By taking the maximum of the right-hand member of this expression with fespect to y, we arrive at the inequality

cEC

x$+461__x 9

3

+8x9

,

3

0Cx 4.

This result was obtained by investigation of univalent functions based on use of extremal properties of certain systems of functions. We point out a few results characterizing the asymptotic behavior of the coefficients in the expansion of univalent functions. Hayman [1955] showed the close connection between the maximum modulus

M(r, f)=maxIzI_rIf (z)I, where 0 < r < 1, of the function f(z)=z +c2z2+... C S and the coefficients of the two functions f (z) and

Js(z)= J (Z9)=z +b3Z3+... E S

,

namely, the limits

lim (1 - r)' M (r, f) = lim r-l n-oo

1 c' n

= lim

I b9n_1 I'== is, C 1,

(10)

n-+oo

where of = 1, exist only for f (z) = z/(1 - e`9z)2 . One result of this is the fact that, for every function f (z) C S, there exists a number Nf for which Bieberbach's inequality Ic,l holds for n > Nf. However, it still remains an open question whether there exists a number N independent of the function f (z) in the class S beginning with which Bieberbach's inequality is satisfied. Hayman [1958a] proved the following: If Cn is the least upper bound of Icnl for all f(z) C S, then the limit

lim nCn-a exists. (This limit is equal to 1 if Bieberbach's conjecture is true.)

S2. UNIVALENT FUNCTIONS IN DISK AND ANNULUS

613

For odd functions, it is well known that n > 1,

ban-t I < A.

where A is an absolute constant. It has also been shown that A > 1. Furthermore, a uniform inequality of the form Ib2R_II < 1 for n > N is impossible (see Chapter IV, § 7). The exact value of the constant A is unknown. By using the bound for the area of the image of the disk I z I < r < 1 under a function in the class S(2) (see the article by Lebedev and Milin [1951)), Gong Sheng (Kung Sun) [19551 obtained the inequality

n > 1.

I b9._t I `\ 2.55.

Let us denote by S(a) and S(2) (a) respectively the subclasses of functions in theti classes S and S(2) for which (10) holds with fixed of = a, where 0 < a Y2, the well-known sharp bound on the increase in the coefficients t+

where A (a) depends only on a. (See, for example, Hayman [1958] 3.3; an analogous inequality is valid for p-valent functions.) For a < Y2, Pommerenke [1961/62]


617

obtained the following result: If f (z) = z + n=2 CnZn C S and If (z)I < M/(1 - IzI)a in IzI < 1, then

a-

cnM'A(a)aC Furthermore, for a < Y2,

I cnl=o(n '/') as n -ioo. Analogous relations hold for functions that are p-valent in mean with respect to area. As Pommerenke [1962a] has shown, for starlike univalent functions, the condition M(r) = O((1 - r)_a), where 0 < r < 1 and M(r) = max,Z _r If (z)I, leads to the relation

IcnI=O(,'+°) for all a in the interval 0 < a < 2. Clunie and Pommerenke [1966] proved this same result for functions that are close to convex. With regard to the problem of coefficients for special classes of univalent functions, we mention the following results: In the class S*, the problem of the coefficients has been solved. Specifically, Hummel [1957] determined in explicit form the set Vn in (n - 1)-dimensional complex Euclidean space consisting of all points (c2, , cn) that are put in correspondence with at least one function S. Here, Vn is determined from its twof(z) = z + c2 z2 + C + cnzn + dimensional cross section Cn , for which c 2, ,c1 are fixed. If (c 2, ,c_1)

is an interior point of Vn _1, then Cn is a disk whose radius does not exceed 2/(n - 1). This is a sharp bound since it is attained when c2 = C 3 = = cn-1 = 0 , where IE I = 1. In particular, by a function of the form f (z) = z/(1 the boundaries of the cross-sections C* and C3 are determined by the equations Ezn-1)2/(n-1)

c2 = 2eand c3 =.3cz/4+ e`9(4- Ic21)2/4. In the class K (see p.619) of functions f(z) = z + c2z2 + to convex in the disk IzI < 1, Bieberbach's conjecture

that are close

ICJ 1, Pommerenke [1962] has obtained

and-0(n) Supplementary to the results given above, on the growth of adjacent coefficients for univalent functions, Pommerenke [1963] proved the following theorem: If f(z) = z + 2n=2Cn z' E K, then

Ilcn+tl-IcnH
1.

Pommerenke also indicated the form of all functions for which this inequality is sharp with respect to the order of dependence on n. In particular, for all functions in S* except the functions .f (Z) =

z (1 - ei°'z) (1 - e1B3z)

where 01 and 02 are real, there exists a S = 6(f) > 0 such that

C.+1 I -I C.1=0 4°. Structural formulas for various classes of analytic functions. A large number of results have been obtained for classes of functions that can be represented by some "structural formula" or other, in particular, by a Stieltjes integral. It is well known that the class C of functions f (z) = 1 + c 1z + that are regular in the disk I zI < 1 and satisfy the condition Rf (Z) > 0 in that disk can be represented by the Stieltjes integral

.f (z) = S e,t+z dµ (t),

(1)

where IL(t) is a nondecreasing function in [- a , n] such that 1L67)-1A-i7)= 1. On the basis of a simple connection between functions in the classes C and T, Robertson [1935] and Goluzin [1950a] used this formula to obtain an integral representation for the class T (see Chapter XI, §9). Beginning with this representation, Goluzin [1950a] obtained in the class T a number of distortion and rotation theorems and also bounds on the mean moduli of f (Z) and f '(z) on the

circle IZI=r 0, where g (z) is some function that is independent of f and regular and convex in jzl < 1. This class was introduced by Kaplan [19521. Every close-to-convex function is univalent in I z I < 1 and f (z) is close to convex if and only if f '(z) A 0 and 9y

11

Zf (z) de

- t
- e ,

SN{ _x

f' (z) }d0 = 2pts

for f (z) E Se(p) in the annulus 0 < p < I zI < 1, where p depends on the function

z=re`B,-rr 2, in the class T. As an example, we present the following theorem, which is directly related to the result given above for the class T (c 2). Let T (c 2, c 3) de-

note the subclass of all functions in T with fixed coefficients c2 and c3. The range of values SZ (T (c 2, c 3)) of the functional f (z) for fixed z in the disk I z I < 1 in the class V c 2, c 3) when ` z 0 is a convex lune bounded by

arcs of two circles (these are explicitly defined). In the case lz = 0, the range of 1l (T (c 2, c 3)) degenerates into a segment of the real axis. In connection with the results given above, we mention that Lozovik [1963a] presented a geometrical method of determining the range of values of the system of functionals that can be represented by a Stieltjes integral and also the subset of that set that satisfies given additional conditions. As an illustration, let us find the range of values of the functional f '(z) (where z is a fixed point in the C3 with disk I zj < 1) defined on the class of functions f (z) = z + c 2z 2 +


627

, cn. The article by Nosenko [1963] deals with the same fixed coefficients c2, question. that Now, let C denote the class of functions of the form f (z) = 1 + c 1z + are regular and satisfy the inequality Rf (z) > 0 in the disk I z I < 1 and let CN denote the subclass of functions in C of the form N f (z) _

I Xhk k-I

"Pk

1 + ze 9o-jib I

'

where the 0k are arbitrary real numbers and the Ak are arbitrary nonnegative numbers satisfying the equation AI +A2 + + AN = 1. Robertson [1962, 1963] used a variational method that he constructed to show that a number of extremal problems in the class C are satisfied by functions belonging to the class C2. Zmorovie proved [1965] in a very simple way the following theorem, which contains, in particular, the corresponding theorem of Robertson: Let cF (C; w) denote a real function of the complex variables t;' and w that is single-valued and bounded in the domain l > 0, lwl < oc. Suppose that, for arbitrary fixed t; in RC > 0, this function attains its greatest lower bound or its least upper bound in any disk 1w - woI < R on the boundary of that disk. Then, the quantity

J= extr

extr (f (z), zf'(z)) fEC 1z1 -r 0, Iwkl < - for k = 1, 2, , n and one studies its extrema

onadisk Izl=r a in the disk Izl < 1, where 0 < a < 1. For a = 0, we then obtain a theorem proved

SUPPLEMENT

628

by Aleksandrov and Cernikov [19631 by a different procedure (see §2, subsection 2° of this supplement). Aleksandrov and GutljanskiT [19661 proved a number of results regarding boundary functions of ranges of values of functionals of a general form that are de-

fined on the class K and its subclasses and also on the class C. They used internal variations in these classes. As an application, they found in explicit form the range of values of the functional f '(z) on the class K. This range of values had been found earlier by Krzyi in 1965, who started directly with an integral representation of the class K. Aleksandrov and GutljanskiT [19651 obtained other quite general results regarding boundary functions of ranges of values of weakly differentiable functionals and systems of functionals defined on classes of functions that can be represented as a sum of Stieltjes integrals. They did so by using variational formulas in these classes of the same type as the variational formulas of Goluzin (see Chapter XI, N. We present one of the results obtained by these , z , (for p = 1, 2, ) denote fixed points in authors for the class C. Let z 1, the disk Izl < 1. Define um = f (m) (zi) and vm = um (where r n= 0, 1, , si and j = 1, p).// Let >

I

(f)=J

>

1

u01, Vol, ..., us11vs11; ...; u0p, v0p, ..., uspp, vspp)

denote a weakly differentiable function defined on the class C. The only boundary functions in the range of values of the functional 1(f) defined on the class C are functions in the class CN, where N < s 1 + + SP + 2p - 1. It follows, in particular, that the only boundary functions of the functional 1(f) = J (f (z), r (z), f '(z), f '(z)) defined on the class C are functions in the class CN for N < 2. We mention some of the other recent results that are based directly on a parametric representation of these classes of functions. Gel'fer [19651 applied Goluzin's variational formulas in the class T to the problem of finding the maximum in that class of tle°'zf'(z)/f(z)j for fixed z and y (where I z I < 1 and y is real). Galperin [19651 obtained an integral representation of the class of functions f(z)= CO + c lz + that are regular in the disk Izl < 1 and satisfy there the conditions If (z)I < 1 and f (z) 0. Using its simple connection with the class C, they arrived at a number of inequalities in that class. Gel'fer and Kresnjakova [19651 conthat are sidered the class H 8 (where 8 > 0) of functions f (z) = co + c 1 z +

§3. FUNCTIONS ANALYTIC IN MULTIPLY CONNECTED DOMAINS

629

regular in the disk Izi < 1 and satisfy for arbitrary r in the interval 0 < r < 1 the condition 2%

2zc

S

I f (rele) JA d9 ` 1.

A parametric representation of the class H S follows from the representation found by Smirnov [1932] for the broader class of functions that are regular in the disk IzI < 1 and satisfy for arbitrary r in the interval 0 < r < 1 the condition 2i

127t } I f (reie) Ia d6)'18 1, denote arbitrary distinct points in the domain B. Let aO,j , al,j , .. . ... , a Pj.l , where j = 1, , s, denote arbitrary coefficients (not all zero) such

that Is =1 a0 j = 0. Define s

P

a*,j

S (z;

+ ao.1 log (z

J=1 k=1

j

l

- cj)J

(a function of the singularities). Under these conditions, we have the theorem: For an arbitrary given angle 0 in the interval -zr/2 < 0 < zr/2 and an arbitrary given function of the singularities S(z; C, a), there exists a function, unique up to an additive constant, (D9(z) = S(z; a) + F,9(z) where F9(z) is regulars) in the closure of B, that possesses the following property: on every boundary curve CIU of the domain B, every branch of the function a `Bc0(z) has a constant imaginary part. The functions

P (z) =

[,D

2

R (z) - (Do (z)] and Q (z) = c

2 [(D

(z)

(Do (z)]

solve more general extremal problems than do functions of the same kind that had 1) In what follows, when we speak of regular and meromorphic functions, we mean single-valued regular and meromorphic functions.


631

been considered earlier (see Garabedian and Schiffer [1949]). a), Thus the function P (z) defined in terms of a suitable function S(z; and only this function minimizes the area of the image of the domain B in the class of all functions that are regular in that domain and that have fixed initial segments in their Taylor expansions at given points of B (segments such that this class does not include a constant). This provides a generalization both of the theorem of minimization of area that was given in Chapter V, §4 and of Bergman's theorem 419501, which he obtained by a different method and in a different form, on the minimization of area in the class of all functions that are regular in the domain B with fixed initial segment of their Taylor expansions at a given point of the domain.

We define the outer area A (f) of a function f (z) in a domain B as the quantity (finite or not)

A (f) _ -

litn

-

c

f (z) f (z) dz,

where the C(') are the boundaries of the domains B"), which approximate the ;interior of the domain B as v ---> -. Then the function Q (z), and it alone, maximizes the external area in the class of all functions of the form f (z) = S (z; C, a) + g (z), where S(z; C, a) is a fixed function of the singularities and g (z) is an arbitrary function that is regular in the domain B.

The properties of the function 19(z) can be used to obtain theorems of the distortion-theorem type for functions that are meromorphic and pvalent in mean with respect to area in a multiply connected domain. A very special case of the corresponding theorem on the range of values of a certain functional defined on the class of functions that are meromorphic and p-valent in mean with respect to area in a given multiply connected domain is the following: In the class of all functions f (z) = zP (1 + a1/z + ) that are p-valent and regular in the domain Izl > 1 except for a pole at z = with cut in a neighborhood. of a fixed point z0 oc of that domain and that have a representation of the form f(z) = f(z0) + z0)P + , the range of values of the functional log (f (P) (z0)/p!) is the disk

wI 1 and it is the special case, with n = 1, of Theorem 1 of Chapter IV, §3.

632

SUPPLEMENT

Komatu and Ozawa [1951, 1952] established the extremal properties of the same type as the theorems on the distortion of univalent conformal mappings of a given multiply connected domain onto a plane with cuts along two mutually perpendicular systems of parallel segments, along two systems of radial segments and circular arcs, etc., and they showed that proof of the existence of such mappings can be reduced to a consideration of a domain of considerably lower-order connectedness than the given domain. Jenkins [1963] used a slight modification of the extremal-metric method to find a number of more general extremal properties of certain analogous mappings including some that are not necessarily univalent but with a unique fixed simple pole in the domain in question. He was the first to obtain theorems of this type for functions that are regular in a domain. We present one of his theorems. Let B denote a finite domain in the z-plane that is bounded by closed analytic curves C. for j = 1, . , n + 1. Let P 11 P2, P3, and P4 denote arbitrary points on C1 numbered in the order in which they are encountered as one moves around that curve in the positive direction relative to B. Let m, p, and q denote nonnegative integers such that m + p + q = n. Let L (B; m, p) denote the class of all functions f (z) that are regular and univalent in the domain B and that map that domain onto the domain bounded from without by the contour of some rectangle 0 < 82 f (z) < L, 0 < `` f (z) < 1 under the condition that C 1 is mapped into that con-

tour, the points P1, P2, P3 and P4 are mapped respectively into the points 0, L, L + i, and i, the curves C, , for j = 2, , m + 1, are mapped into horizontal cuts, and the curves Cr . , for j = m + 2, , m + p + 1, are mapped into vertical cuts. Finally, suppose that F (B; m, p) denote the class of all functions f (z) that are regular in B and that possess the following properties: The mapping w = f(z) maps C 1 into the contour of some rectangle of the type indicated with the same correspondence between the points P1, P2, P3, and P4 and the vertices of the rectangle; the functions f (z) are regular on C. for j= 2, , n + 1; on each of these boundary components 0 < 82 f (z) < L, and 0 < f (z) < 1; on Cr , for j = 2, , m + 1, the functions f (z) have a constant imaginary part; on Ci , for j = .

, m + p + 1, they have a constant real, part, It follows from Jenkins's results [1957] that there exists a function l (z) = f (z; B, m, p, 0) E i (B; m, p) such that the mapping w = l (z) maps the boundary curves C , for j = m + p + 2, , n + 1 into the horizontal cuts and a unique function q (z) = f (z; B, m, p, rr/2) (B; m, p) such that in the mapping w = q (z) these boundary curves correspond to the vertical cuts. Let us denote by L 1, L 2' and L the values of the length m + 2,

.


633

corresponding to the functions 1, q, and f respectively. We define A = L area f (B). Under these conditions, we have the following theorem: For arbitrary f3 and nonnegative y such that S + y = 1, the sum A + (f3-y) L is maximized in the class F (B; m, p) only by the function 01 + yq and this maximum is equal to 132 L I - y2L 2. If m = p = 0, then the function 01 + yq is univalent and the preceding result is valid for the class E (B; m, p). In this last case, the function 131 + yq maps B onto a domain for which each finite supplementary continuum has in common with an arbitrary horizontal or vertical straight line no more than a single segment (or point). Alenicyn [1965] established a number of properties of the functions D0(z), P (z), and Q (z) mentioned above for analogously defined functions in the case of mappings of a finitely connected domain onto many-sheeted Riemann surfaces with cuts along two mutually perpendicular systems of parallel segments. In particular, he obtained a generalization of certain of Jenkins's results [1963] that deal with meromorphic functions.

Various area theorems are immediately associated with conformal mappings onto canonical domains. Meschkowski [1952, 1953, 1954] obtained an area theorem generalizing to functions that are univalent in a multiply connected domain the Gronwall-Bieberbach theorems (see Chapter 11, §4) and also a number of distortion theorems on functions that are meromorphic in a multiply connected domain. Abe [1958] particularized Meschkowski's area theorem for the case in which the domain in question is a circular annulus. Alenicyn [1964] generalized Meschkowski's area theorem to functions of a more general form, namely, to functions with singular part S (z; C, a) and with nonnegative exterior area in a given multiply connected domain. In particular, he obtained an area theorem for functions that are meromorphic and p-valent in a disk and have the property that the sum of the orders of the poles in that disk is equal to p, and this gives a generalization of Goluzin's familiar area theorem (see Chapter Xl, §6) for p-valent functions. Milin [1964a, 1964b] discovered a yet more general proposition of the areatheorem type. Specifically, let B denote a finitely connected domain in the extended z-plane with boundary C consisting of closed analytic Jordan curves and suppose that oc E B. Suppose that a so-called Taylor system of functions tq5n(z)}, where n = 1, 2, , is constructed satisfying the following conditions: The functions (b (kn(z) are regular in the closure of B and they have an expansion of the k, where a > 0, for n = 1, 2, form n (z) , in a neighborhood of k=nank z nn

634

SUPPLEMENT

z = Do; the derivative'sOn(z) constitute an orthonormal system in the domain B:

S S k (z) T. (z) da = Skn, k, n = 1, 2, ... , that is complete in the class Lo (B) of all functions that are regular and squareintegrable with a unique inverse in B. Let {cn(z)I, for n= 1 , 2, , denote a system of functions that are regular in the closure of B except for a pole of order n at the point at infinity, in a neighborhood of which they have a Laurent expansion without constant term, and that satisfy on every boundary curve of the domain B relations of the form cn(z) = On(z) + const. The system of pairs of the form (dn(z), $n(z)), where n = 1 , 2, , is called a "Laurent system of functions" for the domain B. The set of all points of B for which the Green's function gB(z, 00) of the domain B satisfies the condition gB(z, -) < log R is called the boundary annulus set B I ,R' where 1 < R. Milin showed that If a function 0 (z) is regular on the boundary annulus set B 1 R, then 1) i/i'(z) can be expanded on that set in a Laurent series 00

00

(z) -n=1 Lj

ArsDn (z)

2] Xn?, (z); n=1

n-1 is defined by the same limit as the quantity A (f) mentioned above. A-1

where A

Thus, we have the following generalization of the area theorem: Let Vi(z) denote a function that is regular on the boundary annulus set of a domain B. If A(qt) is nonnegative, then 00

00

I IX.I1=En=1I'Arcls

n=1

A number of works have been devoted to the question of the existence and extremal properties of conformal mappings of infinitely connected domains onto canonical domains. Grotzsch [1955/561 gave a proof of the existence and uniqueness of a univalent conformal mapping of an infinitely connected domain onto the unit disk with concentric circular cuts that did not involve the condensation principle. Reich and Warschawski [19601 obtained a simple proof of the existence of a conformal mapping of an arbitrary bounded domain onto a disk with concentric circular cuts and, if that domain has at least one nondegenerate boundary


635

component, onto a disk with radial cuts, and they found certain geometric properties of these mappings. We give two fundamental and typical results of their study. Suppose that K is the disk Izi < 1 with concentric circular cuts such that the circular projection of the boundary of the domain K onto each of its radii is of linear measure 0. Let w = f (z) denote a function that is analytic and univalent in K and that maps K onto a bounded domain A such that the circle I zI = 1 is mapped into the exterior boundary component of the domain A and f (0) = 0. Suppose, finally, that F (z) = log [f (z)/z1, F (0) = log f '(0), and A is the area of A. Then, f' (o) I exp

27;

I F' (z) I9 dx

dy

In particular I f '(0)I < A/rr, with equality holding only for a function of the form f (z) = az, where a is a constant. From this we get the following theorem. Let SZ denote a bounded domain in the z-plane that includes z = 0. Let F denote the class of all functions f(z) that have the following two properties: 1) f (z) is regular and univalent in 12; 2) f (0) = 0, f'(0) > 0, and If (z)I < I for all z E 11. Then, a function w = c(z) such that 0'(0) = maxfeF f'(0) maps the domain SZ onto the unit disk with concentric circular cuts. 2°. Extremal properties of analytic functions in multiply connected domains. The necessary and sufficient conditions found by Grunsky [19391 for univalence of a function that is regular in a multiply connected domain except for a simple pole at z = oo were then obtained by Bergman and Schiffer [19511 (in a somewhat different form) by a considerably simpler procedure, one that starts from the theory or orthonormal families. They developed a theory of kernels of the first and second kind of a given domain, they established a relationship between these kernels and univalent conformal mappings of a multiply connected domain onto canonical domains, and they obtained a number of new extremal properties of functions that are univalent in a multiply connected domain. The more fundamental of these results are treated in the article by Schiffer [19531.

Bergman and Schiffer chose as their point of departure for finding these necessary and sufficient conditions for univalence of a function an inequality which they obtained, a simple special case of which is the inequality I {f (z), z} + 6zcl (z, z) I 0 and IEII = IE21 = 1.

Starting with an examination of suitable Dirichlet integrals, one can carry over (see Alenicyn [19661) the results of Bergman and Schiffer [19511 mentioned above to the case of several functions that are univalent and have no common values in a multiply connected domain. Specifically, we have the theorem: Let B denote a bounded and finitely connected domain in the z-plane bounded by closed analytic Jordan curves. Let fk(z), for k = 1, , n denote functions that are regular and univalent in the domain B and suppose that they have no common values in that domain. Let kµ denote arbitrary points in B and let ak denote arbitrary constants for µ = 1, , N and k = 1, , n. Then, N

n

akµa'kv [Uk (CkW Ckv) + 10 Gkµ, CkA

{

µ,r=1 k=1 R

N

f!

2

alµakv (fI

l=l0 is satisfied, this function is unique. It follows that the problem of finding max , m, If (a)I is satisfied only by a function f (z, a, C1), v = 1, R(1) (B, a,

for which F'(a, a) = maxv=l,...,m1'(a, a, Cu). We denote this function by F(z, a). P Obviously, the function F (z, a) in general is not unique. If the domain B is the disk I z I < 1, then F (z, a) = F (z, a) = (z - a)/(1- iz) and, consequently, F'(a, a) = F'(a, a) = 1/(1 - lal z) Let f(z) denote a function that is meromorphic in a domain B. We shall say that f(z) is subordinate in B about a point z1 E B to a function g(r) that is meromorphic in the disk r I < 1 if, for every z in B, we have f(z) = g (tv (z)), z 1). We indicate this relation by where w (z) is a function in the class I

R(1)(B,

.f (z) -< g ('t), z E B.

(1)

zl

If in addition W (z) E R(1)(B, z 1), we shall say that the function f (z) is univalently subordinate to the function g (r) and we shall indicate this by f (z) -< 9 (,), z E B. zl

(2)

If a function g (r) is univalent in the disk I rl < 1, the relation (1) means simply that the function f(z) does not assume in the domain B those values that the function g (r) does not assume in the disk I r 1 < I and that `(z 1) = g (0). The relation (2) means in addition that f(z) is univalent in B. If the domain B is the unit disk I z I < 1 and zl = 0, then (1) and (2) are respectively the familiar relations f (z) -< g (z) and f (z) -< g (z), I z I< l. From the definitions given above, we immediately get the following subordination principle: If f(z) - 0) in the disk IwI < Me x but it is not always true that this function does not have simple uncovered points in a larger disk. When the domain B is the unit disk, this theorem yields Landau's familiar theorem [19291.

Certain results from Robinson's paper [19431 were presented in Chapter XI, §4. We shall now elaborate on this question, using the remaining results of that paper and Alenicyn's paper [19611. Let Bq denote a circular annulus q < IzI < 1, where q > 0. We know from Robinson's paper [19431 that, for arbitrary given a1 E Bq , there exists a unique function H (z; a1) that is regular in the closure of the annulus Bq, that is univalent in Bq itself, and that satisfies the conditions

H(a1; a,)=- 0, IH(z;a,)I1=vIa1I I H(z; a,)I1,z1=1=1, H(l;a1)=1. By using the corresponding results of Robinson [19431, we find, after some easy transformations,

§3. FUNCTIONS ANALYTIC IN MULTIPLY CONNECT ED DOMAINS

647

A

co

H(z;

z

a)=et!,

at

`\

1 -qsna,z' (1 _ qsn a,z )

1-a12n-1I1-gsnalzqsn= 1

,

atz /

where a is defined by the condition H(1; a1) = I. For arbitrary given a E Bq and z0 E Bq , we set a0(z0) _ (q/IaI) e ' arg z0 H(z, 1; a0(z0), a) = z-' H(z; a0(z0))H(z; a). Then, for an arbitrary given point z0 in the annulus Bq and an arbitrary function f (z) in the class Rt a), we have I

(zo) I c I H (zo, 1; ao (zo), a) I

(3)

with equality holding (when z0 a) only for f(z) = eH(z, 1; a0(z0), a), where IEI = I. In the case of the annulus Bq, we have (Alenicyn [19581) the following relationship between the functions exhibited by Ahlfors and Robinson:

F(z, a)=e`Pz 1Hlz;-?)H(z; a), a

3=-argH'(a, `

a

a)

Starting with Robinson's inequality (3), which is obviously a generalization of the first of the inequalities in the Schwarz lemma (see Chapter VII, §0 to the case of a circular annulus, we can obtain theorems of a general nature that enable us to carry over to the case of a circular annulus certain distortion theorems with which we are familiar for a circle. We present one of these general theorems and its simplest consequences. For brevity, we write H (z, 1; a0(z), a) = H W. Let lfl denote a class of functions g (z) that are meromorphic in the disk IzI < 1 and let Tla(B) denote the class of all functions f(z) that are meromorphic in the domain B and that are subordinate in that domain about a point a in it to one of the functions in the class TI.. We indicate this relationship between these two classes by writing 3J (B) -< ,,. Let us suppose that I (w) is a real function defined for all values of w assumed in the unit disk by functions in the class R*. Let M1(IzI) (M2(lzI)) denote a decreasing (increasing) function of IzI for IzI n. If u (z) _ 1k =1 =1 pki pk are constants, then n < m - 2 + I'= Nk. The problem considered in this theorem generalizes, in particular, the problem (ak)1' (where ak B and the Ak are arbitrary of finding supfE(1)(B) k=1Ak1 I f'(a)I. The first of complex constants) and the problem of finding these problems was investigated by Garabedian [1949]; the second was solved by 1

IF'r

Ahlfors.

With regard to Havinson's generalization [19611 of the inequality in the Schwarz lemma to arbitrary domains, the question is as follows: let C denote a closed bounded set in the z-plane, let B denote whichever of the domains complementary to C includes the point z = oo, and let y denote an arbitrary contour encircling C. The problem is to investigate the properties of the extremal functions in the problem of finding supfER(1)(B) (1/27r) I ff (z) dzI, which is the analogue, for the case a = -, of the problem of finding supfER(1)(B) If`(a)I when a -. Havinson showed that the extremal function f*(z) is unique up to a factor eyO (where a is a real constant), that all its zeros except a zero at the point z = oo lie in the convex envelope of the boundary set C, and that this function assumes in B all values in the unit disk except possibly a set of values of analytic capacity zero (see op. cit.). Because of the complexity of the formulation of the other properties of the extremal function f*(z), we cannot give them here.

BIBLIOGRAPHY FOR THE SUPPLEMENT H. Abe

1958/59 On univalent functions in an annulus, Math. Japon. 5, 25-28. MR 21 #4248.

N. I. Ahiezer and M. G. Krein 1938 Some questions in the theory of moments, Naucn. Tehn. Izdat. Ukrain. Harkov; English trans., Transl. Math. Monographs, vol. 2, Amer. Math. Soc., Providence, R. I., 1962. MR 29 #5073. L. V. Ahlfors 1930 Untersuchungen zur Theorie der konformen Abbildung and der ganzen Funktionen, Acta Soc. Sci. Fenn. A. 1, no. 9, 1-40. L. V. Ahlfors and A. Beurling 1946 Invariants conformes et problemes extremaux, C. R. Dixieme Congres Math. Scandinaves, 341-351, Jul. Gjellerups Forlag, Copenhagen, 1947.

MR 9, 23.

I. A. Aleksandrov 1958 The range of values of the functional I = w"f'(w)m/f(w)o lf (w)I 1 in

the class S, Tomsk. Gos. Univ. Ucen. Zap., no. 32, 41-57. (Russian) 1958a Conditions for convexity of the image region under mappings by functions regular and univalent in the unit circle, Izv. Vyss. Ucebn. Zaved. Matematika no.6(7), 3-6. (Russian) MR 23 #A3834. 1959 On the star-shaped character of the mappings of a domain by functions that are regular and univalent in the circle, Izv. Vyss. Ucebn. Zaved. Matematika no. 4 (11), 9-15. (Russian) MR 26 #2601. 1960

1963

Domains of definition of some functionals on the class of functions that are univalent and regular in a circle, in Series on Modern Problems in the Theory of a Complex Variable, Fizmatgiz, Moscow, 39-45. (Russian) MR 22 #5744. Boundary values of the functional J = J (f, f, f', f ') on the class of

holomorphic functions univalent in a circle, Sibirsk. Mat. Z. 4, 17-31. (Russian) MR 26 #3892. 1963a Variational formulas for univalent functions in doubly connected domains, Sibirsk. Mat. Z. 4, 961-976. (Russian) MR 27 #5902.

651

652

BIBLIOGRAPHY FOR THE SUPPLEMENT

1963b Variation of non-univalent analytic functions, Trudy Tomsk. Gos. Univ. Ser. Meh.-Mat. 163, 155-159. (Russian) 1963c Extremal properties of the class S(w0), Trudy Tomsk. Gos. Univ. Ser. Meh.-Mat. 169, 24-58. (Russian) MR 29 #3627. 1964 Geometric properties of schlicht univalent functions, Trudy.Tomsk. Gos. Univ. Ser. Meh.-Mat. 175, 29-38. (Russian) MR 32 #2571. The range of systems of functionals, Trudy Tomsk. Gos. Univ. Ser. 1965 Meh.-Mat. 182, 59-70. (Russian) MR 33 #7514. I. A. Aleksandrov and V. V. Cernikov 1963 Extremal properties of star-like mappings, Sibirsk. Mat. Z. 4, 241267. (Russian) MR 26 #6387. I. A. Aleksandrov and V. Ja. Gutljanskii 1965 Extremal problems for classes of analytic functions having a structural formula, Dokl. Akad. Nauk SSSR 165, 983-986 = Soviet Math. Dokl. 6 (1965), 1531-1535. MR 33 #5900. 1966 Extremal properties of close-to-convex functions, Sibirsk. Mat. Z. 7, 3-22. (Russian) MR 33 #4270. I. A. Aleksandrov and V. I. Popov 1965 Solution of a problem of 1. E. Bazilevic and G. V. Korickii on starshaped arcs of level curves, Sibirsk. Mat. Z. 6, 16-37. (Russian) MR 30 #3202.

Ju. E. Alenicyn 1956 On univalent functions in multiply connected domains, Mat. Sb. 39(81), 315-336. (Russian) MR 18, 292. 1956a A contribution to the theory of univalent and Bieberbach-Eilenberg functions, Dokl. Akad. Nauk SSSR 109, 247-249. (Russian) MR 18, 293. 1958 On functions without common values and the outer boundary of the domain of values of a function, Mat. Sb. 46(88)9 373-388. (Russian) MR 20 #6532. 1961

1962

1964

An extension of the principle of subordination to multiply connected regions, Trudy Mat. Inst. Steklov. 60, 5-21; English transl., Amer. Math. Soc. Transl., (2) 43 (1964), 281-297. MR 25 #1281. On the ranges of variation of systems of coefficients of functions representable as a sum of Stieltjes integrals, Vestnik Leningrad. Univ. 17, no. 7, 25-41. (Russian) MR 25 #2179. Con formal mappings of a multiply connected domain onto many-sheeted canonical surfaces, Izv. Akad. Nauk SSSR Ser. Mat. 28, 607-644. (Russian) MR 29 #1315.


653

Conformal mappings of multiply connected domains onto surfaces of several sheets with rectilinear cuts, Izv. Akad. Nauk SSSR Set. Mat. 29, 887-902. (Russian) MR 32 #2587. 1966 On univalent functions without common values in a multiply connected domain, Dokl. Akad. Nauk SSSR 167, 9-11 = Soviet Math. Dokl. 7 (1966), 305-307. MR 33 #7515. V. A. Andreeva, N. A. Lebedev and A. V. Stovbun 1961 On the range of certain systems of functionals in various classes of regular functions, Vestnik Leningrad. Univ. 16, no. 7, 8-22. (Russian) 1965

MR 24 #A243.

G. K. Antonjuk 1958 On the covering of areas for functions regular in an annulus, Vestnik

Leningrad. Univ. 13, no. 1, 45-65. (Russian) MR 20 #2434. I. Ja. Anevic and G. V. Ulina 1955 On ranges of values of analytic functions represented by a Stieltjes integral, Vestnik Leningrad. Univ. 10, no. 11, 31-42. (Russian) MR 17, 599.

I. E. Bazilevic 1951 On distortion theorems and coefficients of univalent functions, Mat. Sb. 28(70), 147-164. (Russian) MR 12, 600. 1951a On distortion theorems in the theory of univalent functions, Mat. Sb. 28(70), 283-292. (Russian) MR 13, 640. 1957 Regions of the initial coefficients of bounded univalent functions of p-fold symmetry, Mat. Sb. 43(85), 409-428. (Russian) MR 20 #3991. 1959 On an estimate of the mean modulus in a class of bounded univalent functions, Mat. Sb., 48(90), 93-104. (Russian) MR 22 #1679. 1961 On an estimate of the mean modulus and coefficients of univalent functions, in Series on Modern Problems in the Theory of Functions of a Complex Variable, Fizmatgiz, Moscow, pp. 7-41. (Russian) 1961a An asymptotic property of the derivatives of a class of functions that are regular in a disk, in Series on Modern Problems in the Theory of Functions of a Complex Variable, Fizmatgiz, Moscow, pp. 216-219. 1964 1965

(Russian) Generalization of an integral formula for a subclass of univalent functions, Mat. Sb. 64(106), 628-630. (Russian) MR 29 #3625. On the spread of the coefficients in the expansions of univalent functions, Mat. Sb. 68(110), 549-560; English transl., Amer. Math. Soc. Transl. (2) 71 (1968), 168-180.

654


I. E. Bazilevic and G. V. Korickii 1953 On some properties of univalent conformal mappings, Mat. Sb. 32 (74), 209-218. (Russian) MR 14, 632. 1962 Some properties of level curves on univalent conformal mappings, Mat. Sb. 58(100), 249-280. (Russian) MR 26 #2600. I. E. Bazilevic and N. A. Lebedev 1966 On the dispersion of the coefficients of mean p-valent functions, Mat. Sb. 71(113), 227-235; English transl., Amer. Math. Soc. Transl. (2) 72 (1968), 107-117. MR 33 #7516. S. Bergman

The kernel function and conformal mapping, Math. Surveys, no. 5, Amer. Math. Soc., Providence, R. I. MR 12, 402. S. Bergman and M. Schiffer 1951 Kernel functions and conformal mapping, Compositio Math. 8, 205-249. 1950

MR 12, 602.

A. Bielecki and Z. Lewandowski 1962 Sur certaines majorantes des fonctions holomorphes daps le cercle unite, Colloq. Math. 9, 299-303. MR 27 #2625. M. Biernacki

1936 1956

Sur les fonctions univalentes, Mathematica (Cluj) 12, 49-64. Sur les coefficients tayloriens des fonctions univalentes, Bull. Acad.

Polon. Sci. Cl. III, 4, 5-8. MR 17, 957. S. Bochner

1922

Uber orthogonale Systeme analytischer Funktionen, Math. Z. 14, 180-207.

E. Bombieri 1963

Sul problema di Bieberbach per le funzioni univalenti, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 35, 469-471. MR 29 #3622.

C. Caratheodory

Uber den Variabilitatsbereich der Fourier'schen Konstanten von positives harmonischen Funktionen, Rend. Circ. Mat. Palermo 32, 193-217. T. Carleman 1922/23 Uber die Approximation analytischer Funktionen durch lineare Aggregate von vorgegebenden Potenzen, Arkiv Mat. Astronomi Fysik, 17. V. V. Cernikov 1962 Extremal properties on certain classes of analytic functions with real coefficients, Dissertation, Tomsk State University. (Russian) 1911


655

Z. Charzynski and W. Janowski 1959 Domaine de variation des coefficients A2 et A3 des fonctions univalentes et bornees, Bull. Soc. Sci. Lettres ,Lodz, 10, no. 4. MR 23 #A3841.

Z. Charzynski and M. Schiffer 1960 A new proof of the Bieberbach conjecture for the fourth coefficient, Arch. Rational Mech. Anal. 5, 187-193. MR 22 #5746. 1960a A geometric proof of the Bieberbach conjecture for the fourth coefficient, Scripta Math. 25, 173-181. MR 22 #9615. J. Clunie

1959

On meromorphic schlicht functions, J. London Math. Soc. 34, 215-216. MR 21 #5737.

1959a On schlicht functions, Ann. of Math. (2) 69, 511-519. MR 21 #6438. J. Clunie and C. Pommerenke 1966 On the coefficients of close-to-convex univalent functions, J. London Math. Soc. 41, 161-165. MR 32 #7734. L. E. Dunducenko 1956 Certain extremal properties of analytic functions given in a circle and in a circular ring, Ukrain. Mat. Z. 8, 377-395. (Russian) MR 19, 25. P. L. Duren 1963 Distortion in certain conformal mappings of an annulus, Michigan Math. J. 10, 431-441. MR 28 #199. P. L. Duren and M. Schiffer 1962 A variational method for functions schlicht in an annulus, Arch. Rational Mech. Anal. 9, 260-272. MR 25 #179. 1962a The theory of the second variation in extremum problems for univalent functions, J. Analyse Math. 10, 193-252. MR 27 #284. Ja. S. Fel'dman 1963 On certain extremal domains connected with univalent functions, Vestnik Leningrad. Univ. 18, no. 2, 67-85. (Russian) MR 26 #6391. I. M. Galperin 1965 Some estimates for functions bounded in the unit circle, Uspehi Mat. Nauk 20, no. 1 (121), 197-202. (Russian) MR 30 #2150. P. R. Garabedian, G. G. Ross and M. M. Schiffer 1965 On the Bieberbach conjecture for even n, J. Math. Mech. 14, 975989. MR 32 #207. P. R. Garabedian and H. A. L. Royden 1954 The one-quarter theorem for mean univalent functions, Ann. of Math. (2) 59, 316-324. MR 15, 613.

656


P. R. Garabedian and M. Schiffer 1949 Identities in the theory of con formal mapping, Trans. Amer. Math. Soc. 65, 187-238. MR 10, 522. 1955 A coefficient inequality for schlicht functions, Ann. of Math. (2) 61, 116-136. 1955a A proof of the Bieberbach conjecture for the fourth coefficient, J. Rational Mech. Anal. 4, 427-465. MR 17, 24. S. A. Gel'fer 1954 The variation of multivalent functions, Dokl. Akad. Nauk SSSR 98, 885-888; English transl., Amer. Math. Soc. Transl. (2) 26 (1963),

1-4. MR 16, 459; 1956

1958 1960

MR 27 #1581.

The method of variations in the theory of p-valent functions, Uspehi Mat. Nauk 11, no.5 (71), 60-66; English transl., Amer. Math. Soc. Transl. (2) 18 (1961), 37-43. MR 18, 648; MR 23 #A1789. On the maximum of the con formal radius of the fundamental region of a given group, Mat. Sb. 44(86), 213-224. (Russian) MR 20 #1782. On the maximum con formal radius of a fundamental domain of a group of fractional-linear transformations, Mat. Sb. 52(94), 629-640. MR 22 #11112.

1962

1964

An extension of the Goluzin-Schiffer variational method to multiplyconnected regions, Dokl. Akad. Nauk SSSR 142, 503-506 = Soviet Math. Dokl. 3 (1962), 70-73. MR 24 #A3286. Typically real functions, Mat. Sb. 64 (106), 171-184. (Russian) MR 29 #1317.

1966

Variational method in the theory of functions of bounded type, Mat. Sb. 69 (111), 422-433; English transl., Amer. Math. Soc.

Transl. (2) 72 (1968), 51-64. MR 33 #1464. S. A. Gel'fer and L. V. Kresnjakova 1965 A variational method in the theory of analytic functions with bounded mean modulus, Mat. Sb. 67(109), 570-585. (Russian) MR 33 #288. N. V. Genina (Semuhina) 1962 On an extension of a variational method of G. M. Goluzin to doubly connected domains, Tomsk Gos. Univ. Ucen. Zap. 44, 226-240. (Russian) E. G. Goluzina 1962 On typically real functions with fixed second coefficient, Vestnik Leningrad. Univ. 17, no. 7, 62-70. (Russian) MR 25 #2180. 1965 On ranges of values of certain systems of functionals in the class of typically real functions, Vestnik Leningrad. Univ. 20, no. 7, 45-62. MR 32 #4277.


657

Gong Sheng (Kung Sheng, Kung Sun) 1955

Contributions to the theory of schlicht functions. II, The coefficient problem, Sci. Sinica 4, 359-373.

A. W. Goodman 1950 On the Schwarz-Christoffel transformation and p-valent functions,

Trans. Amer. Math. Soc. 68, 204-223. MR 11, 508. 1958 Variation formulas for multivalent functions, Trans. Amer. Math. Soc. 89, 129-148. MR 20 #3293. 1958a Variation of the branch points for an analytic function, Trans. Amer. Math. Soc. 89, 277-284. MR 20 #5873a. 1958b On the variation formulas for univalent functions, Trans. Amer. Math. Soc. 89, 285-294. MR 20 #5873b. 1958c On the critical points of a multivalent function, Trans. Amer. Math. Soc. 89, 295-309. MR 20 #5874. A. W. Goodman and E. Reich 1955 On regions omitted by univalent functions. II, Canad. J. Math. 7, 83-88. MR 16, 579. A. Grad

The region of values of the derivative of a schlicht function, in the appendix to the book by Schaeffer and Spencer [1950]. MR 11, 508. T. H. Gronwall 1914/15 Some remarks on conformal representation, Ann. of Math. 16, 72-76. H. Grotzsch 1928 Uber einige Extremalprobleme der konformer Abbildung, Ber. Verh. Sachs. Akad. bliss. Leipzig, 80, 367-376. 1928a Uber einige Extremalprobleme der konformer Abbildung. II, Ber. Verh. Sachs. Akad. Wiss., Leipzig, 80, 497-502. 1955/56 Zum Haufungsprinzip der analytischen Funktionen, Wiss. Z. MartinLuther Univ., Halle-Wittenberg Math.-Natur. Reihe 5, 1095-1100. 1950

MR 18, 726.

H. Grunsky 1932

1939

Neue Abschatzungen zur konformen Abbildung ein- and mehrfach zusammenhangender Berichte, Schr. Math. Seminars Inst. Angew. Math. Univ. Berlin 1, 93-140. Koeffizientenbedingungen fur schlicht abbildende meromorphe Funk-

tionen, Math Z. 45, H. 1. 29-61. S. Ja. Havinson 1953

On extremal properties of functions mapping a region on a multi-sheeted circle, Dokl. Akad. Nauk SSSR 88, 957-959. (Russian) MR 14, 967.

658


1953a On some nonlinear extremal problems for bounded analytic functions, Dokl. Akad. Nauk SSSR 92, 243-245. (Russian) MR 15, 515. 1955 Extremal problems for certain classes of analytic functions in finitely connected regions, Mat. Sb. 36(78), 445-4788 English transi., Amer. Math. Soc. Transl. (2) 5 (1957), 1-33. MR 17, 247; MR 18, 728. The analytic capacity of sets, joint nontriviality of various classes of 1961 analytic functions and the Schwarz lemma in arbitrary domains, Mat. Sb. 54(96), 3-50; English transl., Amer. Math. Soc. Transl. (2) 43 (1964), 215-266. MR 25, #182. W. K. Hayman

Symmetrization in the theory of functions, Tech. Rep. no. 11, Navy Contract N6-ori- 106, Task Order 5 (No. R-043-992), O.N.R., Washington. MR 12, 401. 1951a Some applications of the transfinite diameter to the theory of functions, J. Analyse Math. 1, 155-179. MR 13, 545. 1955 The asymptotic behaviour of p-valent functions, Proc. London Math. Soc. (3) 5, 257-284; Russian transl., Matematika 2 (1958), no. 1, 55-80. MR 17, 142. 1958 Multivalent functions, Cambridge Univ. Press., Cambridge; Russian transl., IL, Moscow, 1960. MR 21 #7302. 1958a Bounds for the large coefficients of univalent functions, Ann. Acad. Sci. Fenn. Set. A, I. Math. no. 250/13. MR 20 #3292. 1963 On successive coefficients of univalent functions, J. London Math. Soc. 38, 228-243. MR 26 #6382. 1964 Paper read at seminar on theory of functions of a complex variable, Moscow University, March, 1963; Russian transl., Matematika 8 (1964), no. 1, 142-150. Coefficient problems for univalent functions and related function 1965 classes, J. London Math. Soc. 40, 385-406. MR 31 #3592. G. Ja. Hazalija 1958 On a covering theorem for functions that are regular in doubly connected domains, Trudy Kutaissk. Gos. Ped. Inst. 18, 251-258. (Russian) E. Hille 1949 Remarks on a paper by Zeev Nehari, Bull. Amer. Math. Soc. 55, 552-553. MR 10, 697. J. A. Hummel 1957 The coefficient regions of starlike functions, Pacific J. Math. 7, 1381-1389. MR 20 #1780. 1951


659

J. A. Jenkins 1949 1951

Some problems in conformal mapping, Trans. Amer. Math. Soc. 67, 327-350. MR 11, 341. On a theorem of Spencer, J. London Math. Soc. 26, 313-316. MR 13, 338.

1953

On values omitted by univalent functions, Amer. J. Math. 75, 406-408. MR 14, 967.

1953a Symmetrization results for some conformal invariants, Amer. J. Math. 75, 510-522. MR 15, 115. 1954 On a problem of Gronwall, Ann. of Math. (2) 59, 490-504. MR 15, 786. 1954a On the local structure of the trajectories of a quadratic differential, Proc. Amer. Math. Soc. 5, 357-362. MR 15, 947. 1955 On circumferentially mean p-valent functions, Trans. Amer. Math. Soc. 79, 423-428. MR 17, 143. 1956 Some theorems on boundary distortion, Trans. Amer. Math. Soc. 81, 477-500. MR 17, 956. 1957 Some new canonical mappings for multiply connected domains, Ann. of Math. (2) 65, 179-196. MR 18, 568. 1957a On a conjecture of Spencer, Ann. of Math. (2) 65, 405-410. MR 19, 25. 1958

1960

Univalent functions and conformal mapping, Springer-Verlag, Berlin, 1958; Russian transl., IL, Moscow, 1962. MR 20 #3288. On univalent functions with real coefficients, Ann. of Math. (2) 71,

1-15. MR 22 #5741. 1960a An extension of the general coefficient theorem, Trans. Amer. Math. Soc. 95, 387-407. MR 22 #8126b. 1960b On certain coefficients of univalent functions. II, Trans. Amer. Math. Soc. 96, 534-545. MR 23 #A309. 1960c On the global structure of the trajectories of a positive quadratic differential, Illinois J. Math. 4, 405-412. MR 23 #A1794. 1963 On some span theorems, Illinois J. Math. 7, 104-117. MR 27 #278. 1963a An addendum to the general coefficient theorem, Trans. Amer. Math. Soc. 107, 125-128. MR 26 #5154. 1965 On Bieberbach-Eilenberg functions. III, Trans. Amer. Math. Soc. 119, 195-215.

MR 31 #5969.

J. A. Jenkins and D. C. Spencer 1951 Hyperelliptic trajectories, Ann. of Math. (2) 53, 4-35. MR 12, 400.


660 W. Kaplan 1952

Close-to-convex schlicht functions, Michigan Math. J. 1, 169-185.

MR 14, 966. A. Kobori and H. Abe

Une remarque sur un theoreme de M. Hayman, Japan. J. Math. 29, 32-34. MR 23 #A3832. L. I. Kolbina Some extremal problems in conformal mapping, Dokl. Akad. Nauk SSSR 1952 1959

84, 865-868. (Russian) MR 14, 35. 1952a On the theory of univalent functions, Dokl. Akad. Nauk SSSR 84, 1127-1130. (Russian) MR 14, 35. 1955

Conformal mapping of the unit circle onto mutually nonoverlapping domains,

Vestnik Leningrad. Univ. 10, no. 5, 37-43. (Russian) MR 17, 26. Y. Komatu 1943

Untersuchungen fiber konforme Abbildung von zweifach zusammen-

hangender Gebieten, Proc. Phys.-Math. Soc. Japan (3) 25, 1-42. MR 7, 514.

On the coefficients of typically-real Laurent series, Kodai Math. Sem. Rep. 9, 42-48. MR 19, 404. 1958 On analytic functions with positive real part in a circle, Kodai Math. Sem. Rep. 10, 64-83. MR 20 #3998. 1958a On analytic functions with positive real part in an annulus, Kodai Math. Sem. Rep. 10, 84-100. MR 20 #3999. 1957

Y. Komatu and M. Ozawa 1951 Conformal mapping of multiply connected domains. I, Kodai Math. Sem. Rep., 81-95. MR 13, 734. 1952 Conformal mapping of multiply connected domains. II, Kodai Math. Sem. Rep., 39-44. MR 14, 461.

G. V. Korickii 1955 On curvature of level lines and of their orthogonal trajectories in conformal mappings, Mat. Sb. 37(79), 103-116. (Russian) MR 17, 26. 1957 Curvature of level lines in univalent conformal mappings, Dokl. Akad. Nauk SSSR 115, 653-654. (Russian) MR 19, 845. 1960 On the curvature of level curves of univalent conformal mappings, Uspehi Mat. Nauk 15, no.5, (95), 179-182. (Russian) MR 23#A1793. L. V. Kresnjakova 1961 Analytic functions with bounded mean modulus, Izv. Vyss. Ucebn. Zaved. Matematika, no. 1 (20), 98-103. (Russian) MR 24 #A3298.


1962

661

On regular functions with bounded mean modulus, Dokl. Akad. Nauk SSSR 147, 290-293 = Soviet Math. Dokl. 3 (1962), 1590-1594. MR 27 #1603.

1963

Some estimates for regular functions with bounded mean modulus, Izv. Vysg. U°cebn. Zaved. Matematika, no. 1 (32), 94-97. (Russian) MR 26 #6422.

J. Krzyz 1964

Some remarks on close-to-convex functions, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 12, 25-28. MR 28 #5174. J. Krzyi and Z. Lewandowski 1963 On the integral of univalent functions, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys.11, 447-448. MR 27 #3791. T. Kubo 1954

Symmetrization and univalent functions in an annulus, J. Math. Soc.

Japan 6, 55-67. MR 15, 948. 1954a Kelvin principle and some inequalities in the theory of functions. I, Mem. Coll. Sci. Univ. Kyoto Ser. A. Math. 28, 299-311. MR 16, 122. Kelvin principle and some inequalities in the theory of functions. II, Mem. Coll, Sci. Univ. Kyoto Ser. A. Math. 29, 17-26. MR 16, 914. 1955a Kelvin principle and some inequalities in the theory of functions. III, Mem. Coll. Sci. Univ. Kyoto Ser. A. Math. 29, 119-129. MR 20 #3977. 1958 Hyperbolic transfinite diameter and some theorems on analytic functions in an annulus, J. Math. Soc. Japan 10, 348-364. MR 21 #3551. P. P. Kufarev 1943 On one-parameter families of analytic functions, Mat. Sb. 13(55), 1955

87-118. (Russian) MR 7, 201. On integrals of a very simple differential equation with movable polar singularity in the right-hand member, Tomsk. Gos. Univ. Ucen. Zap. no. 1, 35-48. (Russian) 1947 A theorem on the solutions of a certain differential equation, Tomsk. Gos. Univ. Ucen. Zap. no. 5, 20-21. (Russian) 1947a On a special family of one-sheeted domains, Tomsk. Gos. Univ. Ucen. Zap. no. 5, 22-36. (Russian) 1947b On a method of numerical determination of the parameters in the Schwa rz-Chris toffel integral, Dokl. Akad. Nauk SSSR 57, 535-537. (Russian) MR 9, 277. 1947c A remark on integrals of Lowner's equation, Dokl. Akad. Nauk SSSR 57, 655-656. (Russian) MR 9, 421. 1946


662

1947d On the theory of univalent functions, Dokl. Akad. Nauk SSSR 57, 1948

751-754. (Russian) MR 9, 507. On a system of differential equations, Tomsk. Gos. Univ. Ucen. Zap. 8, 61-72. (Russian) MR 11, 21.

1950

On conformal mapping of complementary regions, Dokl. Akad. Nauk

SSSR 73, 881-884. (Russian) MR 12, 401. Remark on extremal problems in the theory of univalent functions, Tomsk. Gos. Univ. Ucen. Zap. no. 14, 3-7. (Russian) 1954 On a property of extremal regions of the problem of coefficients, Dokl. Akad. Nauk SSSR 97, 391-393. (Russian) MR 16, 122. 1955 Remark on the problem of coefficients, Tomsk. Gos. Univ. Ucen. Zap. 25, 15-18. (Russian) MR 19, 404. 1956 On a method of parametric representations and the variational method of Goluzin, Proc. Third All-Union Math. Congress. Vol. 1, Moscow, pp. 85-86. (Russian) 1956a Methods and results in the theory of univalent functions, Proc. Third All-Union Math. Congress. Vol. 2, Moscow, pp. 29-31. (Russian) 1956b On a certain method of investigation of extremum problems in the theory of univalent functions, Dokl. Akad. Nauk SSSR 107, 633-635. (Russian) MR 17, 1069. 1958 Certain methods and results in the theory of univalent functions, Proc. 1951

1963

Third All-Union Math. Congress. Vol.3, Moscow, pp. 189-198. (Russian) On G. M. Goluzin's variational formula, Trudy Tomsk. Gos. Univ. Ser. Meh.-Mat. 163, 58-62. (Russian)

P. P. Kufarev and A. E. Fales 1951

On an extremal problem for complementary regions, Dokl. Akad. Nauk

SSSR 81, 995-998. (Russian) MR 14, 262. P. P. Kufarev and N. V. Semuhina 1956 On the extension of G. M. Goluzin's variational method to doubly connected regions, Dokl. Akad. Nauk SSSR 107, 505-507. (Russian) MR 17, 1193. M. P. Kuvaev

Generalizations of an equation of the Lowner type for automorphic functions, Trudy Tomsk. Gos. Univ. Ser. Meh.-Mat. 144, 27-30. (Russian) 1959a A new derivation of Lowner's equation for doubly connected regions, Trudy Tomsk. Gos. Univ. Ser. Meh.-Mat. 144, 45-55. (Russian) 1959


663

M. P. Kuvaev and P. P. Kufarev 1955 An equation of Lowner's type for multiply connected regions, Tomsk. Gos. Univ. Ucen. Zap. 25, 19-34. (Russian) MR 19, 401. G. V. Kuz'mina 1962 Some covering theorems for univalent functions, Dokl. Akad. Nauk SSSR 142, 29-31 = Soviet Math. Dokl. 3 (1962), 21-23. MR 24 #A13831965 Covering theorems for functions which are regular and univalent in the circle, Dokl. Akad. Nauk SSSR 160, 25-28 = Soviet Math. Dokl. 6, (1965), 21-25. MR 30 #3204. E. Landau 1929

Uber die Blochsche Konstante and zwei verwandte Weltkonstanten, Math. Z.

30, H. 4, 608-634. M. A. Lavrent'ev 1934 On the theory of conformal mappings, Trudy Fiz.-Mat. Inst. Akad. Nauk SSSR 5, 195-246. (Russian) M. A. Lavrent'ev and B. V. Sabat 1958 Methods of the theory of functions of a complex variable, GITTL, Moscow, 1951; 2nd rev. ed., 1958; German transl., Mathematik fur Naturwiss. and Technik, Band 13, VEB Deutscher Verlag, Berlin, 1966. MR 14, 457; MR 21 #116. N. A. Lebedev The method of variations in conformal mapping, Dokl. Akad. Nauk 1951 SSSR 76, 25-27. (Russian) MR 12, 491. 1951a Certain estimates and problems on an extremum in the theory of conformal mapping,, Dissertation, Leningrad State University. (Russian) 1955 Majorizing region for the expression I = ln(z'A f '(z)I -A/f (z)') in the

class S, Vestnik Leningrad. Univ. 10, no. 8, 29-41; English transl., Amer. Math. Soc. Transl. (2) 22 (1962), 43-57. MR 17, 248. 1955a Certain estimates for functions regular and univalent in the unit disc, Vestnik Leningrad. Univ. 10, no. 11, 3-21; English transl., Amer. Math. Soc. Transl. (2) 22 (1962), 59-80. MR 17, 599. 1955b On the theory of conformal mappings of a circle onto nonoverlapping regions, Dokl. Akad. Nauk SSSR 103, 553-555. (Russian) MR 17, 250. 1955c On a parametric representation of functions regular and univalent in a ring, Dokl. Akad. Nauk SSSR 103, 767-768. (Russian) MR 17, 356. 1955d On domains of values of functionals defined on classes of analytic functions, Dissertation, Leningrad State University, (Russian)

664


On the domain of values of a certain functional in a problem of nonoverlapping domains, Dokl. Akad. Nauk SSSR 115, 1070-1073. (Russian) MR 19, 951. 1961 An application of the area principle to non-overlapping domains, Trudy Mat. Inst. Steklov. 60, 211-231. (Russian) MR 24 #A1384. 1966 Application of the area principle to the problems on non-overlapping finitely connected regions, Dokl. Akad. Nauk SSSR 167, 26-29 = Soviet Math. Dokl. 7 (1966), 323-327. MR 33 #5865. N. A. Lebedev and I. M. Milin 1951 On the coefficients of certain classes of analytic functions, Mat. Sb. 1957

1965

28(70), 359-400. (Russian) MR 13, 640. An inequality, Vestnik Leningrad. Univ. 20, no. 19, 157-158. (Russian) MR 32 #4248.

Z. Lewandowski

1958 1960

Sur l'identiti de certaines classes de fonctions univalentes. I, Ann. Univ. Mariae Curie-Sklodowska, Sect. A. 12, 131-146. MR 24 #A217. Sur l'identiti de certaines classes de fonctions univalentes. II, Ann.

Univ. Mariae Curie-Sklodowska, Sect A. 14, 19-46. MR 28 #200. Sur les majorantes des fonctions holomorphes daps le cercle I zl < 1, Ann. Univ. Mariae Curie-Sklodowska, Sect. A, 15, 5-11. MR 26 #314. 1961a Starlike majorants and subordination, Ann. Univ. Mariae Curie Sklodowska, Sect. A. 15, 79-84. MR 25 #2186. Li En-pir 1953 On the theory of univalent functions on a circular ring, Dokl. Akad. Nauk SSSR 92, 475-477. (Russian) MR 15, 516. 1953a On typically real functions on a circular ring, Dokl. Akad. Nauk SSSR 92, 699-702. MR 15, 516. 1953b Certain questions in the theory of univalent and typically real functions on a circular annulus, Dissertation, Leningrad State University. (Russian) K. Loewner 1923 Untersuchungen fiber schlichte konforme Abbildung des Einheits1961

kreises. I, Math. Ann. 89, 103-121. V. G. Lozovik 1963 On a class of functions which are univalent in the unit circle, Izv. Vyss. Zaved. Matematika, no. 2 (33), 63-69. (Russian) MR 26 #5150. 1963a Functionals defined on certain classes of analytic functions, Ukrain.

Mat. Z. 15, 95-99. (Russian) MR 27 #1575.


665

Ju. D. Maksimov

Extremal problems in certain classes of analytic functions, Dokl. Akad. Nauk SSSR 100, 1041-1044. (Russian) MR 16, 810. 1955a On locally c-convex and locally c-starlike multivalent functions, Dokl. Akad. Nauk SSSR 103, 965-967. (Russian) MR 17, 357. 1961 Extension of the structural formula for convex univalent functions to a multiply connected circular region, Dokl. Akad. Nauk SSSR 136, 284287 = Soviet Math. Dokl. 2 (1961), 55-58. MR 22 #8121. 1955

M. Marcus 1964

Transformations of domains in the plane and applications in the theory of functions, Pacific J. Math. 14, 613-626. MR 29 #2382.

H. Meschkowski 1952 ' Einige Extremalprobleme aus der Theorie der konformen Abbildung, Ann. Acad. Sci. Fenn. Ser. A I. no. 117, 1-12. MR 14, 367. 1953 Verzerrungssatze fur mehrfach zusammenhangende Bereiche, Compo1954

sitio Math. 11, 44-59. MR 15, 116. Verallgemeinerung der Poissonschen Integralformel auf mehrfach zusammenhangende Bereiche, Ann. Acad. Sci. Fenn. Ser. AL 166.

1962

MR 15, 695. Hilbertsche Resume mit Kernfunktion, Die Grundlehren der Math Wissenschaften, Bd. 113, Springer-Verlag, Berlin. MR 25 #4326.

I. M. Milin 1964

The area method in the theory of univalent functions, Dokl. Akad. Nauk SSSR 154, 264-267 = Soviet Math. Dokl. 5 (1964), 78-81. MR 28 #1283.

1964a Closed orthonormal systems of analytic functions in domains of finite connectivity, Dokl. Akad. Nauk SSSR 157, 1043-1046 = Soviet Math. Dokl. 5 (1964), 1078-1082. MR 34 #4469. 1964b The area method in the theory of univalent functions, Dissertation, Leningrad State University. (Russian) A bound for the coefficients of schlicht functions, Dokl. Akad. Nauk 1965 SSSR 160, 769-771 = Soviet Math. Dokl. 6 (1965), 196-198. MR 30 #3206.

Ja. S. Mirosnicenko 1951 On a problem in the theory of univalent functions, Ucen. Zap. Donetsk.

Ped. Inst. 1, 63-75. (Russian)

666


1. P. Mitjuk

Univalent conformal mappings of multiply connected domains, Dopovidi Akad. Nauk Ukrain. RSR, 158-160. (Ukrainian) MR 23 #A3252. 1961a Some theorems on univalent conformal mappings of multiply connected regions, Dopovidi Akad. Nauk Ukrain. RSR, 420-423. (Ukrainian) 1961

MR 24 #A2030.

1961b A generalization of some theorems on univalent conformal maps of doubly connected domains, Dopovidi Akad. Nauk Ukrain. RSR, 11151118. (Ukrainian) MR 24 #A2660. 1964 A generalized reduced module and some of its applications, lzv. Vyss. Ucebn. Zaved. Matematika, no. 2 (39), 110-119. (Russian) MR 29, #2376. 1964a The symmetrization principle for multiply connected domains, Dokl. Akad. Nauk SSSR 157, 268-270 = Soviet Math. Dokl. 5 (1964), 928930. MR 29 #2375. The inner radius of a domain and certain of its properties, Ukrain. 1965 Mat. Z. 17, no. 1, 117-122. (Russian) MR 32 #7730. 1965a The principle of symmetrization for multiply connected regions and

certain of its applications, Ukrain. Mat. Z. 17, no.4, 46-54. (Russian) MR 33 #7510. 1965b Certain properties of functions regular in a multiply connected region, Dokl. Akad. Nauk SSSR 164, 495-498 = Soviet Math. Dokl. 6 (1965), 1252-1255. MR 32 #5862. 1965c The symmetrization principle for an annulus and certain of its appli-

cations, Sibirsk Mat. Z. 6, 1282-1291. (Russian) MR 35 #4396. 1965d Certain extremal problems in the geometric theory of functions, Dissertation, Kiev State University. (Russian) P. T. Mocanu 1957 Une generalisation du theoreme de la contraction dans la classe S de fonctions univalentes, Acad. R. P. Romine. Fil. Cluj. Stud. Cerc. Mat. 8, 303-312. (Romanian) MR 21 #5735a. 1958 Sur une generalisation du theoreme de contraction dans la classe des fonctions univalentes, Acad. R. P. Romine. Fil. Cluj. Stud. Cerc. Mat. 9, 149-159. (Romanian) MR 21 #5735b. M. A. Monastyrskii 1959 On an application of the method of positive functionals for C-functions, Ucen. Zap. Sahtinsk. Gas. Ped. Inst. 2, no. 6, 109-117. (Russian) Z. Nehari 1949 The Schwarzian derivative and schlicht functions, Bull. Amer. Math. Soc. 55, 545-551. MR 10, 696.

BIBLIOGRAPHY FOR THE SUPPLEMENT, 1953

667

Some inequalities in the theory of functions, Trans. Amer. Math. Soc. 75, 256-286. MR 15, 115.

H. Nishimiya

On a coefficient problem for analytic functions typically-real in an annulus, Kodai Math. Sem. Rep. 9, 59-67. MR 20 #101. 1959 On coefficient-regions of Laurent series with positive real part, Kodai Math. Sem. Rep. 11, 25-39. MR 21 #7310. O. S. Nosenko 1963 On the range of Stieltjes functionals with restrictions of equality type, 1957

Dopovidi Akad. Nauk Ukrain. RSR, 1563-1567. (Ukrainian) MR 29#5996. M. Ozawa 1952

On functions of bounded Dirichlet integral, Kodai Math. Sem. Rep. 4,

95-98. MR 14, 967. 1965

On the sixth coefficient of univalent function, Kodai Math. Sem. Rep. 17, 1-9. MR 31 #2394.

E. Peschl Zur Theorie der schlichten Funktionen, J. Reine Angew. Math. 176, 61-96. V. V. Pokornyi 1951 On some sufficient conditions for univalence, Dokl. Akad. Nauk SSSR 79, 743-746. (Russian) MR 13, 222. 1936

C. Pommerenke 1961/62 Uber die Mittelwerte and Koeffizienten multivalenter Funktionen,

Math. Ann. 145, 285-296. MR 24 #A3282. Uber einige Klassen meromorpher schlichter Funktionen, Math. Z. 78, 263-284. MR 28 #1285. 1962a On starlike and convex functions, J. London Math. Soc. 37, 209-224. 1962

MR 25 #1279.

On starlike and close-to-convex functions, Proc. London Math. Soc. (3) 13, 290-304. MR 26 #2597. 1964 Linear-invariante Familien analytischer Funktionen. 1, Math. Ann. 155, 108-154. MR 29 #6007. 1964a Lacunary power series and univalent functions, Michigan Math. J. 11, 219-223. MR 29 #4883. 1963

V. 1. Popov 1965

The range of a certain system of functionals on the class S, Trudy Tomsk. Gos. Univ. Set. Meh.-Mat. 182, 107-132. (Russian) MR 33 #7521.


668

H. Prawitz 1927/28 Uber Mittelwerte analytischer Funktionen, Arkiv Mar. Astronom.

Fysik 20A, no.6, 1-12. B. N. Rahmanov On the theory of univalent functions, Dokl. Akad. Nauk SSSR 78, 2091951 1953

211. (Russian) MR 12, 816. On the theory of univalent functions, Dokl. Akad. Nauk SSSR 91, 729732. (Russian) MR 15, 413.

M. O. Reade 1955

On close-to-convex univalent functions, Michigan Math. J. 3, 59-62.

MR 17, 25. M. I. Red'kov 1960

On the range of values of a certain functional defined on certain classes of bounded univalent functions, Tomsk Gos. Univ. Ucen. Zap. no. 36, 33-50. (Russian)

1962

The range of the functional I = ln(w)L\'(w) I -'LS6'(0) v/c6 (w)A) in the

class SI[I0(w)I], Izv. Vyss. Ucebn. Zaved. Matematika, no.2 (27), 119-129. (Russian) MR 25 #4086. 1962a The range of the functional I = ln(wh0'(w)1-N'(0) v/S6 (w)''I0 (w)I' ) in the class S1, Izv. Vyss. Ucebn. Zaved. Matematika, no. 4 (29), 134-142. (Russian) MR 25 #4087. 1963 The range of values of the functional 1= 1(f(w), f(w), f'(w), f'(w), f'(0)) in the class S1[If(w)1], Trudy Tomsk. Gos. Univ. Set. Meh.-Mat. 169, 59-68. (Russian) MR 29 #1324. E. Reich and M. Schiffer 1964 Estimates for the trans finite diameter of a continuum, Math. Z. 85, 91-106. MR 30 #4921. E. Reich and S. E. 'Warschawski 1960 On canonical conformal maps of regions of arbitrary connectivity,

Pacific J. Math. 10, 965-989. MR 22 #8120. M. P. Remizova On regions of values of analytic functions represented by the sum and 1959

product of Stielt/es integrals, Ukrain. Mat. Z. 11, 175-182. (Russian) MR 22 #771. E. Rengel 1933

Uber einige Schlitztheoreme der konformen Abbildung, Schr. Math. Semund Inst. Angew. Math. Univ. Berlin 1, no. 4, 141-162.


669

F. Riesz Sur certains systemes singuliers d'equations integrales, Ann. Sci. Ecole Norm. Sup. 28, 33-62. M. S. Robertson 1935 On the coefficients of a typically-real function, Bull. Amer. Math. Soc. 41, 565-572. 1936 Analytic functions star-like in one direction, Amer. J. Math. 58, 465-472. 1962 Variational methods for functions with positive real part, Trans. Amer. Math. Soc. 102, 82-93. MR 24 #A3288. 1963 Extremal problems for analytic functions with positive real part and applications, Trans. Amer. Math. Soc. 106, 236-253. MR 26 #325. R. M. Robinson 1943 Analytic functions in circular rings, Duke Math. J. 10, 341-354. 1911

MR 4, 241. W. Rogosinski 1932 Uber positive harmonische Entwicklungen and typisch reelle Potenz-

reihen, Math. Z. 35, 93-121. S. Sato

Two theorems on bounded functions, SGgaku 7, 99-101. 1955a On values not assumed by bounded functions, Kenkju hokoku. Sidzen Kacaku, Liberal Arts J. Natur. Sci. 6, 1-6. (Russian) A. C. Schaeffer and D. C. Spencer Coefficient regions of schlicht functions, Amer. Math. Soc. Colloq. 1950 Publ., vol. 35, Amer. Math. Soc., Providence, R. 1. MR 12, 326. 1955

M. Schiffer 1938 A method of variation within the family of simple functions, Proc.

London Math. Soc. (2) 44, 432-449. 1938a On the coefficients of simple functions, Proc. London Math. Soc. (2) 44, 450-452. 1943 Variation of the Green function and theory of p-valued functions, Amer.

J. Math. 65, 341-360. MR 4, 215. Some recent developments in the theory of con formal mapping, appendix to Russian transl. of R. Courant, Dirichlet's principle, conformal mapping and minimal surfaces, Interscience, New York and London, 1950. (Russian) M. Schiffer and O. Tammi 1965 The fourth coefficient of a bounded real univalent function, Ann. Acad. Sci. Fenn. Set. A I, No. 354. MR 35 #4394. 1953

670

BIBLIOGRAPHY FOR THE SUPPLEMENT 1965a

On the fourth coefficient of bounded univalent functions, Trans. Amer. Math. Soc. 119, 67-78. MR 32 #2579.

V. Singh

Grunsky inequalities and coefficients of bounded schlicht functions, Ann. Acad. Sci. Fenn. Set. A I, no. 310. MR 26 #1444. G. G. Slionskii 1958 On the extremal problems for differentiable functionals in the theory of univalent functions, Vestnik Leningrad. Univ. 13, no. 13, 64-83. (Russian) MR 20 #4652. 1959 On the theory of bounded schlicht functions, Vestnik Leningrad. Univ. 14, no. 13, 42-51. (Russian) MR 22 #2704. 1962

V. I. Smirnov

Sur les formules de Cauchy et de Green et quelques problemes qui s'y rattachent, Izv. Akad. Nauk SSSR Set. Mat. 7, 337-372. D. C. Spencer 1941 On mean one-valent functions, Ann. of Math. (2) 42, 614-633. 1932

MR 3, 78. O. V. Stepanova 1963 1965

Certain properties of level curves for univalent conformal mappings, Mat. Sb. 61 (103), 350-361. (Russian) MR 27 #1579. A property of level lines in univalent con formal mappings, Dokl. Akad. Nauk SSSR 163, 1330 = Soviet Math. Dokl. 6 (1965), 1124-1125. MR 32 #2577.

G. Szego

On a certain kind of symmetrization and its applications, Ann. Mat. Pura Appl. (4) 40, 113-119. MR 17, 1074. P. M. Tamrazov 1962 Relative boundary distortion under a schlicht conformal mapping of a doubly connected domain, Dopovidi Akad. Nauk Ukrain. RSR, 338340. (Ukrainian) MR 26 #38831962a On the theory of schlicht conformal mappings of doubly connected domains, Dopovidi Akad. Nauk Ukrain. RSR, 563-566. (Ukrainian) 1955

MR 26 #3884.

1962b On univalent conformal mapping of doubly connected domains, Dopovidi Akad. Nauk Ukrain. RSR, 1142-1145. (Ukrainian) MR 27 #276. 1963

Some estimates in the theory of univalent conformal mappings of doubly connected domains, Dopovidi Akad. Nauk Ukrain. RSR, 11601163. (Ukrainian) MR 29 #6008.


671

Theorems on the covering of lines under a con formal mapping, Mat. Sb. 66(108), 502-524. (Russian) MR 31 #2392. 1965a Certain extremal problems of the theory of univalent con formal map-

1965

pings, Mat. Sb. 67(109), 329-337. (Russian) MR 33 #7523. 1965b A con formally metric theory of doubly-connected regions, and a generalized Blaschke product, Dokl. Akad. Nauk SSSR 161, 308-311 = Soviet Math. Dokl. 6 (1965), 432-435. MR 31 #327. 1965c Univalent functions and conformally metric theory of multiply con-

nected domains, Dissertation, Kiev. (Russian) 0. Teichmiiller 1938

Untersuchungen fiber konforme and quasikonforme Abbildung,

Deutsche Math. 3, 621-678. G. C. Tumarkin and S. Ja. Havinson 1960 Qualitative properties of solutions of certain types of extremal problems in Series on Modern Problems of the Theory of Functions of a Complex Variable, Fizmatgiz, Moscow, pp. 77-95. (Russian) MR 22 #11136.

G. V. Ulina

On the domains of values of certain systems of functionals in classes of univalent functions, Vestnik Leningrad. Univ. 15, no. 1, 34-54. (Russian) MR 22 #5745. Xia Dao-xing (Hsia Tao-hsing) 1956 On the functions univalent in a circular ring, Acta Math. Sinica 6, 598-618. (Chinese) 1957 Goluzin's number (3 - V5)/2 is the radius of superiority in subordination, Sci. Record 1, 219-222. MR 20 #6530. 1957a On the radius of superiority in subordination, Sci. Record 1, 329-333. 1960

MR 20 #6531.

Some covering properties of convex domains in the theory of conformal mapping, Sci. Sinica 7, 816-828. MR 21 #2046b. Xia Dao-xing (Hsia Tao-hsing) and Zhang Kai-ming (Chang K'ai-ming) 1958 Some inequalities in the theory of subordination, Acta Math. Sinica 8, 408-412 = Chinese Math. Acta 9 (1967), 116-120. MR 21 #7306. V. A. Zmorovic 1952 On certain variational problems of the theory of univalent functions, 1958

1953

Ukrain. Mat. Z. 4, 276-298. (Russian) MR 15, 301. On some classes of analytic functions univalent in a circular ring, Mat. Sb. 32 (74), 633-652. (Russian) MR 14, 1075.

672


On some special classes of analytic functions univalent in a circle, Uspehi Mat. Nauk 9, no. 4 (62), 175-182. (Russian) MR 16, 459. 1956 On certain classes of analytic functions in a circular ring, Mat. Sb. 40(82), 225-238. (Russian) MR 18, 648. 1958 On a generalisation of Schwarz's integral formula on n-connected circular domains, Dopovidi Akad. Nauk Ukrain. RSR, 489-492. (Ukrainian) MR 20 #5277. 1959 Theory of special classes of univalent functions. 1, Uspehi Mat. Nauk 14, no.3 (87), 137-143. (Russian) MR 22 #3819. 1959a Theory of special classes of univalent functions. 11, Uspehi Mat. Nauk 14, no. 4 (88), 169-172. (Russian) MR 22 #3820. 1965 On a class of extremal problems associated with regular functions with positive real part in the circle I zI < 1, Ukrain. Mat. Z. 17, no.4, 12-21; English transl., Amer. Math. Soc. Transl. (2) 80 (1968), 215226. MR 33 #5901. 1954

SUBJECT INDEX

Classes: C 563, 575; Eg 528; Ep 438; Hp 402, 498; hp 385; L 563; L 563;

Accessible boundary points 35

Analytic curve 9 Anharmonic ratio 330 Approximation in mean 448-449 Asymptotic value 351 Automorphic function 64, 258 Bieberbach conjecture 605-606, 608-612 generalized 589

LP 388; N 393; R 563; W 563; S 110, 563; SM 563; Sa 152; Sp 488; S(1)

563; SR 563;

S(k) 563; 5" 530, 563;

S 563; T 563; Tr 541; 1 110, 563; Im 563; Yp 488; Y-O 563; I(k) 563;

Blaschke function 398 Bloch constant 363

V 563;

563; $2 563

Closedness condition 451

Boundary continua 7 Bounds for area of an annulus 605 the image of a disk 595

Conformal mapping of a multiply connected domain onto a disk 254-255 Conformal mapping of Riemann surfaces 461-467 Conformal radii of disjoint domains 156, 583, 584, 587 Conformal radius 29, 313 Convergence of a sequence of domains

the star of the image of a disk 180 Bounds for bounded functions 514-526 Bounds for coefficients of p-valent functions 490, 492, 493

Bounds for coefficients of univalent functions 187, 197, 199-204, 605-618 Bounds for curvature of level curves 599-603 Bounds for mean modulus of a function

to a kernel 54, 228-229 Convergence of sequences of analytic functions 11-14 Convergence of sequences of harmonic

functions 19-22

595

Convex domain 166, 201 Convexity of level curves, conditions

Bounds for the growth of adjacent coeffi-

cients of close-to-convex functions 617 p-valent functions in mean 615 univalent functions 193, 195, 614, 615

for 601-602 Correspondence of boundaries under conformal mapping 31-46, 262-265, 417-428 Covering theorems 73-75 89, 494 of arcs of a circle by image of disk and annulus 595, 597, 605 of segments 174, 177., 597, 604

Brouwer theorem 252 Capacity of a set 310

Caratheodory convergence theorem 55 Caratheodory-Fejer problem 497 Caratheodory-Fejer theorems 500, 502 Cauchy integral formula 408, 435-441 Cauchy integral theorem 408 Cauchy singular integral 431 Cebysev constant 294, 295 Cebysev polynomial 294, 295 Circular domain 234, 235 Circular polygon 78

Cut 9 Dirichlet problem 267 Entire function of finite order 352 Equivalent points 260 Family of polynomials complete in a domain 449 Function of bounded variation 385 673

SUBJECT INDEX

674

Gauss's differential equation 84 General theorem of Jenkins coefficients 573

Gluing theorems 454, 456 Green's formula 273 Green's function 269, 310, 311 Gronwald problem 590 Grotzsch principle 171 Grotzsch theorem, generalized 603 Hadamard three-circle theorem 344 Harmonic function 19, 266 Harmonic measure 315, 342 Harnack's theorem 21 Heine-Borel covering theorem 6 Helly theorem 385 Hilbert theorem 213-214 Holder inequality 388 Hyperbolic disk 332 Hyperbolic distance 331 Hyperbolic metric 332, 336, 361 Hyperbolic transformation 259 Hyperconvergence 356 Inner measure 420

Integral formula, Cauchy 408, 435-441 Integral representation of classes of functions regular in a disk (annulus) 529, 574, 619-623 Integral theorem, Cauchy 408 Jenkins coefficients, general theorem of 573

Jensen formula 323 Jensen-Schwarz formula 323 Jordan content 300, 302 Jordan theorem 8 Schoenflies supplement to 36 Kellogg theorem 426 Kernel of sequence of domains 54, 228 Koebe covering theorem 49 generalized and sharpened 597, 598,

602

Koebe lemma 31

Landau constant 363 Landau theorem 338 Laplace differential equation 19 Laurent system of functions 634 Lavrent'ev formula 156 Lavrent'ev theorems, generalized 638, 639, 642

Lemmas on mean moduli 183, 184, 186 Limiting value of Cauchy integral 431 Limits of convexity 166, 598

Limits of generalized starlikeness 167170, 599

Limits of starlikeness 167, 599 Lindelof lemma 33 Lindelof principle 30, 339 Lipschitz condition 413, 426 Loewner differential equation 91 Majorant, univalent 368 Mapping, conformal univalent 23 continuous 239 quasiconformal 456

topological 239 Mapping a disk onto mutually disjoint domains 157, 583-586 Mapping a doubly connected domain 205210, 593-594 onto a circular annulus 208

Mapping a multiply connected domain onto

a disk with cuts along arcs of logarithmic spirals 242-243 a finite surface 629, 630 a multiply covered disk 648-649 a plane with cuts along arcs of logarithmic spirals. 220, 243 a plane with rectilinear cuts 242 a plane with rectilinear parallel cuts 213-214,277 an annulus with cuts along arcs of logarithmic spirals 243 mutually disjoint domains 638-641 Mapping an n-connected domain onto an

n-sheeted disk 278, 468 Mapping the exterior of the unit disk onto a plane with cut along arc of ellipse (hyperbola) 135 Mapping the interior of a circular polygon onto the unit disk 78-81 Mapping the interior of a regular n-sided circular polygon onto the unit disk 83-87 Mapping the interior of a rectilinear polygon onto the unit disk 76-78 Mapping the interior of a regular n-sided polygon onto the unit disk 77-78 Maximum principle generalized 267 Mean p-valent function 587, 588 Measurable set 420 Meromorphic function 23, 321

SUBJECT INDEX

Method of continuity 234, 239-243 Method of contour integration 225, 283, 284, 567 Method of extremal metrics 572 Method of integral representations 574576

Method of parametric representations

89-99, 110-122, 568, 579, 591-592 Method of quadratic differentials 572-574 Method of symmetrization 574 Metric, continuous 360 Metric, regular 360 Metric, support 361 Minimal property of Dirichlet integral 637 Minimization of area 30, 227, 631 Minimization of the maximum modulus 29 Minkowski inequality 389 Modulus of doubly connected domain 209 Modulus of n-connected domain 244 Modular function 64 Modular grid 63 Modular Riemann surface 64

n-sheeted disk 277 Natural boundary 64 Necessary and sufficient condition for univalence of a function 609 Nontangential path 382, 428 Normal family of analytic functions 67 Normality test 68, 70 Order of entire function 352 Orthogonal polynomials 451 Outer area 631 Outer measure 420 p-valent function 487 Parabolic transformation 259 Parametric representation of univalent functions 89-99 Picard's theorem 72 Poincare's theorem 462 Poisson formula 20, 268 Poisson integral 380 Poisson-Stieltjes integral 384 Prime ends 40 Principle, hyperbolic metric 336 Principle of areas 47, 490, 565-567, 587 Principle of compactness 19 Principle of condensation 14-18 Principle of extension 342-343 Principle of length and area 572

675

Principle of subordination for annulus 648 for disk 368-369, 643-644 for multiply connected domains 645647

Principle of symmetrization 588, 603 generalized for multiply connected domains 642 Privalov's uniqueness theorem 428 Problem of coefficients 605, 617 Quasiconformal mapping 456 Range of a functional on a given class 117

Range of values of functionals and systems of functionals on classes of univalent functions 117, 133, 136, 139, 580-583 representable by means of Stieltjes integrals 624-628

Riemann surface, constructive definition 461-462 Riemann surfaces of elliptic, hyperbolic and parabolic types 462 Riemann theorem 25-29 Robin's constant 310 Scalar product of regular functions 226 Schoenflies supplement to Jordan theorem 36

Schottky function 287 Schottky theorem 337-338 Schur's theorems 498, 505 Schwarz derivative 636 Schwarz formula 92, 322 Schwarz invariant 79, 580 Schwarz lemma 25, 329, 332; generalized and sharpened 329-330, 333, 361, 373, 648-649 Schwarz-Christoffel formula 77 Spread of coefficients of univalent func-

tions 614 Star of a domain 177 Starlike domain 167, 202 Starlikeness generalized 167, 599

Starlikeness of level curve 599-602 Subordinate function 368, 369 Summable function 431 Symmetrization method of Marcus 603 Symmetrization principle 588, 603;

generalized 642

676

SUBJECT INDEX

Theorems of area 47; of p-valenc functions 490 generalized 565, 566, 578, 586 generalized for multiply connected domains 567, 633, 634, 635 Theorems of distortion 51, 134; of chords 577

generalized 587-590 Theorems of rotation 51, 110 Transfinite diameter 294 Typically-real function 540 Uniform boundedness in the interior of a domain 15 Uniform convergence in the interior of a domain 11

Uniqueness theorem of Privalov 428 Univalent function 23 Universal covering surface of n-connected domain 261 Variation of univalent functions 100-108 Variational formulas in classes E9 527,

528; S and 2 106, 107; Sa 108, 109 Variational method 99-108, 128-165, 526, 569, 572, 592 Variational-geometric method of Lavrent'ev 569

Variational-parametric method of Xufarev 571

Vitali's theorem 18 Weierstrass's theorem 12

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Geometric theory of functions of a complex variable (Translations of Mathematical Monographs, Vol. 26)

Geometric Theory of Functions of a Complex Variable (Translations of Mathematical Monographs)

Geometric Theory of Functions of a Complex Variable (Translations of Mathematical Monographs)

Theory of Functions of a Complex Variable

Theory of Functions of a Complex Variable

Theory of Functions of a Complex Variable ; VOL II

Theory Of Functions Of A Complex Variable - Vol 1

Mathematical Monographs no. 11: Functions of a Complex Variable

Theory of Functions of a Complex Variable, Vol. 2

Functions of a complex variable

Records: Mathematical Theory (Translations of Mathematical Monographs)

Elements of the Theory of Functions of a Complex Variable

Distribution of Zeros of Entire Functions (Translations of Mathematical Monographs)

Functions of one complex variable

Functions of one complex variable

Functions of One Complex Variable

Distribution of Zeros of Entire Functions (Translations of Mathematical Monographs)

Functions of One Complex Variable

Introduction to The Theory of Functions of a Complex Variable

Theory of Functions of a Complex Variable, Volume Two

Theory of functions of a complex variable, 1st edition

Function Theory in Several Complex Variables (Translations of Mathematical Monographs)

Theory of Functions of a Complex Variable Volume 1

Theory of Functions of a Complex Variable Volume 1

Theory of Functions of a Complex Variable Volume 2

Theory of Functions of a Complex Variable, Volume 3

Fewnomials (Translations of Mathematical Monographs)

Theory of Functions of a Complex Variable, Volume 3

Theory of Functions of a Complex variable, Volume One

Theory of Functions of a Complex variable, Volume One

Theory of Functions of a Real Variable,

Geometric theory of functions of a complex variable (Translations of Mathematical Monographs, Vol. 26)