Semigroups in Geometrical Function Theory by
David Shoikhet Department of Mathematics, Technion-Israel Institute of Technology, Haifa, Israel
KLUWER ACADEMIC PUBLISHERS DORDRECHT/BOSTON/LONDON
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ISBN 0-7923-7111-9
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Contents Preface
vii
Preliminaries 0.1 Notations and notions . . . . . . . . . . . . . 0.2 Holomorphic functions of a complex variable 0.3 Convergence of holomorphic functions . 0.4 Metric spaces and fixed point principles . . 1 The 1.1 1.2 1.3 1.4 1.5
Wolff-Denjoy theory on the unit disk Schwarz-Pick Lemma and automorphisms . Boundary behavior of holomorphic self-mappings Fixed points of holomorphic self-mappings . . . . Fixed point free holomorphic self-mappings of~The Denjoy-Wolff Theorem. . . . . . . . . . . . . . . . . . . . Commuting family of holomorphic mappings of the unit disk.
1 1 4 6 7 9 9 17 25 32 36
2 Hyperbolic geometry on the unit disk and fixed points 2.1 The Poincare metric on~ . . . . . . . . . . . . . . . 2.2 Infinitesimal Poincare metric and geodesics . . . . . . . . 2.3 Compatibility of the Poincare metric with convexity . . . 2.4 Fixed points of p-nonexpansive mappings on the unit disk
39 39 44 46 52
3 Generation theory on the unit disk 3.1 One-parameter continuous semigroup of holomorphic and p-nonexpansive self-mappings . . . . . . . . . . . . . . . . 3.2 Infinitesimal generator of a one-parameter continuous semigroup 3.3 Nonlinear resolvent and the exponential formula . . 3.4 Monotonicity with respect to the hyperbolic metric . . . . . . . 3.5 Flow invariance conditions for holomorphic functions . . . . . . 3.6 The Berkson-Porta parametric representation of semi-complete vector fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
4 Asymptotic behavior of continuous flows 4.1 Stationary points of a flow on ~ . . 4.2 Null points of complete vector fields v
59 62 67 79 83 95
101 101 104
CONTENTS
vi
4.3 4.4 4.5 4.6 4.7
Embedding of discrete time group into a continuous flow . . . . Rates of convergence of a flow with an interior stationary point A rate of convergence in terms of the Poincare metric . . . . . Continuous version of the Julia-Wolff-Caratheodory Theorem . Lower bounds for p-monotone functions . . . . . . . . . .
109 113 120 124 135
5 Dynamical approach to starlike and spirallike functions 5.1 Generators on biholomorphically equivalent domains . . . 5.2 Starlike and spirallike functions . . . . . . . . . . . . . . 5.3 A generalized Visser-Ostrowski condition and fanlike functions 5.4 An invariance property and approximation problems . . 5.5 Hummel's multiplier and parametric representations of starlike functions . . . . . . . . . . . . . . . . . . . . . . 5.6 A conjecture of Robertson and geometrical characteristics of fanlike functions . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Converse theorems on starlike, spirallike and fanlike functions 5.8 Growth estimates for spirallike, starlike and fanlike functions 5.9 Remarks on Schroeder's equation and the Koenigs embedding property . . . . . . . . . . . . . . . . . . . . . . . . . .
153 154 157 163 166
Bibliography
205
Author and Subject Index
216
List of figures
221
172 176 186 194 198
Preface
Historically, complex analysis and geometrical function theory have been intensively developed from the beginning of the twentieth century. They provide the foundations for broad areas of mathematics. In the last fifty years the theory of holomorphic mappings on complex spaces has been studied by many mathematicians with many applications to nonlinear analysis, functional analysis, differential equations, classical and quantum mechanics. The laws of dynamics are usually presented as equations of motion which are written in the abstract form of a dynamical system: dxjdt+ f(x) = 0, where xis a variable describing the state of the system under study, and f is a vector function of x. The study of such systems when f is a monotone or an accretive (generally nonlinear) operator on the underlying space has been recently the subject of much research by analysts working on quite a variety of interesting topics, including boundary value problems, integral equations and evolution problems (see, for example, [19, 13] and [29]). In a parallel development (and even earlier) the generation theory of oneparameter semigroups of holomorphic mappings in en has been the topic of interest in the theory of Markov stochastic processes and, in particular, in the theory of branching processes (see, for example, [63, 127, 48] and [69]). Later, such semigroups appeared in other fields: one-dimensional complex analysis ([17, 28]), finite-dimensional manifolds [2, 5], Banach spaces geometry [12, 142], control and optimization theory [64], and Krein spaces [147, 148]. At the same time very rapid developments in multi-dimensional complex analysis, functional analysis and a variety of techniques and methods have diverted attention from the roots of one-dimensional dynamic approaches. Furthermore, some interesting results on evolution equations, presented in various papers in journals on nonlinear analysis, abstract analysis, and differential equations, have remained unnoticed by experts in one-dimensional complex variable. One of the first applicable models of the complex dynamical systems on the unit disk arose more than a hundred years ago from studies of the dynamics of stochastic branching processes. In 1874 F. Galton and H.W. Watson [151] in treating the problem of the extinction probability of family names, formulated a mathematical model in terms vii
PREFACE
viii
of the probability generating function: 00
F(z)
=
LPkZk,
lzl::; 1,
k=O
where z is a complex variable, Po, Pl, ... , Pk are nonnegative numbers (probabilities) such that E~o Pk = 1, and its iterations: p(O)(z)
= z,
p(n+l)(z)
= p(n)(F(z)).
The first complete and correct determination of the extinction probability for the Galton-Watson process as the limit points of the iteration sequence was given by J.F. Steffensen in 1930 [139]. Since then the interest in this model has increased because of connections with chemical and nuclear chain reactions, the study of the multi plication of electrons in the electron multiplier, the theory of cosmic radiation, and many other biological and physical problems. Detailed description of classical results concerning branching processes can be found in the books of T.E. Harris [63] and B.A. Sevastyanov [127]. We only note here that while the original Galton-Watson process has been related to discrete time branching process (i.e., it is described by an iteration process of a single probability generating function) the further development involved also the consideration of continuous time branching processes based on one-parameter semigroups of analytical self-mappings of the unit disk. One of the problems in analysis is, given a function F(z) find a function F(z, t), with F(z, 1) = F(z) satisfying the semigroup property F(z,t+s)=F(F(t,z),s),
t,s;:=:o,
where z is a complex variable. Since this formula expresses the characteristic property of iteration when t and s are integers we may consider F(z, t) as a fractional iterate ofF, when t is not an integer. G. Koenigs (1884) [78] showed how this problem may be solved, ifF is analytic self-mapping on the unit disk with an interior fixed point z 0 = F(z 0 ), such that 0 < IF'(zo)l < 1, by considering the convergence of the sequence {F(n)(z)} to zo, as n ~ oo in a neighborhood of the point z0 . These and other problems led to the following general question: Let D be a bounded domain in C and let F : D ~----> D be a holomorphic mapping of D into itself. Does the sequence of iterates {F(n)(z)}~=l converge uniformly on compact subsets of D to some holomorphic mapping h : D ~----> C? In 1926 J. Wolff (see [155]-[157]) and A. Denjoy [31] solved this problem for D = ~, the unit disk in C. Applying Schwarz's and Julia's Lemmas they proved the following remarkable results: Let F : ~ ~----> ~ be a holomorphic mapping of the unit disk of C into itself. lf F is not an automorphism of~' with exactly one interior fixed point, then {F(n)(z)} converges uniformly on compact subsets of~ to a holomorphic mapping h:~t--->~.
Moreover, if F is not the identity then h is a constant.
PREFACE
ix
This result has given a powerful thrust to the development of different aspects of complex dynamical systems on the unit disk, the complex plane, or, more generally, hyperbolic Riemann surfaces. In 1930 G. Julia [72] in publishing of geometrical principles of analysis characterized the dynamics of analytic motions in the unit disk. Over the last 40 yean; these results have been developed in at least three directions. In 1960 V.P. Potapov [110] extended the classical lemma of G. Julia to matrixvalued holomorphic mappings of a complex variable. Next I. Glicksberg [49] and K. Fan [43] established the version of Julia's lemma for function algebras. Also K. Fan [44, 45] extended a result of J. Wolff [155] on iterates of self-mapping of proper contraction operator in the sense of functional calculus. Recently K. Wlodarczyk [152]-[154] and P. Mellon [94] motivated by the research ofV. Potapov and K. Fan generalized these results to holomorphic mappings on J* -algebras. In general, they used operator theoretic methods. Finite-dimensional generalizations are to be found in G.N. Chen [24], Y. Kubota [80], B. MacCluer [91], M. Abate [2, 5], P. Mercer [96], among others. In all cases it appears that some sort of 'finiteness' or 'compactness' is required. E. Vesentini [144, 145], P. Mazet and J.P. Vigue [92, 93] have used spectral analysis for the case when a holomorphic self-mapping of a domain in a Banach space has an interior fixed point. Further observations of the Wolff-Denjoy Theory (even for one-dimensional cases) yield extensions to those mappings which are not necessarily holomorphic but are nonexpansive with respect to a hyperbolic metric on a domain. This approach has been developed by C. Earle and R. Hamilton [34], K. Goebel, T. Sekowski, T. Kuczumov, A. Stachura, S. Reich, I. Shafrir including others (see, for example, [50, 52, 53, 52, 82, 83, 129, 130]) and it is based on the study of the so called approximating curves. These results can also be considered as an implicit analogues of the DenjoyWolff Theorem. It is also remarkable that the asymptotic behavior of the approximating curves is actually nicer than that of the usual iterative process. Moreover, these implicit methods may be useful not only for a self-mapping of a domain, but also for a wider class of mappings which satisfy certain flow invariance conditions. Actually, the study of these methods shows their deep connections with continuous time dynamical systems. In fact, in 1964 F. Forelli [46] established a one-to-one correspondence of the group of linear isometrics of a Hardy space HP, p > 0, p =/= 1, and the group of Mobius transformations on the unit disk. E. Berkson and H. Porta in [17] established a continuous analog of the Denjoy-Wolff Theorem for continuous semigrou ps of holomorphic self-mappings of the unit disk. This approach has been used by several mathematicians to study the asymptotic behavior of solutions of Cauchy problems (see, for example, [16, 2, 5, 28] and [117]). Berkson and Porta [17] also apply their continuous analogue of the DenjoyWolff Theorem to the study of the eigenvalue problem of composition operators in Hardy spaces. Similar approaches were used by A.G. Siskakis [136] to study the Cesaro operator on HP [136], F. Jafari and K. Yale [70], Y. Latuskin and M. Stepin [85] for weighted composition operators and dynamical systems on HP. See
PREFACE
X
also the book of C. C. Cowen and B. MacCluer [28] and references there. These arguments motivate us to give a systematic exposition of the WolffDenjoy theory and its application to dynamical systems. In fact, it does not seem possible to cover the extensive literature concerning this subject in a single book. Nevertheless, I believe that the first step in this direction should be the understanding of dynamic processes on the unit disk of the complex plane. Actually, one can see that multi-dimensional generalizations may often be obtained by the reduction to the one-dimensional case, or by generalizations of attendant notions to higher dimensions. The study of images of domains under holomorphic (or biholomorphic) transformations links some of the most basic questions one can ask about semigroups with nice classical results in geometric function theory. As a part of analytic function theory, research on the geometry of domains in the complex plane is of old origin, dating back to the 19th century. The basic methods of geometrical function theory include the square principles, methods of contour integration, variational methods, extreme metrics, and integral representation theory. Good introductions in these topics can be found in the books [55, 87, 11] and [57]. An exhaustive bibliography on geometrical function theory was compiled by S.D. Bernardi [18] before and until 1981. This book is not intended to be a survey of the theories mentioned above. Our primary focus will be only on the material investing old theorems with new meanings which are related to applications of evolution equations to the geometry of domains in the complex plane. For example, it is a well known result, due to R. Nevanlinna (1921) [102], that if h is holomorphic in [z[ < 1 and satisfies h(O) = 0, h'(O) # 0, then h is univalent and maps the unit disk onto a starshaped domain (with respect to 0) if and only if Re[zh'(z)/h(z)] > 0 everywhere. This result, as well as most of the work on starlike functions on the unit disk, can be obtained from the identity f)
ae arg h(re
iiJ )
= Re
{rei 8 h'(re;8 )} h(reiiJ) .
This idea does not readily extend to a higher-dimensional space. Moreover, such an approach is crucially connected with the initial condition h(O) = 0. Much later Wald [150] characterized of those functions which are starlike with respect to another center (sometimes these functions are called weakly starlike [65, 66]). Observe that although the classes of starlike, spirallike and convex functions were studied very extensively, little was known about functions that are holomorphic on the unit disk t:l. and starlike with respect to a boundary point. In fact, it was only in 1981 that M.S. Robertson [121] introduced two relevant classes of univalent functions and conjectured that they were equal. In 1984 his conjecture was proved by A. Lyzzaik [90]. Finally, in 1990 Silverman and Silvia [134], using a similar method, gave a full description of the class of univalent functions on t:l., the image of which is starshaped with respect to a boundary point. However, the approaches used in their work have a crucially one-dimensional character (because of the Riemann Mapping Theorem and Caratheodory's Theorem on Kernel Convergence). In addition, the conditions given by Robertson and by Silverman and
PREFACE
Xl
Silvia, characterizing starlikeness with respect to a boundary point, essentially differ from Wald's and Nevanlinna's conditions of starlikeness with respect to an interior point. Hence, it is difficult to trace the connections between these two closely related geometric objects. Therefore, even in the one-dimensional case the following problem arises: To find a unified condition of starlikeness (and spimllikeness) with respect to an interior or a boundary point. By 1923 K. Lowner [89] described the problem of continuous deformations of a domain of certain class by using a first order differential equation au at
au k(t)
+u
= az k(t) - u u,
u(O, z) = z, z E ~ = {lzl < 1}, 0 < t < T, lk(t)l = 1, the right hand of which is time dependent and has a moving polar singularity. This equation is central in the theory of parametric extensions of univalent functions and its application (see, for example [11] and references there). In particular, L. de Branges [30] used it to solve the famous Bieberbach conjecture. It seems that the idea to use autonomic dynamical systems was first suggested by Robertson [120] in 1961 and developed by Brickman [20] in 1973, who introduced the concept of Ill-like functions as a generalization of starlike and spirallike functions (with respect to the origin) of a single complex variable. Suffridge, Pfaltzgraff [104, 105, 140, 141, 106] and Gurganus [58] (see also [26] and [109]) developed a similar approach in order to characterize starlike, spirallike (with respect to the origin), convex and close-to-convex mappings in higher-dimensional cases. Since 1970 the list of papers on these subjects has became quite long. Nevertheless, it seems that there is no extension of Wald's, as well as of Silverman and Silvia's, results to higher dimensions. In the last chapter we demonstrate how one can study starlike and spirallike functions by using the behavior of trajectories defined by related dynamical systems and their characterizations. For example, in the description of starlike functions, this approach, roughly speaking, consists of the following observation: If we consider the flows defined by the evolution equations dujdt + h(u) = 0 and dvjdt+v = 0, the condition of starlikeness (with respect to a point in the closure of a domain), translates to the fact that h takes integral curves of the first evolution equation into those of the second one. So the problem is to find a suitable condition to describe those holomorphic functions which generate a converging semigroup of holomorphic self-mappings of a domain. Note that such approach can be useful also for higher dimensions (see, for example, [140, 141, 104, 105, 106, 58, 26, 109, 56, 36] and [41]). My purpose has been to write a tutorial rather than a scientific book. And though many results are previously contained only in journal papers, I have tried to facilitate the understanding of their proofs to students and nonspecialists. Certain chapters of this book were used in my lectures on Geometrical Theory of Complex Variables at Ort Braude College, Karmiel. More advanced sections were presented in a special course for the College teachers and at the Seminars on Functional Analysis and Nonlinear Analysis at the Technion, Haifa.
xii
PREFACE
It seems that the book will be useful to students and postgraduate students in engineering who may use Complex Dynamic Systems. Furthermore, I hope that it will be of interest to mathematicians specializing in complex analysis and differential equations. My joint work, as well as numerous talks, with Simeon Reich, Mark Elin aud Dov Aharonov convinced me that Generation Theory may serve as a showcase for the Classical Geometrical Function Theory. Simeon Reich has given a series of lectures on Complex Analysis and Hyperbolic Geometry in the Complex Plane. His initiative has stimulated me to design a course on the relationships of various topics related to one-dimensional analysis. I would like to thank ORT Braude College, Karmiel and the Technion, Haifa for their support throughout the project. I am very grateful to my colleagues Mark Elin, Giora Enden, Yakov Lutsky, Ludmila Shvartsman for their help. Special thanks go to Mark Elin who examined the manuscript in detail and whose proofreading and astute observations led to significant improvements in Chapter V. I am deeply indebted to my wife Tania for typing the final version.
Preliminaries In this short chapter we compile the series of very basic notions and results, probably familiar to most readers. As we mentioned in the Preface, only a very modest preliminary knowledge is required to read the following material. Nevertheless, certain fundamental topics, such as those related to integral representations, convergence theorems, and fixed point principles, will be used throughout the text, and therefore should be presented at least an auxiliary material.
0.1
Notations and notions
Throughout the book we shall use the following notation: lR- the set of all real numbers (real axis); lR + = { x E lR : x ~ 0} - the set of all nonnegative real numbers (the nonnegative real half-axis);
C - the set of all complex numbers z = x + iy, x, y E JR, i 2 = -1 (i.e., the complex plane). If z = x + iy E C then z = x- iy is its conjugate complex number, Re z := x, Imz := y and [z[ = Jx2 + y2. As usual, if z E Cis represented in the form z = [z[ei 9 , then () = arg z. II+ = {z E C : Re z ~ 0} ~ the right half plane. ~r(a) = {z E C: [z- a[< r} -the open disk of radius r > 0 centered at a E C. ~ = ~r(O) = {z E C: [z[ < 1}- the open unit disk in C. Definitions of a few technical terms will be given here. An open connected subset D ofC is called a domain in C. The symbol aD will denote the boundary of D. In particular, a~= {z E C: [z[ = 1} is the unit circle. When z represents any number of a set D C C, we call z a complex variable. The set D is usually a domain in C. If to each value z in D we assign a second complex variable w, then w said to be a function of the complex variable z on the 1
2
set D: w
= f(z).
The term 'function' signifies a single-valued function unless specified differently. When w = f (z) and w and z are complex variables some information about f may be conveniently illustrated graphically, however, two separate complex planes for the two variables z = x + iy and w = u + iv are required.
f
Figure 0.1: The function w = f(z). The correspondence w = f (z) between points in the two planes is called a mapping or transformation of points (or sets) given by the function f.
-~
-2
Figure 0.2: The translation f(z)
= z +a, a= 4 + 2i.
We use the following geometrical terms in the form: translation: f(z) = z +a, a E C, (Figure 0.2) rotation: f(z) = ei 0 z, BE (0, 2n), (Figure 0.3) contraction: f(z) = kz (Figure 0.4). It is sometimes convenient to consider the mapping as a transformation in a single plane. Sometimes to conform our terminology with the theory of dynamical systems we will refer to the 'vector field' f : D ~----> C, as a vector f(z) whose origin is at z E D (cf., Figure 0.1 and Figure 0.5).
PRELIMINARIES
3
-1
-2
Figure 0.3: The rotation f(z) = ei-rroz, 8 = -2rr/3.
0
-
::2
:L
:L
Figure 0.4: The contraction f(z)
= kz,
k
= 1/3.
f(c;
Figure 0.5: The veclu1 f1dd w = f(z).
4
0.2
Holomorphic functions of a complex variable
Iff is a complex-valued function defined on an open subset D of C we say that f is holomorphic in D if for each zo E D the quotient
f(z) - f(zo) z- zo has a limit inC as z approaches z0 . When this limit exists it is called the derivative off at zo and is denoted as f'(zo). Iff is holomorphic in D and w = f(z) maps D into the set n C C we say that f maps D holomorphically into n and write f : D ~--+ n. The set of all holomorphic functions (mappings) in D with values in a set n c C will be denoted by Hol(D, f!). If such a mapping is one-to-one on a domain D a function f E Hol(D; C) is said to be univalent on D. In particular, Hol(D, C) is the vector space of all holomorphic functions in D. We will simply use Hol(D) to denote the set Hol(D, D) of holomorphic selfmappings of D. IfF and G are in Hol(D), then If»= FoG is also in Hol(D), where o denotes the composition operation on this set, i.e., IP(z) = F(G(z)). A fundamental property of holomorphic functions is the Cauchy Integral
Formula: If f E Hol( D, C), then for each closed contour "' in D and each z E D \ "' _1
27ri
J
f(()d( = (- z
{ f(z), 0,
if z inside "' otherwise.
(0.2.1)
'"'(
This formula leads to the most basic and important results in the theory of holomorphic functions. In particular, iff E Hol(D,C), then f'(z) also belongs to Hol(D,C), hence by induction, f is infinitely differentiable. We adopt the notation ~:t(z), for the n-th derivative off at the point zED. The following integral formula is known as the general Cauchy formula
dn f n! dzn (z) = 27ri
J
f(()d( ((- z)n+l'
n = 0, 1, 2, ... ,
(0.2.2)
'"'(
where "' here is any closed positively oriented contour in D, such that z is inside "f.
In this formula as well in the Cauchy formula (0.2.1) the contour "' can be replaced by the oriented boundary aD of a domain D c C, whenever f admits a continuous extension to an. Furthermore, if f E Hol(D, C) then for each point zo E D there is a disk 6-r(zo) = {z E C : lz- zol < r} in D such that f admits Taylor's power series
PRELIMINARIES
5
extension at zo: 00
f(z)
=
L an(z- zot,
z E D-r(zo),
n=O
where ao = f(zo), That is, the infinite series converges to f. Another very important consequence of the Cauchy integral formula is the so called maximum modulus principle: Iff E Hol(D, C) is not a constant and z 0 E D then every neighborhood U of zo contains points z such that
lf(z)l > 1/(zo)l. In other words, if for f E Hol(D, C) the modulus 1/1 attains a maximum in D, then f is constant. Further, any real function u = u(x, y) of two real variables x and y that has continuous partial derivatives of the second order and that satisfies Laplace's equation {)2u
cPu
8x2
+ 8y2
= 0
throughout a domain Dis called a harmonic function in D. For f E Hol(D,C) the functions u = Ref(x + iy) and v = lmf(x + iy) are known to be harmonic. The maximum modulus principle imply maximum and minimum principles for harmonic functions: If u = u(x, y) is a nonconstant harmonic function in D, then it cannot attain neither maximum nor minimum in D. The function u = Ref plays a crucial role in the study of holomorphic functions on the unit disk D.. In particular, each f E Hol(D., C) can be expressed in terms of its real part: f(z)
1
r
( +_
= ilmf(O) + 21ri 1-r Ref(() (
z d( z ('
(0.2.3)
where"(= {( : 1(1 = r} is a circle of radius 1 > r > lzl. This formula is often referred to as the Cauchy-Schwarz representation formula. The holomorphic functions fonD. whose image lies in the right half-plane II+, play a special role in further discussions. A remarkable result proposed by (0.2.3) and proved by Herglotz and Riesz (see, for example [9, 57, 122]) is an integral representation of positive harmonic functions on D.: ifp E Hol(D.,q with p(O) = 1 and Rep(z) > 0, zED., then p(z) =
1
1 +z(
- - - dm((), IC:I=l 1 - z(
where m is a probability measure on the unit circle.
zED.,
6
0.3
Convergence of holomorphic functions
The vector space Hol(D, 1 _ (1- IF(O)i2)(1- IF(z)i2) (1 -IF(O)IIF(z)l) 2 (IF(z)I-IF(O)I) 2 (1 -IF(O)IIF(z)l) 2 .
Hol(~).
then it
THE WOLFF-DENJOY THEORY
13
On the other hand, it follows by the Schwarz-Pick inequality that F(z)- F(O) I < I1F(z)F(O) -
Jzl
and IF(z)J-JF(O)J 1 -IF(O)JJF(z)J :::; lzi. Solving the last inequality for JF(z)J we obtain our assertion.
0
Corollary 1.1.3 (see L.A. Harris [61]} IfF E Hol(~) then it satisfies the following estimate 1-IF(O)i2 JF(z)- F(O)J:::; Jzl1 -JF(O)Jizi Proof. Consider the mapping G = m-F(O) oF. Solving this equation for F we obtain: F z - m G z - G(z) + F(O) ( ) F(o)( ( )) - 1 +F(O)G(z)
and F(z) _ F(O)
But G(O)
= G(z) + F(O)- F(O)- G(z)JF(O)i2 = G(z) 1 -JF(O)i2 1 + F(O)G(z) 1 + F(O)G(z)
= 0, hence JG(z)J :::; JzJ,
z E ~.
This implies the required estimate.
D
Remark 1.1.1 Observe also that (1.1.4) implies the estimate:
(1.1.5) for each F E Hol(~) and z E ~. Therefore for each r E (0, 1) we obtain the following uniform Lipschitz condition: 1 IF(z)- F(w)J :S 1 _ .,. 2 Jz- wJ,
(1.1.6)
whenever z and w belong to ~r = {z E ~: Jzl < r}. Now iff E bounded, i.e., Jf(z)J ::; M for all z E ~'then F
1
= Mf
Hol(~,q
is
E Hol(~).
Thus we have that each bounded holomorphic function on Lipchitzian with respect to Euclidean distance in C.
~
is locally uniformly
Remark 1.1.2 From the point of view of geometric function theory on the unit disk, the Euclidean distance on the disk is inappropriate. The pseudo-hyperbolic
14
Chapter I
distance between points defined as follows:
z
and w of /:).. is the function d : /:).. x /:).. d(z, w)
-->
JR+ = [0, oo)
= lm-w(z)i.
It is easy to see that, actually, d is a metric on /:).. that induces the usual Euclidean topology. Thus the Schwarz-Pick Lemma states the contraction property of a mapping F E Hol(/:)..) with respect to this metric: d(F(z), F(w)) ~ d(z, w),
z, wE/:)...
Moreover, F is an isometry with respect to d (i.e., d(F(z), F(w)) and only ifF E Aut(~).
= d(z, w))
if
Exercise 5. Show that for each r E (0, 1) lim
inf d(z.w) = 1.
lwi-+I- lzi:Sr
Consider now a few geometrical properties of the action of the group which will be useful in the sequel.
Aut(~)
y
X
c
r Figure 1.1: An orthogonal circle
to()/:)..
and its image under an automorphism.
Let r be a circle orthogonal to a~, the boundary of~. r n ~ f. 0. If cis the center of r, then lei 2: 1, and Jicl 2 - 1 is the radius of r (see Figure 1.1). Thus the circle r satisfies the equation:
r = {z E C:
izi 2
-
2Re(zc)
+ 1 = 0}.
(1.1.7)
THE WOLFF-DENJOY THEORY
15
Consider now the set 'Y = r n ~ which is the arc of r in ~ with end points A and B (see Figure 1.1). We raise the following question: what is the image of 'Y under F E Aut(~)? The next assertion answers this question.
Proposition 1.1.4 lf r is an orthogonal circle to 8~1 and 'Y = r n ~I then for each FE Aut(~) the image F('Y) of'Y is either the intersection of~ with another orthogonal circle to 8~ 1 or it is a diameter of~Proof. IfF is a rotation of~. i.e. F(z) = ei'Pz, r.p E [0,2rr], z E ~.then by (1.1.7) we obtain: F('Y) n ~={wE C: lwl 2
-
where c1 = ei'Pc and the proof is completed. IfF is a Mobius transformation ma, a E
2Re(wcJ.) ~.
+1=
0},
we have for wE ma('Y) n ~:
or ((lwl 2 + 1)- 2Re(wf)) = 0, where ( = (1 + 2 Re(ca)
+ lal 2) and TJ = c + ca 2 + 2a. If ( #- 0 then lwl 2 - 2 Re(wc2)
+1=
(1.1.8) (1.1.8) implies:
0,
where c2 = TJ / (. Hence ma ('Y) n ~ is an intersection of~ with an orthogonal circle to a~. If ( = 1 + 2 Re(ca) + lal 2 = 0, then by (1.1.7) we have that -a E 'Y n ~- Hence 0 = ma( -a) E ma('Y) n ~ and once again it follows from (1.1.8) that Re (wi))
=
0.
This equation represents a straight line through zero, hence ma ('Y) n ~ is a diameter of~Since each F E Aut(~) has the form ei'Pma, r.p E [0, 2rr], z E ~. (see Proposition 1.1.2.) our proof is completed. D This assertion states that each automorphism of~ preserves the class of subsets of ~ which consists of all diameters of ~ and all arcs of the circles which are orthogonal to the boundary of ~We will see below that this result allows us to define the so called geodesic segments on~. whose 'length' is preserved under automorphisms. Another useful image property of automorphisms is the following. Let the mapping ma : ~ r-+ ~be defined as above by formula (1.1.1).
Proposition 1.1.5 The images of concentric disks ~r = {z E ~: lzl < r}, r E (0, 1) under the Mobius transformations ma, a E ~~ are disks in~ centered at 1
c
1- r 2 r21 a 12 . a
= c( r) = 1 -
(1.1.9)
16
Chapter I
------> Figure 1.2: A Mobius transformation of
~r·
with radius d = d(r) =
1-JaJ 2 J J • r. 1- r 2 a 2
(1.1.10)
Proof. Let nr(a) := ma(~r)· Then ~r = m_a(nr(a)). In other words, nr(a) ={wE~: Jm-a(w)J < r}. So, we need to solve the inequality
w-a - I 0 we have the following inclusion:
F (D((, K)) ~ D(TJ, aK).
Proof. It follows that (1.2.4) and (1.2.5) induce ITJI = 1. For z, w E D. we define the function _ (1 - lwl 2 ) (1 - lzl 2 ) (1.2.6) a ( z, w ) I _ . 1-wz 12
THE WOLFF-DENJOY THEORY
19
It is a simple exercise to show that the Schwarz-Pick inequality (Proposition 1.1.3) can be rewritten in the form
u(z,w):::; u(F(z),F(w)) for all FE
Hol(~)
(1.2. 7)
and z, wE~- In particular, we have:
u(z, Zn) :::;
0"
(F(z), F(zn))
for each z E ~ and all n = 1, 2, .... Writing the latter inequality explicitly we obtain after simple manipulations: 11- F("z.JF(zW (1 -IF(znW)I1- z;;:zl 2 2 (1-IF(z)l ) :::; (1 -lznl 2)(1 -lzl 2) · Letting n go to infinity we obtain (1.2.5). D Exercise 1. Prove formula (1.2.6) directly. Exercise 2. For FE Hol(~) and ( E number of F at () by the formula:
8~,
define the value a((, F) (the Julia
a(( F)= liminf 1 -IF(z)l ' z-+( 1- lzl '
(1.2.8)
where z tends to ( unrestrictedly in ~If a((, F) is finite, it follows by Julia's Lemma, that 11- F(z)iJI 2 < (~" F) 11- z(l 2 1 -IF(z)l2 -a .,, 1 -lzl2 ' where Tf
(1.2.9)
= n-+oo lim F(zn), {z,.} being a sequence along which the lower limit in (1.2.8)
is achieved. Show that if for at least one z E automorphism of ~-
~
we have equality in (1.2.9) then F is an
Remark 1.2.1 Actually inequality (1.2.5) means that for each K > 0, under
conditions (1.2.3) and (1.2.4), F maps the horocycle with radius R = K
K
+1
centered at - K ( into the horocycle with radius r = ;K centered at KTf +1 a +1 a +1 (see formula (1.2.2)). Therefore, ifF is a nonconstant holomorphic mapping, then the number a in (1.2.3) must be positive. Also, the same conclusion can be obtain by using the following result of the Schwarz-Pick Lemma. Exercise 3. Show that for each FE Hol(~) and z E holds: _1---'---IF,. .:. ,(z.. .:. .:.)l > 1- IF(O)I_ 1- lzl - 1 + IF(O)I
~the
following inequality
20
Chapter I
Remark 1.2.2 Observe also, that if the limit a in {1.2.3) exists for a sequence Zn -+ ( E 8/:l, then so does limit rJ E 8/:l in {1.2.4). Indeed, (1.2.5) implies that IF(zn)l -+ 1 as n -+ oo. Therefore, each subsequence of {F(zn)};:"= 1 has a further subsequence which converges to some unimodular point. Choose two such subsequences and take their limit points, say 'r/1 and "12, lrJ1I = lrJ2I = 1. If 'r/1 f. 'r/2, by decreasing K if necessary, we can find two horocycles D( ry 1 , o:K) and D( ry2 , o:K) whose intersection is empty. But from (1.2.5) for each z E D((, K) the element F(z) must lie in both of them. Contradiction. Hence 'r/1 = 'r/2, and the limit in (1.2.4) exists. Moreover, we will show that, in fact, this limit exists (and equals to ry) for each sequence {zn} which converges to ( along so called nontangential directions. More precisely: Definition 1.2.1 For a point ( on the unit circle 8/:l and"' > 1 a nontangentiai approach region at ( is the set r((, K)
= {z E !l: lz- (I < K(1 -lzl)}.
(1.2.10)
The term 'nontangential' refers to the fact that f((, K) lies in a sectorS in !l at point ( which is the region bounded between two straight lines in !l that meet at (and are symmetric about the radius to(, i.e., the boundary curves off((, K) have a corner at (, with an intersection angle less than 1r (see Figure 1.4).
1
Figure 1.4: A nontangential approach region at a boundary point.
Definition 1.2.2 We will say that a function f E Hol (!l, C) has a nontangentiu (or angular) limit L at a point ( E 8/:l if f(z)-+ L as z-+ (, z E f((, K) for eac, "' > 1. We will write in this case L
= L.lim
z-+(
f(z).
THE WOLFF-DENJOY THEORY
21
Exercise 4. A Stolz angle at ( E 8!::l. is the set
S= {z
E !::l.: larg(1- (z)l
< /3, lz- (: < r, f3 E (O,n/2), r E (0,2cosf3)}
(see, Figure 1.5). Show that f E Hol(!::l., C) has a nontangentiallimit L at a point ( if and only if f(z)---+ Las z---+ (,for each Stolz angleS at(.
1
Figure 1.5: A Stolz angle at a boundary point.
A simple consequence of .Tulia's Lemma is the following. Corollary 1.2.1 Let F be a nonconstant holomorphic self-mapping of !::l., and let Zn which converges to (, such that
( E 8!::l.. Suppose that there is a sequence
. . 1 -IF(zn)l hmmf =a< oo. z,.-+( 1- lznl Then: (i)
Q
> 0;
(i·i) the nontangential limit TJ :=
L lim F(z), z-+(
which is a point of the boundary 8!::l. exists; (iii) for each K > 0 the following inclusion holds
F(D((, K))
~
D(ry, aK).
(1.2.11)
Chapter I
22
Proof. It remains to show only that condition (1.2.11) implies assertion (ii). Indeed, given e > 0 choose K > 0 such that D( ry, aK) is contained in the e-disk centered at ry. Further, let S be a sector in .0. with its vertex at (. Then one can find r5 > 0 such that S1 = S n {z E C: lz- (I
< r5} c D((, K).
Hence by (1.2.5) we have
IF(z)- TJI < e for z E S1 (see figure 1.6) .D
1
Figure 1.6: Boundary behavior of a self-mapping of .0.. Finally, we adduce a much stronger assertion, established by Caratheodory (see [22] and [23]).
Proposition 1.2.2 (The Julia-Caratheodory Theorem) Let FE Hol(.0.) an let ( E 8.0.. Then the following are equivalent: (i)
li~1~~1f 1 ~ ~~? =
a
(
Z-
~ TJ
:= L_F'(() exists for a point 71 E 8.0.;
(iii) L lim F'(z) exists, and L lim F(z) = TJ E 8.0.. z-.(
z-
0 in {i); (b) the boundary points TJ in (ii} and (iii} are the same; (c) L lim F'(z) = LF'(() = a(ry. z~(
The value LF'(() is called the angular derivative ofF at ( (see also Section 4.6). There are various proofs of the Julia-Caratheodory Theorem (see for example, the papers [157, 86, 143, 126, 54, 123] and the books [23, 103, 131, 28]. In fact, we will prove below (see Section 4.6) in detail several general assertions which extend Proposition 1.2.2. Nevertheless, in order to demonstrate a direct method and for the completeness we give here a proof which is based on two important classical statements. Proposition 1.2.3 (Lindelof's Principle, [88]) Let ( E 86. and let function f E Hol(6., C) be bounded on each nontangential approach region at(. If for some continuous curve 'Y E 6. ending at ( there exists the limit
L = lim f(z), z_,(
z E "(,
then the angular limit
L lim f(z) = L z_,(
also exists.
Proposition 1.2.4 (see [107], p. 79) Let ( E 86. and let f E Hol(6., C). Suppose that the limit L = L lim f(z) := f(() z_,(
exists (finitely). Then
Llim f(z)-f(() Z- (
z_,(
exists if and only if L lim J'(z) z_,(
exists and both coincide.
For a proof of this proposition see also Section 4.6. Proof of Proposition 1.2.2. First we note that the implication (iii) => (ii) is obvious due to Proposition 1.2.4. In turn, (ii) => (i) because of the inequality 1 -IF(r()l ITJ- F(r()l < ' 1- r I(- r(l
--~~~
which holds for all r E (0, 1).
Chapter I
24
Now if (i) holds, then by Corollary 1.201 there exists
L lim F(z) := TJ E 8!:::.0
(1.2012)
Z-+(
Thus we need to show that (i) and (1.2012) imply (ii)o To this end we observe that for each K > 1 and z E f((, K) we have by Julia's Lemma -2
11- F(z)i/1 2 1 -IF(z)l2
< o:ll-z(l =o:lz-(llz-(1 1 - lzl2
1 - lzl2 1-lzl
< o: lz- (I"' 1- lzl2 = o:Kiz- (lo On the other hand
IF(z)- TJI (1 -IF(z)l)
IF(z)- TJI 1 + IF(z)l
1 -IF(z)l 2 11- F(z)i/1 2 1- IF(z)l2 0
< This implies that
~~~z~ ~ITJI
:::; o:"' (1 + IF(z)l) :::; 2o:K, 0
whenever z E f((, K)o In other words, the functwn f(z) :=
F(z)-TJ
z-(
0
IS
bounded
on each nontangential approach region at (o Therefore, to complete the proof, it is sufficient to show (by using Proposition 1.203) the equality 0
Inn
r--+1-
TJ-F(r() (-
r(
= o:(ryo
or, equally,
11m TJ- F(r() 0
r--+1-
1-
1"
=
O:TJo
(1.2013)
Indeed, since o: is the lower limit in (i) we have lim 117- F(r()l > lim 1 -IF(r()l > O:o r--->1-
1-
1"
-
r-+1-
1-
1"
-
Ou the other hand, setting .:: = r( in (1.205) we obtain
10 1111 r-+1-
117- F(r(W (1
-
1" )"-
e behavior::> we first as::>ume that F, defined by (1.3.6), has two different fixed points, say (1 and (z, (1 =!= (z, on 86.. Set >. := 1 - a(z = (z- a . (2. (1.3.10) 1 - a(1 ( 1 - a (I Since (j = F((j ), j = 1, 2, we have -
1- a(1 = e
i
cp
(j -a --c;-·
Hence
>. = 1 - a(z = 1 -a(!
(2 - a . (1 (I -a (2
=
(2 - a . (2. (I -a (1
Comparing the latter expression with (1.3.10) we conclude that >. = X is a real number. Further, it is clear that >. =!= 1, since a =/= 0 and ( 1 =/= ( 2 . Also >. =/= -1, because otherwise 2 l(t + (21 = lal > 2 = l(tl + 1(21· In other words, we have shown that 1>-1 =/= 1. Exercise 2. Show that
>., defined by (1.3.10), is positive.
29
THE WOLFF-DENJOY THEORY
0
\ F( 3 )(zo),/
\\ ····---...
F( 2) ( zo)
_.//
............ _____________......
Figure 1.9: Parabolic automorphism.
If we now introduce the fractional linear transformation L : C the formula:
t---+
C defined by
then, by using relation (1.3.9) we obtain:
L (F(z))
= >-.L(z),
and consequently:
F(z) = L - 1 (>-.L(z)). This implies by induction that
p(n)(z)=L- 1 (>-.nL(z)),
n=0,1,2, ... ,
where p(n) are iterates ofF: F( 0 l(z) = z, p(n)(z) = F(F(n- 1 l(z)), n = 1, 2, .... Thus, if 1>-.1 < 1 then for each z E ~the sequence {F(n)(z)}:'=o converges to
L- 1(0) =
(1- If
1>-.1 > 1 then this sequence converges to
(2. Since
{p(n)}~=O
is a
normal family on ~. this convergence is uniform on each compact subset of ~. So ifF is a hyperbolic automorphism of~' i.e., F has exactly two (distinct) fixed points on the boundary of~' then the sequence of itemtes { p(n)} ;:"= 0 converges uniformly on compact subsets of~ to one of them. Moreover, one can estimate such a convergence.
Exercise 3. Prove that for each z E
~
the following rate of convergence holds:
30 where ( 1>-1 > 1.
Chapter I
=
(t and
E
= 1>-1 if 1>-1 < 1, or, respectively, ( = (z and
E
= 1>-l- 1 if
Now we consider the case when F is a parabolic automorphism, i.e., F has exactly one fixed point ( on the boundary of Ll. In this case ( is a double root of equation (1.3.6) and we have:
1 - ei"'
2(=-_-. a
Since 1(1 = 1 and lal < 1 it follows that ei'P cannot be -1. Then by direct calculations one can verify the relations:
( F(z)-(
ei"'- 1 ( + -ei'P+1 z-(
---..,.--'--- = - - -
and
( ei"' - 1 = n--p(n)(z)-( ei'P+1
=-:-....,...,......,..---
(
+z-(
for all z ELl and n = 0, 1, 2, .... The latter equality implies that p(n)(z) converges to ( E ail as n tends to infinity. To summarize our considerations we formulate the following assertion.
Proposition 1.3.1 If F E Aut(Ll) is not an elliptic automorphism, then the sequence of iterates { p(n)} is convergent. Moreover, ifF is not the identity, then the limit of this sequence is a unimodular constant, which is a boundary fixed point of F.
:"=o
This result is the first step in the proof of a more general assertion on the asymptotic behavior of holomorphic self-mappings of the unit disk, called the Denjoy-Wolff Theorem. 2. Iterates of holomorphic self-mappings oft. with an interior fixed point.
Proposition 1.3.2 Let F E Hol(t.) have a fixed point ( E Ll. Then: (i) for each r E (0, 1) and n = 0, 1, 2, ... , the following invariance condition holds:
p(n)(flr(()) ~ flr((), where nr(() is defined by (1.3.3); (ii) ifF is not the identity then the point ( E Ll is a unique fixed point ofF in Ll. Moreover, the following are equivalent:
:"=o
(the orbit) converges to ( as n (a) For each z ELl, the sequence { p(nl(z)} goes to infinity. (b) The mapping F: Ll ~--> Ll is not an automorphism of Ll. (c) IF'(()I < 1.
THE WOLFF-DENJOY THEORY
31
Proof. By the induction method condition (i) is an immediate consequence of inclusion (1.3.2). Also, (1.3.2) implies that ( E b. is a unique fixed point of F, if F is not the identity. Indeed, if we assume that F has two different fixed points, say ( 1 and (2, ( 1 =f. ( 2 , then one can choose r 1 and r 2 in (0, 1) such that (2 ~ Drt((I), ( 1 ~ Dr 2 ((2) and Drt((I) nDr 2 ((2 ) = !1 =f. 0. It is clear that !1 is a convex closed subset of b. and F(D) ~ !1. Then it follows by Brouwer's Fixed Point Principle (see Section 0.4) that there exists ( 3 = F((3 ) in n which is obviously different from ( 1 and (2. Repeating these arguments we can find a converging sequence {(n}:'= 1 C b. such that (n = F((n)· By the uniqueness property this implies that F(z) = z, for all z. To prove the second part ofthe assertion we first note that implications (a)=>(b) and (b)=>(c) follow directly from the Schwarz-Pick Lemma. Therefore it is enough to prove the implication (c)=>(a). Since F'(z) is a continuous function on b. there is a disk b.r(() C b. centered at (with radius r > 0 such that
IF'(z)l < 1
(1.3.11)
for all z E b.r((), the closure of b.r((). In turn, (1.3.11) implies that F satisfies the Lipschitz condition (1.3.12) IF(z)- F(w)l ::; q lz- wj, where q = max { IF'(z)J, z E b.r(() }· In addition, we have from (1.3.12) that F maps b.r (() into itself:
IF(z)- (j ::; qjz- (J. So F is a self-mapping of b.r(() which is a strict contraction. It then follows by the Banach Fixed Point Theorem (see Section 0.4) that {F(nl(z)}:'=o converges to ( for all z E b.r((). Using the Vitali theorem (see Section 0.3) we prove our assertion. 0 Combining Proposition 1.3.2. with Brouwer's Fixed Point Principle we obtain the following sufficient condition of existence and uniqueness of an interior fixed point for holomorphic self-mappings of the unit disk.
Corollary 1.3.1 Suppose that FE Hol(b.) maps b. strictly inside, i.e., for some rE(0,1)
IF(z)i::; r for all z E b.. Then F has a unique fixed point ( E b., j(J ::; r, and for each z E b. the orbit { pCnl(z)} :'=o converges to ( as n goes to infinity.
1.4
Fixed point free holomorphic self-mappings of .6.. The Denjoy-Wolff Theorem.
:=
We will say that F is power convergent if the sequenceS= { p(n)} 1 converges uniformly on any subset strictly inside D.. If the limit of this sequence is a constant ( E D. then it is called an attractive point of S. Clearly, if ( is an interior point of D. then it is a unique fixed point of F. In this section we intend to study the dynamics of holomorphic self-mappings of .6. with no fixed points inside. A simple case of this situation occurred in the previous section where we saw that an automorphism of D. with no fixed points has to be either hyperbolic or parabolic, with its fixed point on the boundary of D.. The content of a remarkable result which was essentially obtained simultaneously by J. Wolff and A. Denjoy is that this fact continues to hold for any holomorphic self-mapping of D. with no fixed point inside. In other words, each F E Hol(D.) which has no fixed points in D. is power convergent to its boundary fixed point in the following sense: lim p(nl(z) = ( E 86.. n-+oo
and lim F(r() = (.
r-1-
We have already mentioned that a holomurphic self-mapping of .6. may have many fixed points on the boundary 86.. of D.. So, an additional question is how to recognize which of these is attractive. The key to the answer arrives from Julia's Lemma and the Julia-Caratheodory Theorem where the value of the angular derivative defines such a point (see Proposition 1.4.2 below ). Note also that a consequence of the Schwarz-Pick Lemma (Proposition 1.3.2) tell us about the invariance condition in neighborhoods of an interior fixed point of .6.. For mappings with no fixed points a similar result was established by Wolff [157] where pseudo-hyperbolic disks were replaced by horocycles at a certain boundary point of D..
Proposition 1.4.1 (Wolff's Lemma) Let F E Hol(D.) have no fixed points in D.. Then there is a unique unimodular point ( E 86.., such that for each K > 0 and n = 0, 1, 2, ... , the horocycle
J1- z(J2 } D((,K) = { zED.:
JR.+,
c.p(t)
51
= p((l- t)z + tu, (1- t)w +tv), is not
1];
+ w(w- v)] < Re z(w- v) + w(z- u). 1-
lwl 2
1 - zw
-
Proof. It is clear that (a) implies (b) which, in turn, implies (c). The equivalence (c) and (d) follows from (2.3.12) and (2.3.13), because f.L ~ 0 if and only if (d) holds. Now, we will show that (c) implies (a). Assume that c.p'(O+) > 0. We claim that O+
zero, it is also right-differentiable at zero. Proof. Step 1. Let ¢ E Hol(b..) and { ¢Ck)} be its iteration family. Let Ar denote the disk centered at zero with radius r. Suppose that there are positive r1 < r2 < 1 and 0 < J.l < r2 - r 1 and an integer p ~ 1 such that:
\z- ¢Ckl(z)l
o, Ft(z) = exp( -ta)z, a E C, with the infinitesimal generator f : C f-+ C defined bY . z-Ft(z) f(z) := hm = az.
t
t-+O+
Conversely, for a given number a E C two following equivalent definitions of the exponential function u(t,z) =exp(-ta)z usually are employed in the classical analysis. The first one is based on the solution of the linear Cauchy problem:
{
Du(t,z) ( ) at +au t,z = 0 (3.3.1) u(O,z) = z E ~'
a E C, while the second one uses the exponential formula:
u(t, z) = lim n~oo
(1
+.!.a) -n z. n
(3.3.2)
On the other hand, if for given z E C, a E C, and r 2:: 0, we solve the linear equation (3.3.3) w+raw = z, we have that its solution
w = lr (z) = (1
+ ra)- 1 z
can be considered as a linear mapping defined on C. Then the exponential formula (3.3.2) can be rewritten in the form u(t, z) = lim Jt(/n\z), n-+oo
n
(3.3.4)
where J$nl denotes the n-fold iterate of the mapping lr, r 2:: 0. In addition, it is clear that S = { exp( -ta)} is a p-nonexpansive action on the unit disk ~ if and only if Rea 2:: 0. This is, in fact, equivalent to the property that for each r 2:: 0 the mapping lr : C f-+ C is a self-mapping of~' hence is p-nonexpansive.
69
GENERATION THEORY Now, if we consider, for example, a (nonlinear) analytic function defined by
f : .6.
~-----+
C
f(z)=z-z 2 , we see that the solution of the Cauchy problem
()u~; z) + f (u(t, z)) = 0, { u(O, z) = z .6.,
(3.3.5)
E
is a well defined semigroup (even a flow) of holomorphic (hence p-nonexpansive) mappings of .6.: u(t, z)(= Ft(z)) =
z t
( t
e-ze-1
)
(see Exercise 4 in Section 3 .1). But in this case the similar formula (3.3.2) makes no sense with respect to (3.3.5). At the same time, it turns out, that formula (3.3.4) continues to hold with Jr : .6. ~-----+ .6., r 2: 0 defined as the solution w of the equation w
+ r(w- w2 )
= z
(3.3.6)
(compare with equation (3.3.3)).
Exercise 1. Show directly that for each z E .6. and r > 0 the quadratic equation (3.3.6) has a unique solution w = Jr(z) E .6. which holomorphically depends on z E .6.. Show that lim Jt(/n) ( z) = Ft ( z). n--+oo
n
In general such an approach, which we call the resolvent method, is very useful in the study of generated p-nonexpansive semigroups and their properties. In addition, by using this method one can give different characterizations of the class of functions which generate a flow of p-nonexpansive or holomorphic mappings, or in other words, to give sufficient and necessary conditions for the global solvability of the Cauchy problem. To continue we need the following definition.
Definition 3.3.1 Let f : .6. ~-----+ C be a continuous function. We will say that f satisfies the range condition (RC) if for each r > 0 the nonlinear resolvent Jr =(I+ rf)- 1 is well defined on .6. and belongs to Np(.6.). In other words, the equation
f satisfies the range condition (RC) if for each r > 0 and;; w+rf(w)=z
has a unique solution w = Jr(z) in .6., such that
whenever
w 1 , w2
belong to .6..
E .6.
(3.3.7)
Chapter III
70
Proposition 3.3.1 A continuous {holomorphic) function f on~ is an infinitesimal generator of a one-parameter semigroup S = {Ft} C Np(~) {respectively, Hol(~)) defined on [0, T), T > 0 if and only if it satisfies the range condition (RC). Moreover, the semigroup S = {Ft} is unique and can be continuously extended to a flow of p-nonexpansive (holomorphic) self-mappings defined on JR+ = [0, oo) by the following exponential formula
z E ~' t > 0.
Ft(z) = n-+oo lim J'1n)(z) t n
(3.3.8)
Since the proof of this assertion is rather long we will give it step by step using several lemmata which will also be needed independently in the sequel. Lemma 3.3.1 Let p be the Poincare metric on~ and let {Gt}, t E [0, T), T > 0 be a family of p-nonexpansive mappings of~' i.e.,
p(Gt(z), Gt(w)) :S p(z,w) for all z, wE~' t E [0, T). Suppose that for each z E
(3.3.9) ~
there exists the limit:
. 1 f(z) = hm -(z- Gt(z)) t-.o+ t
(3.3.10)
and assume that f is continuous on each compact subset r > 0 and each w E ~ the equation w
has a unique solution z
Proof.
of~-
Then for each
+ r f(w) = z
= Jr (w) and Jr : ~
f-->
(3.3.11)
~
is also p-nonexpansive.
Given t E (0, T) denote: 1
(3.3.12)
ft =-(I- Gt) t and consider the equation: w
+ r ft(w)
= z,
z E ~'
7'
> 0.
(3.3.13)
This equation can be rewritten in the form: r r+t
w = --Gt(w)
t
+ --z.
(3.3.14)
r+t
For fixed z E ~ the mapping defined by the right-hand side of (3.3.14) is a strict contraction with respect to the metric p (because of (3.3.9) and Proposition 2.3.1(ii)). Therefore, for each z E ~ and r ~ 0, this equation has a unique solution Wt = lrt(z) E ~- In addition, this solution can be obtained by the iteration method:
Wn+l(= Wn+l(z))
=
r r+t
--Gt(wn)
t
+ --z, r+t
(3.3.15)
GENERATION THEORY
71
where z0 is an arbitrary element in b.. Setting wo(z) = z we have by induction and Proposition 2.3.1(iii) that p(wn+l (z1), Wn+l (z2))
< max {p( Gt(Wn (z1) ), Gt(Wn (z2)) ), p(z1, z2)} < max{p(wn(zl),wn(z2)),p(zl,z2)}
~
p(z1,z2).
It means that all wn(·) are p-nonexpansive and so is Jr,t(·) = limn-+oo wn(·). :\ote, in passing, that if Gt E Hol(b.) then Jr,t : b.~---+ b. defined as the solution of !13.14) is also holomorphic on b.. ~ow we want to show that for some r > 0 and z E b. the net {Jr,t(z)}tE(O,T) converges to Jr(z), as t tends too+ and its limit is a solution of equation (3.3.11). To do this we first claim that this net lies strictly inside b.. The latter is equivalent to the inequality:
p(z, Jr,t(z)) ~ M (= M(z)) < oo, as t---+ o+. Indeed, since
r Jr t(z) = -Gt(Jr t(z)) ' r+t '
(3.3.16)
t
+z, r+t
we have by Proposition 3.3.1(ii):
p(Jr,t(z) , _r_z + _t_z) r+t r+t
(-r+ _t_lzl) p(Gt(Jr t(z)), z) r+t r+t ' (-r+ _t_lzl) [p(Gt(Jr t(z), Gt(z)) + p(Gt(z), z))] r+t r+t ' (-r+ _t_lzl) [p(Jr t(z), z) + p(Gt(z), z)]. r+t r+t '
< <
0 such that for alll;::: l 0
p ( Jt 0 and z E B(a, R), define 11 > 0 as in Lemma 3.3.3. Then we have by this lemma that there are n 0 > 0, m 0 > 0 and L > 0 such that for all n > no and m > mo (m)
)
(n)
)
p ( lt;n(z), Jtfnm(z) and P ( lt;m(z), Jtfnm(z)
0 the sets Fix( lr) and Null (f) in ~ coincide; (ii) for each z E ~ the net {Jr(z)} r2:0 is convergent as r ~ oo. Moreover, (a) if W =Null(!)
in~
is not empty, then the limit mapping F = lim lr is T--->00
a p-nonexpansive retraction on W; (b) if W =Null(!) in~ is empty, then the limit mapping F = limr_, 00 lr is a unimodular constant ( which is the sink point for each lr, r > 0. In other words, for each K > 0 the horocycle
D((, K) = internally tangent to
8~
{ z E ~: 0 we have W := Null(!) = Fix (lr)· Consider now the mapping Gt := (I+ s(I- J.)- 1 ) that was introduced in the previous lemma. By the construction of this mapping we have, in turn, that for each pairs, t > 0 W =Fix (J.) =Fix (Gt)· Moreover, it follows by Propositions 2.4.3 and 2.4.4 that for each z E ~ there exists the limit F(z) = lim Gt(z) E ~. More precisely, if W =Null(!) in ~ is not empty, then t->oo
by Proposition 2.4.4 F is a p-nonexpansive retraction on W. Otherwise, F is a unimodular constant ( which is the sink point for J•. But in both cases equation (3.3.29) implies: J. (Gt(z))- Gt(z) ~ 0 as t ~ oo for all z E ~. Setting now r = (1 + t)s and letting t to oo we obtain by Lemma 3.3.5 that F = lim lr. Finally, condition (b) follows from the resolvent identity T--->00
(Lemma 3.3.4) and we are done. 0
GENERATION THEORY
3.4
79
Monotonicity with respect to the hyperbolic metric
Our main goal in the following considerations is to find analytical characterizations and parametric representations of the classes IJNp(fl) and g Hol(fl), respectively. It turns out, that a good tool in these investigations is the notion of monotonicity with respect to the hyperbolic Poincare metric on the unit disk. To motivate the definition below, we recall that a mapping f : IR 2 ~----> IR2 is said to be monotone with respect to the Euclidean norm of IR2 if for each x, y in a domain of definition f (x- y, f(x)- f(y)) 2:: 0,
(3.4.1)
where by(·,·) we denote the inner scalar product in IR2 . Since each complex-valued function f : e . 2
1 - lwl
Substituting u = f(z) + z and v = f(w) characterization of a p-monotone function: Re [ f(z)z
1-
Substituting now z
lzl 2
1 - zw
-
+ w into this inequality we obtain a
] R zf(w) + wf(z) + 1f(w)w > e . 2 - lwl 1 - zw
(3.4. 7)
= 0 into (3.4.7) we obtain the condition: Ref(w)w ~ Ref(O)w(1-lwl 2 ).
(3.4.8)
for all wE~Now we will show that condition (3.4.8) implies the solvability of the equation: w+rf(w)=z
(3.4.9)
for each r 2:: 0 and z E ll. To this end we will establish a more general assertion which we will call the numerical range lower bound (cf., [60] and [62]). Lemma 3.4.1 Let a : [0, 1] a(O) ::; 0 and the equation
f--->
lR be a continuous function on [0, 1] such that s+m(s)=t
has a unique solutions= s(t) E [0, 1) for each t E [0, 1) and r ~ 0. Suppose that f : ~ f---> C be a continuous function on ~ which satisfies the following condition: Ref(w)w ~ a(lwl) lwl, wE~Then for each z E such that
~
and r 2:: 0 equation (9} has a unique solution w = w(z)
lw(z)l ::; s(t), whenever Iz I ::; t < 1.
(3.4.10)
(3.4.11)
GENERATION THEORY
81
Proof. Fix t E (0, 1) and z E .6., izl :S t < 1, and consider the equation: s+m(s)=izj.
It follows from our assumption that this equation has a unique solution s 0 = so (izl). Then setting -y(s) = s+ra(s) -lzi we have -y(O) < 0 and ')'(so)= 0. Note that')' must be monotone on [0, 1]. Hence, for an arbitrary 0 < o< 1- so we can find e > 0 such that -y(so + o) 2': e. Taking now wE .6. such that lwl =so+ owe have by (3.4.10) for those w:
Re(w+rf(w)-z)w > lwl 2 +m(lwl)lwl-lwllzl lwi'Y (iwi) 2': lwle > 0. Then it follows by the Bohl-Poincare theorem (see, for example, [79]), that equation (3.4.9) has a unique solution w = w(z) such that lwl < s 0 + o. Since o is an arbitrary sufficiently small number, we must have lwl :S so :S s(t). D To complete the proof of Proposition 3.4.1 we first note that the function
a(s) = -lf(O)I (1- s2 ) satisfies the conditions of the above Lemma. Therefore inequality (3.4.8) and this Lemma imply that for each r 2': 0 the resolvent Jr, defined by Jr(z) = w(z)- the solution of (3.4.9) is a single-valued self-mapping of .6.. It remains to show that this mapping is p-nonexpansive on .6.. Indeed, for a pair u and v in .6., setting z = Jr(u) and w = Jr(v) we obtain z + rf(z) = u and w + rf(w) = v. Since f is p-monotone we obtain finally
p(u, v)
= p (z + r f(z), w + r f(w)) 2': p(z, w) = p (Jr(u), Jr(v)).
The Lemma is proved. D We already mentioned in the proof of Proposition 3.4.1 that, in fact, condition (3.4.7) is equivalent to the property of a continuous function f on .6. to be pmonotone. Thus combining these assertions with Proposition 3.3.1 and Lemma 3.3.1 we can formulate a summary assertion for this chapter which characterizes the property of a continuous function on D. to be in class QNp(.6.).
Proposition 3.4.2 ([116]) Let f be a complex-valued continuous function on .6.. Then the following conditions are equivalent:
(i) f E QNp(.6.), i.e., it is a generator of a continuous fiow S = {Ft}t:=::o of p-nonexpansive self-mappings of .6.; (ii) there is a family {Gt} 0 :S t :S T, of p-nonexpansive self-mappings of D. such that f(z) = lim z- Gt(z) t--->0+ t for each z E .6.;
82
Chapter III (iii} the Cauchy problem
{
ou(t,z) at
+ !( u (t, z ))-0 - ,
u(O, z) = z E
~
has a unique solution u(t, z) E ~ for all t ~ 0 and z E ~. such that for each t ~ 0 u(t, ·) E Np(~); (iv) f satisfies the range condition (RC}, i.e., for each r ~ 0 and z E ~ the equation
w+zf(w)=z has a unique solution w = Jr(z), such that Jr E (v) f is a p-monotone function on ~; (vi) f satisfies the condition:
Re [ f(z)z 1 - lzl 2 for each pair z, w in
Np(~);
+ f(w)w ] > Re zf(w) + wl[Z) 1 -lwl 2 1 - zw
~;
Remark 3.4.1 Condition (vi) plays a crucial role in our further considerations. Inequalities of such a type (which characterize the classes of generators of flows) are called flow in variance conditions. For the class of holomorphic mappings a simpler inequality (3.4.8) is also a flow invariance condition, since it characterizes the class of holomorphic generators. Indeed, as we saw in the proof of Proposition 3.4.1 (or in Lemma 3.4.1) this condition is sufficient for the existence of the (nonlinear) resolvent (I +r f)- 1 which maps~ into itself. In addition, it follows by the Implicit Function Theorem that this mapping is holomorphic (hence, p-nonexpansive) in ~. so f belongs to g Hol(~). The necessity of this condition follows directly from condition (vi). Thus for f E Hol(~, C) inequalities (3.4.7) and (3.4.8) are equivalent.
We will see below that these conditions can be considered as forms of the Schwarz-Pick inequalities for the classes QNp(~) and g Hol(~) of generators of flows of p-nonexpansive and holomorphic mappings, respectively. A geometric nature of these conditions for holomorphic functions will be explained in the next section. Here we note that an immediate consequence of these flow invariance conditions is the following (cf., Remark 3.3.1): Corollary 3.4.1 The sets 9Np(~) and g Hol(~) are closed (with respect to the topology of uniform convergence on compact subsets of~) real cones. Exercise 1. Show that the set QHol(~) n (-QHol(~)) is precisely the set of all generators of one-parameter groups of automorphisms of~- Hence, this set is a real vector space. Exercise 2. Describe the set
9Np(~)
n ( -QNp(~)).
GENERATION THEORY
3.5
83
Flow invariance conditions for holomorphic functions
In this section we study several flow invariance conditions for the class of holomorphic functions. We will use these conditions to obtain parametric representation of functions of the class Q Hol(~), to study their dynamic transformations and the asymptotic behavior of the flows generated by them. In the first step we give a simpler explanation of the necessity of the flow invariance condition (3.4.8) for a function f E Q Hol(~) and later we study it in greater detail. In addition, we will see below that this condition can be improved by a more qualified condition (see Proposition 3.5.3) which has some additional applications. Also, note that in the case of holomorphic functions one uses a special terminology which comes from the theory of bounded symmetric domains (see, for example, [142, 12, 32]).
Definition 3.5.1 A function f E Hol(~, q is said to be a semi-complete vector field on ~ if the Cauchy problem (*):
{
au~~ z) + f
(u(t, z)) = 0,
u(O,z) = z E ~' has a unique solution u(t, z) E
~for
all z E
~
and all nonnegative t, i.e., t E
JR+ = [0, oo). If the Cauchy problem (*) has a solution u(t, z) defined for all real t, i.e., t E lR = (-oo, oo), then f is said to be complete (or integrated) (see, for example, {32, 142}). Thus f is semi-complete if and only if it is an infinitesimal generator of a one-parameter semigroup; f is complete if and only if it is a generator of a oneparameter group. Exercise 1. Show that the mapping semi-complete vector field on ~-
f :~
f-+
C, defined as
Exercise 2. Show that f : ~ f-+ C, defined by f(z) semi-complete vector field. Find the flow generated by f. Exercise 3. Show that for each a complete vector field.
E
C the mapping
=
f (z)
=
z- 1 +
f : f(z)
z - z 2 is a
J1="Z
is a
= a- az 2 is a
We begin with a characterization of all complete vector fields. Note again that if f is a complete vector field, then it generates a one-parameter subgroup S = {Ft} tElR of the group Aut(~) of all automorphisms of~- Consequently, each
84
Chapter III
Ft E S is a fractional linear Mobius transformation. Thus Ft has a holomorphic continuation on a neighborhood of b. and so does f, because of the equality:
f
=
lim I - Ft. t-.o+
t
(3.5.1)
The family of complete vector fields on b. will be denoted by aut(b..). As we mentioned above, this family is a real vector space, which can be described as follows. Since IFt(z)l::; 1, for z E b. we have by (3.5.1) that:
Ref(z)z 2:0 for all z E 8/::i. At the same time the function fore we also obtain that
f is also a complete vector field on
-Ref(z)z 2:0 for all z E 8!1.
(3.5.2) b.. There-
(3.5.3)
Comparing (3.5.2) and (3.5.3) we obtain the necessary boundary condition for
f
E aut(b..):
Re f(z)z = 0 for all
z E 8!1
(3.5.4)
(see Figure 3.1)
····················...
Figure 3.1: Boundary condition for f E aut(b..). Actually, this condition is also sufficient for f E Hol( b., C) to be complete. Indeed, suppose that f E Hol(b.., C) and satisfies (3.5.4). Rewriting f in the Taylor series form: we have for
z E
8/::i:
Ref(z)z = Reg(z) = 0,
GENERATION THEORY where g(z) = a1
85
+ (ao + a2)z + a3z 2 + ... E Hol(-6., C).
It follows by the maximum principle for harmonic functions that:
Hence f(z) is actually polynomial of the second order at most: f(z) = ao
+ a1z + a2z 2
(3.5.5)
with (3.5.6)
Now it can be shown by direct computation that the Cauchy problem:
{ au~; z) + f
(u(t, z)) = 0,
u(O,z) = z E b.,
with f satisfying (3.5.5) and (3.5.6) has a unique solution u(t, ·) E Hol(b.) for all t E R Moreover, for fixed t E lR the mapping Ft = u(t, ·) is a Mobius transformation of the unit disk. So, we have proved the following result (see, for example, [16, 8]). Proposition 3.5.1 (Boundary group invariance condition) A mapping f E Hol(-6., C) is a complete vector field on b. {i.e., f E aut( b.)) if and only if it has a continuous extension to b. and Ref(z)z = 0
for all z E 8.6..
This is equivalent to the statement that f is a polynomial of the second order at most: f(z) = ao + a1z + a2z 2 with coefficients ao, a 1, a2 which satisfy the conditions:
Corollary 3.5.1 The family aut(.6.) is a real vector space of entire functions. Moreover, this space has the following decomposition: aut(b.) =auto( b.) El7 P2, where
{! E aut(.6.) : f(O) = 0} {! E Hol(b., C) : f(z) = bz, Reb= 0}
auto (.6.)
is the subspace of linear functions, and
{! E aut(b.), f'(O) = 0}
P2 =
{!
E Hol(b., C): f(z) = i i - az 2 , a E
is the subspace of so called 'transvections '.
C}
Chapter III
86
Now we will turn to the general case when f E Hol(b., C) is a semi-complete vector field on the unit disk. To explain the nature of the condition (3.4.8) we adduce first some heuristic grasps. Assume temporarily that f E QHol(b.) is holomorphic in the neighborhood of b.. In this case we will write just f E Hol(b., C). Then, again we have the following boundary flow invariance condition: Re f (z) · z 2:: 0 for all
z E ab.
(see Figure 3.2).
Figure 3.2: Boundary flow invariance condition.
It implies that
Re(f(z)- f(O))z 2: - Re f(O)z,
z E aD..
Dividing the left hand side of this inequality by lzl 2 Re ( f(z)
~
f(O)) 2::- Ref(O)z,
= 1 we obtain:
z E aD..
Now again it follows by the maximum principle for harmonic functions that the latter inequality holds also for z E b.. Multiplying it by lzl 2 =f. 0, z E b., we obtain: (3.5.7) Ref(z) · z 2:: Ref(O) · z · (1 -lzl 2 ), zED.. However, there are holomorphic mappings on the unit disk which are semicomplete, but have no holomorphic extension to D., the closure of b.. Consider, for example, f(z) = z- 1 + v/1 - z. Nevertheless, we will see that even f E Hol(D., C) does not extend continuously to D., condition (3.5.7) is necessary and sufficient for f to be in QHol(D.). The
87
GENERATION THEORY
necessity can be shown also by the following simple considerations which are useful, however, to obtain a parametric representation of the class g Hol(~). As we already know this class is a real cone. Therefore, if we present f E g Hol(~) in the form:
f(z) = g(z) + h(z),
(3.5.8)
where
g(z) = f(O)- f(O)z 2 is a transvection (i.e., g E P2 , see Corollary 3.5.1), we have
+ (-g(z))
h(z) = f(z)
E QHol(~)
(3.5.9)
and (3.5.10) h(O) = 0. Now, conditions (3.5.9) and (3.5.10) imply that there is a semigroup Sh = {Ht}t~o of holomorphic self-mappings Ht of~. such that Ht(O) = 0, for all t ~ 0. Hence by the Schwarz Lemma we have:
[Ht(z)[ ::::; [z[,
for all
Since
h(z) = lim
t-->0+
z E ~-
~(z- Ht(z)) t
we obtain:
Reh(z)z
~ 0
for all z E
~-
(3.5.11)
In addition, note that
Reg(z)z = Re f(O)z · (1 -[z[ 2 ).
(3.5.12)
Then by (3.5.11), (3.5.12), and (3.5.8) we obtain (3.5.7) and the necessity of this condition is proved. The sufficiency of condition (3.5.7) for f E Hol(~, q to be a semi-complete vector field was established in Lemma 3.4.1 (see, also, Remark 3.4.1). However, for the case of holomorphic functions one can make this lemma more precise. Lemma 3.5.1 Let a E g Hol(~) be such that a ([z[) E Hol(~, q satisfy the following condition:
Re f(z)z
~a
([z[) [z[,
zE
~,
z E
~.
and let f E
~-
Then: (i) f is a semi-complete vector field on ~; (ii) if Sf = { Ft}, t ~ 0, is the semigroup generated by f, then for all t and x E ~
fFt(x)[ ::::; ,Bt([x[), where f3t is the solution of the Cauchy problem:
{
d,Bt(s) + a(B (s)) = 0 dt
,Bo(s)
.
=
s,
t
s E [0, 1).
,
~
0
Chapter III
88
Proof. It follows by Proposition 3.3.1 that the function a : b. f-+ 0}; 1- w(z) ( ) , where w E Hol(A) is such that lw(z)l ~ c < 1 for all (b) p(z) = 1 +w z
z EA. Exercise 6. Show that under the conditions of Exercise 5 the function p satisfies the strong Harnack inequality:
c:z: ~ Rep(z) ~ Rep(O) 11+cz - c:z:,
Rep(O) 1 + 1-cz
0
~ c < 1.
Finally, observe that from the representation off E Q Hol(A):
f(z) = f(O)- f(O)z 2 + z · p(z), we deduce that: p(O)
= f'(O).
Thus we have proved the following assertion.
Rep(z) ~ 0
(3.5.17)
GENERATION THEORY
91
Proposition 3.5.3 ([6]) A function f E Hol(~, q is a semi-complete vector field on ~ if and only if Ref' (0) 2 0 and the following inequality holds:
7
2 Ref'(O) lzl (1 \zl) + Ref(O)z(1 -lzl 2) 2 Ref(z)z 1- z
2 Re f(O)z(1 - lzl 2) + Re f'(O)
lzl:(~ lzllzl).
(3.5.18)
Moreover, the equality in (3.5.16} holds if and only ifRef'(O) = 0. Corollary 3.5.2 ([7]) A mapping f E Q Hol(~) belongs to
aut(~)
if and only if
Rej'(O) = 0.
Corollary 3.5.3 ([7]) Iff E Q Hol(~) is given with f(O) = f'(O) = 0, then =.0.
f
This assertion can be considered as a tangential version of the Schwarz Lemma. Indeed, ifF E Hol(~) is a holomorphic self-mapping, then f = I-F is a semicomplete vector field. Therefore, if F(O) = 0 and F'(O) = 1, then f(O) = f'(O) = 0. Therefore, by the Corollary 3.4.2 we obtain that F(z) = z. This is the statement of the second part of the Schwarz Lemma. Recently M. Abate [5] established another condition, which characterizes a semi-complete vector field f by using the estimate for its derivative f'. To establish his condition we first prove the following characterization of the class P := {p E Hol(~, IC): Rep(z) 2 0, z E ~}.
Lemma 3.5.3 ([7]) Let p E
Hol(~,
q.
Then condition (3.5.14}:
Rep(z) 2 0, holds for all z E ~. if and only if there is a positive function 'lj;: [0, 1) that: Re (zp'(z) + 'lj;(lzl)p(z)) 2 0 for all z E
1--4
JR+ such (3.5.19)
~.
Proof. Let p E Hol(~, q satisfy (3.5.14). Define F = (p- 1)(p + 1)- 1 which maps ~ into itself, F E Hol(~). Applying the Schwarz-Pick Lemma to F we obtain the inequality:
l(
p-1)'1 2lp'l lp+11 2 -IP-11 2 p+1 =l1+pl2:::; lp+112(1-lzl2)'
which implies: I '( )I p z
:::;
2 Rep(z) 1- lzl2 .
Chapter III
92 Consequently,
Re( -zp'(z)) :::; lzp'(z)l :::;
2lzl Rep(z) 1 + lzl 2 :::; Rep(z). 1 - lzl2 1 - lzl2
1 + t2 Setting here 1/J(t) = - -2 we obtain (3.5.19). 1-t Conversely. Suppose that (3.5.19) holds with a positive Setting z = Tei 6 we have:
1/J: [0, 1) f-+ JR+.
zp'( z ) = r ap ar and (3.5.19) becomes: ( Re
(r:~) +1/J(r)Rep(z)) ~ 0,
z = rei 8 E !:J..
(3.5.20)
Assume that there exists z 0 = r 0 ei!Jo in !:J. such that Rep(zo) < 0. But (3.5.20) implies that Re p(O) ~ 0, hence there is r 1 E [0, r 0 ) such that Rep(r 1 ei80 ) = 0 due to continuity. Then one can find r2 E ( r 1 , ro) such that Rep(r2ei 80 ) < 0 and Re
:~ heiBo)
< 0.
But these inequalities contradict (3.5.20). Thus it follows that Rep(z) everywhere and we are done. D
~
0
Now it is easy to verify that condition (3.5.19) with 1/J(r) = 1 + r: is equivalent 1-r to the condition: Re [2f(z)z + f'(z)(1 -lzl 2)] ~ 0. Thus we can summarize the assertions of this section in the following result. Proposition 3.5.4 Let f E Hol(!:J., q. Then the following are equivalent:
(i) f E QHol(!:J.,C), i.e., f is a semi-complete vector field on l:J.; (ii) Ref(z)z ~ Ref(O)z(1-lzl 2); (iii) Ref'(O) ~0 and Re [f(O)z(1 -
lzl2)
+ f'(O)Izl21 + lzll 1-lzl
> Re f(z)z > Re [f(O)z(l -lz1 2) + f'(O)Izl 21 - lzl] . 1 + lzl '
(iv) Re [2f(z)z + f'(z)(1 -lzl 2)] ~ 0;
93
GENERATION THEORY (v) f(z) = f(O)- f(O)z 2
+ z · p(z),
with p E Hol(~, C), Rep(z) ;::: 0.
Moreover, if for some z 0 E ~ the equality in one of the conditions {ii) or (iv) holds, then it holds for all of these conditions and for all z E ~. In this case the function p(z) in (v) is constant and f is actually a complete vector field.
Remark 3.5.3 The class offunctions of the form: h(z) = z·p(z), where Rep(z);::: 0, z E A usually referred to as class N. The class M consists of all elements of N which are not linear functions, i.e.,
M ={hE
Hol(~,
C): h(z) = z · p(z), Rep(z) > 0, z E
~}.
Note that N = auto(~) E9 M. Thus condition (v) of the above proposition means that the class of the semi-complete vector fields on ~ admits the following decompositions: (3.5.21) QHol(~) = P2 +N =aut(~) E9 M. (see Corollary 3.5.1). In addition, it is well known (see, for example, [57] and [122]) that for each holomorphic function p on ~ with values in the closed right half-plane II+ (i.e., Rep(z) ;::: 0) there exists a positive increasing finite function /-lp on the unit circle 8~, such that 1 + z( . (3.5.22) p(z) = ---df..lp(() + zb 1- z(
J
a.c::..
with some real b. This formula is called the Riesz-Herglotz representation of functions in P = {p E Hol( ~.C) : Rep( z) ;::: 0}. It establishes a linear one-to-one correspondence between the set of all positive measures on 8~ and P. We will call the function /-lp : 8~ 1--4 lR the measure characteristic function for pEP.
Thus by (3.5.21) and (3.5.22) we have the following integral parametric representation for f E Q Hol(~) : f(z) =a- az 2
+ izb +
J
a.c::..
1 + z( z---df..l((), 1- z(
(3.5.23)
where a E C, b E lR and J.l is a positive function on 8~. Another parametric representation of the class g Hol( ~) which is determined by the location of null points off E QHol(~) is due to E. Berkson and H. Porta [17]. We will give it in the next chapter.
Exercise 7. LethE 0, z E
Hol(~,q
belong to class N: h(z) = z · p(z), Rep(z);:::
~.
(a) Show directly that the Cauchy problem: {
8 u~; z) + h (u(t, z)) u(O,z)
=
z E
~.
= 0,
(3.5.24)
94
Chapter III
has a unique solution u(t, z) E 6. for all t ~ 0, and u(t, 0) (b) Show that lim u(t, z) = 0
= 0, t
~
0.
t-->oo
if and only if hEM, i.e., Rep(z) > 0, z E 6.. (c) Show that if hEM with h'(O) = 1, then the limit lim etu(t, z) := F(z)
(3.5.25)
t-->oo
exists for all z E 6. and F E Hol(6., q. Hint: Define the function Pl E M by p 1 ( z) = 1/p( z) and show that the Cauchy problem (3.5.24) is equivalent to the following integral equation
1
d(
u(t,z)
In [etu(t,z)] = lnzwhich
impl~s
t~~ etu(t, z)
(Pl(() -1) ( '
z
t
d(
= zexp(Jo (p 1 ( ( ) - 1) ().
(3.5.26)
(d) Show that F E Hol(6., q defined by (3.5.25) (or (3.5.26)) satisfies the following inequality:
zF'(z)] Re [ F(z)
(3.5.27)
> 0.
(It is well known due to R. Nevanlinna [102] that the latter inequality characterizes all univalent F functions on 6., normalized by F(O) = 0, F'(O)-# 0, whose image F(6.) is starlike with respect to zero. See Chapter 5.)
f, defined by:
Exercise 8. Show that the function
h(z) = z belongs toM, and h'(O)
= 1.
1 + zeiiJ
.9 ,
1- ze'
() E
[0, 27r]
Find explicitly the function F(z) in (3.5.25). 00
Exercise 9. Show that if the function f defined as: f(z) is a semi-complete vector field on 6. and Re a 1
= 0,
then a0
= ao+a1z+ z::::akzk
= ak = 0 for
k=3
all k
~
3.
95
GENERATION THEORY
3.6
The Berkson-Porta parametric representation of semi-complete vector fields
An important consequence of Proposition 3.4.1 and Corollary 3.4.1 is the following representation of semi-complete vector field, which is originally due to E. Berkson and H. Porta [17] (see, also [6]).
Proposition 3.6.1 A mapping f E Hol(D., C) is a semi-complete vector field on D. if and only if there is a point T E D. and a function p E Hol( D., C) with Rep( z) ~ 0 everywhere such that: (3.6.1) f(z) = (z- r)(l- zr)p(z). Moreover, such a representation is unique and
D., or the boundary sink point of the resolvent lr
T
:=
is either a null point off in , r > 0.
(I+ r f) - l
Proof. Firstly, let f be a semi-complete vector field on D. with a null-point TED.. Then it follows by formula (3.4.7) that the following inequality holds: Ref(w)w ~ (1
f(w)r
-lwl 2 ) Re -_ 1-WT
(3.6.2)
or,
~W l2 - ~) 1- WT ~ 0.
Ref(w) ( 1-
(3.6.3)
We calculate
w
w - lwl 2 7 -
7 + rlwl 2 lwl 2 )(1 - wr) w-r (1 - lwl 2 )(1 - wr) lw-rl 1 (1 -lwl 2 ) . (w- r)(1- wr) ·
7
-....,...-=- - 1 -lwl 2 1- WT
(1 -
(3.6.4)
Now by identifying w E D. with z E D. we obtain from (3.6.3) an'd (3.6.4):
Re (Z -
f(z) T) ( 1 -
>0
(3.6.5)
ZT) -
for all z E D.. Denoting:
p(z = f(z) ) (z-r)(1-z7)
(3.6.6)
we have (3.6.1). Now suppose that f E QHol(D.) has no null point in D.. Then it follows by Proposition 3.3.2 that in this case there is a unique boundary point T, such that for each w E D., the net { Zr (w)} r>O defined as the solution of the equation
Zr(w)
+ r f(zr(w))
=
w
(3.6.7)
Chapter III
96
converges toT, as r----+ oo. (Indeed, for each r 2: 0 the value zr(w) is just the value of the resolvent Jr = (I + r f) - l at the point w E .6). Fixe > 0 and consider the mapping fe E Hol(.6, C) defined as fe(z) = e · z + f(z). It is clear that fe converges to f as e goes to zero. Since QHol(.6) is a real cone, it follows that f" E Q Hol(.6) for each e 2: 0. In addition, the equation
fe(z)
= 0
is a particular case of (3.6.7) with r = 1/e and w = 0. Hence fe has a unique null point T,;; E .6 and the net {T,;; }00 converges to T as e tends to zero. Since fe satisfies the inequality: Re
fe(z) >0 (1-ZT,;;)(z-T,;;)- '
letting e tend to zero we have the same (inequality (3.6.5)) for f(z), which in turn implies representation (3.6.1). Conversely. Suppose that f E Hol(~) admits representation (3.6.1) with T E .6 and Rep(z) 2: 0 everywhere. If Rep(z) = 0 for some z E .6, then by the maximum principle it follows that p(z) = im for some mER In this case:
f(z)
(z- r)(1- zf)im = (z- T - z 2 f -imT- z 2 fim + (1 + ITI 2 )imz.
Denoting -imT :=a,
(1
+ ITI 2 )im := b we obtain:
f(z) =a- az 2 + bz, i.e.,
+ ziTI 2 )
Reb= 0,
f is a complete vector field (see Proposition 3.5.1). Therefore, we have to consider only the case when Rep(z) > 0. Let us present f E Hol(.6) in the form: f(z) =a- az 2
+ z · q(z),
(3.6.8)
where a= f(O). Comparing (3.6.1) with (3.6.8) we have f(O)
q(z) = (1- f'z
+ ITI 2 )q(z)-
q(z)- q(O) z
=
-rq(O) and
T- zTq(O).
(3.6.9)
To proceed we need the following lemma which will be also useful in the sequel.
Lemma 3.6.1 (cf., [6]) Let T be in .6 and let p and q be those holomorphic functions in .6 which satisfy equation (3.6.9). Then Rep(z) > 0 if and only if Req(z)>O. Moreover, the values of p lie strictly inside IT+ = {Re w > 0} if and only if the values of q lie strictly inside IT+.
97
GENERATION THEORY
Proof. First we note that assuming one of the functions p or q to be holomorphic in ~ we have that the second one is holomorphic on ~ too. Observe also that it is enough to prove our assertion under the above stronger assumption. Indeed, for the general case one can use an approximation argument: given p (or q, respectively) with Rep(z) > e: ~ 0, set Pn(z) = p(rnz) for r E (0, 1), rn--> 1-. So we assume that both these functions are holomorphic on ~ Then substituting z = eiiJ, () E JR., in (3.6.9) we calculate
+ ITI 2 - fei 0 )p(ei0 ) - Te-i 0 p(ei0 )} (1 + ITI 2 - 2 Re fei 0 ) Rep( ei 0 )
Re{(1
Req(ei0 ) =
11- fei 0 12 Rep(ei 0 ).
Since by our assumptions, T E ~. and the functions Rep and Re q are harmonic, we see that Rep(z) > e: ~ 0 if and only if Req(z) ~ 8 > 0. Moreover, if e: is positive, then 8 can be chosen positive too, and conversely. D Returning to the proof of the Proposition 3.6.1, we see that if T E ~ then > 0, hence it is semi-complete (see Remark 3.5.2). If Tin (3.6.1) belongs to{)~ we just apply again the following approximation argument. We choose any sequence Tn E ~ such that Tn --> T and set fn(z) = (z- Tn)(1- zfn)p(z). It is obvious that {fn}~=l converges to f uniformly on each compact subset of~- Since we already know that each fn belongs to Q Hol(~), we have that so does f, and we have completed the proof. D f defined by (3.6.1) admits representation (3.6.8) with Req(z)
Remark 3.6.1 Thus this proposition implies that a semi-complete vector field f on ~ has at most one null point in ~- If such a point exists it must be T in representation (3.6.1). For T E ~ we will denote by Q Hol(~, T) the class of functions f E Q Hol(~) with f(T) = 0. Thus, Q Hol(~, T)
= {f
E Hol(~, 0 the horocycle
D(T,K) = inter·nally tangent to{)!:;,. at
zfl 2 } { z E ~: 'Pr(z) := 111 -lzl 2 < K T,
is Ft-invar'iant, i.e.,
Ft(D(r,K))
~
D(r,K).
Also, it can be shown (see Section 4.6) that T E in the following sense: lim f (rr) = 0. r----.1-
{)~
is the limit null point off
Chapter III
98
However, it happens that f E g Hol(A) may have more than one null point on the boundary of A (consider, for example, f(z) = 1 - z 2 ). So the question is which one of them is the sink point of the semigroup {Ft}t>o generated by f. Another question relates to the asymptotic behavior of such a-semigroup. In the next chapter we intend to answer these questions as much as to find the best rates of the exponential convergence. At the end of the section we will add some preparatory material concerning this matter. To clarify our further reasoning we first summarize briefly different characterizations of semi-complete vector fields.
Summary Let f E Hol(A, C). The following are equivalent: {i} f E QHol(A), i.e., f is a semi-complete vector field; is a well defined holomorphic (ii} for each r > 0 the mapping Jr = (I+ r self-mapping of A; (iii} f is p-monotone with respect to the Poincare hyperbolic metric on A, i.e., for each pair z, w E /::). p(z+rf(z),w+rf(w)) ~p(z,w),
n-l
whenever z + r f ( z) and w + r f (w) belong to A for some positive r; (iv) f admits the following parametric representation f(z) =a-
az 2 + zq(z)
for some a E C and q E Hol(A,C) with Req(z) ~ 0, z E A; (v) f admits the Berkson-Porta parametric representation f(z) = (z- r)(l- zf)p(z) for some TEA andp E Hol(A,C) with Rep(z) ~ 0, z EA. In addition, different flow invariance conditions given in terms inequalities are presented in sections 3.4 and 3.5. The simplest one can be formulated as follows (vi) there exists a number m E lR (in fact, m :::; 0) such that Ref(z)z ~ m(l-lzl 2 ),
z EA.
In the study of asymptotic behavior of the semigroup generated by f E g Hol(A) with an interior null point, a few stronger conditions than (i)-(vi) will be relevant.
Definition 3.6.1 (cf., [39]) A function f : A f-+ C is said to be strongly pmonotone if for some c: > 0 and for each pair z, w in A there exists J = J(z, w), such that p(z + rf(z), w + r f(w)) ~ (1 + rc:)p(z, w), whenever
0:::; r < J.
Of course, a strongly p-monotone holomorphic function in A is semi-complete. Moreover, such a function must have a unique null point in A. Indeed, by definition
99
GENERATION THEORY we have that (at least for r :2: 0 small enough) the resolvent lr E contraction with respect to the hyperbolic metric pin ~:
Hol(~)
1 p(Jr(z), lr(w)) :::; --p(z, w). 1 +rc
is a strict
(3.6.10)
Thus lr has a unique fixed point T in ~ because of the Banach Fixed Point Principle. This point is a null point of f. Further, putting w = T in (3.6.10) and differentiating it with respect to r at the point r = o+ we obtain
or Rep(z) :2: c(1
-ITI 2 )- 1 > 0,
where pis the factor in the Berkson-Porta representation (see (3.6.1)). Now Lemma 3.6.2 enables us to conclude that q(z) in representation (3.6.8) has a real part strictly separated from zero, i.e., Req(z) > c 1 for some c 1 > 0 and all z E ~- In turn, the same formula (3.6.8) implies
for all z close enough to a~, the boundary of~ Thus we have proved the following assertion. Proposition 3.6.2 Let f E
Hol(~,
q
be strongly p-monotone
in~-
Then:
(i) f admits the representation f(z)
= (z- T)(1- zr)p(z)
with T E ~ and Rep(z) > E for some E > 0; (ii) there exist positive numbers 8 and TJ such that Ref(z)z :2: TJ for all z in the annulas {1- 8
0
lzl < 1}.
Again from Lemma 3.6.1 it can easily be seen that (ii) implies (i). Thus these conditions are equivalent. As a matter of fact, we will see below (Section 4.5) that (ii) implies the strong p-monotonicity of a function f E Hol(~, q.
Chapter 4
Asymptotic behavior of continuous flows In this chapter we want to trace a connection of the iterating theory of functions in one complex variable and the asymptotic behavior of solutions of ordinary differential equations governed by evolution problems. Therefore our terminology is related to both these topics.
4.1
Stationary points of a flow on
~
Quoting M. Abate [2], note that E. Vesentini seems to be the first person who suggested an analog of the Denjoy-Wolff Theorem for continuous time semigroups. In fact, in 1938 J. Wolff [158] himself initiated the consideration of dynamical systems determined by holomorphic functions. However, the first general continuous version of the Wolff-Denjoy Theory was given by E. Berkson and H. Porta [17] in their study of the eigenvalue problem for composition operators on Hardy spaces.
Definition 4.1.1 A point ( E {Ft}t>D C Hol(~), if
~
is said to be a stationary point of a flow S
lim Ft(r()
r-1-
= (
=
(4.1.1)
for all t > 0. In other words, ( E ~ is a stationary point of S if it is a common fixed point of all Ft E S. Note that the family S = {Ft}t~o is commuting, that is, FtoF5 = F.oFt = Ft+s for all t, s ~ 0. Hence, it follows by the Shield theorem [132] that if each Ft had 101
102
Chapter IV
been continuously extended to {)jj. the boundary of /j., then the stationary point set of S would not be empty. As a matter of fact, it is enough to require the existence of an interior fixed point only for one t > 0 to ensure the existence of such a point for the whole semigroup. Indeed, if for at least one t > 0 the mapping F1 E S has an interior fixed point ( E jj. then it is a unique fixed point for F1 , and for each s 2: 0 we have: i.e., (is also a fixed point of Fs E S, s 2: 0. Henceforth this fixed point is a unique stationary point of S.
Exercise 1. Show that if ( E jj. is a stationary point for a semigroup S = {Ft}t>O• F1 E HoJ(Ij.), t > 0, then (Ft)'(a) = e-at is a contraction linear semigroup, i.e., Rea 2: 0. Hint. Use the chain rule and the Schwarz-Pick lemma. Naturally, the strategy now is to study the convergence of a semigroup to its stationary point. The foregoing result is the first step in the study of the asymptotic behavior of a flow in /j.,
Proposition 4.1.1 ([81]) Let S = {Ft}t>O C Hol(lj.) be a flow on jj., Then this net converges uniformly on compact subsets of jj. to a holomorphic mapping FE Hol(fj., C) if and only if for at least one to the sequence {Ft 0 n};;c'= 0 converges uniformly on compact subsets of jj., Moreover, if F 10 is not the identity then F is a constant with the modulus less or equals to 1. Proof. The necessity is obvious. To prove the sufficiency we assume that Ft is not the identity for t > 0, otherwise the assertion is trivial. Then the limit:
. IIm
n--+oo
r.
rton
1'1m p(n) = n-+oo t0
is a constant mapping, say ( (=((to)) E 1.3.2 that
/j.,
If ( E
jj.
then it follows by Corollary
Consequently, the chain rule and the semigroup property imply that for all
t > 0: (4.1.2)
(see also Exercise 1). Hence (is an attractive fixed point for each Ft. t
. 11m
p(n)(~)- r· t "' - ..,,
zE
> 0, i.e., (4.1.3)
/j.,
11---+00
If (
(=
lim F1~"))
11--+00
l
E f)jj.,
then F10 has no fixed point inside
/j.,
In this situation,
as we mentioned above, each F1. t > 0, must be fixed point free on /j., Then the Denjoy-Wolff Theorem implies that for each integer m > 0 the sequence of iterates
{ Ft~fm} ~=O converges uniformly on compact subset of
~
to a point (m E
~.
ASYMPTOTIC BEHAVIOR
103
But (m = lim p(nm)(z) = lim Ft(n)(z) = (,
n-+oo tofm
n-+oo
0
and it follows that (m = ( does not depend on m. So, in both cases (either ( E ~or ( E 8~) we have the following equality lim Ft(n/) (z) = (
n~oo
(4.1.4)
om
for all z E ~~ and each mEN. Now we will show that (4.1.4) implies that for each z E ~the net Ft converges to (, as t tends to infinity. Indeed, for a given £ > 0 and z E ~ one can choose 8 > 0 such that JFt(z)-Ft(w)l < e/2 for all t > 0, whenever wE~ and lw-zi < o. For such take m E N so large that jF8 (z) - zi < whenever s E [0, t 0 /m]. Finally, for such m and t > 0, large enough settings= [tm/to] we have by (4.1.4):
owe
o
IFtonfm(z)Noting that s such t > 0:
=
Cl = JFt~n)m(z)- cJ
< ~·
t- ton/m E [0, t 0 /m] and setting w = F 8 (z), we obtain for
IFt(z)-
(I
IFton/m(Fs(z))-
cl
o c Hol(~) be a flow on ~. If for at least one to the mapping Fto is not the identity and is not an elliptic automorphism of~. then the net {Ft}t~o converges to a constant ( E ~ as t ----> oo uniformly on each compact subset of~. Remark 4.1.1 Since every continuous semigroup S = {Ft}t~o of holomorphic self-mappings Ft of~ is differentiable (by parameter t ;::: 0), it is natural to describe 1
its asymptotic behavior in terms of the generator f = lim - (I- Ft). This becomes t-+0 t more desirable when such a semigroup is not given explicitly, but is defined as the solution of the Cauchy problem:
{
8u(t, z) at +f(u(t,z))=O, (4.1.5)
u(O,z) = z We set here Ft(z) = u(t, z).
E ~.
Chapter IV
104
Note also that iff is holomorphic in a neighborhood of the point ( E D. then it follows by the uniqueness of the solution of the Cauchy problem that f(() = 0 if and only if ( is a stationary point of S = {Ft}t~O· In particular, an interior null point of a semi-complete vector field is a stationary point of the generated semigroup. However, this fact is no longer true for a boundary null point.The following example shows that even a semi-complete vector field f has a continuous extension to D.; it may have two null points in D. (one of them on aD.), while the semigroup generated by f has a unique stationary point in D. (which is the interior null point of f). Example 1. Set f(z) = z - 1 + vfl=Z. It is clear that f(O) = f(1) = 0. At the same time, solving the Cauchy problem (4.1.5) one can find the solution explicitly:
u(t,z) = 1- [1-e-t/ 2 +e-t1 2
Jf=Zr
Setting Ft = u(t, ·), it is easy to verify that Ft E Hol(D..), hence complete. But for all t > 0: lim Ft(r·)
r--+1-
= 1- [1- e-t/ 2 ] 2
O· Nevertheless, as we will see below (see Section 4.6), if
f has no null point in
D. then it must have a boundary null point on aD. which is an asymptotic limit of the semigroup generated by
4.2
f.
Null points of complete vector fields
In this section we deal with a one-parameter group of automorphisms of D... Since the generator of a one-parameter group is a complete \'ector field, it is a polynomial at most of the second order, aucl hence holomorphic in C. Now the assertion follows: Proposition 4.2.1 The stationar-y point set of a one-parameter groupS = {Ft} tEIR' Ft E Aut(D..), has either one or two points in D.. which are exactly the null points of the function f: f(z) =a+ ibz- az 2 , (4.2.1)
ASYMPTOTIC BEHAVIOR where a= _ aFt(O) at
I
and
then z 1
= 1/ z0
+ ibz -
~ a2 Ft (z) I i
Exercise 1. Show directly that if iiz 2
b=
t=o
z 0 =1-
a = 0,
105
ataz
.
t=O,z=O
0 is a solution of the equation b E IR,
a E C,
(4.2.2)
is also a solution of this equation.
Thus equation (4.2.2) has at least one solution in b.. If one of the solutions of abo. the second one (if it exists) lies on abo. also. Moreover, given a complete vector field f one can characterize the group of automorphisms generated by f.
(4.2.2) lies on
Proposition 4.2.2 Let f E aut(b..) be a complete vector field on b. and let S = {Ft}, t E IR be the group of automorphisms generated by f. The following assertions hold: 1) If 21 f (0) I < I!' (0) I then S has a unique stationary point zo in b. which is actually in b. and S consist of elliptic automorphisms of b.. In this case Ft(z) does not converge to zo for each z E b., z =1- z 0 . 2} If 2lf(O)I = lf'(O)I then S has a unique stationary point zo in b. which actually lies on abo. and S consists of parabolic automorphisms of b.. In this case for all z E b.: lim Ft(z) =ZoE abo.. t-oo
3) lf2lf(O)I < lf'(O)I then S has exactly two different null points z1 and z2 in b.. Both of them lie on abo., and S consists of hyperbolic automorphisms of b.. Consider in more details the group of hyperbolic automorphisms. In this case its generator f has the form: f(z) =a+ ibz- az 2 ,
(4.2.3)
2lal > lbl.
(4.2.4)
with a E b., b E IR, such that Using (4.2.4), by direct calculations one obtains that the null points off are
(4.2.5) It is clear that z1 =1- z2 and lz1l = lz2l = 1. Let now S = {Ft}tEIR be the flow generated by f. Since for a fixed to E IR+ the points z1 and z2 are fixed points of Ft 0 , therefore one of them is a sink point for Ft 0 and thus this point is the limit of the net { Ft} when t goes to infinity. It is easy to understand that the second null point is the sink point for the mapping Ft~ 1 = F-to· Therefore it is the limit of the net {Ft} when t---> -oo.
Chapter IV
106 Write Ft in the form:
Ft(z) = ei'Pt z- at ' 1 - atz
(4.2.6)
where at E b., 'Pt E JR., t E R It is easily seen that for each t E JR. and z E b.:
Ft(z)- z1 Ft(z)- Z2
1- atZ2 Z - Zl ·-1- atZl Z - Z2
This equation may be written in the form
L(Ft(z)) = AtL(z),
(4.2.7)
where L is the fractional linear transformation defined by
L(z)= z-z1 Z - Z2
(4.2.8)
and (4.2.9) At the same time, since
Zi,
i = 1, 2 are fixed points of Ft we have by (4.2.6):
It follows that (4.2.10) In addition note, that (4.2.5) implies that
(It is also clear because of the equality zizi = 1, i = 1, 2). Comparing (4.2.9) and (4.2.10) we obtain that:
so the result that At is real is established for all t E R Furthermore, it follows by the group property and (4.2. 7) that
i.e., So At = ekt for some k E R If k > 0 we have again by (4.2.7):
Ft(z) = L - 1 (ekt L(z)),
ASYMPTOTIC BEHAVIOR
107
or (4.2.11) Hence lim Ft(z) t--+-00
= z 1 while
lim Ft(z)
t--+-00
= z2
for all z E
~.
In case of k < 0 we obtain similarly: lim Ft(z) =
z1
t--+-00
and lim Ft(z)
t--+-00
= z2.
Therefore we need to recognize k. Since u(t, z)
= Ft(z)
satisfies the equation
au(t,z) at +f(u(t,z))=O,
with the initial data: u(O,z)
= z,
we obtain by solving this equation: Ft(z)-zl _ e -ii(z,-z 2 }·t ·Z-Zl _:........:.... ___ --. Ft ( z) - Z2 z - Z2
Comparing this formula with (4.2.7) we obtain that k must be -a(z 1 By using (4.2.5) we calculate:
k
-
z2).
=
So k is negative in our case, and we have the second situation. Now we are ready to formulate our result.
Proposition 4.2.3 Let f(z) = az 2 + ib- a such that bE lR and 2lal > lbl. Then f generates a flow of fractional linear transformations:
which converges to z
1 -
-
2a
J 4lal2 -
and to z -
2-
J
2a 4lal2 - b2
b2 - ib'
+ ib'
as t
as t
-->
-->
oo,
-oo.
Chapter IV
108
In addition,
where
k
= -J4Ial2- b2
and
Exercise 2. Let
f
E aut(.6.). Show that the fractional linear transformation
G, defined by the formula:
where c =
j)~j
z-c G(z) = -1- zc'
has the same fixed points as the null points of f.
Exercise 3. Show that if f E aut(.6.) generates a one-parameter group of parabolic automorphisms of .6. then f has the form:
f(z) = T'(z- T)('Fz- 1), where T E 8.6. and T' E IC, ReT' generated by f and show that
=
lim Ft(z)
t--+oo
0. Find explicitly the group S
= {Ft} tEIR
= t___.-oo lim Ft(z) = T.
Exercise 4. Show that iff E aut(.6.) has a null point Tin .6., i.e., f generates a one-parameter group S = {Ft} tEIR of elliptic automorphisms, then Ref' (T) = 0. Show that the group S = {Ft} tEIR can be presented in the form:
where and
Rea= 0.
ASYMPTOTIC BEHAVIOR
4.3
109
Embedding of discrete time group into a continuous flow
A classical problem of analysis is: given a self-mapping F(z) (FE Hol(A)). Find a semigroup {Ft(z)}t>o C Hol(A) with F 1 (z) = F(z). In this case we say that F satisfies the embedding property on A. In general, however, this problem cannot be solved globally even if F E Hol(A) is a fractional linear transformation (see Example 4.3.1).
Example 1 Consider the fractional linear transformation defined as follows:
F(z)= ~ 1- bz
l
with a= exp(i3.1) and b = ~· Since lal+lbl = 1 we have that F is a self-mapping of the unit disk A. Therefore its iterations F(n) map A into itself. Moreover, F(n) converge to zero uniformly on compact subsets of A as n goes to infinity. At the same time the semigroup
atz
Ft
:=
b
1 - -1-a · (1 - at)z
is a continuous extension of the discrete time semigroup {F{n}} because of the equality F = F1. However, for some t > 0 (which is not integer) Ft does not map A into itself. If, for example, we choose z = 0.99 + 0.1i E A then by calculation we obtain that
IF~ (z) I > 1 (see Fig.
4.1).
Nevertheless, by using some geometric methods we will give in Section 5.9 a complete characterization of those fractional linear transformations of the unit disk into inself that satisfy the above embedding property. In this section we show how to solve this problem, when FE Aut(A). Suppose first that F is a hyperbolic automorphism of A, i.e., F has exactly two distinct fixed points ZI and Z2 on the boundary aA. As we saw above (see section 1.1. 3), in this case if F = r 'P • ma, then
F(z) = L- 1 (>-.F(z)), where
A=
~
-
QZ 2 QZI
(4.3.1)
E JR+ \ {0, 1}
and L is defined by
L(z)=z-zl. Z - Z2
(4.3.2)
Chapter IV
110
Figure 4.1: Fractional iterations of the self-mapping F(z). In addition, if z 1 is the sink point ofF, then 0 < A < 1. Denote k
= log A,
a
= __k_, Z2 -
Since lz1l Indeed,
= lz2l = 1, and z1
a k -=--· c
z1-z2
Z2 - Z1 z1·z2k
b=
=I= z2
=
~ z 1 + z 2 , and c = Zl · z 2 t Zz -
Z]
Z2
Z2 -
it is easy to verify that
Z2 - Z1 (z2-zl)z1z2
=
a=
k.
Z1
-c, and bE R
Z2 - Z1 =-1 z1iz2l 2 -z2izll 2 ·
In addition, Reib
i.e., bE R Therefore the mapping
f:
f(z)=az 2 +ibz+c
is a complete vector field. It follows by the Vietta formulas that z1 and z2 are the roots of the equation f(z) = 0. Hence as above (see section 4.2) the group of automorphisms generated by f has a form:
where Lis defined by (4.4.2).
ASYMPTOTIC BEHAVIOR
111
For t = 1 we have:
because of (4.4.1). This gives us the desirable embedding. As a matter of fact, we have proved somewhat more:
Proposition 4.3.1 Let z 1 =/:- z 2 are two arbitrary points on the boundary 8!:1 of the unit disk !:1. Then for each k E JR, k =/:- 0, there is a one-parameter group S = {Ft}, t E JR, such that the following assertions hold: i = 1, 2. (a) Ft(zi) = zi, dFt(z) k kz1+z2 2 (b) - - lt=o= -(az + ibz- a) where a=-.- - , b = --:---. dt Z2- Z1 t Z2 - Zl (c) If k < 0 then z 1 is a sink point of {Ft}t>o; if k > 0 then z2 is a sink point of {Ft}t>O· (d) In particular, if z 1 and F(z)
are fixed points of a hyperbolic automorphism
z2
= ei'f'
z +a 1 + O:z'
a E
!:1
c.p E JR,
'
then for
we have the equality:
F1(z)
= F(z),
z E !:1.
Now we turn to a parabolic automorphism F E Aut(/:1). In this case F has a unique fixed point z0 E 8!:1 which is a sink point of F. Moreover, as we already know F satisfies the equation: z0 -=-:---:-'---
F(z)-zo
ei'f' - 1
z0
e''f'+1
z-zo
= -.-- + - -
(4.3.3)
for some c.p E JR, c.p =/:- 1r + 2n7r (see section 1.3.1). It is clear that in such a situation we must solve the following Cauchy problem:
8u(t,z) {
at
( )2 _ +a z- zo - 0 ' (4.3.4)
u(O, z) = z E !:1, with suitable a E C in order to find a one-parameter group {Ft(·)}, Ft such that: F1(z) = F(z). Solving (4.3.4) we have that Ft satisfies the equation: 1 Ft(z)-zo
-=-:--:---
1 z-zo
= at + - - .
= u(t, ·), (4.3.5)
Chapter N
112
Comparing the latter equality with (4.3.3) and (4.3.5) we obtain:
ei"'- 1 1 a=---·ei'P + 1 zo ·
(4.3.6)
Now we only need to check that the mapping f defined by:
f(z) = a(z- zo) 2
(4.3.7)
is a complete vector field. In fact, substitute (4.3.6) in (4.3.7) we obtain:
f(z) =a z 2 + ibz + c, where 2
c = az0
ei"' - 1 = -. -- · zo = -a-
e'"' + 1
and -2 ei"'-1 41mei"' b=-·-.--=E!R.. i e•'P + 1 jei'P + 1j 2
Thus we have proved the following assertion:
Proposition 4.3.2 Let zo and T =f. -1 be two unimodular points. Then the mapping f defined by (4.3. 7} with T-1 a= --zo r+1 is a generator of the one-parameter groupS= {FthelR E Aut(Ll), such that each Ft is a parabolic automorphism of Ll, and the point zoE 8Ll is a sink point of S. If, in particular, FE Aut(Ll), is a parabolic automorphism of Ll: F(z)=ei"'z+a
1 +za
such that F(zo) = z0 , then forT= eicp we have: F 1 (z) = F(z). Finally, we consider the simplest situation where F E Aut(Ll) is an elliptic automorphism of Ll, i.e., F has a unique fixed point z0 E Ll. In this case F can be presented as: (4.3.8) where
z
mzo(z) is a Mobius transformation of Ll, i.e.,
r"'(z) = ei"'z, Setting in this case
+ zo
= 1 + zoz
mz 0 (0) = z 0 , and r"' is a rotation around zero,
cp E JR.,
k = 0, 1, 2, ....
ASYMPTOTIC BEHAVIOR
113
and (4.3.9) we have that S = {Ft}, t E lR is a one-parameter group of elliptic automorphisms of~ such that Ft(zo) = zo for all t E lR and Ft(z) = F(z), z E ~-
Exercise 1. Prove that group (4.3.9) is generated by the mapping f(z) =
-ic.p I l 2 (z- zo)(1- z 0 z). 1- zo
Exercise 2. Prove directly that the mapping plete vector field.
f: (4.3.10)
f defined by (4.3.10) is a com-
Finally, we formulate the result:
Proposition 4.3.3 Let zo E ~ and c.p E R, c.p = 21rk, k = 0, 1, 2 .... Then the mapping f defined by (4.3.10) generates the one-parameter groupS= {Ft}, t E lR of elliptic automorphisms defined by formula (4.3.9). In particular, ifF E Aut(~) is an elliptic automorphism, defined by (4.3.8), then Ft(z) = F(z), z E ~-
4.4
Rates of convergence of a flow with an interior stationary point
Let f E Hol(~, C) be a semi-complete vector field on ~. and let us assume that Null f n ~ =f:. 0. Actually, it means that if f is not zero identically, then f has exactly one null point in ~ (otherwise the semigroup S = { Ft}t~o generated by f should have more than one stationary point in ~. and a contradiction results). A :standard que::;tion of dynamical ::;y::;tems aualysis is: given a vector field J, describe the asymptotic behavior of the flow defined by the evolution equation dujdt + f(u) = 0, u(O) = z, in a neighborhood of its singular point T (J(T) = 0). In this section we intend to answer this question for a semi-complete vector field, as much as to find global rates of convergence of the generated flow to its interior stationary point. For our purpose we need the following definition.
9 Hol(~) is said to be a strongly semicomplete vector field if it has a unique null point T in ~ which is a uniformly attractive stationary point for the flow S = { Ft}t~o generated by f, that is, the net {Ft}t~o converges toT uniformly on each compact subset of~ as t--+ oo.
Definition 4.4.1 ([40]) A function f E
Chapter IV
114
We begin with establishing a simple assertion which is a (nonlinear) analog of the Lyapunov stability theorem.
Proposition 4.4.1 Let f E Hol(D., C) be a semi-complete vector field with f(r) = 0 for some T E D.. Then: (i) Ref'(r) ~ 0; (ii) Re f'(r) > 0 if and only iff is strongly semi-complete.
Proof. Let S = {Ft}t~o be the flow generated by f. Since Ft(T) = T for all t ~ 0, it follows by the chain rule, that (Ft)'(r) = At satisfies the semigroup property At+s =At· A., Ao = 1, hence At= ekt, with k = f'(r) E C. In addition, it follows by Corollary 1.1.1 that
But this inequality holds if and only if Re k = Re !'( r) ~ 0 and assertion (i) follows. Furthermore, Re k = Ref' (T) > 0 if and only if for all t 2: 0
and then assertion (ii) is a consequence of Proposition 4.1.1. and Corollary 3.1.1. D
Exercise 1. Show that if Ref'(r) = 0 then the semigroup S = {Ft}t~o is actually a group of elliptic automorphisms. Thus, iff E
9 Hol(D.) is not a complete vector field of the type f(z) = (z- r)(l- 'fz)a
with TED. and Rea= 0, then the semigroup S = {Ft}t~o generated by f does not consist of elliptic automorphisms and converges uniformly on each compact subset of D. to a point in D.. If this point belongs to D., then it is a unique uniformly attractive stationary point of the semigroup, and the question of finding a rate of convergence arises.
Remark 4.4.1 We already known that for a holomorphic function fonD. the property of being semi-complete is equivalent to the property of being monotone with respect to the Poincare metric p on the unit disk. Also, it can easily be seen that iff is strongly p-monotone then it is strongly semi-complete. For this case one can obtain a rate of convergence in terms of the hyperbolic metric on D. (see Section 4.5). The converse, however, does not hold in general: a strongly semi-complete vector field is not necessarily strongly p-monotone. Therefore in this section we will establish some rates of convergence in terms of the Euclidean and pseudo-hyperbolic distance on D.. First we consider the case when f(O) = 0, i.e., zero is a stationary point of the semigroup S = {Ft}t>D·
ASYMPTOTIC BEHAVIOR
115
Proposition 4.4.2 (see (58] and (109]) Let f E g Hol(~) be a strongly semicomplete vector field with f(O) =·O and .A= Ref'(O) > 0, and letS= {Ft}t~o be the semigroup generated by f. Then there exists c E [0, 1] such that for all z E ~ and t :2: 0 the following estimates hold: (a) IFt(z)l
~ lzl·exp( -.A~~~:::t).-
IFt(z)l < lzl (b) (1- c1Ft(z)l)2 - exp( -.At) (1- clzl)2.
Proof. The procedure that is to follow is based on general flow invariance conditions established in sections 3.4 and 3.5. We start with the consideration of an auxiliary Cauchy problem: 8u(t,z) {
8t u(O,z)
+p
= z,
. ( )·1-cu(t,z)_ 0 u t, z ( ) - ' 1 + u t, z
zE
(4.4.1)
~-
1-z Noting that the function cx(z) = p · z - - satisfies the condition 1+z
for all z E ~. we conclude that ex E g Hol(~) and the Cauchy problem above has a unique solution u(t, z) E ~ for all z E ~ and t :2: 0. In addition, u(t, 0) = 0 for all t :2: 0, because of cx(O) = 0. Then by the Schwarz Lemma we obtain that lu(t, z)l ~ lzl,
z E ~-
(4.4.2)
= ps 1 -
s, defined on the interval [0, 1] 1+s is real we have by the uniqueness property that the differential equation:
Furthermore, since the function cx(s)
(4.4.3)
with the initial data:
v(O)
=s
(4.4.4)
has a unique solution v(t, s) for all t :2: 0 and s E [0, 1) which coincides with u(t, s) restricted on the real interval [0, 1). Consequently, by (4.4.2) we obtain:
v(t,s) < s for all t :2: 0 and s E [0, 1). Also, it follows from (4.4.3) that: dv 1- v = -p--dt. v l+v
-
(4.4.5)
Chapter IV
116 Hence, by (4.4.4):
v = s exp ( / -p
~ ~ ~ dt) .
Finally, inequality (4.4.5) implies the estimate: (4.4.6) Recall that by Proposition 3.4.5 inequality:
f E QHol(.6.) with f(O) 1-lzl
Ref(z)z 2:: Ref'(O)Izl 2 1 + lzl = a(lzl) ·lzl,
0 satisfies the
z E .6..
Then by Lemma 3.4.1 and (4.4.6) we have: IFt(z)l
~ u(t, lzl) ~ lzl· exp (- Ref'(O) ~ ~
:::t).
This proves (a). To prove (b) we return to (4.4.3) and rewrite it in the form: 1 +v 1 - - · -dv = -pdt
1- v v
(we substitute a(v) = vp 1 + v into this equation). Integrating the latter equality 1-v in [0, t] we obtain: v(s,t) _ e-pt s (1 - v( s, t) )2 (1 - s )2,
[ ) s E 0, 1 .
Again, settings= lzl and using the relation
IF(t, lzl)l ~ v(s, t), we obtain (b). D Note that estimate (a) with c = 1 is due to K. Gurganus, whilst estimate (b) was established by T. Poreda. Now we are prepared to consider a more general case, when f E 9 Hol(.6.), with f( r) = 0 for some T E .6., which is not necessarily zero.
Proposition 4.4.3 Let .6. equipped with the pseudo-hyperbolic distance
d(z,w)
w-z I , = I--_1-wz
and let Sf = {Ft }t~o be the semigroup generated by f E E .6. the following conditions are equivalent:
T
9 Hol( .6.).
Then for some
ASYMPTOTIC BEHAVIOR ~
{i) f(r) = 0 with Ref'(r) {ii} for some c E [0, 1]:
d (Ft(z), r)
p
~
117
0;
~ d(z, r) exp ( -p ~: ~~~:: ~~ t)
for all t ~ 0 and z E !::.. In addition, iff is bounded and strongly p-monotone then the number c in {ii} can be chosen as strictly less than 1.
Proof. Define the Mobius transformation Mr : !::. T-Z
Mr(z) = -1 _ , -TZ
z
E
t---4
!::. by:
!::..
(4.4.7)
It is clear that (4.4.8) and it follows that Mr is an involution, i.e., (4.4.9) Consider the family
{Gt}t~o
defined as follows: {4.4.10)
It is obvious that {Gt} is also a semigroup of holomorphic self-mappings of !::., and for all t ~ 0 we obtain {4.4.11) Gt(O) = 0, t ~ 0, since Ft(r) = r, for all t ~ 0. Let g be the generator of the semigroup {Gt}t>o , i.e.,
g(z) =
-dGt(z) dt
(4.4.12)
lt=O+ •
By direct calculations we obtain:
d
dMr
dFt
dt (Mr(Ft(Mr(z)))) = ----;J:;- (Ft(Mr(z))} · dt(Mr(z))
=
d~r (Ft(Mr(z))) · (- f(Ft(Mr(z))),
z
E
!::..
Hence by (4.4.12)
g(z) =
d~r (Mr(z)) · f(Mr(z)),
z
E
!::..
(4.4.13)
Note that the latter formula expresses the direct connection between generators
f and g. Also, (4.4.8) and (4.4.13) imply: g(O)
= 0,
(4.4.14}
Chapter IV
118
which is equivalent to (4.4.11). In turn, differentiating (4.4.13) we calculate 1
9 (z) =
d 2Mr ( dMr ) dz 2 (Mr z)) · dz(z · J(Mr(z))
dMr df dMr ( +dz(Mr(z)) · dz (Mr(z)) · d z z).
(4.4.15)
Substituting here z = 0 and using that f(r) = 0 we obtain: g'(O) = dMr (r). df (r). dMr (O). dz dz dz
(4.4.16)
Since dMr d ( T- z ) lrl 2 - 1 d z = dz 1- TZ = (1- Tz)2
relation (4.4.16) becomes: g'(O) =
It 1/ -1 f'(r) · (lrl 2 -
1)
= f'(r).
(4.4.17)
So by Proposition 4.4.2 (a) we obtain:
IGt(z)l
~ lzl exp (- Reg'(O) · ~ ~ :~:
·
t),
and using (4.4.10) and (4.4.17) we obtain from the latter inequality:
I(Mr o Ft
o M r) ( z) I
For arbitrary w, setting I(Mr o Ft)(w)l ~
~ Iz Iexp (- Re f' (T) · ~ ~ :~:
z =
·
t) .
(4.4.18)
Mr(w) in (4.4.18) we obtain:
IMr(w)l exp
(
-Ref
1
1 -IMr(w)l ) (r) · 1 + IMr(w)l · t ·
(4.4.19)
-I r-~: I t) '
(4.4.20)
Rewriting (4.4.19) in the form:
I ( - Re !'(
Ft(w) < T -w exp 1-Ft(w)r- 1--rw T-
T
)· 1 1
+1r---r:l
·
we obtain (ii). Conversely, assuming that (ii) holds we first observe that f(r) should be zero. Of course, this is an immediate consequence of (ii) if p > 0, hence Ref'(r) > 0. Indeed, in this case lim Ft (z) = r, t-oo
so, Tis the stationary point of St = {Ft}t~o, whence, a null point of f. But even if Re/'(0) = 0, condition (ii) means that each pseudo-hyperbolic ball f2r(7) (see Section 1.3) is F 1-invariant for all t ~ 0, and the same conclusion follows.
ASYMPTOTIC BEHAVIOR
t
=
119
To arrive at the second condition of (i) we just need to differentiate (ii) at o+ to obtain:
Letting z -+ T we obtain immediately that Re f' (T) ~ p as desired. Finely, note that our last assertion follows from the strong Harnack inequality (see Section 3.5 and Proposition 3.6.2). D Remark 4.4.2 Since IFt(w)l < 1 one may obtain from (4.4.20) (setting c = 1) the following estimate
IFt(w)- rl
~
1 + lrl ( , 11 - rwl -lr- wl ) jw- rl1 -lrllwl exp -Ref (r). 11- rwl + lr- wl t .
(4.4.21)
This formula gives an estimated rate of convergence of the semigroup { Ft}t>O to its stationary point r in the usual Euclidean metric in C. As a matter of fact, this estimate can be slightly improved by using the following considerations. Return to formula (4.4.19). Denoting Ft(w) = J and Mr(J) = u we have Mr(u) = J and Mr(O) = T. Now, it follows by Corollary 1.1.3 of the Schwartz Lemma (see section 1.1.1), that:
or
IFt(w)- rl
< lr-wl - 11-'fwj
(4.4.22)
It is clear that (4.4.22) implies (4.4.21 ). Remark 4.4.3 Note that the above proposition may be considered as an analogy of the Schwarz-Pick inequality. Moreover, for the continuous case this inequality has a more precise form because of the exponential factor 1- d(z, r) ) exp ( -p1+d(z,r)t ·
However, the property of this factor depending also on z E ~ is sometimes inconvenient for applications. In some situations it would be preferable to find a metric on ~ in which such a factor becomes uniform. In addition, as in the discrete time case, condition (ii) in our proposition cannot be useful in the study
Chapter IV
120
of the boundary asymptotic behavior of flows with no stationary point inside ~. Therefore we will treat the last question separately in Section 4.6 by using a continuous extension of the Julia-Wolff-Caratheodory Theorem. However, it turns out that one can define a non-Euclidean distance on ~ (which, in fact, is not a metric on ~' but induces the original topology of ~) such that the rates of convergence for interior and boundary points have some unified form. This makes it possible to study the dynamic behavior of evolution equations when their stationary points approach the boundary. Moreover, we will show in Section 4.6 that in terms of such a distance, the question about a uniform rate of convergence of the exponential type may be solvable too. Finally, we note that a deficiency of the rates of convergence given by formulas (4.4.20) and (4.4.21) is the presence there the value of the derivative f' at the null point T of a generator f. In fact, if f has a complicated form even the existence of its null point is unclear a priori. Therefore it would be nice to establish a rate of convergence by using another condition which will guarantee also the existence of an interior null point of a semi-complete vector field. The next section is devoted to this matter.
Remark 4.4.4 From the geometrical function theory point of view there is a need to point out a lower bound estimate of the flow behavior. Exactly as in Proposition 4.4.2, by using the left side inequality of (3.5.18) and Lemma 3.5.1 one again obtains: if f(O) = 0 and Ref'(O) =A 2:0, then for some c E [0, 1] (a') IFt(z)l 2: lzl exp( -A 1 + c:z: t); 1- c z
,
IFt(z)l
lzl
(b) (1-c1Ft(z)l) 2 2:exp(-At) (1+clzl) 2 ' Moreover, iff is bounded strongly p-monotone then c can be chosen strictly less then 1.
4.5
A rate of convergence in terms of the Poincare metric
In this section we will give several sufficient conditions for f E Hol(~, q to be strongly p-monotone, hence strongly semi-complete on the open unit disk ~ and obtain rates of convergence for the semigroups generated by such functions in terms of the Poincare hyperbolic metric p on~.
ASYMPTOTIC BEHAVIOR
Proposition 4.5.1 ([40]} Let
f
121
E Hol{A, C) satisfy the condition:
Ref(z)z
a(Jzi)izl,
;::=:
z E
A
(4.5.1)
for some real continuous function a( l) on the interval [0, 1], such that:
a{1) =a> 0.
{4.5.2)
Then f is strongly p-monotone, hence a strongly semi-complete vector field on T E A which is uniformly attractive for the semigroup S = { Ft}, t ;::=: 0, generated by f. Moreover, for each pair z, w E A
A. Thus f has a unique null point
at
~ exp { -
p( F1 ( z), F1 ( w))
2} p( z, w).
(4.5.3)
}p(z, w).
{4.5.4)
In particular,
p(F1(z), T)
~
exp { -
Proof. Consider the mapping g. : A
g8 (z) = z
C defined as follows:
f-+
+ sf(z)- w,
at
2
where
w E A, s
Let Ar be the disk centered at zero with radius r z E oAr= {z E C: lzl = r} we have by {4.5.1):
Reg.(z)z
izl 2 r (r
0
;::=:
~
0.
r < 1. For all
+ s Re f(z)z- Rewz ;::=: r 2 + sa(r)r- 1·Jwl + sa(r) -lwl).
(4.5.5)
Since a{1) > 0 it follows that for s > 0 small enough the equation:
cp 8 (r) = r
+ sa(r)
= 1
has a solution r 8 E {0, 1). In fact, cp 8 (0) = sa(O) ~ 1 for 0 < s < 1/a{O) and cp 8 {1) = 1 + sa{1) follows by (4.5.5) that for such fixed n and z E OArn,
Reg.(z) · i
;::=:
(4.5.6)
;::=:
1. It
0.
The latter inequality implies that the equation:
g.(z) = z
+ sf(z)- w = 0
has a unique solution z = J.(w) :=(I+ sf)- 1 (w) EAr. for each wE A. In other words, the resolvent mapping J8 maps A into Ar,· Thi~ mean~ that ] 8 has an interior fixed point T E A which is also a unique null point of f in A. Furthermore, we consider the function h(z) = h8 (z) defined as follows:
hs(z) = Js(z)
a( rs)
+ s -2-
{Js{z)- Js(w)),
Chapter IV
122
where w is an element We obtain:
of~.
and r 8 is the solution of (4.5.6).
zE ~
~-
Since h 8 (w) = J.(w) E ~ it follows by the maximum principle that hs maps into itself. Therefore by Corollary 1.1.1 lh~(w)l
0 because of its continuity. Therefore there exist some positive 8 and f such that for all s E (0, 8) we have o:(rs) ?: f > 0. Then for such s we obtain the inequality
p (J.(z), J 8 (w))
~
1 - - p(z, w), 1 +Sf
which means that f is strongly p-monotone. Now settings= t/n in (4.5.7) and using the exponential formula:
Ft(z) = lim J(n)(z) n--->oo tfn we obtain by induction: lim
The proposition is proved.
0
1
k . o(;n))
h--->oo (
1+
exp {-
o:t 2 } p(z, w).
n
p(z,w)
ASYMPTOTIC BEHAVIOR Example 1. Let
123
f E Hol(~, q be defined by 1- cz f(z) =a- az 2 + bz - - , 1 + cz
where a, b E 0, and 0 :::; c < 1. a(s)
=
If we introduce the function
-lal(l- s 2 )
1 - cs
+ (Reb)s--, 1 + cs
then we obtain
Reg(z)z;;::: a(izl)izl and
1-c a(l) =Reb--> 0. l+c Hence f(z) is a strongly semi-complete vector field on~Remark 4.5.1 Note that iff E Hol(~, q is known to be a semi-complete vector field on~' then condition (4.5.2) can be replaced by a slightly more general condition, namely, a(l) > 0 for some l E (0, 1], which still ensures f to be strongly semi-complete. The above arguments can be employed in the disk ~~- This note leads us to the following simple sufficient condition yielding the existence of an interior null point of a semi-complete vector field and its attractiveness. Corollary 4.5.1 (cf., EM-RS-SD-2000c) Let
f E QHol(~) be such that:
Re !' (0) > 41/(0) I· Then f has a unique null point verges to r as t goes to infinity.
T
(4.5.8)
E ~ and the semigroup Sf = { Ft}t2':D con-
Proof. Consider the function: a(r) = -a(1- r 2 )
1-r
+ br--, 1+r
where a= l/(0)1, b = Ref'(O). It follows by Proposition 3.5.3 that
Re f(z)z;;::: a(izi)izl,
zE
~-
Then we have under condition (4.5.8), that a(O) = -a < 0, a'(l) < 0. Hence, there islE (0, 1), such that a(l) > 0. 0
(4.5.9) a(1) = 0, and
Remark 4.5.2 Note that conditions (4.5.8) and (4.5.9) imply that for some ~r with r E (1- 8, r] is invariant for the semigroup {Ft} generated by f. However these conditions are not sufficient to ensure the validity of the strong p-monotonicity of f.
8 > 0 each disk
Chapter IV
124
Remark 4.5.3 Observe also that the above Proposition 4.5.1, Lemma 3.6.1, and Proposition 3.6.2 imply that the property off E Hol(~, C) being strongly p-monotone is equivalent to the strong flow invariance condition
Ref(z)z
~ Tf
>0
for all z in the annulus {1- 8 < lzl < 1} for some 8 > 0. In terms of the Poincare metric this property is equivalent to the global uniform exponential convergence (see formula (4.5.3)) of the semigroup {Ft}t~o generated by f on all of ~, whilst the property of f to be strongly semi-complete is equivalent to local uniform exponential convergence on each compact subset of~ (see Proposition 4.4.3).
4.6
Continuous version of the Julia-Wolff-Caratheodory Theorem
Let now f E Hol(~) be a null point free mapping. We already know that in this case the semigroup Sf = {Ft}t>o generated by f is convergent (see Section 4.3). Moreover, there is a point T E 8 ~ (a sink point of Sf), such that for each z E ~, lim Ft(z) = T. However, the study of rates of convergence in this case is more t--+00
complicated than in the case when f has a null point in ~- Indeed, as we saw above, in the latter case the asymptotic behavior of the semigroup generated by f is completely determined by the value of f'(r). Namely, {Ft}t>O is convergent if and only if Ref'(r) > 0. Therefore it is natural that the rates of convergence obtained in Section 4.4 are connected with this value. Iff E QHol(~) is null point free, i.e., Sf= {Ft}t~O has a sink point T E 8~, one cannot use the same approach, since !' (T) is not defined in general. In addition, the above approach of using a Mobius transformation is also unfeasible in this situation. Therefore we need another method to study the asymptotic behavior of the semigroup generated by a null point free holomorphic function. A complete characterization of convergence of a semigroup to its sink point can be done by using the so called nontangential derivative of its generator in the spirit of the Julia-Wolff-Caratheodory Theorem (Proposition 1.4.2). Here we establish a continuous version of this result (Proposition 4.6.2) which will provide another proof of the classical one. Note in passing that if F E Hol(~) is a holomorphic self-mapping of~, then f =I-F is semi-complete. We first prove some auxiliary assertions.
ASYMPTOTIC BEHAVIOR
Hol(~, C)
Lemma 4.6.1 Let f E
(finitely) then: (i) L lim f(z) z-.e
(ii)
and let e E
125
8~.
If (3 := L lim f(z) exists z-e
z- e
= 0;
the angular limit L lim
z-.e
f' (z) also exists and equals to (3.
Proof. (i) is trivial. To prove (ii) let us assume that e the form f(z) = (3(z -1) + h(z).
= 1 and present f in (4.6.1)
Then we have
Llimh(z)=O Z-->1
and L lim h(z) = 0. Z-+1 Z -
(4.6.2)
1
We wish to show that
L lim h'(z) = 0.
(4.6.3)
z-+1
To this end take two sectors SandS in~ both with vertex at e = 1, so that S C S. For z E S we denote by f(z) the circle with its center at z, such that f(z) is tangent to the boundary of S. By(} we denote the angle between segments [0, 1] and [z, 1] and let 2Bs and 2Bs
be the angles of the sectors S and Then . sm(Bs- Bs)
S, ~
respectively, at vertex e = 1 (see Fig. 4.2).
. sm(B_s;- B)
=-
r(z)
(4.6.4)
--, 1
1-z
1
where r(z) is the radius of the circle f(z). Now observe that when z converges to 1 in the sector S, all points w E 8f(z) converge to 1 in the sector S. Therefore using (4.6.2) we obtain that for each e > 0 there is a point z close to 1 such that
lh(w)l < e, lw -11
wE f(z).
It then follows by the Cauchy formula and (4.6.4) that
h'(z)
=
_1
2rri
J
h(z) dw < .!...._ (w- z)2 - 2rr
r(z)
1-
(r-l)e
exist. We will denote it by j f'(e). It is clear that iff has the angular derivative at a point e E 8tl then it has a radial derivative at this point and
i f'(e) = L.f'(e). We will show that for a null point free semi-complete vector field a somewhat converse statement is also true at the sink point of the flow generated by f. First we establish the following assertion. Lemma 4.6.2 Let f be a semi-complete vector field on tl. Suppose that for a
ASYMPTOTIC BEHAVIOR
127
point e E 8b. there exists the radial limit derivative (3
-
-
. l liD
r-+1-
f(re)
(r -1)e,
such that
Re (3 2: 0.
(4.6.5)
Then f has no null point in b.. Moreover, the point e must be the sink point of the semigroup generated by f.
Proof. Without loss of generality let as assume that e the Berkson-Porta representation
= 1,
and write
f(z) = (z- r)(1- zf)p(z),
f (z) by (4.6.6)
where Rep( z) 2: 0 everywhere, and T is a point of b.. We wish to show that T in (4.6.6) is equal to 1. First we assume that Re(3>0 and conversely suppose that Re(J
= Re
T
-=f. 1. Then we have by (4.6.6) and (4.6. 7)
/(r))
( lim r-+I-
(4.6.7)
Re [lim (r- r)(1- rt) p(r)] r-+Ir- 1
r- 1
[1- 7[ 2 Re [lim Rep(r)] > 0. r-+I- r - 1 On the other hand for all r E (0, 1) p(r) 1 Re - - = - - Rep(r):::; 0, r-1
r-1
which is a contradiction. To complete our proof for the general case we consider the mapping fe : b. f-> C defined by (4.6.8) fe(z) = f(z) + c:(z 2 - 1) with c: > 0. Since c:( z 2 complete and
-
1) is a complete vector field, the vector field /" is semiRe [T /~(1)]
= Re(3 + 2c: > o.
(4.6.9)
Therefore by the previous step, for each c: > 0 the mapping fe has no null points in b.. Now, if Tin (4.6.6) belongs to b., that is, f has an interior null point, then by the Rouchet theorem it follows that one can choose a small enough E: such that !e has an interior null point T 0 close to T. Once again a contradiction. Moreover, (4.6.9) and the step proved above show that for each E: > 0 the point 1 must be a sink point off". Thus there are functions Pe E Hol(b., C) with Re Pe 2: 0, z E b., such that
Chapter IV
128
Comparing the latter formula with (4.6.8) we obtain
( ) _ (z- 7)(1 - z'F)p(z) Pc: z - (1 - z )2
z +1 +e1 - z.
Since Re Pc: ;:::: 0 for all e > 0 and Pc: (z) converges to Po ( z)
( ) _ (z- 7)(1- z'F)p(z) paz-(1-z)2 , of~
uniformly on each compact subset
as e goes to 0, we obtain that
Repo;:::: 0. But
-(1- z) 2 po(z)
= (z- 7)(1- z'F)p(z) = f(z),
hence contradicting the uniqueness of the Berkson-Porta representation. Thus po(z) = p(z) and 7 = 1. D
Proposition 4.6.1 If for a point e E
i
8~
lim f'(z) = (3
z-e
the radial limits
and
i
lim f(z) = 0
z-e
exist with Re f3 ;:::: 0, then
0::; (3 and e =
7
=
Lf'(e)
is a sink point of the flow generated by f.
Proof. Indeed, since lf'(re)l < M < oo we obtain
J 1
lf(re)l =
f'(te) dt ::; M(1- r).
r
Hence lim inf ( f(re~ exists. Let e r-->1r- 1e
= 1 and u(z) = Ref(z).
Then we obtain
J 1
u(r) = -
u'(t)dt = u'(ry)(r- 1),
r
where 17 E [r, 1). Hence lim ( u(r)) = Re(J;:::: 0. D r- 1
r-->1-
To establish the converse assertion to Lemma 4.6.2 we will use the RieszHerglotz integral representation of functions of the class P = {p E Hol(~. C) : Rep(z);:::: O}(see Section 0.2):
p(z) =
J+
1 z( 1- z(
- - - dJ-Lp(()
at::.
.
+ tlmp(O),
(4.6.10)
ASYMPTOTIC BEHAVIOR
129
where f..Lp : 8/:l. ~---+ lR (the measure characteristic function for p E P) is a positive increasing finite function f..Lp on the unit circle 8/:l., such that
J
df..Lp(() = Rep(O).
at:. Lemma 4.6.3 Let f E Hol(b., C) be a semi-complete vector field on b. with no null point in b.. Then there is a point T E 8/:l., such that f has an angular derivative at T. Moreover, if T is a sink point of the flow generated by f then Lf' (T) exists and it is a positive real number which is equal to 2t-Lp( T), where
pz)= f(z) ( (z- T)(1 - ZT) Proof. Of course, it is enough to prove only the second assertion of the Lemma. Let us again present f by the Berkson-Porta formula
j(z) = (z- T)(1- zr)p(z), where Rep(z);::: 0,
-(f(z) --) = z- T
z E b.. Then by (4.6.10) one can write
J at:.
1 + z( (1 - zr) - - - df..Lp( () - (1 - zr)i Im T f(O). 1- z(
(4.6.11)
Let Zn E b. be a sequence of points which converges to T nontangentially. Then if we write Zn in the form Z11 = T(Xn + iyn), we can find 0 < K < oo, such that
IYnl
~
(4.6.12)
K(l- Xn)
(see Fig. 4.3). Further, consider the sequence of functions g11 : 8/:l.
~---+
C defined by
gn := gn(() = (1- TZn)(1 + Zn(). 1- Z 11 (
To estimate 9n on the unit circle we calculate
or Zn( (1 - TZn
This implies that
+ 9n) = gn -
(1 - Tz,.).
( 4.6.13)
Chapter IV
130
Figure 4.3: The nontangential convergence to or
Consequently, _ I
9n
(1- fzn)(l + lznl 2) 12 1 -lznl2
Finally,
I_
(1- fzn)(l
9n
+ lznl 2)
1 -lznl2
1--
21 Zn
111- fznl
1 -lznl2'
and we obtain
lwn(()l < 4 ll- fznl = 4 J(l- Xn)2 + 1 - lznl 2 (1 - Xn) 2 y~ Taking (4.6.12) into account we obtain
Since for n large enough 1 + :rn
1
-IYniK > 2'
y;.
T.
ASYMPTOTIC BEHAVIOR
131
we have for such n
l9n(()l ~ 8V1 + K 2 • In addition, it follows by (4.6.13) that
9n(()
--->
0,
(
#- T.
Then by the Lebesgue Theorem (see, for example, [125]) and (4.6.11) we obtain
. l lm Zn~T
f(zn)
-Zn- T
J(
1 - fzn)( 1 + Zn() dJ-Lp(()- i Im T f(O) · lim (11- Zn( Zn-+T
Znf)
OA
2J-Lp(r), and the Lemma is proved. D Our next goal is to establish relations between angular derivatives of semicomplete vector fields and the asymptotic behavior of their generated semigroups. Lemma 4.6.4 Let F E Hol(D., D.) be a holomorphic self-mapping of D. with no null point in D., and letT E fJD. be its sink point. If Zn converges toT nontangentially, then so does the sequence F(zn)· Proof. 1.4.2) that
It follows by the Julia-Wolff-Caratheodory Theorem (Proposition
0 ~a= L.F'(r) ~ 1 and 'Pr (F(z)) ~ O:'Pr ( z ),
(4.6.14)
where
If z E r(r, k) = {z E D. : 1.2.1) then (4.6.14) implies
lz- rl < k(1- lzl)}
IF(z)-ri
1 (see Definition
IF(z)l
1 + lzl
(4.6.15)
In turn, The Julia-Caratheodory Theorem states that L.lim F(z)-r =a>O. z,-+T
Z-T
Therefore the third factor in (4.6.15) is bounded. Thus we can find k 1 such that F(z) E f(r, k!) and we are done. D
~
k,
Finally we are able to formulate our main assertion of this section which is a dynamical analog of the Julia-Wolff-Caratheodory Theorem.
132
Chapter IV
Proposition 4.6.2 (see [42]) Let f E QHol(D.,C) be a semi-complete vector field, and let S = {Ft}t~o be the semigroup generated by f. The following are equivalent: (i) f has no null point in D.; (ii) f admits the representation f(z)
=
-'f(z- r) 2 p(z)
for some T E aD. and Rep(z) ~ 0 everywhere; (iii) there is a point T E aD., such that
i J'(r) = f3 exists and Re f3 ~ 0; (iv) there is a point
T
E aD., such that
Lj'(r) = f3 exists andRe f3 ~ 0; (v) there is a point
T
E aD., such that the following limits exist
L lim j'(z) = f3
and
Z~T
with Ref3 ~ 0; (vi) there are a point
T
L lim f(z) = 0 Z~T
E aD. and a real positive number"(, such that
IFt(z)- r[ 2 < e -t-r [z- r[ 2 . 1 -1Ft(z)[ 2 1 -[z[ 2
.:....__...:...,..:,=-:-...,...:..,.
Moreover, (a) the points rEaD. in (ii)-(vi) and the numbers f3 in (iii)-(v) are the same; (b) f3 is, in fact, a nonnegative real number which is the maximum of all "f ~ 0 which satisfy (vi).
Proof. Equivalence of (i)-(iv) has been proved in Lemmata 4.6.2-4.6.3, while equivalence (iv) and (v) follows from Lemma 4.6.1 and Proposition 4.6.1. Thus it is enough to show that (v)==>(vi) and (vi) implies one of the conditions (i)-(v). Suppose that (v) holds. We already know that f has no null point in D. and e = T is the sink point of its generated semigroup. In addition, if lr E Hol(D.), r ~ 0, is a resolvent off: lr(z) +rf(Jr(z)) = z, zED., (4.6.16) then T is also the sink point for Jr, r > 0. It follows by Lemma 4.6.4 that if z converges nontangentially tor, then so does lr(z) for each r ~ 0. Hence (4.6.17) L lim f'(Jr(z)) = (3. Z-->T
On the other hand, it follows by Julia's Lemma that for each r > 0 there is a number O:r, 0 < O:r :::; 1, such that L lim J;(z) z-->r
= O:r
(4.6.18)
ASYMPTOTIC BEHAVIOR
and
llr(z)- rl 2 1 -llr(z)l 2
133
iz- rl 2 ~ O!r 1 -lzl 2 ·
(4.6.19)
At the same time, differentiating (4.6.16) we obtain
or
J;(z) = 1 + r/(Jr(z)) Using (4.6.17) and (4.6.18) we obtain (4.6.20) Substituting (4.6.20) into (4.6.19) and applying the exponential formula we obtain
J~~ 'Pr(Jt(/l(z))
0, while Ref'(zk) < 0, where z;: = -1, k = 1, 2, ... , n. Hence if {Ft}t~o is the semigroup generated by f, then the point T = 1 is an attractive point of this semigroup and the following rate of exponential convergence holds
for all z E ~ and t :::: 0. As we mentioned above, ifF is a self-mapping of~ then the function f(z) = z- F(z) defines a semi-complete vector field on~- As a matter of fact, this fact holds even if F E Hol(~, C) is not necessarily a self-mapping of ~, but satisfies the following one-sided estimate: lim F(rz)z
r--+ 1-
~
1 for all z E
8~.
(4.6.27)
Thus we have the following version of the Julia-Wolff-Caratheodory Theorem:
ASYMPTOTIC BEHAVIOR
135
Corollary 4.6.1 Let F E Hol(~, ~) satisfy (4.6.27) and z E following statements are equivalent:
a~.
Then the
{i) F has no fixed point in~; {ii) for some w E a~ there exists the angular limit L lim Z-+W
F(z)- w Z- W
:=a
with Rea=:::; 1; {iii) F admits the representation F(z) = z + (z- w) 2 wp(z) for some w E a~ andp E Hol(~,C) with Rep(z);:::: 0. Moreover, a in (ii) is actually a real number and the boundary points w in {ii) and (iii) are the same.
4. 7
Lower bounds for p-monotone functions
We have seen in previous sections that the asymptotic behavior of semigroups generated by holomorphic functions can be described in terms of their derivatives. If f E g Hol(~) and T E Null(!) then T is (globally) attractive if and only f'(r), the derivative off at T, lies strictly in the right half-plane. Moreover, f E QHol(~) has no null point in~ if and only if for some TEa~ the angular derivative L/'(r) = {3 exists (finitely) with {3 ~ 0. In addition, if S = {Ft}t;;::o is the flow generated by
f then
and the point TEa~ is a (globally) attractive sink point of 8 (even if {3 = 0). However, if Lf'(r) = {3 = 0 the latter formula does not help to establish a rate of convergence of the semigroup to its sink point. Note also that if f E QNp(~) is not holomorphic the characteristics of the derivatives are not relevant. Actually, we will show that for f E QHol(~) the number {3 = Lf'(r) is equal to (4.7.1) inf 2Ref(z)z*, zELl.
136
Chapter IV
where z
*
=
Z
1 -lzl 2
- -T 1- ZT
It turns out that even iff E (}Np(Ll) is not holomorphic, expression (4.7.1) can serve as a characterization of the asymptotic behavior of flows of p-nonexpansive mappings in Ll both in the cases of an interior stationary point and a boundary sink point. In this section we will mostly follow the material of [39]. For a fixed T E Ll, the closure of Ll, and an arbitrary z E Ll, we define a non-Euclidean 'distance' between z to T by the formula:
II- zrl2 Il 1- z 2
d,.(z) =
(4.7.2)
(1- a(z, r)),
where
(see Section 2.3).
Exercise 1. Show that the sets
s > 0,
E(r, s) = { z ELl: d,.(z) < s }, have the following geometric interpretation: (a) If TELl, then these sets are exactly the p-balls
E(r,s) centered at (b) If T
r E E
= B(r,r) = { z ELl: p(z,r)
0,
are horocycles in Ll which are internally tangent to the unit circle 8Ll at
T.
Now for fixed TELl and z E 8E(r,s) = { z ELl: d,.(z) = s }, s > 0, z =/:-
T,
consider the nonzero vector *
1
z = 1- a(z, r)
(
1-
1
lzl 2
1
z- 1 - zf
7
) ·
(4. 7.3)
Exercise 2 ([6]). Show that z* is a so called support functional of the smooth convex set E(r, s) at the point z E Ll, d,.(z) = s, i.e., for all w E E(r, s) the following inequality holds:
Re(wz*)
~
Re(zz*).
ASYMPTOTIC BEHAVIOR
137
In order to classify the asymptotic behavior of a flow generated by f E QNp(D.) for a point r E D. we consider two real nonnegative functions on (0, oo ):
wt>(s) :=
inf
2Ref(z)z*,
s > 0,
(4.7.4)
inf
2Ref(z)z*,
s > 0,
(4.7.5)
dT (z)~s
and
w"(s) :=
dT(z)=s
where z* is defined by (4.7.3). If TED. is a stationary (or sink) point for the flow generated by f E QNp(D.), then it follows that Re f(z )z* 2: 0 by p-monotonicity off (see Section 3.4). Hence
w"(s) 2: wt>(s) 2: 0 and wb(s) is clearly decreasing on (O,oo). Let M (0, oo) denote the class of all positive functions w on (0, oo) such that ~ is Riemann integrable on each closed interval [a, b] c (0, oo) and
J
ds
.. IS d1vergent.
w(s)s
o+
Note that for each wE M(O,oo) the function d7
O(s) :=
n defined by
(z}
J
d)..
(4.7.6)
w(>-.)>-.
s
is a strictly decreasing positive function on (O,dr(z)] and maps this interval onto [O,oo). We denote its inverse function by V: [O,oo) ~----> (O,dr(z)].
Definition 4.7.1 We will call a function w E M(O, oo) an appropriate lower bound for f E QNp(D.) if
w(s) ~ wU(s) =
inf dT(z)=s
2 Ref(z)z*,
s > 0.
Exercise 3. Show that if wt> defined by (4.7.4) is not zero then it is an appropriate lower bound for f E QNp(D.). Exercise 4. Let p E Hol(b.,C) be such that Rep(z) 2:0, zED. and L.lim(1- z)p(z) = z-+1
f3 2:0.
Show that iff E Q Hol(D.) is defined by the Berkson-Porta formula (see Section 3.6)
f(z) then for the sink point r
= 1 the
=
-(1- z) 2p(z),
function Wt> is constant which is equal to (3.
Chapter IV
138
Proposition 4.7.1 Let f E QNp(!::J.) be continuous and letS = {Ft}t~o be the flow generated by f. Given a point T E !::J. and a function w E ..A-1(0, oo), the following conditions are equivalent: {i) the function w is an appropriate lower bound for f; {ii} for any differentiable function W on [0, oo) such that V(t) ~ W(t), V(O) W(O) and V'(O) = W'(O), dr(Ft(z)) ~ W(t),
=
z E !::J., t;::: 0,
v
where = n-l and n is defined by (4. 7.6). In particular, dr ( Ft ( z)) ~ V (t); hence T is a globally attractive stationary point for S.
Proof. Consider the function
w: JR+ x !::J.
t--+
JR+ defined by
= dr(Ft(z)).
w(t, z)
(4.7.7)
By direct calculations we have
~~
(4.7.8)
lt=o+= -2\li(O,z) Ref(z)z*.
We first assume that condition (ii) holds. Since w(O, z) = dr(z) = W(O), we obtain by (4. 7.8) and (ii) that
d
aw
-at it=O+;::: - dt [W(t)Jt=O+
2\li(O, z) Ref(z)z*
d
1
- dt [V(t)Jt=O+ =- fl' (dr(z)) dr(z)w (dr(z)).
Varying z E 8E(T,s) = {z E !::J.: dr(z) = s} we see that the latter inequality immediately implies (i). Conversely, let condition (i) hold. It follows by (4.7.7) and the semigroup property that for all z E !::J. and s, t ;::: 0, w(s
+ t, z)
= w (s,
Hence by (4. 7.8) and the continuity we deduce from (i) and (4.7.8) that:
Ft(z)).
off, W is differentiable at each t ;::: 0 and
aw(t,z) tt at ~ -w(t, z)w (w(t, z)) ~ -w(t, z)w (w(t, z)).
Separating variables we obtain dw w(w)w = f2 (drFt(z));::: t.
ASYMPTOTIC BEHAVIOR
139
This is equivalent to condition (ii). Our assertion is proved. D As we mentioned above, if wb(s) := inf :S s 2 Re f(z)z* is positive then it may dT(z)
be used as an appropriate lower bound for
Example 1. Let
f
E QNp(l:i) (see Example 1).
f : 6. ~----+ O of holomorphic self-mappings of 6. and T = 1 is a sink point of S. If we now set • z 1 z = --2 1- lzl 1- z then we obtain
-
Ref(z)z* =
l1-zl 2 Il 1- z 2
1+zn 1+zn Re - - = d 1 (z) Re -1 > 0. 1- zn - zn
In addition, it can be shown (see Exercise 4 and Proposition 4.7.5 below) that w(s) = wb(s) =
inf
dl(z)~s
In this case
j
2 2 Ref(z)z* =constant=-. n
d1(z)
!l(s) =
n
2
d>..
n
= exp
(-~t)
T = -2
s ln d1(z)
s
and
V(t) =
n- 1
(t)
d1(z).
Thus we have an exponential rate of convergence of the flow S to the boundary point T = 1: 2 2 } 11- zl 2 -'IFt(z)----'--'-:--:::::-:--:-'11-:::-2 < exp { --t . 1 -1Ft(z)l n 1 -lzl 2 )rote also that although f has n + 1 null points {ak : k = 1, 2, ... , n + 1} on the unit circle, only a 1 = 1 is an attractive point of S = {Ft}t~o· The reason is that Ref'(al) > 0, while Ref'(ak) < 0, k = 2,3, ... ,n+ 1 (see Figure 4.4 for n = 3).
Remark 4.7.1. However, examples show (see Example 2 below) that sometimes wb may be zero identically, while wU itself belongs to the class M(O, oo ). Moreover,
we will see below that for a semigroup of holomorphic mappings with a boundary sink point T the function is always a constant which coincides with the angular
w,
Chapter IV
140
y \ \ \ \ \ \ ' ' ' ' ''\1/ '\\\/!/~~' ___ , ,
\\\J\,, ___ \ i } \ \
\ \
1 \ ,...,. ....,
It
I/~
(1/ / / /
I(
lh/~'\ /~/~,,\
i
/--~' '\ ~~'''\\
1/ll\,,'\\\
\\\\'\'\ \l!
Ill\\\~\\~
\ f.l/~·-,, '''''"\~ l '\\ ;,~-~ ' ' ' ' ' ' " I ,,,,/ -----~' I / .---.--" _.._..._..._..__/ / /, \ ' X /////// ~\
I'
///1//l
f{{1~~#~J}
\\\11 11//
,,...._ V / / / / /
-..::::~~rt'~///1
//I~,.._~/.?/
I
I l \ v-..__..//
Figure 4.4: The asymptotic behavior of the flow generated by 1 +z 3 f(z) = -(1 - z) 2 - - . 1- z 3 derivative off at r. Also observe that the same estimate as in Example 1 can be obtained by using Proposition 4.6.2. Nevertheless, even for holomorphic mappings Proposition 4.7.1 becomes an effective tool when the angular derivative L.f'(r) = 0.
Example 2. Let f : .6.
~
C be defined by
f(z) = -(1 - z)2 1 + czn 1- czn with lei< 1. Once again, if we define z* as in Example 1 we have
11- zl 2 1 + czn 1 -lei Rej(z)z* = 1 I 12 Re 1 2: d1(z)-1- 1 > 0. - z - czn 1+ c In this case L.f'(1)
= wb(s) = 0 for all
s E (0, oo) and we cannot use it as an
1-lcl
appropriate lower bound. However, we can define w(s) =as, where a= - -1- 1, 1+ c and we find d 1 (z)
n(s) =
~
a
j
s
dA =
>. 2
~ (~ a
s
1_).
__
d1(z)
Thus we obtain by Proposition 4.7.1 the following rate of nonexponential con-
ASYMPTOTIC BEHAVIOR
141
vergence:
11- Ft(z)i2 1 11- zl 2 2 1-IFt(z)l ::; 1+atdl(z)1-lzl 2 " The reason is that f E g Hol( D.) has no null point inside D. and the boundary sink point r = 1 is equal to zero.
wb ( s)
defined at
Next we consider the case when f E QNp(D.) is not holomorphic on .6.. In this situation the convergence of the flow generated by f may be of nonexponential type even f has an interior null point.
Example 3. Let z
= x + iy E .6.. Define f : D. ~--+ C by f(z) = x7f3
Since Re f(z)z
+ iy7f3.
= xl0/3 + yl0/3
~ 0,
f is p-monotone and the origin is the unique null point of f. Then, setting T = 0, we have lzl2 do(z) = 1-lzl2 and
2 1/3 82/3( 1 + s)l/3.
Since w~(s) E M(O,oo) we can set w(s)
= w~(s) and we have
do(z)
n(s)
=
I
I
do(z)
1 w(>.)).. = 21/3 d)..
s
d)..
)..5/3()..
+ 1)1/3
0
s
Inverting this function we obtain the estimate
do(z)
do(Ft(z))::; V(t) = 24/3
[ - - tdo(z) 213 3
3; 2
+ (do(z) + 1) 2/
The latter inequality is equivalent to the estimate
IFt(z)l ::;
lzl [
(2lzl)4/3
3
. ] 3/4
t
+1
] 3
-
d0 (z)
Chapter IV
142
Note that one can calculate Ft directly by solving the Cauchy problem and obtain
Thus for x
= y we obtain IFt(z)J =
JzJ . (2JzJ)4/3 ]3/4 [ 3 t +1
So the rate of (nonexponential) convergence we have obtained is sharp.
Remark 4. 7.2. We saw above (see Sections 4.4 and 4.5) that a similar phenomenon is impossible for holomorphic mappings: Namely, if a flow of holomorphic self-mappings converges locally uniformly to an interior stationary point then the convergence must be of exponential type. These examples and Proposition 4.7.1 above motivate the following definitions.
Definition 4.7.2 LetS = {Ft}t>o be a flow with a stationary (or sink) point E b.. We will say that the asymptotic behavior of S at T is of order not less than a> 0 if there is a function wE M(O, oo) such that
T
liminf {wl(/s)} > 0 s-->0+
and dr(Ft(z))
~
(1 +
S
(4.7.9)
"'
~w~dr(z)))"' dr(z)
(4.7.10)
for all z E b. and t 2: 0.
Definition 4. 7.3 We will say that the asymptotic behavior of S at T is of wexponential type if there is a decreasing function w E M(O, oo) such that dr(Ft(z)) ~ exp ( -tw(dr(z))) dr(z)
(4.7.11)
for all z E b. and t 2: 0. In particular, if w can be chosen to be a positive constant a then we will say that S has a global uniform rate of exponential convergence: dr(Ft(z)) ~ exp( -ta)dr(z).
(4.7.12)
The following assertion is a consequence of Proposition 4.7.1.
Proposition 4.7.2 LetS= {Fth::=:o be a flow generated by f E QNp(b.) with a null (or sink) point T E b.. Then the asymptotic behavior of S at T is of order not less than a > 0 if and only if there exists an appropriate lower bound w E M (0, oo) for f such that
w(s) . . sl/a zs decreaszng on (0, oo).
ASYMPTOTIC BEHAVIOR
143
Proof. We first observe that condition (4.7.10) with some w E M(O, oo) satisfying (4.7.9) is equivalent to the same condition with a function WI E M(O, oo) which satisfies both (4.7.9) and(**). Indeed, for a given wE M(O,oo) define a function 1-L : (0, oo) ~---+ (0, oo) by . {w(l) ~-t(s) = mf [1/a: l E (O,s] } ,
s > 0.
It is clear that ~-t(s) is decreasing. Setting now wi(s) = si/a · ~-t(s) we clearly see that WI satisfies (4.7.9) and that wi(s):::; w(s). Hence
J o+
is divergent and
WI
ds wi(s)s
E M(O, oo). Then the inequality
proves our claim. Thus we can assume for the rest of the proof that w satisfies (**). It remains to be shown that w is an appropriate lower bound for f. Indeed, defining n: (O,dr(z)]~-+ [O,oo) by (1.7) and using(**) we have
O(s)
Inverting this expression we obtain V(t) :=
n- I (t):::;
(
1 t ) dr(z) := W(t). 1 + ;:;w(dr(z))
It is clear that the function W(t) satisfies all the conditions of Proposition 4.7.1. This completes the proof of Proposition 4.7.2. D Corollary 4.7.1 LetS= {Ft}t~o be afiow generated by f E QNp(!l) with a null (or sink) point T E .6.. Then: (i) The asymptotic behavior of S at
T
is of w-exponential type if and only if
inf{w"(l): l E (O,sJ} > 0,
s > 0.
(4.7.13)
Chapter IV
144
(ii} The flow S has a global uniform rate of exponential convergence if and only if
(4.7.14) for some a> 0.
Indeed, in both cases (i) and (ii) there is one function wE M(O, oo) such that the asymptotic behavior of S at T is of order not less than a for all positive a. In case (i), w can be chosen to be w(s) := inf{ wU(l) : IE (0, s]}
> 0,
s
> 0,
while in case (ii) w can be chosen to be a constant a. The following example shows that for a semigroup of p-nonexpansive (but not holomorphic!) mappings an asymptotic behavior of w-exponential type does not imply, in general, a global uniform rate of exponential convergence.
Example 4. Define a continuous mapping f : D. f(x
~---+
C by the following formula:
+ iy) = x(1- x) 2 + iy(1- y) 2 .
Since Re f(z )z 2': 0 for all z = x + iy E D., it follows that f is a generator of a semigroup S = {Ft}t>O of p-nonexpansive mappings such that each disk
D.r = {z E C:
lzl < r < 1} ls Ft-invariant.
Setting
T
= 0 and z* = izi 2 ( 1 ~ izi 2 ),
we have
It is easy to see that lim wU(s) = 1 while wtt(s) S-->0+
example, y = 0 and x =
J
8
s+1
--+
--+
0 as s
---->
oo (take, for
1).
Remark 4.7.3 For holomorphic mappings, however, condition (4.7.14) holds for some a > 0 whenever condition (4.7.13) holds for a decreasing positive function w. In other words, for holomorphic flows any convergence of w-exponential type implies global uniform exponential convergence. To explain this phenomenon in terms of lower bounds we observe that for a holomorphic generator with an interior null point the function (4.7.15) is bounded from below by a positive number, while for a boundary sink point the function w17 is just a constant.
ASYMPTOTIC BEHAVIOR
145
In both cases (interior stationary point or boundary sink point) the asymptotic behavior of a flow generated by f E g Hol(A) is completely determined by the value wU(O) := liminf wU(s) which is related to the value of derivative off at its null s~o+
point (for the interior case) or the angular derivative (for the boundary case).
Proposition 4.7.3 Let f E QHol(A) and let {Ft}t~o be the flow generated by f. If for some point T E A there is a decreasing function w : (0, oo) ~----+ (0, oo) such that dr(Ft(z)) :S e-tw(dT(z)) dr(z), z E A, t ~ 0, (4.7.16)
then there exists a number p, > 0 such that (4.7.17)
Moreover, {i) ifr E A, then p, can be chosen asp,= wb(0)/4, but p, cannot be larger than wb(O) { = lim wl> ( s )); S-+0+
(ii) if T E oA, then the maximal p, for which (4. 7.17) holds is exactly wi>(O), that is 0 < p,:::; w•(O). The proof of this Proposition is based on the following two lemmata.
Lemma 4.7.1 Let f E QHol(A) with f(O) = 0, and let wl> and wU be defined by (4.1.4) and (4.1.5), respectively. Then:
{i) wU(O) = wi>(O) = 2 Ref'(O) := 2v; (ii) v/2:::; wl>(s):::; 2v. Proof. First we show that (4.7.18) where
v Since in our case
T
= Re f'(O).
= 0, we have Ref(z)z*
1
= lzl 2 ( 1 -lzl 2 ) Ref(z)z.
Now fixing ( E oA, we set z = r(, where r E (0, 1). Then we obtain -
Re f(z)z*
= Re - -1 , -1 f(r() · -(. 1- r- r
Therefore II
1
1
-
w•(s):::; 2 Re - - - f(r() · (, 1- r2 r
where
Letting s (hence, r) tend to zero we obtain
s
r 2 = lzl 2 = - - . s+1
Chapter IV
146
On the other hand, it follows by Harnack inequality that for all z E b.
This implies that 2 lzl2(1-lzl2) Ref(z)z
2 Ref(z)z*
>
2vlzl 2 1 - lzl lzl2(1 - lzl2) . 1 + lzl
2v (1 + lzl)2.
Hence w 0(s)
>
2v
inf
inf
d.,.(z)~s (1 + lzl) 2
2v
lzl 2 ~
.~1
2v (1 + lzl)2
( 2: v /2).
(4.7.19)
(1+ff+;)2
Letting s tend too+ in (4.7.19) we see that w,(O) 2: 2v. Comparing the latter inequality with (4.7.18) and (4.7.15) we obtain (i). On the other hand, substituting now in (4.7.19) v = w 0 (0)/2, we obtain assertion (ii) and we are done. 0 0,
Suppose now that QHol(b.) has a null point different from zero, say, f(r) = E b., T =/= 0. Note that if the automorphism Mr of b. is given by
T
T-Z
Mr(z)= -1 _, -ZT then the following equality holds: (4.7.20)
Now let us consider the flow
{Gt}t~o C
Hol(b.) defined by (4.7.21)
and let g E QHol(b.) be its generator, i.e.,
g(z) = - :tGt(z)lt=O+ = [(Mr)'(z)r 1 f(Mr(z)). Then Gt(O)
= 0 for
all t 2: 0 and g(O)
Lemma 4. 7.2 The functions wU ( s) and mations (4. 7.21} and (4. 7.22}.
(4.7.22)
= 0. W 0 ( s)
are invariant under the transfor-
ASYMPTOTIC BEHAVIOR
147
Proof. We have seen already in (4.7.7) and (4.7.8) that
~
vt
[dr(Ft(z))]
=
-2dr(z) Re f(z)z*
t=O+
= -2
dr(z) Re [/(z) ( z 2 - ~)]. 1 - lzl 1- zf" 1 - O"(z, r)
(4.7.23)
On the other hand, by (4.7.20) and (4.7.21) we have
(4.7.24) Since
do(w)
w
(
-~ 2 Re g(w) 1 -lwl2
)
dr(z) ( W ) (1 -lrl2)1wl2 2 Re g(w) 1 -lwl2 ' we obtain from (4. 7.24) and (4. 7.23) the required equality 1
[
1- O"(z, r) Re f(z)
(
z
f
1- lzl 2 - 1- zf
)]
w 2] · = lwl1 2 Re [ g(w) 1 -lwl
The Lemma is proved. D Let the flow {Gt ( z) h>o C Hol( 6.) and its generator g E Q Hol( 6.) be defined by (4.7.21) and (4.7.22). By Lemmata 4.7.2 and 4.7.1 we have
. mf do(w)=s
w,(O) v Reg(w)w* 2: - - = -. 4
2
Then by Corollary 4.7.1(ii) we have
do(Gt(w)):::; do(w)exp(v/2) Finally, setting w
=
for all wE 6..
Mr(z) and using (4.7.20) we conclude that (4.7.25)
This enable us to point out the following rates of convergence.
Proposition 4. 7.4 Let {Ft} t>o be a flow generated by f E Q Hol( 6.) and let f(r) = 0, T E 6.. Then the folTowing estimates are equivalent:
148
Chapter IV
(i)
z E ~' t 2: 0;
(ii)
z E ~' t 2: 0,
(iii)
where the numbers J..L in (i) and (ii) can be chosen to be one and the same such that 0 ~ v /2 ~ J..L ~ 2v and v in (iii) is defined by v
= ~wi>(O) = ~wU(o) = Re/'(T).
(4.7.26)
Proof. First we note that inequalities (ii) and (iii) are equivalent to the following ones: (ii*) (iii*)
IGt(w)l ::::; lwl· exp ( -J..L 1 IGt(w)l
~wl 2
t),
~ lwl· exp ( -v~ ~ ::Jt),
t 2: 0,
t 2:0,
where w = Mr(z) E ~ and the flow {Gt}t;::::o is defined by (4.7.21). First let us suppose that estimate (i) holds. By using (4.7.20) for this flow we have ~
do (Gt(w))
do(w) exp ( -tJ..L).
Rewriting the latter inequality in the form
we obtain by direct calculations
IGt(w)l 2 ~lwl 2 l w l2 + ( 1 -
1
1w 12
)
(
exp -tJ..L
)
~lwl 2 ·exp(-tJ..L(1-Iwl 2 )).
which coincides with (ii*). Now we will assume that inequality (ii) (and hence (ii*)) holds. Differentiating both sides of this inequality with respect to t at t = o+ we obtain 1
-~ Reg(w)w ~ -lwiJ..L
1 -lwl 2 . 2
(4.7.27)
This implies that w~(s) ~ J..L. Thus the function w(s) = J..L is an appropriate lower bound and the implication (ii) => (i) follows by Proposition 4.7.2. Let us suppose now again that inequality (ii) (hence, (ii*) and (4.7.27)) holds with some number J..L > 0. Setting in (4.7.27) w = r(, ( E 8~, r E (0, 1) and
ASYMPTOTIC BEHAVIOR
149
letting r to zero (cf., the proof of Lemma 4.7.1) we obtain Reg'(O) 2: 11/2 > 0. A direct calculation shows that
g'(o)
= [(M,.)'(o)r 1 J'(r)
(M.,.)' (O)
= J'(r).
and so v > 0. Therefore, again by Harnack inequality, we have
1-lwl 1+ lwl -
1-lwl 1 + lwl
> Reg'(O)-- > vlwl 2- - .
Reg(w)w
-
On the other hand, 8ln IGt(w)l
8t
1 [ -] = -IGt(w)i2 Re g(Gt(w))Gt(w) .
Also it follows by the Schwarz Lemma that 8ln IGt(w)l
IGt(w)l :s; lwl.
Thus we have
1 -lwl
----'-::--'--'----'-'- 0. So one can set w(s) = 'Y/2 and the asymptotic behavior of Sat Tis seen to be of exponential type. In addition, it follows by Proposition 4.6.2 that 'Y:::; {3 := L.j'(T). Finally, we observe that lim Re f(sr)(sr)*
s--->1-
lim Re f (sr) ( r--->1-
sf' 1- S 2
-
lim Ref(sr)T' _1_ = s-1 1+s
s--->1-
and the assertion is proved. D
1 -T S )
!3/ 2,
Chapter 5
Dynamical approach to starlike and spirallike functions This chapter is devoted to showing some relationships between semigroups and the geometry of domains in the complex plane. Mostly we will study those univalent (one-to-one correspondence) functions on the unit disk whose images are starshaped or spiralshaped domains. Several important aspects, however, had to be omitted, e.g. convex and close-to-convex functions (see, for example, [57, 55]), and other different classes of univalent functions. We have selected the forthcoming material according to the guiding principle that the demonstrated methods may be generalized to higher dimensions. For example, the celebrated Koebe One Quarter Theorem states that the image of a univalent function h on ~ normalized by the condition h(O) = 0 and h'(O) = 1 contains a disk of radius This theorem is no longer true at higher dimensions. Nevertheless, the dynamical approach analogues of the Koebe theorem have been recently established and used for subclasses of starlike (or spirallike) functions (see, for example [141, 109, 26, 56, 14]).
t·
Our second objective is to study the dynamics of starshaped (or spiralshaped) domains when the origin is pushed out to the boundary. For example, a domain which is starshaped with respect to a point may fail to be starshaped with respect to another point. We will study inter alia some unified conditions which describe starlike (or spirallike) functions which are independent of the location of their null points.
153
Chapter V
154
5.1
Generators on biholomorphically equivalent domains
Although the studies in the previous chapters were carried out on the unit disk, one can translate them to any simply connected domain (which differs from C) of the plane as the consequence of the Riemann Mapping Theorem. First we recall some notions and definitions in classical function theory. Definition 5.1.1 Let D be a domain in C. A function h E Hol(D, C) is said to be univalent on D if for each pair of distinct points
h(zl)
z1
and
z2
in D we have
=f. h(z2).
The set of all univalent functions in a domain D C C will be denoted by Univ(D). For h E Univ(D) one can define the inverse mapping h- 1 : n ~--+ D, where n = h(D). The content of the Open Mapping Theorem (see, for example, [122]) is that n = h(D) is also a domain (open connected subset) in C. In addition, h- 1 E Hol(O, D). In other words, h is one-to-one and h- 1 is also holomorphic on h(D). In this case f is also called a (globally) biholomorphic mapping on D. A mapping hE Hol(D, C) is said to be locally biholomorphic on D if for each point z E D there is a neighborhood V CD, of this point such that hE Univ(V). It is well known that h E Hol(D, C) is locally biholomorphic on D if and only if h'(z) =f. 0 everywhere (see, for example, [122, 128]). Two domains D and n in C are called biholomorphically (or conformally) equivalent if there exists hE Univ(D) such that 0 = f(D). The fundamental Riemann Mapping Theorem states that every simply connected domain n in c (but not c itself) is biholomorphically equivalent to the open unit disk 6. in C. Moreover, for each a E 0 there is a unique h E Univ(6.) with 0 = f(6.) such that h(O) =a and h'(O) > 0.
For the special case when D = 6. is the open unit disk in C, the subset of Univ(D) normalized by the conditions h(O)
= 0 and
h'(O)
=1
will be denoted by S(6.). This notation conforms to the one used in the classical geometric function theory. In this case we simply write S (= S(6.)) ={hE Univ(6.): h(O) = 0 and h'(O) = 1}.
In other words, the class S C Hol(6., C) consists of all the mappings h E Univ(6.) such that h has the following Taylor series at the origin: 00
h(z) = z + Lakzk. k=2
STARLIKE AND SPIRALLIKE FUNCTIONS
155
Thus, referring to some geometrical properties of a simply connected domain 0 containing the origin, if we are permitted to shrink or expand it we can find a function hE S (= S(~)) for which the domain fi = h(~) is similar to fl. (Of course, we can translate a domain, if necessary, so that the origin would be its interior point). However, in this way we may sometimes loose some features of the dynamical transformation of a domain n if its geometrical characteristics are related to a certain given fixed point in C. In particular, it happens if such a point lies on the boundary of n. The following simple assertion is the key to our further considerations.
Proposition 5.1.1 (Main Lemma) Let D and 0 be two domains in C, such that 0 = h(D) for some biholomorphic (univalent mapping) h. Then there is a linear invertible operator T on the space Hol(O, q onto the space Hol(D, C), which takes the set QHol(O) C Hol(O,C) onto the set QHol(D) (i.e., QHol(D) = T(QHol(O)). Moreover, such an operatorT: QHol(O) ............ GHol(D) can be given by the formula: (5.1.1) T(t(h(z)). d 0,
z E b.}.
(5.2.3)
In 1978 Wald [150] characterized starlike functions with respect to another center. Using our notions, his result can be reformulated in the following way. Proposition 5.2.2 Suppose that h E Hol(b., C) is either of the form h(z) = z + 2:::;:'= 2 akzk or of the form h(z) = 2::::::'= 1 bkzk with h'(7) = 1 for some 7 E b.. Then the function h(z)- h(7) belongs to Star( b., 7) if and only if Req(z) > 0, where h(z) q(z) = h'(z) (z -7)(1- zr)'
z E b.\ {7 },
and
We will see below that Proposition 5.2.2 (as well as Proposition 5.2.1) can be easily obtained by using a different approach in more general settings. Different applications of this assertion are presented in Wald's thesis [150] (see also [57]). Definition 5.2.2 A set n in C is said to be spiralshaped (with respect to the origin) if there is a number f-L E C with Re 11 > 0 such that for each w E n and t ~ 0 the point e-tp.w also belongs to n. (see Figure 5.3} If D is a domain in C, then a univalent function h E Hol(D, C) is said to be spirallike on D if the closure of its imagen = h(D) is a spiralshaped set. Once again, if the origin belongs to n then we will say that h is a spirallike function on D with respect to an interior point. Otherwise (if the origin belongs to the boundary of n) we say that his spirallike with respect to a boundary point.
an
The set of all biholomorphic mappings on D which are spirallike on D will be denoted by Spiral(D). If, in addition, there is 7 E D such that h(7) = 0,
then we will write h E Spiral(D, 7). Note also that if in Definition 5.2.2 the number f-L is a real positive number, then n is actually starshaped, i.e., Star(D) c Spiral( D). Consequently, Star(D, 7) C Spiral(D, 7). If h E Spiral(D) has no null point in D (i.e., h is spirallike with respect to a boundary point) we will write that h E Snail(D). (We will see below that the class Snail(D) is quite narrow: in particular, each h E Univ(b.) which has a biholomorphic extension to b. is, in fact, in Fan(b.), i.e., h(b.) is starshaped).
Chapter V
160
Figure 5.3: The spiralshaped domain (0 E of!): e-t1-1w E f! for each w E fl. It seems that the first occurrence of the class Spiral(6., 0) = { h E Spiral(6.) : h(O) = 0} appeared when condition (5.2.2) was modified analytically by inserting the factor ei 8: Re [eiO zh'(z)] > 0 (5.2.4) h(z) (see Montel [101] and Spacek [137]). Actually, this definition is compatible with our Definition 5.2.2 (see also, Remark 5.2.2 below) Proposition 5.2.3 LethE Hol(6., C) have the form (X)
h(z)
= z + Lakzk
(5.2.5)
k=2
(i.e., h(O) = 0 and h'(O) = 1). Then h E Spiral(6., 0) if and only if condition (5.2.4) holds for some() E (1rj2, 7r/2). Of course, Proposition 5.2.1 follows from Proposition 5.2.3 if we set there()= 0. To prove the latter proposition as well as Proposition 5.2.2 we will need another more general assertion which is a direct consequence of the Main Lemma (Proposition 5 .1.1). Proposition 5.2.4 Let h E Hol(D, C) be a spirallike (respectively, starlike) function on D. Then there exists a generator f E g Hoi( D) of a continuous flow on D such that h satisfies the following differential equation:
J.Lh(z)
= h'(z)f (z),
zED,
(5.2.6)
for some J.L E C (respectively, J.L E IR) with Re J.L > 0. Proof. Let fl = h(D) be a spiralshaped (respectively, starshaped) domain. Then for some J.L E C (respectively, J.L E IR) with Re J.L > 0 the mapping
STARLIKE AND SPIRALLIKE FUNCTIONS
161
g = J.d, (g(z) = J-Lz), belongs to QHol(f2). Indeed, in this case the integral curve defined by the Cauchy problem {
av(t,w) av(t,w) at + g(v(t, w)) = at v(O,w) = w, wEn,
+ J-LV(t, w)
= 0,
(5.2.7)
has the form {v(t, w) = e-J.Ltw, t ~ 0, wE f2} and belongs to f2 by Definition 5.2.2 (respectively, Definition 5.2.1). Then Proposition 5.1.1 implies that f E Hol(D, C) defined by: (5.2.8) f(z) = (Tg)(z) = J-L [h'(z)r 1 h(z) is a generator of a flow on D, and we are done. D To establish a converse assertion we may omit the requirement on h to be univalent.
Proposition 5.2.5 Suppose that h E Hol(D) satisfies equation (5.2.6}, with J-L E C, (respectively, J-L E IR) ReJ-L > 0, and f E QHol(D). Then the set n = h(D)
is spiralshaped (respectively, starshaped}. Moreover, if D is bounded and h has a null point TED with h'(T) =1- 0, then h is univalent, hence spirallike (respectively, starlike) with respect to an interior point. Proof. Fix wE f2 and find zED such that h(z) = w. Since f E QHol(D) generates a continuous flow s, = {Ft(z)}t>o on D, setting v(t, z) = h (Ft(z)), we have that v(t, z) E n for all t ~ 0. That is for a fixed z E D, the family {v(t,z), t ~ 0} determines a continuous curve inn, such that v(O,z) = w. In addition, it follows by the chain rule that
av~~ z) = h' (Ft(z)) a~~z) = -h' (Ft(z)) f
(Ft(z)).
On account of formula (5.2.6) we obtain that
av(t,z) at = -J-Lh(Ft(z)) = -J-Lv(t, z).
(5.2.9)
Integrating the latter relation with the initial data
v(O,z) = w
(5.2.10)
we obtain
v(t, z) = e- 11 tw E fl
for all t ~ 0.
Since w E f2 was arbitrary the first assertion of our proposition follows. Assume now, in addition, that h has a null point T E D and h'(T) =1- 0. This implies that there is a neighborhood U C D of the point T such that h is univalent on U and V = h(U) C n is a neighborhood of the origin. We want to show that, in fact, h is univalent on the whole of D. Indeed, assuming the contrary, suppose that for some w E n there are two distinct points Zl and Z2 in D such that h(zl) = h(z 2 ) = w.
Chapter V
162
Observe also that, as a result of equation (5.2.6), the conditions h(r) = 0 and h'(r) # 0 imply that f(r) = 0 and f'(r) = J.L with ReJ.L > 0. Since Dis bounded the family {Ft (·)} t>o is a normal family in D. Then exactly as in Proposition 4.4.1 one can conclUde that T E D is an attractive stationary point of the flow Sf = {Ft(·)}t>o generated by f. In particular, we have that Ft(zi) and Ft(z2) converge toT as t----> 00. Note that Ft(ZI) and Ft(Z2) are different for all t ~ 0. As above, now define v(t, z) = h (Ft(z)), t ~ 0, z E D and choose to > 0 such that Ft(z 1 ) and Ft(z2) belong to U for all t ~to. Then for such t the curves v(t, z;), i = 1, 2, lie in V = h(U) C n. But v(O, z 1 ) = v(O, z2) = w and we have that v(t, z;), i = 1, 2, are the same as the solutions of the differential equation (5.2.9) with the same initial data (S.2.10). Consequently Ft(zi) = Ft(z2) for all t ~to. That is a contradiction. D Sometimes we will say that h E Univ(D, C) is J.L-spirallike if it satisfies equation (5.2.6) with J.L E C, Re J.L > 0, and f E Q Hol(D). Remark 5.2.1. Thus h E Spiral(D) is spirallike with respect to an interior point (that is h E Spiral(D, r) for some T E D) if and only if the generator f in (5.2.6) vanishes at T E D. Moreover, iff is defined, then J.L = f'(r), and Tis an attractive stationary point of the flow S 1 = {Ft (·)} t>O generated by f. In fact, it can be shown (see Section 5.7) that for each f E Q-Hol(D), normalized by the conditions f(r) = 0 andRe f'(r) > 0 there is a unique solution h E Spiral(D, r) of the equation (5.2.6) with J.L = f'(r) normalized by h'(r) = 1. In addition, if J.L is a purely real number, then h defined by (5.2.6) is a starlike function on D. Similarly, we can say that h E Snail(D) if and only iff has no null point in D. In this case the flow S 1 = { Ft (·)} t>o generated by f for each z E D converges to a boundary point T E aD. Since for the special (but most important) case when D = b. is the unit disk we have a complete description of the class Q Hol(b.), the proved propositions imply several corollaries. In particular, applying the Berkson-Porta parametric representation of the class Q Hol(b.) we obtain the following: Corollary 5.2.1 Let h : b. ~---> C be a univalent holomorphic function on b.. Then h(b.) is spiralshaped if and only if the following equation is fulfilled:
J.Lh(z) = h'(z)(z- r)(1- zf)p(z), where T E b., J.L E C with ReJ.L z E b..
>
z E b.,
(5.2.11)
0, andp E Hol(b.,C) with Rep(z) ~ 0 for all
Remark 5.2.2 Thus, if T E b., then h E Spiral( b., r) (i.e., is spirallike with respect to an interior point); if T E 86, then h E Snail(b.) (i.e., spirallike with respect to a boundary point). Separating these two cases we conclude: A locally biholomorphic function h on ~ belongs to Spiral(~, r) if and only if there exist T E b. and J.L E C with Re J.L > 0 such that
R h'(z)(z- r)(l- zf) 0 e J.Lh(z) > '
A
z Eo.
STARLIKE AND SPIRALLIKE FUNCTIONS
163
Indeed, equation (5.2.11) can be rewritten in the form
Re h'(z)(z- 7)(1 - zf) 11h(z)
= Re _1_ > 0 p(z) -
If 7 E b., then differentiating (5.2.11) at
z =
z E b.. '
(5.2.11')
7 we obtain p(7)
which means that inequality (5.2.11') is actually strict. If we normalize 11 by the condition 1111 = 1, we obtain that T and p are uniquely determined by h. Of course, the converse assertion is also true. Moreover, if 7 = 0 then setting 0 = - arg 11 we obtain a description of the set Spiral( b., 0) which coincides with the classical description (Proposition 5.2.3). Letting 11 = 1 and 7 = 0 in (5.2.11') we arrive immediately at Nevanlinna's condition (Proposition 5.2.1) and Wald's condition if 7 E b., 7 ::f. 0 (Proposition 5.2.2). Similarly, we conclude from (5.2.11): A univalent function h on b. belongs to Snail( b.) (respectively, Fan( b.)) if and only if for some T E {)b. and 11 E 0. Contradiction. So T = 1. Once again, since J.Lh(z) =
(;~~) h'(z)(z- 1)
we obtain J1. = L lim ( f(z)) Q(z) = (3L lim Q(z), z-+1 z - 1 z-+1
and our assertion follows. 0 Thus the class of properly spirallike (i.e., not starlike) functions with respect to a boundary point contains neither conformal nor isogonal mappings at this boundary point.
Remark.5.3.1. We will show below (see Section 5.6) that the number v = L lim Q(z) yields an important geometrical characteristic of hE Fan(.D.), namely, Z-+'T
the angle (} = 1rv is the smallest one such that h(.D.) lies in the wedge of the angle 0. Thus, in fact, for h E Fan(.D.) with lim h(r) = 0 we will show that r-+l-
L lim Q(z)::; 2. Z-+'T To do this we need an approximation process which is based on Hummel's representation of the class Star( .Do, ·) of starlike functions with respect to interior points. We will give this representation in Section 5.5.
5.4
An invariance property and approximation problems
The following question naturally arises in approximation function theory: given h E Hol(.D., C) such that n = h(.D.) is spiralshaped (respectively, starshaped) and a sequence nn c n spiralshaped (starshaped) domains such that nn = n, find the sequence hn such that lim hn(z) = h(z) for each z E .Do, and hn(.D.) = !1n. n->oo Theoretically, under certain conditions this problem can be solved because of the Riemann Mapping Theorem and Caratheodory's Kernel Convergence Theorem.
u
STARLIKE AND SPIRALLIKE FUNCTIONS
167
However, in general, to find such a sequence implicitly has not seemed plausible, even the sequence nn is well described. At the same time one can define, in a sense, a dual approximation problem which is related to some invariance property of spirallike (respectively, starlike) functions. Namely, if h E Spiral(D.) (respectively, Star(D.)), find a 'nice' sequence of domains Dn in D. such that UDn = D. and h continue to be spirallike (respectively, starlike) on each Dn. In view of Propositions 5.2.4 and 5.2.5 an answer to the above question is provided by the following observation. If h E Spiral(D.) (respectively, Star(D.)), then for some J1. E C (respectively, J1. E IR) with ReJ-L > 0 the function f(z) = J-Lh'(z)[h(z)]- 1 is a semi-complete vector field on D.. Then these propositions yield that h continues to be spirallike (respectively, starlike) on a domain D C D. if and only if this domain is invariant for the semigroup S = {Ft}t~O generated by f. Thus, due to Proposition 5.2.4 we can formulate the following assertion: Proposition 5.4.1 Let h E Spiral(D.) (respectively, h E Star(D.)). Then there is a unique point TED. such that for all K > 1- ITI 2 the sets h(D(T, K)), where D(T, K) =
zfl 2 } {zED.: 111 -lzl 2 < K
,
are spiralshaped (respectively, starshaped).
We recall that forTE D. the sets D(T,K) ={zED.:
Z-T
1----1 < 1- TZ
r}, K =
(1 -ITI 2 )(1- r 2 )- 1 (see Section 1.1, Exercise 6) are pseudo-hyperbolic balls in D. while for T E 8D. these sets are horocycles internally tangent to 8D. at T. In particular, forT= 0 we have that hE Spiral( D., 0) (respectively, Star( D., 0)) is spirallike (respectively, starlike) on each disk D.r = {lzl < r, 0 < r ~ 1} concentric with D.. (Note that for the functions of the class
S* ={hE Star(D.): h(O) = 0 and h'(O) = 1} this result was obtained independently by Takahashi and Seidel [18] as an extension to Nevanlinna's theorem (see Proposition 5.2.1). A simple proof of this fact with the use of the Schwarz Lemma can be found in [33]). One may expect that even when T i- 0 and h E Spiral( D., T) (respectively, h E Star( D.)) on the unit disk, then for at least r close enough to 1 it continues to be spirallike (starlike) on the disks D.r. However, examples show that in general this conjecture has been answered in the negative.
Example 1. Consider the function h( z) = h 0 ( M 1 ( z)), z E D., where 2
z
ho(z) = (1- z)2
168
Chapter V
is the so called Koebe function and
l-z M1;2(z) = -2- 11- 2 z is a Mobius involution transformation of the unit disk taking the origin to T = ~. It is easy to see that the Koebe function belongs to Star(b.,O) (evenS*) and it follows that hE Star(b., T), because of the relations h(b.) = h0 (b.), h(T) = 0. We claim that there is a sequence {zn} C b., lznl----+ 1 such that
R [ zh'(zn)] O e h(zn) < · Indeed, calculating
Re [zh'(z)] h(z)
=
~ Re z(1- z)(1 + z)(z- !)(1- !z) 4 11 + zl 2 1z- !1 2 11- ~zl 2
and setting z = x + iy we obtain (after several technical manipulations) that the numerator N(z) of the right hand side of the latter equation has the form
Now it can easily he seen that for any sequence {zn} E b., lznl ----+ 1 such that D = { z = x + iy : (x-! )(x 2 + y 2 - 2x) > 0} the expression N(z) is negative (see Figure 5.4). Thus the claimed assertion is proved.
Zn
E
Figure 5.4: The set D.
STARLIKE AND SPIRALLIKE FUNCTIONS
169
Nevertheless, it has been shown in [66] that if T is close enough to zero, then the answer to the above question (concerning starlike functions) is affirmative.
Proposition 5.4.2 Let h E Star(~, T) with ITI < 2- J3. Then there exists c: > 0 such that for each 1 - c: < r ~ 1 the function h belongs to Star( ~r). By using the results in Sections 3.6 and 4.5 we can formulate some sufficient conditions for h E Spiral(~, T) (respectively, Star(~, T)) to be spirallike (respectively, starlike) on each ~r for r close enough to 1, which do not depend on the location of T E ~We will say that a function h E Hoi(~, C) is strongly spirallike if for some f.Lh(z)z f.L E 0 the function f ( z) := h' ( z) is strongly p-monotone on ~.
Proposition 5.4.3 Let h E Hol(~, q be a strongly spirallike function. Then there is T E ~ and e > 0 such that for each 1- e < r ~ 1, hE Spiral(~r) and h(T)
= 0.
In particular, by using a criterion of strong p-monotonicity we have the following:
Corollary 5.4.1 Let a be a real continuous function on [0, 1] such that a(1) > 0 and let h E Hol(~, C) be such that for some f.L E 0 the following condition holds f.Lh(z)z Re h'(z) ;::: a(lzl)lzl, Then there exists T E
~
and c:
hE
> 0 such that for each 1 - c: < r
Spiral(~r)
Exercise 1. Let f(z) = az 2
-
z E ~-
and h(T)
~
1
= 0.
a+ bz for some positive a and b.
(i) Show that if equation (5.2.6) is solvable for some f.L E 0, then f.L is, in fact, a positive real number, that is, the solution h of (5.2.6) is, actually, a starlike function on ~- Find f.L. (ii) Show that there exists c: > 0 such that for each r E (1-e, 1], h E Star( ~r ). (iii) Set a = 2 and b = 3. Show that e in (ii) can be chosen arbitrarily in the interval [0, ~).
1- cz
= az 2 - a+ b z - 1 +cz Reb> 0 and lei < 1.
Exercise 2. Let f(z)
with some complex numbers a, b
and c such that (i) Show that there are /-L E 0 such that equation (5.2.6) has a solution hE Spiral(~r, T) for some T E ~and each r E (1- e, 1]. (ii) Find relations between a, b and c such that f.L is real, i.e., h is starlike. (iii) Setting a= 0 and b > 0, show that h defined by (5.2.6) with an appropriate f.L
> 0 is starlike of order a (see Section 5.2) with a
1-lcl . 1 + 1c 1
= b - --
Chapter V
170
Finally, note that if hE Hol(A, C) is known to be in Spiral( A), then sometimes information on values of h and its first and second derivatives at only one point may be useful for solving the dual approximation problem on disks Ar concentric with A. Indeed, using Corollary 4.5.1 and Remark 4.5.2 we can easily arrive at the following sufficient condition: Proposition 5.4.4 Let h E Spiral(A) (respectively, Star(A)), satisfy equation (5.3.6) with some f. L E C (respectively, f. L E IR) with Ref..L > 0 and f E Hol(A,C). Then . h" (O)h(O) (z) Re f. L [h'(0)]2 < Re f..L,"
. . . h(O) [ h" (O)h(O)] (n) if 41 h'(O) I< 1- [h'(0)]2 then for some
E
> 0 and each r E (1- E, 1]
the function hE Spiral(Ar) (respectively, Star(Ar)). Remark 5.4.1. Of course, the problem above is not relevant for the class Fan(A). Indeed, it is clear that if h E Star(A) has no null points in A then there is no disk Ar concentric with A such that h( Ar) is starshaped. In this case another approximation problem arises: given h E Fan(A); find a sequence hn E Star( A, ·) of starlike functions with interior null points such that hn converge to h uniformly on compact subsets of A as n---+ oo. It looks as if the following procedure should work in solving this problem. If, for example, hE Fan(A) is isogonal at its boundary null point, say T = 1, then it satisfies the equation f..Lh(z) = h'(z)f (z), z E A, (5.4.1) for some
f
E
9 Hol(A): f(z) = -q(z)(1- z) 2 ,
with Req(z) > 0, z E A, and f. L = L.f'(1) > 0. Define an approximation sequence fn to f by using the Berkson-Porta representation formula fn(z) = (z- Tn)(1 fnz)q(z), where {rn} C A is any sequence which converges to 1 as n tends to oo. If, in addition, we can choose this sequence such that the numbers f~ (T n) are real, then we may try to solve the equations
f..Lnhn(z)
= h~(z)fn(z),
z E A,
(5.4.2)
with f..Ln = f'(rn) in order to define a sequence of univalent functions hn which are starlike with respect to interior points (h(rn) = 0). However, in this way there is a risk that hn may not converge to h. Indeed, putting it otherwise, we obtain that the numbers f..Ln = f~(rn) = (1 -lrnl 2 )q(rn) must converge to
f. L = L.lim f'(z) = L. lim f(z) = L.lim(1- z)q(z). z-+1
On the other hand, if T n achieves
Z--+ l
T
Z -
1
Z--+ 1
= 1 along the real axis we have
STARLIKE AND SPIRALLIKE FUNCTIONS
171
That is a contradiction. Also it is easy to construct a counter example of the latter relation.
Example 2. Hoi(~,
q
Consider a semi-complete (even complete) vector field defined as follows:
f(z)
= z2 -
f
E
1.
There are two boundary null points z = 1 and z = -1 of f. Since f' (1) = 2 > 0 it follows by Proposition 4.4.1 (a continuous version of the Julia-WolffCaratheodory Theorem) that z = 1 is a sink point of the semigroup generated by f. Therefore, one can present f by the Berkson-Porta formula:
f(z)
=
21 -(1- z) -
+z
1-z
with q(z) = 1 + z. Obviously, q admits real values if and only if z
1-z
So, we have to choose a sequence {Tn} E ~. Tn procedure to be real. Further, if we define
fn(z) we obtain fn(Tn)
= 0,
=
--->
E
~
are real.
1, in the above approximation
1+z (z- Tn)(1- Tnz)--, 1-z
and fn(z)---> f(z)
in~.
while
As a matter of fact, it can be shown that if hn are solutions of (5.4.2) normalized by the condition hn(O) = 1, then they converge to a fanlike function h which is a power of h defined by (5.2.1). However, in general, we do not know whether the numbers JLn are real (i.e., whether the functions hn are starlike). Thus a procedure of using the Berkson-Porta multiplier (z- Tn)(1- Tnz) has been shown to be inappropriate for the approximation of starlike functions with respect to a boundary point by starlike functions with respect to interior points. Nevertheless, it turns out that a modification of this multiplier in the spirit of J.A. Hummel makes such a procedure very effective. Hummel's multiplier has been used by A. Lyzzaik to prove a conjecture of M.S. Robertson on a description of starlike functions with respect to a boundary point, the images of which lie in a half plane. The next section is devoted to this approach.
Chapter V
172
5.5
Hummel's multiplier and parametric representations of star like functions
Hummel's multiplier is a (meromorphic) function of the form
H ( ) _ (z- r)(1 - zf) T Z ' z
z E
A,
where T is a given complex number with JrJ :::; 1. For the case of JrJ < 1, J.A. Hummel showed in [65] and [67] that this function plays a special role in the study of starlike functions. In fact, it turns out that by the multiplication operation this function translates the set Star( A, 0) onto the set Star( A). Moreover, if T E A then Star( A, 0) is translated onto Star( A, r). More precisely: Proposition 5.5.1 Let h E Hol(A, q, h(O) = 0, and g E Hol(A, q be two functions related by the formula
g(z) = HT(z) · h(z).
(5.5.1)
Then g(A) is starshaped if and only h(A) is starshaped.
In this section we will treat only the case when T E A. In this situation it is more convenient (for some symmetry) to consider the meromorphic function WT on A defined by
W ( )= 1 H ( )= 1 (z- r)(1 - zf) Tz 1 - JrJ2 T z 1 - JrJ2 z '
z EA.
(5.5.2)
Of course, it is sufficient to prove Proposition 5.5.1 replacing HT in (5.5.2) by WT. Moreover, in this case the inverse translation is just the same multiplier composed with a Mobius involution transformation.
Proposition 5.5.2 LethE Hol(A, q and g E Hol(A, 0.
(1 - ei'Pr)(1 - ei'Pf) 1-lrl2
(ii) By direct calculations we have: 1 [(1 - 2::) (1 1 -lrl 2 z
w~(z)
=
1 1 -lrl2
zr)]
[~(1-zf)-(1-2::)r] z2 z
1 [r lrl 2 _ lrl 2] 1 ( r -) 1 - lrl2 z2 - ----;- - r + ----;- = 1 - lrl2 z2 - r .
Again setting z = ei'P we obtain: Re zlll~(z)
Wr(z)
=
Re
e2 i'P (~-f) r- fe 2i'P . . =Re 2 (e''P- r)(1- e''Pr) ei'P 11- e-i<prl 1
- - - .-----,. 2 •
11- e-•'Prl
(iii) Substitute Mr(z)
Wr(Mr(z))
. . Re( re-•'P - fe''P)
= 0.
= r- z_ to Wr instead of z. We obtain: 1- ZT
=
(1- zf) . ( ~ -
1-lrl 2 (r- z- r
r) (1- ~-f) r- z
+
zlrl 2 )(1-
zf
+ zf-
(1 -lrl 2)(r- z)(1- zf) -z(1 - lrl 2) 1
(r- z)(1- zf)
Wr(z)"
lrl 2)
Cbapter V
174 The lemma is proved. D
Following J.A. Hummel [65, 67] we will say that a meromorphic function g on satisfies the condition (A) if for each c > 0 there is p E (0, 1) such that for all z in the annulus p < izi < 1 the following condition holds:
~
zg'(z) Re g(z) > Star(~,
Lemma 5.5.2 Let g E (A).
(5.5.4)
-£.
T) for some T E
~-
Then g satisfies condition
Proof. Since g(~) is starshaped and the origin is an interior point of the function g satisfies the equation:
g(~),
g(z) = g'(z)(z- T)(1- z'f)p(z) with some T E ~.such that g( T) = 0 and p E Then by Lemma 5.5.1(iii) we obtain: ( 1 -ITI
2
).
z · g'(z)
g(z)
Hol(~,
q
with Rep> 0 everywhere.
1
(5.5.5)
= lllr(z) · p(z)
Denoting w = Mr(z) we have in turn from (5.5.4):
z · g'(z) g(z)
(5.5.6)
= Wr(w) · q(w),
1
= (1 -ITI 2 )p(Mr(w)) (recall, that Mr is an involution). Req(w) > 0 for all wE~- Now for each wE~ the right
where q(w)
Hence (5.5.5) can be presented as:
Ill ( ) . ( ) = (w- T)(1- w'f) q(w) = f(w) r w qw (1 - jTj2)w w ,
hand side of
(5.5.7)
(w- T)(1- w'f) q(w) (1 _ ITI 2 ) defines a semi-complete vector field by the Berkson-Porta representation. At the same time, it follows by Proposition 3.4.4 that f satisfies the condition: where f(z)
=
Ref(w) · w;::: Ref(O)w(1-lwl 2 ). Therefore for each c > 0 one can find PI E (0, 1) such that:
f(w) 1 Re - - = - Ref(w)w > w lwl2 -
-£
,
(5.5.8)
whenever PI < lwl < 1. Note now, that if w runs in the annulus PI < lwl < 1, then z = Mr(w) runs in the set A=~\ B(T, r) where B(T, r) C ~is a hyperbolic ball centered at T. But this ball is also strictly inside ~. hence there is p E (0, 1) such that the annulus p < izi < 1 lies in A (see Figure 5.5).
STARLIKE AND SPIRALLIKE FUNCTIONS
175
A=!'!..\ B(.,-,r) Figure 5.5: Condition (A). Thus for such z : p < The lemma is proved. D
lzl
' hence g E Star( D., 0). This sketches another proof that an h E Hol(D., q satisfying (5.6.1) ought to belong to Fan(D.). At the same time, if we keep in mind that (5.6.1) characterizes those h E Fan( D.) with h(O) = 1, such that h(D.) lies in the right half-plane, we reason that a stronger conjecture, namely that (5.6.1) is equivalent to (5.6.3), should be refuted. So it might be worthwhile to replace (5.6.3) by a more qualified condition as well as to replace (5.6.1) by a generalized inequality, which are both related to the same geometrical location of the image h(D.). Following [134] for A E [0, 1) we define the class G>. of nonvanishing holomorphic functions h : .6. f-+ C with h(O) = 1 which satisfy the condition:
1 zh'(z) Re [ 1 _ A ~
+
1+z] 1_ z
> 0 for all z
E D..
(5.6.4)
First we will study some properties of the classes G >.,
Lemma 5.6.1 For each A E [0, 1) the set
is dense in G >. in the topology of uniform convergence on compact subsets of D..
Proof. Since the function
p(:.:) = _1_ zh'(z) 1-A h(z)
+ 1+z 1-z
belongs to the class of Caratheodory: {p E Hol( D., C), p( 0) = 1, Rep( z) > 0, z E D.}, it follows from the Riesz-Herglotz representation of this class that:
p(z) =
1.
1 + z(
- - - dm((),
1(1=1 1- z(
z E .6.,
STARLIKE AND SPIRALLIKE FUNCTIONS
179
where m is a probability measure on the unit circle. Approximating the left hand side of the latter equation by the integral sums: n
L D.mj = 1,
(j E aD.,
j=l
and solving the initial value problem _1_ zh~(z) _ 1- >. hn(z) -
8
( ) z '
hn(O) = 1,
where
s z)(
~
-~
(1 + z(j - 1 + z) D.m·- 2 ~ (j -1 D.m· 1 1-z(j 1-z ~(1-z( )(1-z) 1 '
1
we obtain our assertion with >.1
= 2(1 - >.)D.m1 ,
j
= 1, ... , n. 0
To continue we recall that a holomorphic function 9 : D. starlike of order >. E [0, 1) if 9(0) = 0, 9'(0) = 1 and Re (
~---+
C is said to be
z:~~~))
> >. for all
zED. (see Section 5.2). The set of such functions will be denoted by S*(>.).
The foregoing simple but important fact is due to Silverman and Silvia [134]. Lemma 5.6.2 Let>. E [0, 1) and let 9 and h be holomorphic functions in D. related by the equation 9(z) := z(1- z) 2J..- 2h(z). (5.6.5)
Then 9 E S*(>.) if and only if hE G;... Proof. This fact follows immediately by the equation
z9'(z) ->.=( 1 ->.){-1_zh'(z) + 1+z} 9(z) 1->. h(z) 1-z and the relations g(O) = 0, g'(O) 2mm
= h(O).
(5.6.6)
0
Proof. Assuming that h E G ;.. 2 we obtain by Lemma 5.6.2 that g defined by (5.6.5) with >. = >.2, belongs to S*(.A 2). Let us now denote
9(z) g(z) = (1- z)2(J..2-J..J) · We have by (5.6.5) g(z) = g(z) := z(1- z) 2J.. 2- 2h(z) = z(1- z) 2J.. 1 - 2h(z).
(5.6.7)
180
Chapter V
Thus, again in view of Lemma 5.6.2 we have to show that on account of (5.6.7) we calculate
g E 8*(>, 1 ).
Indeed,
R ( zg'(z)) e
g(z)
g(O) = 0, and we have completed the proof. 0
Exercise 1. Show that Re _z_ < ~' z E z-1-2
~
Exercise 2. Prove that if 0:::; A1, ..\2 < 1, then hE 0>. 1 if and only if
A natural question related to the last lemma is: a given h of the class G.\ for some A E [0, 1) find the maximal ,.\ = ).. *, such that h also belongs to G>.·. In other words, G >.• should be the minimal class which contains h. It turns out, that this question is closely connected to another one: if h is a starlike function with respect to a boundary point, how does one determine the minimal angle () such that h(~) lies in the wedge of this angle. In [37] it was discussed how to resolve the above questions.
Proposition 5.6.1 Let h: ~ ~----+ C be holomorphic and let A E [0, 1). If h is not a constant and h(O) = 1, then the following conditions are equivalent. . [ 1 z h' ( 1+ {z) Re 1 _A h(z) + 1 _ z > 0 for all z E ~-
z)
z]
{ii} There exists a starlike function g : ~ h(z) =
(1- z) 2 z
{iii} The function h belongs to
2 >.
Fan(~)
~----+
g(z)
C of order ).. such that ,
z E ~-
with h(1) := lim h(r) = 0 and
h(~)
r--->1-
lies in a wedge of the angle 211"(1- >.). {iv) The function his a univalent function on~ such that f(z) := h(z)/ h'(z), z E ~, is a semi-complete vector field with
L.lim Re f(z) > - 1z---> 1
Z -
1 - 2 - 2).. '
where the limit in the left hand side of the latter inequality exists finitely. Moreover, the equality in (iv) can be reached if and only if)..=)..*, where G>.• is the minimal class of G.\ which includes h, and if and only if the wedge of the angle 211"(1- )..*) is the smallest one which contains h(~).
STARLIKE AND SPIRALLIKE FUNCTIONS
181
Proof. The equivalence of conditions (i) and (ii) is the content of Lemma 5.6.2. Our next steps are slightly simpler than in [37]. First we claim that condition (ii) and Lemma 5.6.3 imply that hE G>.. \ {1} must be univalent. Indeed, setting in this lemma .>. 1 = 0, .>. 2 = .>. ~ 0 we obtain that h admits a similar representation as (5.6.3):
h(z) = (1- z)2g(z)
(5.6.3')
z
with some
gE
S*. Then, as mentioned above, the approximation process:
) h n( Z ) _- (Tn- z)(1- z'Tn")-( gz, z
implies the desired claim. Next we want to show that h belongs to this fact will be proved once we show that Re ((1- z) 2
Fan(~).
~((:n ~ 0,
To this end let 0 < r < 1, and define hr : ~
t-+
In view of Proposition 5.2.2
z E
~-
(5.6.8)
..) hr(z) := h(rz) ( - , 1- rz
z E ~-
If we use the corresponding function g E S*(.X) we can write equivalently that
hr(z)=
(g(~z)) (~)
(1-z) 2 (l->..)_
This last representation of hr shows that it belongs to G>... Its definition shows that hr ----> h as r ----> 1- and that hr is continuous on the closed disk ~- Therefore the claimed inequality will follow if we inspect it for hr and for z = eicp E 8~. Indeed, for such z we have Re
((1=
z) 2
h~(z))
hr(z)
Re [(1-z) 2 z
(zh~(z) + (1 --\) 1+z) + (1-z) 2 ( 1 --\) 1+z] hr(z)
1-z
= Re [ ( z - 2 + z) ( zh~~) + (1 - .>.)
~ ~ :) + (1 -
zh' (z) 1 + z] = 2(coscp- 1) Re [ hrr(z) + (1- .X) 1 _ z
1-z
z
.>.) (z - z)]
~ 0,
as claimed. The fact established in (5.6.8) and the Berkson-Porta formula mean, actually, that f : ~ t-+ 0 for some f3J. n
Denoting
2:.:: Ajf3J by {3, we have for each
z E
.6.
j=l
largei,'3
n
n
j=l
j=1
IJ w~ 1 I= II: Aj (argeif
31 wJ)
I
n
0 1- z
'
z E ~,
and this inequality no longer holds when >. * is replaced with any number ,\ * < >. < 1. By the Riesz-Herglotz representation theorem we can write
_1_zh'(z) 1-,\1 h(z)
+
1+z = 1-z
J 1+z~dm((), 1-z(
1.·) exp (-2(1- >.*)
j
log(1- z()dm(()) .
1.) exp (-2(1- .A)
j 1.= 1- (1- ).*)(1- a). If a > 0 we reach a contradiction because >. > >. *. Thus a = 0 and m = v. Let g = hjh'. Then f is semi-complete by Proposition 5.2.4. Using (5.6.8) or (5.6.9) we see that z-1
f(z)
=
2(1- >.*)
J
1-(
---dv((), 1- z(
z E
Do.
1(1=1 Let { z,} be any sequence in f(1, k) ={zED.: [1- z[
~
k (1- [z[), k > 1}.
which tends to 1. Consider the functions hn : 86.
hn(() :=
1-( 1- Zn(
,
f-+
C, n = 1, 2, ... , defined by
( E 86..
Since the function hn maps the unit circle 86. onto the circle where Cn = (1- z11 )/(1- [zn[ 2 ), n = 1, 2, ... , we obtain that
[~
-
Cn [
= [en [,
[hn(()[ ~ 2[cn[ ~ 2k. Using (5.6.10) and applying Lebesgue's Bounded Convergence Theorem we now obtain z-1 L.lim - !Z( ) Z-+1
T
2(1- >.*) lim { hn(()dv(() n-+OO }1(1=1 2(1- >.*) ~ 2(1- >.).
(5.6.11)
In other words, condition (d) holds. Finally, we show the implication (iv):::;.(iii). Note that by Proposition 4.6.2, = 1 is the sink point of the semigroup generated by f and L.lim f'(z) is, in fact, z-+1
a real number. Therefore L. lim f(z) > 1 z-+1 Z - 1 2(1 -A).
h(z)
Moreover, f(z) = h'(z) = -(1- z) 2 p(z), where p: .6.
f-+
C is holomorphic with
Rep( z) 2 0 for all z E .6.. Again applying Proposition 5.2.4 and repeating the arguments as in the proof of (5.6.9) we obtain that h is starlike with respect to a boundary point with lim h(r) = 0. Let the smallest wedge in which h(D.) lies be r-+1
of an angle 27r(1 - >.*), where ).* E [0, 1). As we saw in the proof of implication (iii):::;.(iv), it follows that L.lim f(z) = 1 z-+1 Z - 1 2(1 -A*)
(5.6.12)
Chapter V
186
Comparing the latter equality with (5.6.11) we see that>. :::; >.*. Thus h(A) lies in a wedge of an angle 211"(1 - >.), as claimed. This concludes the proof of our assertion. D
Remark 5.6.1. Thus given h E Fan(A) with lim h(r) = 0, formula (5.6.12) r--+1
infers that the value of smallest angle () such that h(A) lies in its wedge is multiplied by the angular limit of the Visser-Ostrowski quotient:
() _
/ 1.
-7l"Llm z->1
7r
(z- 1)h'(z) h() . Z
Corollary 5.6.1 If hE Fan(A) with lim h(r) = 0 satisfies the Visser-Ostrowski r--+1 condition: L. lim (z- 1)h'(z) = 1 z--+1 h(z) (in particular, if h is conformal, or, more generally, isogonal at 1}, then the smallest wedge which contains h(A) is precisely the right half-plane
II+
= {z
E C : Re z :;::: 0} .
Remark 5.6.2 Since G;.. C Go, it follows by the above proposition that h E Fan( A) with lim h(r) = 0 if and only if it satisfies the equation r---+1-
h(z) = -(1- z)2 g(z), z
where g E Star( A, 0). This proves Proposition 5.5.1 (Hummel's representation formula) for the case T = 1. Geometrically this fact should be understood as h(A) lies in the wedge of angle 271".
5. 7
Converse theorems on starlike, spirallike and fanlike functions
In this section we consider inter alia the following converse problem: given f E g Hol(A) find conditions such that the equation h(z) = h'(z)f(z)
(5.7.1)
STARLIKE AND SPIRALLIKE FUNCTIONS
187
has a global solution in b. which is univalent (consequently, starlike) in b.. If f(r) = 0 for some T E b. and we are looking for the solution of (5.7.1) satisfying the initial conditions: h(r) = 0 and h'(r)
i
(5.7.2)
0,
then the necessary restriction is that f' (T) = 1. More generally, if f'(r) = J.L -f. 0 then instead of equation (5.7.1) we must consider equation: (5. 7.3) J.Lh(z) = h'(z) · f(z) with initial conditions (5.7.2), which determines a spirallike function h. Proposition 5. 7.1 Let f E Q Hol(b.) be such that:
f(O) = 0
(5. 7.4)
j'(O) = f.L·
(5.7.5)
and Suppose that hE Hol(b.) is a solution of {5. 7.3}. Then: h(z) = h'(O) lim e~-'t · Ft(z),
(5.7.6)
t->CXJ
where {Ft}t>O is the semigroup, generated by f. Proof. Let {Ft}t~o = Sf be a semigroup generated by (5.7.3) that
or
f. Then it follows by
J.LGt(z) =- 8G~t(z)'
where Gt(z)
= h(Ft(z)), t
~ 0,
(5.7.7)
z E b. and
Go(z) = h(z).
(5.7.8)
Solving (5.7.7) with initial data (5.7.8) we obtain:
Gt(z) =
e-~-'
t · h(z)
or
h(Ft(z)) = e-1-' t · h(z).
(5.7.9)
Consequently, to prove our assertion, it is sufficient to show that lim e~-'t [h(Ft(z))- h'(O)Ft(z)] = 0 t->CXJ
for all z E b..
(5.7.10)
Chapter V
188
Note, that by Taylor's formula we have that for each w E
1 dnh n! dzn (0) wn = h(w).
00
h(w)- h'(O)w =
~
L n=2
If M
= sup h(w) then the Schwarz Lemma (see Section 1.1) implies lwl 0 we have
\eJLth(Ft(z))\
= exp(Re!-Lt) \h(Ft(z))\ Mlzl2
::=; ( 1 _ lzl) 4
as t
-+
·
exp(- Re ~Lt)
-+
0
oo and the result follows. 0
Thus if equation (5.7.3) has a solution hE
Hol(~)
satisfying the condition (5.7.12)
h'(O) = k-::/:- 0, then there is no other solution satisfying the same condition. Conversely, if the limit
h(z) := k lim eJLt Ft(z) t--+00
exists it is easy to see that h(z) satisfies (5.7.3) with (5.7.12). Indeed, in this case
h'(z)f(z)
. t 8Ft(z) t k hm e JL · -~- · f(z) = k lim e JL f(Ft(z)) t--+oo
u
z
t->oo
k lim etJL [ !-LFt(z) t---oo
l
1 dnJ +'"~ """' --(O)(Ft(zt) n! dzn 00
n=2
.
STARLIKE AND SPIRALLIKE FUNCTIONS
189
Again, as above, it can be shown that
.
oo1dnf
n
hm " " - -d (O)(Ft(z) ) t-+oo L.....J n.1 zn
= 0.
n=2
Hence, we obtain:
h'(z)f(z)
= t-tk t-+oo lim eti-L Ft(z) = t-th(z)
and we have completed the proof. 0 Thus to solve the problem (5.7.3)-(5.7.12) we only need to show that the limit in (5.7.6) exjsts. To this end we define for all t 2: 0 and z E /::i.
u(t, z) = eti-L Ft(z) Then for fixed
z E
L\,
au(t,z) = t-tei-Lt Ft(z)- e~-Ltf(Ft(z)) 8t = t-tu(t,z) -ei-Ltf(e-1-Ltu(t,z)) = ei-Lt (t-te-1-Lt u(t, z)- f(e-1-Ltu(t, z))) and u(O, z) = z. Therefore the function u(t, z): IR+
{
8u(t, z) at
--+
C satisfies the equation:
+ eti-Lg(e-1-Ltu(t, z)) =
0, (5.7.13)
u(O, z) = z, where 00
g(z) = f(z)
-j.LZ
=
L
1 dn J n! dzn (0) zn.
n=2
In turn, (5.7.13) implies: t2
u(tb z)- u(tz, z)
=-
J
eti-Lg(e-1-Ltu(t, z)) dt
tt
for each pair t1, t2 E JR+ and z E L\. It follows by (5.7.14) that g'(O) = 0. Hence, for each z E L\: Jg(z)J ~ M1 Jzl 2 , where M1 = supzE.o.Jg(z)J. Therefore
Jg(Ft(z))l ~ MlJFt(zW
< M e-2Re~J.t 1
JzJ 2 (1-JzJ)4"
(5.7.14)
Chapter V
190
Using (5.7.6) we obtain for each z E
~:
as t 1 and t 2 tend to oo. Therefore the limit lim u(t,z) =lim et~'Ft(z)
t -·HX>
t-+- CXJ
exists. D Thus we have proved the following assertion. Proposition 5. 7.2 Let f E Q Hol(~) be a semi-complete vector field which satisfies the conditions: f(O) = 0, and f'(O) = f-l· Then the equation
f-lh(z) = h'(z) · f(z) has a unique solution hE Hol(~) which satisfies the conditions: h(O) = 0, h'(O) = k "# 0. This solution is a univalent spirallike function and it has the form:
where { Ft} is the semigroup, generated by f. In particular, when f-l is a real number h( z) is a starlike function. Turning to a more general situation, assume now that
f(z) = (z- r)(1- zf)q(z), with Req(z) > 0 everywhere. We already know that for each an interior null point T representation (5.7.15) holds. In this case:
(5.7.15)
f
E QHol(~) with
(5.7.16)
If we define
g(z)
=
(Mr)'(Mr(z))f(Mr(z)),
we obtain g(O) = 0 and f-l = g'(O) = (1 Therefore the equation
with the initial conditions h(O) = 0,
-ITI 2 )q(r)
h~ (0)
= f'(r) and g E Q Hol(~).
"# 0 has a
unique solution:
h1(z) = h~(O) ·lim e~'tGt(z), t~oo
where {Gt(z)}t>O = S9 is the semigroup generated by g. Define now (5.7.17)
from which we obtain h(r) = h 1 (0) = 0 and h'(z) =
h~(Mr(z))
· (Mr)'(z).
STARLIKE AND SPIRALLIKE FUNCTIONS
191
In particular, (5.7.18)
In addition,
fl-hl(Mr(z)) = fl- · h~(Mr(z))g(Mr(z)) fl-hl(Mr(z)) · (Mr)'(z)f(z) = fl-h'(z)f(z).
11-h(z)
(5.7.19)
Note now that if {Ft}t>o = St is the semigroup generated by f, then Gt(z) = Mr(Ft(Mr(z))) is the semigroup generated by g (recall, that Mr is an involution), and we have two symmetric relations:
(i) Mr(Gt(z)) = Ft(Mr(z)); (ii) Mr(Ft(z)) = Gt(Mr(z)). Since, h(z) = h 1 (Mr(z)) and h 1 (z) h~(O)
h(z)
= h~(O) · t-->CXJ lim e~-'tGt(z),
we obtain:
· lim e11 tGt(Mr(z)) t-->CXJ
h'(r)(irl 2 -1) · lim e11 t Mr(Ft(z)) t-->CXJ
t r-Ft(z) -----="":-'-:1 - r Ft ( z) h'(r) lim e11 t(Ft(z)- r). 2
h'(r)(irl - 1) · lim e11 t-->CXJ
t-->CXJ
This formula gives us the solution of (5.7.3) under the conditions h(r) 0, h'(r) =1- 0. D So, we arrive at the following conclusion:
Proposition 5.7.3 Let f E
QHol(~)
have the form {5. 7.15)
f(z) = (z- r)(1- zf)q(z) with Re q(z) > 0, 1' E
~
and q(O) =1- 0. Then the equation 11-h(z) = h'(z) · f(z)
with 11- = (1 -lrl 2 )q(O) has a unique solution h(z) satisfying the conditions: h(r) = 0,
h'(r) = k =1- 0.
This solution is a univalent spirallike function on the formula:
~
which can be defined by
If, in particular, q( 1') is real then h is a starlike function on
~.
Now we consider the case when f E g Hol(~) has no null point in ~. It looks like this case is simpler, because equation (5.7.3) has no singularity in~' and one
Chapter V
192
can define a local solution of (5.7.3) for each p, E C under a corresponding initial condition, say (5.7.20) h(O) = 1, and the boundary condition (5.7.21)
lim h(r7) = 0,
r-+1-
where 7 E 8!:l. is the sink point of the semigroup generated by f (see Sections 5.2 and 5.3). But, on the other hand, it is not clear why such a solution has an extension to all of !:l. to be a univalent (spirallike) function (with respect to a boundary point). Moreover, if we wish this solution to satisfy the Visser-Ostrowski condition at the point 7 E 8!:::. we must require for the number p, in (5.7.3) to be real.
Proposition 5. 7.4 Let f be a semi-complete vector field in !:l. with no null point inside, and let 7 E 8!:l. be the sink point of the semigroup generated by f. If (3 = L lim f' (z) then for a real p, > 0 the problem Z-+1"
J1h(z) = h'(z)f(z), h(O) = 1
(5.7.22)
has a unique univalent solution in !:l. if and only if
(5. 7.23)
J1, ::; 2(3.
This solution is a fanlike function (i.e., a starlike function with respect to a boundary point), the image of which lies in the wedge of angle () = 1rp,j {3.
Proof. Without loss of generality we can assume that 7 = 1. If h E Univ(!:l.) satisfies (5.7.22) with some real p, > 0, then its image h(!:l.) must be starshaped by Proposition 5.2.4. Furthermore, if we define fi(z)
f3I
:= L lim Z-+1"
=..!:.{t f(z)
we have
f~ (z) = ~jJ, > 0
and h satisfies the equation
h(z) = h'(z)fi(z) with h(O) = 1. Then it follows by Proposition 5.6.1 that there is >. E [0, 1) such that h belongs to the class G:.. and (31 := L lim f~(z):::: ~:::: -12 . This implies Z-+1" 2- 2/\ (5. 7.23). In addition the angular limit of the Visser-Ostrowski quotient is
v = L lim h'(z)(z- 1) = _!_ = !!.. z-+1h(z) (31 (3'
7f·
and the smallest wedge which contains h(!:l.) is of the angle Conversely. To solve (5.7.22) we first consider the problem:
(Jh(z)
= h'(z)f(z), h(O) =
1,
(5.7.24)
STARLIKE AND SPIRALLIKE FUNCTIONS where
193
f3 = L lim f'(z). z~l-
Consider the functions:
fn(z)
1 = -z + f(z), n
Firstly, it is known that fn E
n
g Hol(t..)
= 1, 2, ....
for all n
(5. 7.25)
= 1, 2, ... ,
since the class
g Hol(t..) is a real cone. Secondly, for each n = 1, 2, ... the equation fn(z) = 0
(5.7.26)
has a unique solution Tn E t. such that Tn ----> 1-, as n----> oo. Indeed, equation (5.7.26) is equivalent to the following one
z+nf(z)=O, which defines the values of the resolvent ln at the point zero, i.e., Tn = ln(O).
If we denote Jln = .!_ + J'(rn) we obtain that Jln----> f3 as n----> oo and f~(rn) = n Jln· Therefore, by the above Proposition 5.7.3, for each n = 1, 2, ... the equation (5.7.27) has a univalent solution h 11 (z) determined by h~,(rn) 1=- 0. Since Tn 1=- 0 for all n = 1, 2, ... and hn is univalent we have h 11 (0) 1=- 0 for all n = 1, 2, .... Therefore, we can define the functions:
which also satisfy equation (5.7.27), with h~(rn) = hn1(o) In addition, h,.(O) = 1 n = 1,2, ....
·
h~(rn) (5. 7.28)
Now for each r E [0, 1) we can find N > 0 such that for all n > N, lrnl > r, that is fn do not vanish on the disk lzl ~ r. Therefore for such z (i.e., lzl ~ r) we can write by using (5.7.27) and (5.7.28):
Furthermore, siuce f(O) 1=- 0 there is a neighborhood U of the point z = 0 in which (5.7.24) has a unique solution:
-h(z)
= exp
{JL • Jor f(z) dz } .
(5.7.29)
Since Jln ----> f3 and fn(z) ----> f(z) for all z E t.., we have that hn(z) converges to h(z) in this neighborhood. It then follows by the Vitali theorem that hn(z)
194
Chapter V
converges to h(z) on all of the disk JzJ < r. Since r is arbitrary we obtain that ~(z) is well defined on all of~. By the Hurwitz theorem (see, for example, [55]) h is univalent on ~. Now again by Proposition 5.6.1 we have that h is a fanlike function the image of which lies in the wedge of angle 1r. Therefore the function h defined as -
h(z) = [ h(z)
] p.f f3
is a univalent function on~ whenever (5.7.23) holds. On account of (5.7.24) it is easy to see that h satisfies (5.7.22). D
Remark 5.7.1 In the proof of the above proposition we have used an approximation process for the generator f (see formula 5.7.25).:_ In turn, this p_:ocess induces an approximation sequence of univalent functions hn converging to h with interior null points defined by (5.7.27). However, in general we can claim only that these functions hn are spirallike, but not necessarily starlike. In fact, we do not know whether the numbers /-Ln in (5.7.27) are real.
5.8
Growth estimates for spirallike, starlike and fanlike functions
The famous Koebe distortion theorem asserts: if hE S ={hE
Univ(~):
(1
h(O) = 0, h'(O) = 1} then
JzJ < Jh(z)l < + Jzl) 2 -
(1
JzJ -lzJ) 2
for all
z E
~.
Equality holds for the Koebe function 00
hi.) (starlike of order>.), i.e.,
, R zg'(z) e g(z) > "· Using the latter inequality it is easy to see that
198
Chapter V
where g 1 belongs to S* (cf., Section 5.6). Then again by (5.8.3) we obtain
[
l1-zl]2-2>. [l1-zl]2-2>. + lzl) :::; lh(z)l:::; (1 -lzl)
(1
(5.8.9)
In particular, we obtain that his bounded in each nontangential approach region at T = 1. 0 Thus on account of Proposition 5.6.1 (see also Remark 5.6.1) and the obvious inequality 1-lzl :::; 11- zl :::; 1 + lzl, we obtain from (5.8.9) the following distortion theorem. Proposition 5.8.3 (see (134]) Let h be a fanlike function on .6. normalized by the conditions h(1) = 0, h(O) = 1,
and let the image of h lie in the wedge of angle Then the following estimates hold :
1r11,
0
. = 0 then h(cp(z)) = 0 for z E !:!. and h = 0 by the uniqueness theorem. If >. = 1 then h satisfied the equations h(cp(n))
= h,
n
= 1, 2, ...
,
and we have
h(z)
=
lim h(cp(n)(z))
n->oo
= h(a)
because of assumption (5.9.2). Thus we have to assume that >. is neither 0 nor 1. Note that if>.# 1 then h(a) = h(cp(a)) = >.h(a), and we have
h(a)
=0
(5.9.5)
as a solution h of Schroeder's equation (5.9.1). Hence h must have the form 00
h(z) =
L an(z- a)n,
k 2: 1.
(5.9.6)
n=k
For simplicity we put a= 0. Then (5.9.5) and (5.9.1) imply
Since the left hand side of this equality is a constant we obtain, letting z go to zero, that >. = [cp'(O)]k. (5.9. 7) Since >. # 0 we obtain (5.9.3). D Thus we have proved the following assertion.
200
Chapter V
Proposition 5.9.1 Let 'P E Hol(~) have an attractive fixed point a E ~ and suppose that Schroeder's equation (5.9.1} has a nontrivial solution hE Hol(~, C) for some A E C. Then
(i) h(a) = 0; (ii) A= ['P'(O)]k =f. 0 for some positive k; (iii) if h is locally univalent, then A = 1p1 (a) =f. 0. In 1884 Koenigs proved a remarkable result on the solvability of Schroeder's equation.
Proposition 5.9.2 (see [78]) Let 'P E Hol(~) have an attractive fixed point a E ~. such that 'P'(a) :=).=f. 0. Then there is a function hE
Hol(~,
h(1p(z))
C) such that
= Ah(z)
for all z E ~. In addition, if 'P is univalent then so is h. The idea in the original proof of Koenigs's theorem is based on the convergence of the sequence
h (z) = 'P(n)(z) = n A"
z
+ L......, ~ a(n) zk k ,
n = 1, 2, ... ,
k=2
which evidently satisfies the recursion equation (5.9.8) Its limit function his called the Koenigs function and it is normalized by the condition h'(a) = 1.
If such a function can be found one can present (5.9.9) whenever the right hand side of this equality is well defined. Hence
'P(n)(z) = h- 1 (Anh(z)).
(5.9.10)
The latter expression then serves as a definition of fractional iterations of 'P when n is not an integer (cf., section 4.4) and large enough. Of course, if 'P can be embedded in a globally defined continuous semigroup of holomorphic mappings it must be univalent, and so should h. Although the discrete semigroup of iterates of 'P cannot be embedded, in general, in a continuous semigroup of holomorphic self-mappings of ~. depending on what one requires the answer may be yes in some suitable cases (see for example, G. Srekeres [138], J. Hadamard [59], T. Harris [63], C.C. Cowen [27]). The simplest case (though very useful) is, of course, when 'Pis a fractional linear transformation and so is h. We consider now, in some sense, the dual problem.
STARLIKE AND SPIRALLIKE FUNCTIONS
201
Definition 5.9.1 We will say that a mapping cp E Hol(.6.) satisfies the Koenigs embedding property (K.e.p.) if its iterations cp(n) : .6. ~ .6. can be embedded in a continuous semigroup {Ft}t?:O of holomorphic self-mappings of .6., i.e., F 1 = cp. It turns out that the answer to the question of what are the conditions for a univalent self-mapping cp E Hol(.6.) to satisfy the (K.e.p.) is related to some geometrical properties of the solution of Schroeder's equation.
Proposition 5.9.3 Let cp be a univalent self-mapping of .6., and let cp(r) = T for some T E .6. with 0 < lcp'(r)l < 1. Then cp satisfies the Koenigs embedding property if and only if its Koenigs function is J.L-spirallike, with J.L = -log cp1(T). Proof. Sufficiency. Suppose that equation (5.9.10) has a univalent J.L-spirallike solution with J.L = -log A. Then for each z E .6. and t;::: 0 element e-tJ.Lh(z) belongs to h(.6.). If we define
we obtain by (5.9.10) that
Necessity. Let cp satisfy the Koenigs embedding property, i.e., there is a semigroup {Ft}t?:O C Hol(.6.) such that
F1(z) = cp(z). Denote by
f = - ~ lt=O+ the generator of {Ft} and consider the equation J.Lh(z) = f(z) h'(z),
(5.9.11)
where J.L = - log A. Since J.L = f'(r) =f. 0 it follows by Proposition 5.7.2 that equation (5.9.11) has a univalent J.L-spirallike solution h which satisfies the equality
Setting here t
=
1 we obtain (5.9.10). D
If cp has no fixed point in .6. then it follows by the Julia-Cratbeodory Theorem that there is a unique point T E 8.6. such that L lim cp(z) = T and Z--+T
0 < L lim cp'(z) ~ 1. Z--+T
Similarly, as in Proposition 5.9.3, by using Proposition 5.7.3 one proves the following assertion:
Chapter V
202
Proposition 5.9.4 Let
T
=f. 0, where f
= - ddF lt=o+.
t
Then there are .X E C, 0 < I.XI < 1, and h E
Hol(~, C)
such that for all t
~
0
(5.9.13)
h(Ft) = .Xh.
In other words, there is a solution of Schroder's equation which does not depend on t ~ 0.
In addition, the solution of (5.9.12) can be found by solving the differential equations (5.9.11) with J.L = -log .X.
Example 1. Let 0. By (5.9.15) inequality (5.9.17) becomes Re1-1z(1- kz) > 0,
(5.9.18)
since by (5.9.15) lkl ::; 1 the latter condition can be rewritten in the form (5.9.19)
cos arg(11) ::=: k. In particular, this condition always holds when 1-1 is real. Thus we have proved
Proposition 5.9.5 Let cp(z) = ~b with lal + lbl :S 1, and 0 < lal < 1. Then 1- z cp satisfies the Koenigs embedding property if and only if cos ( arg (log
~))
::=: 11
~ a I·
(5.9.20)
In particular, if a is a real number then cp can be always embedded in a continuous semigroup of holomorphic self-mappings of!:::...
If we consider the mapping ¢( z) = ~b , as in Example 1 of Section 4.3, with 1- z a= ~ exp(i3.1) and b = ~'then it is easy to see that lal + lbl = 1 whilst condition (5.9.20) does not hold. Thus ¢ : !:::.. ~-----> !:::.. cannot be embedded in a continuous semigroup of self-mappings of!:::.. (see Figure 4.1).
204
Chapter V Exercise 1. Prove the equivalence of conditions (5.9.17) and (5.9.18).
Exercise 2. Under condition (5.9.19) find the semigroup Ft : A---+ A, t? 0, such that F 1 = cp, where cp has the form (5.9.14).
More detailed discussion of this approach for higher dimensions can be found in [76]. Because of our intention in a modest text to emphasize the dynamical flavor of the subject a big part of the theory which is primarily geometrical or functional analytic in nature has not been included. We refer the reader to books of A.W. Goodman [57], P. Duren [33], and J.H. Shapiro [131], which could be good guides to complete the knowledge in these topics. Also, to advance to futher study on the boundary behavior of holomorphic mappings we mention an excellent book of C. Pommerenke [107]. Finally, we point out that many applicable subjects as branching processes, optimization theory, functional calculus etc., mentioned in the Preface may motivate an investigation in this direction.
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[150] J.K. Wald, On starlike functions, (Ph.D. Thesis), Univ. of Delaware, Newark, Delaware, 1978. [151] H.W. Watson and F. Galton, On the probability of the exinction of families, Jour. Anihrop. Inst. 4 (1874), 138-144. [152] K. Wlodarczyk, Iterations of Holomorphic Maps of Infinite Dimensional Homogeneous Domains, Mh. Math. 99 (1985), 153-160. [153] K. Wlodarczyk, Studies of iterations of Holomorphic maps in j-Algebras and complex Hilbert Spaces Oxford University Press, 1986.
[154] K. Wlodarczyk, Julia's Lemma and Wolff's Theorem for j-Algebras, American Math. Society 99 3 March 1987. [155] J. Wolff, Sur !'iteration des fonctions holomorphes dans une region, et dont les valeurs appartiennent a cette region, C. R. Acad. Sci. 182 (1926), 42-43. [156] J. Wolff, Sur !'iteration des fonctions bornees, C. R. Acad. Sci. 182 (1926), 200-201. [157] J. Wolff, Sur une generalisation d'un theoreme de Schwarz, C. R. Acad. Sci. 182 (1926), 918-920. [158] J. Wolff, L'E~quation differetielle dzjdt = w(z) =fonction holomorphe a partie reelle positive dans un demi-plan, Compos. Math. 6 (1938), 296-304. [159] K. Yosida, Functional Analysis, Springer, Berlin, 1968. [160] P. Yang, Holomorpic curves and boundary regularity of biholomorpic maps of pseudoconvex domains, ?reprint (1978).
Author and Subject Index Abate, M., vii, 101, 205 Alexander, J. W., 158, 205 Behan, D.F., 37, 206 Berkson, E., vii, 63, 93, 95, 101, 206 Bernardi, S.D., viii, 206 Brickman, L., ix, 206
Latuskin, Y., vii, 210 Lyzzaik, A., viii, 171, 177, 211 MacCluer, B., vii, viii, 207, 211 Mazet, P., vii, 211 Mellon, P., vii, 211 Mercer, P., vii, 211 Mantel, P., 6, 211 Nevanlinna, R., viii, 94, 158, 194, 212 Pjaltzgraff, J.A., ix, 212 Pick, G., 9 Poincare, H., 7, 43, 44 Porta, H., vii, 63, 93, 95, 101, 206 Potapov, V.P., vii, 212 Reich, S., vii, 205, 207-210, 212 Riemann, B., 7 Riesz, F., 5 Robertson, M.S., viii, ix, 159, 171, 176,213 Schroeder, E., 198, 213 Schwarz, H.A., 9 Sekowski, T., vii, 208 Sevastyanov, B.A., vi, 213 ShafTi.r, I., vii, 213 Shields, A.L., 36, 213 Silverman, H., viii, 177, 179, 213 Silvia, E.M., viii, 177, 179, 213 Siskakis, A.G., vii, 214 Srekeres, G., 200, 214 Stachura, A., vii, 208, 210 Steffensen, J.F., vi, 214 Stepin, M., vii, 210
Caratheodory, C., 9, 17, 206 Chen, G.N., vii, 206 Cowen, C. C., viii, 200, 207 de Branges, L., ix, 194, 207 Denjoy, A., vi, 9, 32, 35, 207 Earle, C., vii, 207 Egervary, E., 176, 207 Fan, K., vii, 208 Forelli, F., vii, 208 Galton, F., v, 215 Glicksberg, 1., vii, 208 Goebel, K., vii, 208 Gurganus, K.R., ix, 209 Hadamard, J., 200, 209 Hamilton, R., vii, 207 Harris, L.A., 13, 209 Harris, T.E., vi, 200, 209 Herglotz, G., 5 Hummel, J.A., 171, 172, 174, 209 Jafari, F., vii, 209 Julia, G., vii, 9, 17, 209 Koenigs, G., vi, 200, 210 Kubota, Y., vii, 210 Kuczumov, T., vii, 210 Lowner, K., ix, 211 216
AUTHOR AND SUBJECT INDEX Suffridge, T.J., ix, 212, 214 Vesentini, E., vii, 101, 208, 214 Vigue, J.P., vii, 211 Wald, J.K., viii, 159, 215 Watson, H. W., v, 215 Wlodarczyk, K., vii, 215 Wolff, J., vi, vii, 9, 32, 35, 101, 215 Yale, K., vii, 209
admissible curve, 42, 45-47 angular derivative, 23, 32, 35, 135, 140, 145, 150, 164 angular limit, 20, 23, 186 approximating curve, vii, 53, 55, 56 asymptotic behavior, vii, 83, 98, 103, 113, 135, 142-145, 150, 177 automorphism, vi, 9, 15, 19, 26, 30, 35, 56, 83, 146 elliptic, 27, 30, 36, 103, 105, 112, 113 hyperbolic, 27-29, 32, 37, 105, 109, 111 parabolic, 27, 28, 30, 32, 105, 111,112 Banach Fixed Point Principle, 8, 31, 52, 99 Berkson-Portarepresentation, 95, 98, 99, 139, 162, 170, 171, 174, 181 Bieberbach conjecture, ix, 194 Bohl-Poincare theorem, 81 boundary behavior, 17, 163 branching process, v, vi Brouwer's Fixed Point Principle, 8, 31 Caratheodory Kernel Theorem, viii, 6, 166, 177, 183 Cauchy Integral Formula, 4
217 Cauchy problem, 66, 68, 69, 72, 76, 82, 83, 87, 103, 161 Cauchy-Schwarz formula, 5 class of Caratheodory, 89, 178 commuting family, 36, 101 complete metric space, 7, 43, 44 complete vector field, 104, 156 contraction, 7, 17 strict, 8, 31, 52, 70, 99, 122 Denjoy-Wolff Theorem, vii, 30, 32, 36, 101, 102 embedding property, 109, Chapter V Euclidean distance, 7, 13, 114 exponential formula, 68, 70, 97, 122 fixed point, 8, 27, 31, 32, 53, 54, 57, 99, 111 attractive, 32, 35, 156, 199, 200 boundar~ 26, 29, 30, 32, 35 common, 36, 101 free mapping, 33, 37, 54 interior, vi, vii, 25, 30-32, 56 of automorphisms, 26 of holomorphic self-mapping, 25, 31 of nonexpansive mapping, 52 flow invariance condition, vii, 82, 98 fractional linear transformation, 10, 29, 40, 107, 109, 200, 202 function p-monotone, 79-82, 98, 141, 195 strongly, 98, 99, 169 close-to-convex, ix, 153, 177 fanlike, 157, 176, 192, 198 harmonic, 5, 85, 86, 97, 175 holomorphic, 4, 9 spirallike, see spirallike function starlike, see starlike function
218
AUTHOR AND SUBJECT INDEX univalent, ix, 4, 6, 153, 154, 158, 159,161, 162,170, 177,178, 180
generator, 66, 68, 81, 82, 144, 147, 150, 160, 162 of a one-parameter group, 82, 83, 104 of a one-parameter semigroup, 70, 76, 83 geodesic segment, 15, 45, 46 geodesics, 44, 47 group of automorphisms, 82, 104 one-parameter, 60 growth estimate, 194, 196 Harnack inequality, 89, 146, 149 strong, 90, 119 horocycle, 18, 19, 32, 37, 54, 97, 136, 167 Hummel's multiplier, 171, 172, 177 Hurwitz convergence theorem, 6 hyperbolic ball, 136, 174 distance, 45 length, 39, 42, 45 metric, vii, 7, 8, 39, 72, 79, 98, 99, 122 Implicit Function Theorem, 72, 82 infinitesimal generator, 66, 70, 76, 83 involution property, 11 isometry, p-isometry, 52, 60 Julia number, 19, 33, 35 Julia's Lemma, vi, 18, 19, 24, 32, 33, 35, 151 Julia-CaratheodoryTheorem, 22, 23, 32, 35, 150, 201
Julia-Wolff-Caratheodory Theorem, 33, 164, 171, 202 Koebe distortion theorem, 182, 194, 195 Koebe function, 168, 194 Koebe One-Quarter Theorem, 153 Koenigs embedding property, 198, 201203 Koenigs function, 200, 201 Lebesgue's Bounded Convergence Theorem, 185 Lindelof's inequality, 12 Lindelof's Principle, 23, 182 lower bound, 80, 135, 144 appropriate, 137, 140, 142, 143, 148 Mobius transformation, vii, 10, 11, 15,40,46,83, 168,172,173, 195, 196 mapping conformal at the boundary point, 164 identity mapping, 11 isogonal at the boundary point, 164 nonexpansive, 7, 8, 39, 43 p-nonexpansive, 52, 56, 57, 6870, 75-77, 79, 81, 82, 136, 144 fixed point free, 54 fixed point of, 52 maximum principle for harmonic functions, 5, 85, 86, 96 maximum modulus principle, 5, 10, 35, 47 Monte! Theorem, 6
AUTHOR AND SUBJECT INDEX Nevanlinna's condition, ix, 163, 167 nontangential approach region, 20, 23, 150, 182, 198 nontangentiallimit, 20, 21, 34 normal family, 6 null point, 98, 104, 135, 144, 156-158, 161, 171 Open Mapping Theorem, 154 Poincare metric, 7, 8, 39, 43-46, 70, 72, 79, 98 infinitesimal, 44, 45, 122 power convergence, 32, 35, 36, 56, 57 pseudo-hyperbolic ball, 17, 25, 118, 167 disk, 32 distance, 14, 17, 39, 114, 196 metric, 7 range condition, 69, 70, 72, 74, 77, 79, 82, 88 rate of convergence, 29, 35, 98, 113, 135, 139, 142, 144, 147, 149 resolvent, 73-78, 81, 82, 96, 99, 193 nonlinear, 67, 69, 72, 79, 82 resolvent identity, 74-78 retraction, 56, 78 Riemann Mapping Theorem, viii, 154, 166 Riesz-Herglotz representation, 5, 93, 178, 184 Schroeder's equation, 198-201 Schwarz Lemma, vi, 9, 12, 27, 87, 89, 91, 149, 167, 188 Schwarz-Pick inequality, 11, 13, 17, 19, 43, 82 Schwarz-Pick Lemma, 11, 14, 17, 19, 25, 26, 31, 32, 57 boundary version, 35
219 semi-complete vector field, 91, 95, 156, 174, 192, 195 semigroup, ix, 9, 60, 68, 97, 98, 135, 150, 155, 187, 190 one-parameter, v, 70 continuous, 60, 63, 66, 109 discrete, 26, 60, 109 sink point, 33, 35-37, 55-57, 78, 97, 105, 110-112, 135, 139, 142, 144, 149, 150, 171, 182, 185, 192 spirallike function, viii, ix, 159, 160, 167, 190, 191, 195 strongly, 169, 196 with respect to a boundary point, 159, 162 with respect to an interior point, 159, 161, 162 spiralshaped set, 153, 159, 160, 166, 167 starlike function, viii, ix, 157, 158, 160, 162,167,169,170,190, 191, 196 of order .\, 159, 179, 197 with respect to a boundary point, viii, 157, 158, 171, 176, 177, 180, 185, 192 with respect to an interior point, 157, 159, 161, 170, 171, 176, 177 starshaped set, 153, 157, 159-161, 166, 167, 172, 178 stationary point, 101, 104, 136, 138, 142 Stolz angle, 21 strict contraction, 8, 31, 52, 70, 99, 122 support functional, 136 Taylor representation, 5, 10, 154, 194
220 uniform convergence, 6, 54, 142 uniform Lipschitz condition, 13 vector field, 2 complete, 83, 85, 93, 96, 104, 110, 112, 156, 171 semi-complete, 83, 91, 95, 97, 98, 113, 156, 171, 174, 177, 180, 182, 190, 192, 195 Visser-Ostrowski condition, 165, 186, 192 Visser-Ostrowski quotient, 163, 165, 186, 192 Vitali theorem, 6, 31 Wald's condition, ix, 163 Weierstrass convergence theorem, 6 Wolff's Lemma, 32, 33, 35, 54
AUTHOR AND SUBJECT INDEX
List of Figures function w = f(z). . . . . . . . . . . translation f(z) = z +a, a= 4 + 2i. rotation f(z) = ei1r 8 z, (} = -27r/3 .. contraction f(z) = kz, k = 1/3. vector field w = f(z) . . . . . . . . .
0.1 0.2 0.3 0.4 0.5
The The The The The
1.1 1.2 1.3 1.4 1.5 1.6 1. 7 1.8 1.9 1.10
An orthogonal circle to a/:1 and its image under an automorphism. A Mobius transformation of t1r. . . . . . . . . . . . . A horocycle at the point ( E a/:1. . . . . . . . . . . . . A nontangential approach region at a boundary point. A Stolz angle at a boundary point. . . . . . Boundary behavior of a self-mapping of /:1. Elliptic automorphism. . . . Hyperbolic automorphism. . Parabolic automorphism. Uniqueness of a point (. . .
14 16 18 20 21 22 27 28 29 34
2.1
The points of anharmonic relation.
40
3.1 3.2 3.3
Boundary condition for f E aut(/:1). Boundary flow invariance condition. Values of functions of Caratheodory's class.
84 86 90
4.1 4.2 4.3 4.4
Fractional iterations of the self-mapping F(z) . . The circle f(z) and the sectors Sand S. The nontangential convergence toT. The asymptotic behavior of the flow.
110 126 130 140
5.1 5.2 5.3
A starshaped domain (0 En) . . . . . A starshaped domain (0 E an) . . . . The spiralshaped domain (0 E an) ..
157 158 160
221
2 2 3 3 3
LIST OF FIGURES
222 5.4 5.5 5.6
The set D .. . . . . . . . . . . . . . . . . . . Condition (A). . . . . . . . . . . . . . . . . An approximation of a starshaped domain.
168 175
183
