Computational Methods And Function Theory

Lecture Notes in Mathematics Edited by A. Dold, B. Eckmann and E Takens

1435 St. Ruscheweyh E.B. Saff L.C. Salinas R.S. Varga (Eds.)

Computational Methods and FunctionTheory Proceedings of a Conference, held in Valparaiso, Chile, March 13-18, 1989 II I I I I

SpringerM~rlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona

Editors

Stephan Ruscheweyh Mathematisches Institut, Universit~.t W~rzburg 8?00 WL~rzburg, FRG Edward B. Saff Institute for Constructive Mathematics Department of Mathematics, University of South Florida Tampa, Florida 33620, USA Luis C. Salinas Departamento de Matem&tica, Universidad Tecnica Federico Santa Maria Casilla 110-V,Valparaiso, Chile Richard S. Varga Institute for Computational Mathematics, Kent State University Kent, Ohio 44242, USA

Mathematics Subject Classification (1980): 30B?0, 30C10, 30C25, 30C30, 30C70, 30E05, 30E10, 65R20 ISBN 3-540-52768-0 Springer-Verlag Berlin Heidelberg NewYork ISBN 0-38?-52?68-0 Springer-Verlag New York Berlin Heidelberg

This work is subject to copyright.All rightsare reserved,whetherthe wholeor part of the material is concerned,specificallythe rights of translation,reprinting,re-useof illustrations,recitation, broadcasting,reproductionon microfilmsor in otherways,and storagein databanks. Duplication of thispublicationor partsthereofis only permittedunderthe provisionsof the GermanCopyright Law 0fSeptember 9, 1965, in its versionof June 24, 1985,and a copyrightfee must alwaysbe paid.Violationsfall underthe prosecutionact of the GermanCopyright Law. © Springer-VerlagBerlin Heidelberg 1990 Printed in Germany Printing and binding: DruckhausBeltz, Hemsbach/Bergstr. 2146/3140-543210- Printed on acid-freepaper

Preface This volume contains the proceedings of the international conference on 'Computational Methods and Function Theory', held at the Universidad T@cnica Federico Santa Mar/a, Valparaiso, Chile, March 13-18, 1989. That conference had two goals. The first one was to bring together mathematicimas representing two somewhat distant areas of research to strengthen the desirable scientific cooperation between their respective disciplines. The second goal was to have this conference in a country where mathematics as a field of research is developing and scientific contacts with foreign experts are very neccessary. It seems that the conference was successful in both regards. Besides, for many of the non-Chilean participants this was the first visit to South-America and these days left them with valuable personal impressions about the regional problems, an experience which may lead to active support and cooperation in the future. About 40 half- and one-hour lectures were presented during the conference. They are listed on the last pages of this volume. Of course, not all of them led to a contribution for these proceedings since many have been published elsewhere. However, the papers in this volume are fairly representative for the areas covered. To hold such a conference, in a place somewhat distant from the international mathematical centers, obviously requires strong support from funding agencies, and it is the organizer's pleasure to acknowledge those contributions at this point. The local organization was made possible through generous grants from the FundaciSn Andes, Chile, mad from our host, the Universidad T@cnica Federico Santa Mafia. In addition, foreign participants were supported by a special grant of the National Science Foundation (NSF), USA, and by other national agencies such as the Deutsche Forschungsgemeinschaft (DFG), FRG, the German Academic Exchange Service (DAAD), FRG, the British Council, UK, etc. We also wish to thank the Universidad T~cnica Federico Santa Maria for the hospitality on its marvellous campus overlooking the beautiful Bay of Valparaiso, and the many people who did help us with the organization. Especially, we wish to thank Ruth Ruscheweyh, who assisted the organizers during the conference and the hot phase of its preparation, and also was responsible for the typesetting (in ISTEX) of the papers in this volume. Finally, we should like to thank Springer-Verlag for accepting these proceedings for its Lecture Notes series.

For the editors: Stephan Ruscheweyh

Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

III

R.W. Barnard Open Problems and Conjectures in Complex Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

J.M. Borwein, P.B. Borwein A R e m a r k a b l e Cubic Mean Iteration

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27

A . C d r d o v a Y . , St. Ruscheweyh On the Maximal Range Problem for Slit Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

R . Freund On Bernstein T y p e Inequalities and a Weighted Chebyshev Approximation P r o b l e m on Ellipses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45

D.M. Hough Conformal Mapping and Fourier-Jacobi Approximations . . . . . . . . . . . . . . . . . . . . . . . . .

57

J.A. H u m m e l Numerical Solutions of the Schiffer Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71

K.G. Ivanov, E.B. Saff Behavior of the Lagrange Interpolants in the Roots of Unity . . . . . . . . . . . . . . . . . . . . .

81

Lisa Jacobsen Orthogonal Polynomials, Chain Sequences, Three-term Recurrence Relations and Continued Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89

A. Marden, B. Rodin On T h u r s t o n ' s Formulation and Proof of Andreev's Theorem . . . . . . . . . . . . . . . . . . .

103

D. Mejfa, D. Minda Hyperbolic G e o m e t r y in Spherically k-convex Regions . . . . . . . . . . . . . . . . . . . . . . . . . . .

117

D. Minda The Bloch and Marden Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

131

O . F . Orellana On Some Analytic and Computational Aspects of Two Dimensional Vortex Sheet Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

143

N. Papamichael, N.S. Stylianopoulos On the Numerical Performance of a Domain Decomposition Method for Conformal Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G . Schober Planar Harmonic Mappings

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155

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171

T.J. Suffridge Extremal Problems for Non-vanishing H p Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . .

177

VI W.J. Thron Some Results on Separate Convergence of Continued Fractions . . . . . . . . . . . . . . . . .

191

R.S. Varga~ A.J. Carpenter Asymptotics for the Zeros of the Partial Sums of e ~. II . . . . . . . . . . . . . . . . . . . . . . . . .

201

Lectures presented during the conference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

209

Computational Methods and Function Theory Proceedings, Valparm'so 1989 St. Ruscheweyh, E.B. Saff, L. C. Salinas, R.S. Varga (eds.) Lecture Notes in Mathematics 1435, pp. 1-26 (~) Springer Berlin Heidelberg 1990

O p e n P r o b l e m s and Conjectures in C o m p l e x Analysis Roger W. Barnaxd Department of Mathematics, Texas Tech University Lubbock, Texas 79409-1042, USA

Introduction This article surveys some of the open problems and conjectures in complex analysis that the a u t h o r has been interested in and worked on over the last several years. They include problems on polynomials, geometric function theory, and special functions with a frequent mixture of the three. The problems that will be discussed and the author's collaborators associated with each problem are as follows: 1. Polynomials with nonnegative coefficients (with W. Dayawansa, K. Pearce, and D. Weinberg) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

2. The center divided difference of polynomials (with R. Evans and C. FitzGerald) . . . . . . . 4 3. Digital filters and zeros of interpolating polynomials (with W. Ford and H. Wang) . . . . . 5 4. Omitted values problems (with J. Lewis and K. Pearce) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

5. MSbius transformations of convex mapp!ings (with G. Schober) . . . . . . . . . . . . . . . . . . . . . . .

12

6. Robinson's 1/2 conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

7. Campbell's conjecture on a majorization - subordination result (with C. Kellogg) . . . . . 14 8. Krzyi conjecture for bounded nonvanishing functions (with S. Ruscheweyh) . . . . . . . . . . 15 9. A conjecture for bounded starlike functions (with J. Lewis and K. Pearce) . . . . . . . . . . . 16 10. A. Schild's 2/3 conjecture (with J. Lewis) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

11. Brannan's coefficient conjecture for certain power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

12. Polynomial approximations using a differential equations model (with L. Reichel) .... 20

2

R.W. Barnard 1. Polynomials with nonnegative coefficients

We first discuss a series of conjectures which have as one of their sources the work of mgler, Trimble and Varga in [66]. In [66] these authors considered two earlier papers by Beauzamy and Enflo [23] and Beauzamy [22], which are connected with polynomials and the classical Jensen inequality. To describe their results, let

p(z)= ~a3z j= j=O

ajz j, where aj =O, j > rn, j=0

be a complex polynomial ( 5 0), let d be a number in the interval (0, 1), and let k be a nonnegative integer. Then (cf [22], [23]) p is said to have concentration d of degree at most k if k

(1)

oo

~ la, I >_d E fail,

j=O

j=0

Beauzamy and Enflo showed that there exists a constant (~d,k, depending only on d and k, such that for any polynomial p satisfying (1), it is true that (2)

2zr f0

()

l°glp(ei°DIdO-l°g ~'~lajl >- Cd,k. j=0

In the case of k = 0 in (2) the inequality is equivalent to the Jensen inequality 1 f02~ log 2--~

[23],

Ip(e~°)ldO>_log ta01.

Rigler, etc., in [66] considered the extension of this inequality from the class of polynomials to the class of H °~ (of. Duren [36]) functions. For f E H ~ the functional 1

2~

(

J(f) := ~ f0 log tf(ei°)ldO - log ~ lajl

)

j=0

can be well-defined and is finite. They let (3)

Cd.k = i n f { J ( f ) : f C H ~ and f(z) = ~ ajzJ(~ 0) satisfies (1)}. j=0

For a (fixed) d 6 (0, 1) and a (fixed) nonnegative integer k, it was shown that there exists an unique positive integer n (dependent on d and k) such that

l ~ - - ~ ( n ) 1. If this digital signal is to be compared with digital signals based on the smaller sample interval, At, the given digital signal must be interpolated to the smaller sample interval, At. For example, insertion of N - 1 zeros between every bk and bk+l, followed by multiplication of the Z-transform of the result by the interpolating series, ~r~, defined by

(10)

PN(z) = 1 +

(z m + z -'~) sinc --~-, rn=l

leads to (11)

A(z)=

anzn= n = - co

bjz jg \j=-

oo

PN(Z). /

Since the coefficient of z kN, akg, in A comes from products of bj and sinc(rmr/N) such that k N = j N + m, it follows that m - 0 (modN), sinc(rnr/N)=O for nonzero m, and a k y = bk. Thus, A is an interpolation of the given B with coefficients, bj.

Open Problems and Conjectures in Complex Analysis

7

A major goal is to study possible alternatives to the interpolation used in (10) in terms of truncation of the interpolating series in (11). In practice one truncates P to obtain the interpolating polynomial, PN,L defined by (12)

P N , L ( Z ) = Z L-1

1 + Z ( zm + z - m ) sine

,

ra= l

where N > 1. To assure stability and accuracy of evaluation it is important that alternative P ' s have no zeros on the unit circle. It is shown in [12] that all of the zeros of PN,L are of unit modulus when L < N and examples are given showing that when L > N + 1 almost any combination of zeros inside, on, and outside the unit circle can occur. A number of classical results are then combined to give sharp conditions on real sequences {cm : 1 < m < oc} so that the function P*N,L defined by -

-

-

-

(13)

P ~ , L ( Z ) = Z L-1

1 +

~ (z m + z - m ) c m sine m=l

has no zero of unit modulus. In particular, in order to define a useful test to determine if a specific sequence of numbers will work for the cm's in (13) the following theorem was proved in [12]. T h e o r e m 1. I f a real sequence, {bin : 0 0 and f -< F ( F E S) in A, then f _ 0 and f -~ F (F E 14~,) then f ' ~ F' for Izl < a + 1 (a 2 + 2~) 1/2. In [14] the author and Kellogg combined Ruscheweyh's subordination result [68], variational methods, and some tedious computations to verify the conjecture for a = 1, i.e., it is shown that if f'(0) __>0 and f ~ F ( F E S ~) in AI then f ' (( F ' for lzl < 2 - v/3.

8. K r z y i ' s

conjecture

for

bounded nonvanishing functions

Another conjecture that has been investigated by a large number of function theorists is Krzy~'s conjecture. Let B denote the class of functions defined by f ( z ) = ao + alz + • .. + a~z ~ + . . . for which 0 < [f(z)I < 1 for z E A. In 1968 in [49] J. Krzy~ posed the fundamental problem of determining for n > 1 An = sup [a~ I. feb That A1 = 2/e dates back to 1932 (see Levin [51]) and appears explicitly in Hummel, etc. [46] and Horowitz [45]. That A2 = 2/e appears in [46] and A3 = 2/e in [65]. For a fairly complete history of this problem see [46] or Brown [31]. These results suggest what has become known as the Krzy~ Conjecture, C o n j e c t u r e 12. An = 2/e, for all n > 1, , with equality only for the functions [,+1] 1 +2z~+... Kn(z) = exp tz n _ lJ = e e and its rotations e~K,~(e~Vz). A~ is to equal the apocryphal Pondiczery constant, named by Boas in [25]. A sharp uniform bound less than one is expected. However, the bound 2/e ~ .7357.-., is somewhat surprising in view of the fact that the best uniform estimate known to date is 1 4 (1) la,~l _~ 1 - ~ + sin = 0.9998772... given by D. Horovitz in 1978 in [45]. The open problem of Krzy~'s Conjecture is stated in A. Goodman's book "Univalent Functions" [40, page 83]. De Branges' recent solution to the Bieberbach Conjecture gave hope to solving many of these type problems. However, not withstanding the amount of effort by several function theorists to solve the corresponding coefficient problem, Conjecture 12 still remains open. A related conjecture made by Ruscheweyh upon verification would give a much improved uniform estimate for An. Consider f ( z ) = exp[-Ap(z)] for ), > 0 and p E P where P = {p: p(z) = 1 + p l z + ' " , a e p ( z ) > 0,1zl < 1}. Then consider the following: For 0 < r < 1, choose x = x(r) such that

16

R. W. Barnard •

J1 \{ix 1 -2r r 2 ] 2r ~] Jo ~(ix x-r~

(J0, J1 are Bessel Functions) -

~

and define

F(~)

(23)

= ~(r)~

_= 3 are no longer starlike suggested the need for a local variational technique that preserved these properties. This was developed by combining the Julia Variational formula with the Loewner Theory in [6] and in [17]. Let

S~t = { f 6 SM: f ( A ) is starlike with respect to the origin}. It was shown in [6] that the extremal domain maximizing la3t in S~ is the disc AM minus at most two symmetric radial slits. Define DM a s A M minus two symmetric radial slits where 20 is the angle between the 2 slits. Let Aa(M, O) be the third coefficient for the function in S~/mapping A onto the domain DM. From the properties of the extremal domains in the class SM, along with initial computations and the observation that A3(3, 0) = A3(3, 7r/2) = 8/9 led this author to the following: C o n j e c t u r e 14. For ali f 6 S ~ (28)

[a3[ < A3(M, rr/2), 1 < M < 3,

18

R. W. Barnard la3l < Aa(M,O), 3 < M < ee.

(29)

It follows from T a m m i ' s results that (28) holds for 1 < M < e and it was shown by the author and Lewis in [16] that (29) holds for 5 < M < oe. Verifying Conjecture 14 for e < M < 5 remained an open problem. This conjecture was announced at the 1978 Brockport Conference and appeared in the open problem set in the proceedings [60] for t h a t conference. It was announced by J. Lewis at the 1980 C a n t e r b u r y Conference and a p p e a r e d in the open problem set in its proceedings, [27]. It was also announced at the 1985 Symposium on the Proof of the Bieberbach Conjecture. Motivated by the observation that the domain DM is indeed a "gearlike" domain and now having the computer software available, Pearce was able to compute Aa(3, 0) and discovered that A3(3, 0), as a function of 0 from 0 to 7r/2, was convex downward, i.e., it took its minimum at the endpoints. Thus Conjecture 14 was false. Indeed further computations shows that there exists a O(M), 0 < O(M) < ~r/2, such that, for some M0 > 0, m a x [Aa(M, 0), A3(M, 7r/2)] < A3(M, O(M)) for 2.83 < M < M0 < 5.

10. A.

Schild's

2/3

conjecture

Another long standing conjecture that was proved false was the 2/3 conjecture. Let rl = rl(f) be the radius of convexity of f , i.e. r a ( f ) = sup{r : f ( A r ) is a convex domain}. P u t d* = min{If(z)l : Iz[ = rl} and d = inf 1/31 for which f(z) 7~/3. In 1953 in [69], A. Sehild conjectured that d*/d >_2/3 for all functions f G S*. Here equality holds for the Koebe function k(z) = z(1 + z) -2. Schild noted that d*/d _> r I ~ 2 - - V ~ and proved the conjecture for p symmetric functions, p _> 7. He also showed for a certain class of circularly symmetric functions that d*/d > .49. Lewandowski in [52], proved the conjecture true for certain subclass of S*. In [58], McCarty and Tepper obtained the best known lower bound of .38 for all starlike functions. The conjecture was shown false by the author and Lewis in [15] by giving two explicit counter examples. The first example is given simply by the two slit m a p defined by Z

f(z) = (1 - z ) o ( l + z) 2-~ ' where a is sufficiently near 0. It was noted that if d is computed as a function of a, then a'(d) --+ +co as a --+ 0 so that a minimal value of .656 for d*/d was obtained for this function at a ~ .03. The second example is a more complicated function that m a p s A onto the circularly symmetric domain shown in Figure 4.

Open Problems and Conjectures in Complex Analysis

19

i!iiiiiii iiiii ii!ii iii i!~: !i!iii!

! :

Figure 4 An explicit formula for this function g~, determined by Suffridge in [72], is given by

1 } z

~l----z-~

]

+

1-zJ

2

+2log

[(1 + 2az + z2) 'I2 + 1 + z]

'

where a = 2b2 - 1 and d = [(1 + b)l+b(1 -- b)'-b]- " with ¢ = ~r(1 - b). A close approximation to the minimum of d*/d for this function is 0.644 given by a ~ 0.89. Also "~b~ .03zr for this minimum value. The author's work suggests: C o n j e c t u r e 15. inf d*/d = min {d*/d for g~} ~ .644...

.f 6 S *

11. Brannan's

a

coefficient

conjecture

for certain

power

series

An innocent looking, but not so trivial, conjecture was made by Brannan in 1973 in [26] on the coefficients of a specific power series. The problem originated in the Brannan, Clunie, Kirwan paper [28] (later completed by a Aharanov and Friedland in [1]) solving the coefficient problem for functions of bounded boundary rotation. Consider the coefficients in the expansion

(1 + xz)o _ ~ A~O,~(x)z,, (1 - z ) ~

Ixl = 1,~ > o,~ > o.

n_-0

Brannan posed the problem as to when (30)

A(~'~)(x) < A(~'Z)(1).

R. W. Barnard

20 He gave a short elegant p r o o f t h a t (30) held if/3 t h a t for/~ = 1, 0 < a < 1, (30) did not hold for the (30) held for o d d coefficients in a sufficiently small t h a t for 0 < c~ = / 3 < 1, IAf~'~)(x)l _< A~7'~)(1). By

= 1 and a > 1. However, he showed even coefficients a n d t h a t for x = e i°, n e i g h b o r h o o d of 8 = 0. He Mso n o t e d using the e x p a n s i o n

A(~,Z)(x) = (/3)(/3 + 1 ) - - . (/3 + n - 1) n! 2Fl(-n, ~, 1 -/3; - x ) a n d t h e p r o p e r t i e s of 2F1, the h y p e r g e o m e t r i c function, this a u t h o r has s h o w n t h a t (i) (30) holds for a _ 1 and IA(~'~)(x)l < A(~'a)(1) for Ixl = 1, x # 1 a n d n=1,2,3,.... (~,I)

(ii) ]A~:~_~(x)] 1,/3 > 1. Milcetich, in [59], has recently shown t h a t (30) holds for n = 5,/3 = 1 and 2 < c~ < n but does not hold for n o n integer o/s less t h a n n - 1,/3 near zero, for o d d n > 3. T h e basic

Conjecture

16. [A~:~(x)I < A ~ : ~ ( 1 )

is still open.

12. P o l y n o m i a l approximations using a differential equation model A n o t h e r conjecture on special functions arose out of the a u t h o r ' s a n d L. R e i c h e l ' s work on p o l y n o m i M a p p r o x i m a t i o n s using a differential e q u a t i o n model. G i v e n equidis t a n t d a t a (x~, y~) with x~ = 1 - (2i - 1)/M, the p r o b l e m is to best fit a p o l y n o m i a l of given degree N - 1 to M d a t a points. A c o m p a r i s o n is used b e t w e e n the discrete n o r m If" liD, defined b y 1 M

IIIlt2D = -~ ~ If( x, )l 2 t

a n d the continuous n o r m , I1" He, defined by 1 tlfllo =: ~ mea~cIf(x)l • G r a m p o l y n o m i a l s {~j} are used where they are o r t h o n o r m a l in the discrete n o r m with an e x p a n s i o n for p given by

J so t h a t [[p[[~ = ~ a j.2 T h e s e are defined recursively by

Open Problems and Conjectures in Complex Analysis

(31)

~ N ( X ) -----2X~N-I~N-1

21

-- (O~N-1/C~N-2)~N-2(X),

where

U ( N 2 : 1 / 4 ~ 112 aN = ~ - ~,M2 _ N2 ]

•

The asymptotics as M and N --* oo are studied by letting r = N/v/-M and x =

1 - (/M. T h e n the recurrence relation in (31) can be used to obtain qON -- 2TN-, + ~ N - 2 = [T2 - ( 1 / 4 r 2) -- 2¢] ~ON/M + O(1). This in turn can be used to obtain the differential equation model: (32) where t = v - 1 / ~

~"(t) = [t2 - ( 1 / 4 t ~) - 2~] ~(t), and the initial condition as t ~ 0 is defined by CN [1 - e/M] =

2~/-2-~v/t+ O(1/M),

i.e., T(t) ~ v~(t --~ 0). A normalization is made by ~(t) ~ ~/~f2-vt-M where ~ is an odd positive integer if and only if x is a grid point. The solution to (32) is given by

~(t) = tl/~e-t2/21Fl ( ~ 2 ~,1;t2) , where 1F1 is Kummer's confluent hypergeometric function (see Gradshteyn and Ryzhik [411). To find error estimates for least square approximates by these polynomials an application of Brass's result in [29] can be used that gives error estimates for least square norms in terms of the uniform sup norm. But in order to apply this result all the ~0N(1 -- ~/M)'s must have their sup norms occur at the right end point of the interval [ - 1 , 1]. An extensive computer analysis suggested that this does occur. W h a t is needed then is to verify C o n j e c t u r e 17. For all ~ > 0 and reM t we have

Indeed, by converting to the Whittaker functions M~,v(x) see [41], for a more convenient range of variables the conjecture is equivalent to showing that

~'I~,o(X) __0 and x _> 0. We have verified Conjecture 17 for the regions dotted in Figure 5.

R. W. Barnard

22

:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: /~0 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ~ ~::::::::::::::::::: ~i

~i ~i 1 ~i

~!~! ~i

~i!!!i!i!!!iii!iii::

i~~:::::::::::"~"::: ,..:t:i!!i!!~iiii;iiiiii ,--:::::::~::::::::::::::::::: ...:::iiiiiiiiiiiii~iiiiiiii~!~iiiiii ~iiiiiiiiiiiiiiiii i...:::ii !i i i i i i i i i ~i i?!?i~i !i ~i i i

?

ii~i Ti iTi :~ i:~ !

X0

I{ I I I

X

Figure 5

References

[1] D. Aharonov, S. Friedland, On an inequality connected with the coe~icient conjecture for functions of bounded boundary rotation, Ann. Acad. Sci. Fenn. Ser. A.I.

524 (1972). [2] H.S. A1-Amiri, On the radius of univalence of certain analytic functions, Colloq. Math. 28 (1973) 133-139.

[3] R.M. All, Properties of Convex Mappings, Ph.D. Thesis, Texas Tech University.

[4] H. Alt, L. Caffarelli, Existence and regularity for a minimum problem with free boundary, J. Reine Angew. Math. 325 (1981) 105-144.

[5] J.M. Anderson, K.E. Barth, D.A. Brannan, Research problems in complex analysis, Bull. London Math Soc. 9 (1977) 129-162.

[6] R.W. Barnard, A variational technique for bounded starlike functions, Canadiaa Math. J. 27 (1975) 337-347.

[7] R.W. Barnard, On the radius of starlikeness of (z f)' for f univalent, Proc. AMS. 53 (1975) 385-390.

[8] R.W. Barnard, On Robinson's 1/2 conjecture, Proc. AMS. 72 (1978) 135-139. [9] R.W. Barnard, The omitted area problem for univalent functions, Contemporary Math. (1985) 53-60.

[10] R.W. Barnard, W. Dayawansa, K. Pearce, D. Weinberg, Polynomials with nonnegative coe~cients, (preprint).

Open Problems and Conjectures in Complex Analysis

23

[11] R.W. Barnard, R. Evans, C. FitzGerald, The center divided difference of polynomials (preprint). [12] R.W. Barnard, W.T. Ford, H. Wang, On the zeros of interpolating polynomials, SIAM J. of Math. Anal. 17 (1986) 734-744. [13] R.W. Barnard, C. Kellogg, Applications of convolution operators to problems in univalent function theory, Mich. Math. J. 27 (1980) 81-94.

[14] R.W. Barnard, C. Kellogg, On Campbell's

conjecture on the radius of majorization of functions subordinate to convex functions, Rocky Mountain J. 14 (1984) 331339.

[15] R.W. Barnard, J.L. Lewis, A counterexample to the two-thirds conjecture, Proc. AMS 41 (1973) 525-529. [16] R.W. Barnard, J.L. Lewis, Coej~cient bounds for some classes of starlike functions. Pacific J. Math. 56 (1975) 325-331. [17] R.W. Barnard, J.L. Lewis, Subordination theorems for some classes of starlike functions, Pacific J. Math. 56 (1975) 333-366. [18] R.W. Barnard, Lewis, J.L. On the omitted area problem, Mich. Math. J. 34 (1987) 13-22. [19] R.W. Barnard, K. Pearce, Rounding corners of gearlike domains and the omitted area problem, J. Comput. Appl. Math. 14 (1986) 217-226. [20] R.W. Barnard, G. Schober, Mdbius transformations for convex mappings, Complex Variables, Theory and Applications 3 (1984) 45-54. [21] R.W. Barnard, G. Schober, Mdbius transformations for convex mapping8 II, Complex Variables, Theory and Applications 7 (1986) 205-214. [22] B. Beauzamy, Jensen's inequality for polynomials with concentration at low degrees, Numer. Math. 49 (1986) 221-225. [23] B. Beauzamy, P. Enflo, Estimations de produits de polyndmes, J. Number Theory (1985) 21 390-412. [24] S.D. Bernardi, A survey of the development of the theory of schticht functions, Duke Math. J. 19 (1952) 263-287. [25] R.P. Boas, Entire Functions Academic Press, New York, (1954).

[26] D.A. Brannan,

On coefficient problems for certain power series, Symposium on Complex Analysis, Canterbury, (1973) (ed. Clunie, Haymasa). London Math. Soc. Lecture Note Series 12.

[27] D.A. Brannan, J.G. Clunie, J.G. (eds.). Aspects of contemporary complex analysis. (Durham, 1979), Academic Press, London (1980).

24

R. W. Barnard

[28] D.A. Brannan, J.G. Clunie, W.E. Kirwan, On the coefficient problem for functions of bounded boundary rotation, Ann. Acad. Sci. Fenn. Set. A.I. Math. 3, (1973). [29] H. Brass, Error estimates for least squares approximation by polynomials, J. Approx. Theory. 41, (1984) 345-349. [30] J. Brown, A coefficient problem for nonvanishing H p functions, Rocky Mountain J. of Math. 18 (1988) 707-718. [31] J. Brown, A proof of the grzyi conjecture for n = 4, (preprint). [32] D. Campbell, Majorization-Subordination theorems for locally univalent functions, Bull. AMS 78, (1972) 535-538. [33] D. Campbell, Majorization-Subordination theorems for locally univalent functions II, Can. J. Math. 25, (1973) 420-425. [34] D. Campbell, Majorization-Subordination theorems for locally univalent functions III, Trans. Amer. Math. Soc. 198 (1974) 297-306. [35] S. Chandra, P. Singh, Certain subclasses of the class of functions regular and univalent in the unit disc, Arch. Math. 26 (1975) 60-63. [36] P. Duren, Univalent Functions. Springer-Verlag, 259, New York, (1980). [37] R.J. Evans, K.B. Stolarsky, A family of polynomais with concyclic zeros, II, Proc. AMS. 92, (1984) 393-396. [38] A.W. Goodman, Note on regions omitted by univalent functions, Bull. Amer. Math. Soc. 55 (1949) 363-369. [39] A.W. Goodman, E. Reich, On the regions omitted by univalent functions II, Canad. J. Math. 7 (1955) 83-88. [40] A.W. Goodman, Univalent Functions. Mariner (1982). [41] Gradshteyn, Ryzhik. Table of integral, series and products, Academic Press, 1980. [42] R.R. Hall, On a conjecture of Clunie and Sheil-Small, Bull, London Math. Soc. 12 (1980) 25-28. [43] E. Gray, A. Schild, A new proof of a conjecture of Schild, Proc. AMS 16 (1965) 76-77, MR30 #2136. [44] W.K. Hayman, Multivalent functions, Cambridge Tracts in Math. and Math. Phys., Cambridge University Press, Cambridge, 1958. [45] C. Horowitz, Coefficients of nonvanishing functions in H ~, Israel J. Math. 30, (1978) 285-291. [46] J.A. Hummel, S. Scheinberg, L. Zalcman, A coefficient problem for bounded nonvanishing functions, J. Analyse Math. 31 (1977) 169-190.

Open Problems and Conjectures in Complex Analysis

25

[47] J. Jenkins, On values omitted by univalent functions, Amer. J. Math. 2 (1953) 406-408. [48] J. Jenkins, On circularly symmetric functions, Proc. AMS. 6 (1955) 620-624. [49] J. Krzy~, Coefficient problem for bounded nonvanishing functions, Ann. Polon. Math. 70 (1968) 314. [50] L. Kuipers, Note on the location of zeros of polynomials III, Simon Stevin. 31 (1957) 61-72. [51] V. Levin, Aufgabe 163, Jber. Dt. Math. Verein. 43 (1933) p. 113, LSsung, ibid, 44 (1934) 80-83 (solutions by W. Fenchel, E. aeissner). [52] Z. Lewandowski, NouveUes remarques sur les thdor~mes de Schild relatifs d une classe de fonctions univalentes (Ddmonstration d'une hypoth~se de Schild), Ann. Univ. Mariae Curie-Sklodowska Sect. A. 10 (1956). [53] Z. Lewandowski, On circular symmetrization of starshaped domains, Ann. Univ. Mariae Curie-Sklodowska, Sect A. 17 (1963) 35-38. [54] J.L. Lewis, On the minimum area problem, Indiana Univ. Math. J. 34 (1985) 631-661. [55] R.J. Libera, A.E. Livingston, On the univalence of some classes of regular functions, Proc. AMS 30 (1971) 327-336. [56] A.E. Livingston, On the radius of univalence of certain analytic functions, Proc. AMS 17 (1966) 352-357. [57] T.H. MacGregor, Geometric problems in complex analysis, Amer. Math. Monthly. 79 (1972) 447-468. [58] C. McCarty, D. Tepper, A note on the 2/3 conjecture for starlike functions, Proc. AMS 34 (1972) 417-421. [59] J.G. Mileetich, On a coefficient conjecture of Brannan, (preprint). [60] S. Miller, (ed.). Complex analysis, (Brockport, NY, 1976), Lecture Notes in Pure and Appl. Math. 36, Dekker, New York, (1978). [61] D. Moak, An application of hypergeometric function to a problem in function theory, International J. of Math and Math Sci. 7 (1984). [62] Z. Nehari, Conformal Mapping McGraw Hill, (1952). [63] K.S. Padmanabhan, On the radius of univalence of certain classes of analytic functions, J. London Math. Soc. (2) 1 (1969) 225-231. [64] K. Pearce, A note on a problem of Robinson, Proc. AMS. 89 (1983) 623-627. [65] D.V. Prokhorov, J. Szynal, Coefficient estimates for bounded nonvanishing functions, Bull. Aca. Polon. Sci. Ser. 29 (1981) 223-230.

26

R. W. Barnard

[66] A.K. Rigler, S.Y. Trimble, R.S. Varga, Sharp lower bound8 for a generalized Jensen Inequality, Rocky Mountain 3. of Math. 19 (1989). [67] R. Robinson, Univalent majorants, Trans. Amer. Math. Soc. 61 (1947) 1-35. [68] St. Ruscheweyh, A subordination theorem for aS-like functions, J. London Math. Soc. (2) 13 (1973) 275-280. [69] A. Schild, On a problem in conformal mapping of schlicht functions, Proc. AMS 4 (1953) 43-51 MR 14 #861. [70] J. Stankiewiez, On a family of starlike functions, Ann. Univ. Mariae CurieSklodowska, Sect A. 22-24 (1968-70) 175-181. [71] K.B. Stolarsky, Zeros of exponential polynomials and reductions, Topics in Classical Number Theory. Collog. Math. Sec., J£nos Bolyai, 34, Elsevier, (1985). [72] T. Suffridge, A coefficient problem for a class of univalent functions, Mich. Math. J. 16 (1969) 33-42. [73] O. Tammi, On the maximalization of the coefficient a3 of bounded ~chlicht functions, Ann. Acad. Sci. Fenn. Ser. AI. 9 (1953). [74] J. Waniurski, On values omitted by convex univalent mappings, Complex Variables, Theory and Appl. 8 (1987) 173-180. Received: August 30, 1989

Computational Methods and Function Theory Proceedings, Valparafso 1989 St. Ruseheweyh, E.B. Saff, L. C. Salinas, R.S. Varga (eds.) Lecture Notes in Mathematics 1435, pp. 27-31 (~) Springer Berlin Heidelberg 1990

A Remarkable Cubic Mean Iteration J . M . B o r w e i n and P . B . B o r w e i n

Mathematics, Statistics and Computing Science Department Dalhousie University, Halifax, N.S. B3H 3J5, Canada

1.

Introduction

Consider the two term iteration defined by (1.1)

a,~ + 2b,~ a,,+l . - - - ,

a0 := a,

3

and

(1.2)

b0 := b.

3

Then since (1.3)

3

z

%+1 -

b.+l

(as - b.) 3 =

27

'

it follows that, for a,b E (0, co), and for n >_ 1,

la.+l

-

b,~+ll _
0.

As an application we obtain a set of new inequalities for polynomials: T h e o r e m 1.2 I f P E P~(J2), J2 = C \ [1,oo) then for z E D n+l zr IImP(z)l _< - - - ~ cot n +--~'

-

COS22n~2 rr < Re P(z) < c°t2 2n + 2 - sin ~ sin 2--~+2 ip(z)l _< cot~

A11 bounds are best possible.

~r n+2"

On the Maximal Range Problem t'or Slit Domains

35

T h e o r e m 1.3 Let ~? = C \ ( ( - o % - 1 ] t3 [1,oo)), n odd. Then for P E T',~(~) we have in D 1 ,n r,~ ,,ir~er~z)l < sin '~ ' n+l

IP(z)l
r. Then, f o r a certain .~ E R,

(8)

=

_ .(~+r)~

Furthermore, ~. Rep(~k) > O, &. Imp(~T) _< O.

_

A. Cdrdova Ydvenes, St. Ruscheweyh

38

Proo£ Let Q(z) = 1 - z TM. By Lagrange interpolation we obtain P(z) =

Q(z) ~

¢JP(~i)

j+l

and, after differentiation,

~kP'(~k) = ~., ~ k - ~j +

(9)

,~+1 j = l ~jp,(~j___~) 2 j#k

=~ = e(O

(10)

P(~k)

cot X : / ~

+i

+

jCk Taking real parts and using (5), (6) we get n+l

(11)

(n + 1)ReP(#k) = ~ I m P ( ~ j ) c o t (j - k)~ , j=l n+l j#k

k = 1 , . . . , n + 1.

Now let k = r, s. Then using (8) we obtain r -

3)~

(n + 1)ReP((~)

=

ImP(G)cot--

(n + 1 ) R e P ( G )

=

(~ - ~)~ ImP(~r)cot - n+l

(12)

n+l

'

Taking imaginary parts in (12) one gets

(13)

I m P ( G ) = -ImP( O.

But P(z; n) has contact with the tip of its corresponding slit at z = 1 and this determines A. We have proved: T h e o r e m 2.1 For 12 = C \ [1, oo) we have 012.\012=

{ P(e~e; n) P(1;n)

: [8-rr[
__0,

as a general representation for extremal polynomials related to boundary points of f2,~(a) in the upper half plane. The final step is to show that for every a E (cot -2 2~-~,cot 2 ~-7~) there is exactly one system (a, fl, r) which fulfills the remaining conditions for an extremal polynomial + r~ (r+2)~ as pointed out in Theorem 1.1. Using (18) and Lemma 2.1, we get for 0 # n+l, - ,+1 (mod 2rr)

p(eiO)

=

n+l

=

__i (1 - ( - 1 ) % i("+O°) [aft(0)4- #f'+l(' 0 + 7)] • 2n

2-'---~ [aft(0) 4- ~fr+l(O 4- 7)]

(19)

where ^t = ,%-f,~fj(8) = 1/(cos 8 - cos ~,~+1)"Let us now fix r and assume 04 fl # 0. Then

af'~(O) 4- fif;+l( 4 - ? ) # 0 ,

8 = -i~-n+1,J

0,...,n, j # r

and it never changes sign when passing through one of these points. Hence the imaginary part of P(e ~e) cannot change sign in these points as well. But since P(0) = 0, the graph of P(e~°), O E [0, 2~r) has to cross the real axis at least twice, and this must occur at the zeros of g(8) = aft(O) 4- flf'+l(O 4- 7).

A. Cdrdova Ydvenes, St. Ruscheweyh

42

However, the terms of g have different sign only in the i n t e r v a l s / 1 = ( - 7 , 0), I2 = (~r - 7, rr). It is easily checked that h(O)=

frl(8) __ sin8 (cos(8 +_ 7_.)- _cos((___r_+ 1 ) 7 ) ) 2 y'+, (8 + 7) sin(8 + 7) cos 8 -- cos r 7

is monotonically increasing from - o o to 0 in b o t h / 1 a n d / 2 , such t h a t the equation

h(8)=---z

(20)

has exactly two solutions: 81 E / 1 and 82 E/2. But if P is an extremal polynomial of the t y p e described in T h e o r e m 1.1 then P(e ~°~), P(e ~°~) must be the vertices of the two slits otherwise there would be an are between (and including) the images of two zeros of P' without point of contact ( compare Figure la). Since P(e ~°~) > 0 , p(e~0~) < 0 we can identify the corresponding slits. To have really candidates for our normalized situation ~2(a) we therefore have to replace P by P / P ( e ~°~). Then, in the new notation, we have

p(eie2) --_ aft(02) +/~fr+l(82 + 7) ~f~(0,) + ~f~+1(81 + 7)' where we obviously can replace/3 by 1 - a and assume 0 < a < 1. In fact, all considerations r e m a i n valid, in a limiting sense, for a = 1. T h u s all what remains to be done is to show t h a t for each a E (cot -~ ~ ' k cot 2 ~ k ) there is at most one pair (a,r) with 0_ O, An-t - 2A~ >_O, and Ak-1 -- 2Ak + Ak+l _> 0 for k = 1,2 . . . . , n - 1. Then:

A0 }-~Akcos(k~)

(25)

t(~) := T +

> o foraU

~eR.

k=l

3. Results for the weighted approximation problem (15) In this section, we are concerned with the constrained Chebyshev approximation problem (15). In the sequel, it is assumed that n C N, r > 1, and c E C \ gr. Standard results from approximation theory (see e.g. [6]) then guarantee that there always exists a unique optimal polynomial for (15). For the case r = 1 of the unit interval gt = [-1, 1] and c E R \ [-1, 1], Bernstein [1] proved that the resealed polynomial (10)

v,&)

(26)

v~(z; c) - v,~(e)

is the extremal function for (15). For purely imaginary c and, again, r = 1, Freund and Ruscheweyh [4] showed that the optimal polynomial is a suitable linear combination of v,~, v~-l, and v~-2. To the best of our knowledge, these two cases seem to be the only ones for which the solution of (15) is explicitly known. For the rest of the paper, we assume that r > 1. It turns out that, somewhat surprisingly, (26) is also best possible for the general class (15) with complex c as long as c is not "too close" to gr. For the following, it will be convenient, to represent c ¢ g~ in the form

(27)

c=cr(¢)---~(R+~)cos¢+

(R-

)sine,

R>r,

-Tr
In analogy to (19) and (20), it follows that

(2s)

dk := Tk+a/2(c) = Ak cos(k + ~)~b + iBk sin(k +

)¢,

where (29)

1

Ak := ~(R

k+i/2

1 + Rk---T~i~)

and

I

Bk := ~(n

k+l/2

1

Rk+i/2).

Based on Rivlin and Shapiro's characterization [10] of the optimal solution of general linear Chebyshev approximation problems, we next derive a simple criterion for the polynomial (26) to be best possible in (15). Note that the extremal points of v,~(z; c) are just the zl, l = - n , . . . , n, stated in (22). By applying the theory [10] to (15), (26), and by using (23), we obtain the following

On Bernstein T y p e Inequalities ...

51

C r i t e r i o n . v~( z; c) is the unique optimal polynomiaJ for (15) iff there exist nonnegative reM numbers a-n, a - n + 1 , . . . , a~ (not all zero) such that (30)

~

al(--1)tw(zl)q(zl) = 0

f o r all

q e II~

with

q(c) = O.

Clearly, it suffices to check (30) for the polynomials

q(z) - Vk(z) - Vk(c),

k --- 1, 2 , . . . , n.

With (17) and (28), this leads to the following equivalent formulation of (30): (30')

~

al(-1)'(doTk+l/2(z,)

-

dkT1/~(z,))

=

O,

k

=

1, 2 , . . . , n.

It turns out that there are simple formulae for all real solutions crz of (30'). The trick is to use the ansatz (31)

n at = ~'~(#j cos(j~l) + ujsin(j~l)), j=0

~t -

2br 2n + 1 '

l = -n,..

''

n,

where #j, uj E R, j = 0 , . . . , n. Note that such a representation (31) is possible for any collection of c~_~,..., c~n E R. Now we insert (31) into (30') and rewrite Tk+l/2(zt) and T~/~(zl) in the form (19). Then, a routine calculation, making repeatedly use of Lemma 1, shows that (30') reduces to the equations (323)

tLkan-k -- iukb~-k - (d,-k/do)(#nao - iu~bo) = O,

k = 1, 2 , . . . , n - 1,

and

(32b)

2#0a~ - ( d , / d o ) ( p , ao - iunbo) = O.

By determining all real solutions #k, uk of the linear system (323,b) and with (31), one easily verifies that all real numbers satisfying (30') are given by al = wa~, l = - n , . . . , n with r E N arbitrary and a~ defined in (33). Hence, in view of the Criterion, we have proved the following T h e o r e m 2. v~(z; c) is the unique optima1 polynomial in (15) iff the numbers

(Re( 1, the polynomial v~(z; c) is not best possible in (15) for all c E R \ Er. Indeed, numerical tests show that among the corresponding numbers (33), in general, positive and negative a~ occur if c is very close to £~. Finally, we note that Theorem 3 is analogous to the following result for the unweighted approximation problem (14).

a)

T h e o r e m A. (Fischer, Freund [3]). Let c = cn(¢) with R > r > 1 (cf. (27)). Then:

r ~ + 1/r ~ E~(r,c) < R ~ + 1 / R .

(35) b) I f R >_ r(73r 4 - 1)/(r 4

-

1), then

pn(z; c) -- (Rn - 1/Rn)T~(z) + 2i sin(n¢) ( R '~ 1/R~)Tn(c) + 2i sin(n¢) is the unique optimal polynomial for (14) and equality holds in (35). R e m a r k 3. For n = t, (14) was solved completely by Opfer and Schober [7]. From their result, one can deduce (see [3]) that, for the case n = 1, the statement in part b) of T h e o r e m A is true for all R > r >_ 1. Clearly, in view of (13), Theorem 1 is an immediate consequence of Theorem 3, Theorem A, and Remark 3. The proofs of Theorem 3 and 4 will be given in the next section.

On Bernstein Type InequMities ... 4. Proofs

53 of Theorem

3 and

4

P r o o f of Theorem 3. With (17) and (21), it follows that

w(z)V~(z)l _ a, E,,+a/z _ r(33r - 1)/(r - 1). With

1

2 1 R 2"+~ >-2

and

r 4k+l

r4 k + 2 - 1

_
g'T' (," + 1 ) ( R - r) ( R - r - 3 2 r - 1 ) -> 0.

In view of Theorem 2, this concludes the proof of Theorem 3.

Proof of Theorem 4. First we consider the case (i), i.e. assume that (39)

1 1 1 1 c = ~(R + ~ ) _> r + -r - -'2

Then ¢ = 0 in (28), and the representation (33) reduces to a? = An 2 an +

A~-k cos(k~2t)), k=l

l = -n,...,n.

an-k

It follows that a~ = A,t(qpt) where t is the trigonometric polynomial (25) with (40)

An )~0:=-an

and

~k.-

An-k an-k

,

k=l,...,n.

R. Freund

54

Therefore, Theorem 2 together with Lemma 2 implies that vn is best possible in (15) provided that the numbers (40) satisfy the assumptions of Lemma 2. Hence, it remains to verify that the estimates (41)

A-A>2A° 31 ao

and

Ak+l

2A--3-k+Ak-1 _>0,

ak+l

ak

k=l,...,n-1,

ak-1

hold. It is easily seen that the first condition in (41) is equivalent to (39). A more lengthy, but straightforward, computation shows that (39) also guarantees that the remaining inequalities in (41) are satisfied. We omit the details. For the case (ii), c < - r , one proceeds similarly. Now ¢ = 7r in (28), and from (33) we obtain 1 Bn @, Bn-k a~=Bn(~7+~ cosk(~,+,~)), l = - n , . . . , n . k=l an-k

By applying Lemma 2, this time with (42)

B,.,. ~0:=-and an

~k.-

Bn-k

,

k=l,...,n,

an-k

and Theorem 2, we conclude that v~ is the optimal polynomial for (15) if the assumptions of Lemma 2 are satisfied. A lengthy computation shows that the condition c < - r indeed implies that the numbers (42) fulfill the required inequalities. Again, details are omitted here. • A c k n o w l e d g e m e n t . The author would like to thank Dr. Bernd Fischer for performing some numerical experiments which were very helpful for developing the results of Section 3.

References

[1]

S. Bernstein, Sur une classe de polynomes d'dcart minimum, C. R. Acad. Sci. Paris 190 (1930), 237-240.

[2]

C. Frappier, Q.I. Rahman, On an inequality of S. Bernstein, Can. J. Math. 34 (1982), 932-944.

[3]

B. Fischer, R. Freund, On the constrained Chebyshev approximation problem on ellipses, J. Approx. Theory (to appear).

[4]

R. Freund, St. Ruscheweyh, On a class of Chebyshev approximation problems which arise in connection with a conjugate gradient type method, Numer. Math. 48 (1986), 525-542.

[5]

N.K. Govil, V.K. Jain, G. Labelle, Inequalities for polynomials satisfying p(z) = znp(1/z)., Proc. Amer. Math. Soc. 57 (1976), 238-242.

[6]

G. Meinardus, Approximation of Functions: Theory and Numerical methods, Springer Verlag Berlin, Heidelberg, New York, 1967.

On Bernstein Type Inequalities ...

55

[7] G. Opfer, G. Schober, Richardson's iteration for nonsymmetric matrices, Linear Algebra Appl. 58 (1984), 343-361. [8] G. P61ya, G. SzegS, Aufgaben und Lehrsdtze aus der Analysis, Vol. I., 4th ed., Springer Verlag Berlin, Heidelberg, New York, 1970. [9] Q.I. Rahman, Some inequalities for polynomials, Proc. Amer. Math. Soe. 56 (1976), 225-230. [10] T.J. Rivlin, H.S. Shapiro, A unified approach to certain problems of approximation and minimization i J. Soc. Indust. Appl. Math. 9 (1961), 670-699. [11] W. Rogosinski, G. SzegS, Ober die Abschnitte yon Potenzreihen, die in einem Kreise beschrdnkt bleiben, Math. Z. 28 (1928), 73-94. [12] V.I. Smirnov, N.A. Lebedev, Function~ of a Complex Variable, Iliffe Books, London, 1968. [13] G. SzegS, Orthogonal polynomials, Amer. Math. Soc. Colloq. Publ., Vol. 23. Providence, R. I.: Amer. Math. Soc., 4th ed., 1975. Received: May 15, 1989

Computational Methods and Function Theory

Proceedings, Valparaiso 1989 St. Ruscheweyh, E.B. Saff, L. C. Salinas, R.S. Varga (eds.) Lecture Notes in Mathematics 1435, pp. 57-70 (~) Springer Berlin Heidelberg 1990

Conformal

Mapping

and

Fourier-Jacobi

Approximations D a v i d M. Hough 1 IPS, ETH-Zentrum, CH-8092 Ziirich

A b s t r a c t . Let f denote the conformal map of a domain interior (exterior) to a closed Jordan curve onto the interior (exterior) of the unit circle. In this paper, we explain how the corner singularities of the of the derivative of the boundary correspondence function can be represented by Jacobi weight functions, and study the convergence properties of an associated Fourier-Jacobi method for approximating this derivative. The practical significance of this work is that some of the best known methods for approximating f are based on integral equations for either the boundazy correspondence function or its derivative.

1. I n t r o d u c t i o n In this section the mapping problem is defined and a particular boundary integral representation for the conformal map is introduced. The contents of the rest of the paper are outlined at the end of this section. Let 0/2 := UN=I Fk denote a piecewise analytic Jordan curve in the complex plane whose component analytic arcs 11, F 2 , . . . , FN are defined by

rk := {z: z = Ck(t), -1 _< t _< 1},

k = 1,2,... ,N,

where ~k is analytic on a domain containing [-1, 1] and satisfies ¢~(t)#0,

-1 0

, (2 ~ int 052,

c := [f'(cx~)]-' > 0

, (2 -= ext 0(2.

and It is well known that the function f exists uniquely, is analytic almost everywhere in (2 and is continuous on 0(2, its only singularities at finite points in Y2 being branch point singularities at corner points on 0(2 T h e b o u n d a r y correspondence function Ok associated with the arc Fk is defined by

Ok(t) := arg(fo~k(t)),

(2)

where arg m u s t be defined so that each Ok is continuous on [--1, 1]. T h e functions N {Ok}k=1 completely define the m a p f and they, or their derivatives, are the f u n d a m e n t a l quantitites which are to be approximated in a number of integral equation methods for the numerical determination of f; see Henrici [3, §16.6-7]. If z = Ck(t) E 0(2 then it follows immediately from (2) that

folk(t) = exp(iOk(t) ),

(3)

whilst if z E Y2 then there are various boundary integral formulations t h a t m a y be used for the calculation of f(z). T h e b o u n d a r y integral formulation considered here is

(4)

f(z) =

z exp(iw - K(z)) c-' e x p ( K ( z ) )

if (2 = int 0~2, if (2 - ext 0(2,

Conformal Mapping and Fourier-gacobi Approximations

59

where w is a real constant, K : Y2 -+ C is defined by

(5)

N

1

K(z) := ~ j l uk(t) log(z - (k(t)) dt k=l

1

and uk : [ - 1 , 1] -+ R is defined by (6)

:=

o'k(t).'

see [2], [3, §16.61 and [51 for further discussion of representations similar to (4). The domain of definition of K may be extended to ~ provided the correct branch is taken for the logarithm appearing in (5). That is, if z ~ Fk then, as a function of the parameter t, log(z -- ~k(t)) must be continuous on [-1,11 whilst if z = £k(r) E -Fk then

(7)

log((k(r) - (k(t)) :=

lim log(z - ~k(t)) z-¢,(,).,~a 0

if r ¢ t,

if r = t. D

Hence, the formula (4) may be used to calculate f(z) for all z E 9 . Note that, in view of (1), w satisfies (8)

log b + iw = K(O)

and that, since If(z)l = 1 for all z C 0 9 , it follows from (4) that (9)

c = exp(Re{K(~)})

for any chosen point ~ E 0 9 . Also note that an immediate consequence of the definition (6) is that the functions {uk}k=t N satisfy (10)

~ l j F l uk(t)dt = l

In the next section it is explained how the presence of corners at the ends of the arc /'k induces end point singularities in the corresponding density function uk. In particular it is shown that these singularities are represented by a Jacobi weight function

wk(t) = (1 - t ) ~ k ( l + t ) a" , where ak , /3k are related to the interior angles of 9 at 4k(4-1). Also in §2 the FourierJacobi polynomial approximations to vk/wk are introduced and it is shown that these approximations converge almost uniformly on [-1, 1]. In §3 it is proved that the Fourier-Jacobi partial sums produce a uniformly convergent sequence of approximations to the map f . In §4 a brief outline of the collocation m e t h o d for the approximate solution of Symm-type integral equations is given and it is shown how this may be viewed as a method for estimating the Fourier-Jacobi partial sums.

D.M.

60 2. S i n g u l a r i t i e s

of

N {/]k)k=l

Hough

and Fourier-Jacobi polynomial

approximations

In order to avoid the unnecessary use of subscripts in this section, we let F, (, 0 and t, denote respectively a typical arc from the set {Fk}N=I and the associated functions (k, Ok and vk. From (3) and (6), u can be expressed directly in terms of f as

i(f°C)'(t) 2~r/o¢(t) "

v(t)=

Since f o ~ is analytic and non-zero on the open interval ( - 1 , 1) it follows that y is also analytic on ( - 1 , 1). However, since the arc end points ~(=kl) are usually corner points of 0Q, fo~ and hence v are not usually analytic at ±1. Using the results of Lehman [8], who derives the asymptotic expansion of the map f in the vicinity of a corner formed by analytic arcs, we have previously established the corresponding asymptotic expansion for the density v in the vicinity of the end points ±1; see Hough and Papamichael [6]. In order to describe this expansion, let Ar and #Tr denote the angles interior to ~2 at the points ¢(1) and ¢ ( - 1 ) respectively and define a:=-l+A

-1,

j3 := - 1 + #-1.

It is assumed always that {A, #} E (0, 2) so that

Then, from the results given in [6] it follows that there exists a number 6 with 0 < 6 < 1 such that (11)

u(t)= { (1-t)~(aCx(1-t)+x-(1-t)) (l+t)Z(bCz(l+t)+x+(l+t))

ifl-6 1 , c~* : =

irA 0 and/3+ > 0 by ~+

• --

2a + 1 4

/3+

,

. -

2/3 + I 4 '

and define # : [-1, 1] -~ R by ¢~(t) : = (1 - t)~+(1 + t ) O + ¢ ( t ) .

Clearly, from Proposition 2.1, • is H51der continuous on [- 1, 1]. Let s, be the polynomial of degree n formed by the nth partial sum of the Fourier-Chebyshev approximation to on [-1, 1]; i.e., s~ocos is the nth partial sum of the classical Fourier cosine series expansion of #ocos on [-Tr, ~r]. For any t E ( - 1 , 1), ¢(t) - S~(t) =

(1 - t)-~+(1 + t)-0+(4~(t) - s~(t))

+ (i

+

-

-

&(O.

Therefore, if 6 is any fixed number in (0,1) it follows that (17)

max tE[--1+5,1--61

I¢(t) - S~(t)I < 6-°+{ ~' --

+

max

te[--1+6,1--5]

max

t~[-l+&l-6]

l ¢ ( t ) - s.(t)[

t(1

&(t)l

•

But # is Hblder continuous on [-1, 1] and hence the Dini- Lipschitz criterion ensures that s, converges uniformly to # on [-1, 1]; see., e.g., Rivlin [11, §3.4]. Also, ¢ satisfies the conditions of Szeg6's equiconvergence theorem mentioned above, which states that the second term on the right of (17) tends uniformly to O on [-1 + 6,1 - 6]. Hence, given any e > 0 there exists an integer ni = nl(e, 6) such that max

t¢[-~+a,,-~]

I¢(t) - &(t)l


nl. This proves the proposition, since uniform convergence on [-1 + 5, t - 6] certainly implies pointwise convergence on (-1,1). •

The fact that Sn is the best approximation of degree n to ¢ in the norm associated with the inner product (16) is sufficient to guarantee certain uniform convergence properties for the corresponding approximation to the conformal map f , as is explained in the next section.

ConformM Mapping and Fourier-Jacol~ Approximations 3. T h e

uniform

convergence

of approximations

63

to the map

f

In this section we revert to the use of subscripts to identify quantities associated with a particular arc Fk. A number of preliminary definitions are required, as follows. Let wk denote the Jacobi weight function associated with the arc Fk and let Ck be the corresponding quotient function ¢k := v~/wk; see (13), (14). Also, let L~[-1, 1] denote the set of complex valued functions g defined on [-1, 1] such that v/-~lg] EL2[-1, 1]. L~ is a Hilbert space with inner product

wkg]~dt

(g, h)k := 1

and corresponding norm

Itgll~ := VJ-~

wktgpdt.

For functions f which are defined on Y) we also intoduce the uniform norms IlflI~ := s u p l f ( z ) l ,

]]fII0a := sup If(z)l.

zED

zEO,Q

In the case where f is analytic on ~2 with a continuous extension to 0~2, the m a x i m u m modulus principle implies that the above two norms are identical. Given any z E Y2 let l, : 0Y) --- C be defined by

(18)

zz(~) :=

{

log(z-~) 0

-

ifze~Q,~e0f2,z#~,

if z = ~ e 0 ~ ,

where the logarithm branch is the same as that described prior to (7), so that log(z - (k(t)) = Izo(k(t). It can be proved that lzo(k eL~ for all finite z E ~2. Now let us suppose that each function Ck is approximated by a real polynomial Pk,s of degree n and that a corresponding approximation fn to f is generated from (4) by (19)

f~(z)

:=

z exp(iws - Ks(z)) Cn1 exp(K~(z))

if ~2 - int 0/2, if ~2 --= ext OQ,

where, using the above inner product notation, N

(20)

Ks(z)

:= ~(~oCk, Pk,T,)~ • k=l

T h e estimate w,~ and approximate inner radius, b~, are obtained from (8) as (21)

log b,~ + iWs = Kn(O)

whilst c,~ is determined using (9) as

(22)

c~ = exp(~{Ks(~)}).

Observe that in the case ~2 --extO~2, Izo(k ~- logz as z -~ c~ so that

D.M. Hough

64 N

K~(z) ~ (~{1,Pk,~)k)log z . k=l

Thus, it follows from (19), that in order for fn(z) to have a simple pole at oo the polynomials Pk,~ must satisfy N

(23)

E O. Proo£ Any 3' can be viewed as starting at the point a and proceeding first to either 0 or b. If to b, we can deform it from a to 0 and then to b. In either case, we assign a positive sign to the first segment of the path. If it continues to b, we assign a plus 1 to the path from 0 to b when the path does not go around O. Thus the line segment from a to b has signature 1,1 as in Figure 1. f"

"II

f"

"1

!

'

'

I°0

!

! ! I.

*b

ob [

\,

I.,,o

I

*a

L

! J

J

Figure 5

L

J

Figure 6

Every simple arc joining a and b in A\{0, a, b} has a signature nl, n2 with nl and n2 odd and n1 > 0. We need only show that each relatively prime odd pair nl, n2 with nl > 0 determines an arc and that there are no arcs for which the signature is not a relatively prime pair. The Dehn-Thurston theorem [1] asserts that the arc 7 can be represented up to homotopy by a special system of arcs in a decomposition of the region into two pairs of pants and a belt as shown symbolically in Fig. 5 where we decompose A\{0, a, b} into three regions by two disjoint Jordan curves containing a and b in their interior and 0 in their exterior. These curves are represented by the dashed lines in Fig. 5. (The second leg of the outer pair of pants is the boundary of the unit disk, and is not shown.) Except for the special case of the homotopy class of the line segment joining a to b (which will be just that), the Dehn-Thurston theorem tells us that the structure in the inner region consists of two arcs joining a and b to the belt and some number, say k - 1, of disjoint arcs with both end points on the belt, separating a from b. The outer

N u m e r i c a l S o l u t i o n s o f t h e Schiffer E q u a t i o n

77

region will contain k disjoint arcs with both end points on the belt, separating 0 from [w[ = 1. The two sides will be connected by 2k disjoint arcs crossing the belt. Figure 5 shows the case of k = 2 with direct connections across the belt, resulting in a signature of 1,-3. For a general k, direct connection across the belt defines a curve with signature 1 , - ( 2 k - 1). T h e only remaining variable is the number of D e h n Twists in the belt. This can be any integer, positive or negative. Fig. 6 shows the result of a Dehn Twist of +1. Each arc crossing the belt has been moved up one position on the left. The arc connecting to the top position on the left goes around the belt to connect to the b o t t o m left position. The signature is nov,, 3,-1. T h a t is, each integer in the signature has been increased by two. Another Dehn Twist would similarly result in a signature of 5,1. It is easy to see that the result of each Dehn Twist is equivalent to adding an arc going completely around the points a and b. Half of this is the new arc around the belt, while the other half is made up of the totality of the rest of the shifts. A twist in the reverse direction would subtract two from each n u m b e r in the signature. If nl becomes negative, we would no longer have the standard form for nl, n2, but the results remain valid. (For some initial configurations, the results of the positive or negative twists will be reversed, but in every case, a twist in one direction will add two while the reverse twist will substract two.) Let nt, n2 be any pair of odd integers with nl > 0. By interchanging the roles of a and b if necessary, we m a y assume without loss of generality that n2 _< nl. Set j = (nl - 1)/2 and substract 2j from b o t h nl and n2 to give the pair 1 , m where m = n2 - n l + 1. T h e n m < 1. Suppose first that rn < 0. Letting k = (1 - m ) / 2 and making direct connections across the belt as described above there exists a curve joining a to b with signature 1, m. We now m a k e a Dehn twist of + j to obtain a configuration with signature n l , n 2 . The question is, does this configuration represent a single arc joining a to b? We represent the junctures of the arcs on the right and left sides of the belt by/?4 and Li respectively, where i is an integer 0 < i < 2k - 1 and the arcs are n u m b e r e d from the b o t t o m to the top. Thus R0 and Rk are the end points of the arcs from the points a and b respectively. In the left hand region an arc connects Li to L 2 k - l - i for each i, and on the right Ri is connected to R2k-i except when i = 0 or k. Now a Dehn shift of + j results in an arc from Ri to R2j+l+i (mod 2k) by Ri -~ Li+j --+ L 2 k - l - i - j --+ R 2 k - l - i - 2 j -~ -R2j+I+i.

But 2j + 1 = nl, so after the twist of j , we have the arc defined by a --~ Ro --+ Rnl ---+R2nl -~ Ran1 --* . . . -'+ Rpnl --+ b

which terminates when p n l - k (mod 2k). This certainly holds when p = k, since nl is odd, but m a y hold for a smaller p. Indeed if (nl, k) = q, then let p = k / q . We see that p a l - k ( m o d 2k) and this will be true for no smaller p. If nl and n2 are relatively prime, then (nl, k) = 1 and the resulting arc joining a to b includes all of the pieces crossing the belt and we have constructed an arc with signature n l , n2. If (nl, n2) = q > 1, then since 2k = nx - n2, (nl, k) = q also, and the curve system will contain an arc joining a to b with signature n l / p , n2/q and one or more closed curves. So, if ( n l , n 2 ) = 1, there exists an arc with signature nl, n2 while if we start with a standard system of arcs

J.A. Hummel

78

connected directly across as described above with signature 1, m (m < 0), then no set of Dehn Twists will result in an arc with a signature nl, n2 for which (nl, n2) > 1. Next, suppose that there exists an arc joining a to b which has signature nl, n2 with nl > 0, nl _> n2 and (hi, n2) > 1. Do a Dehn Twist of - j where 2j = nl - 1. This results in an arc system having signature 1, m with ra _< 1 and containing an arc joining a to b. If m = 1, then a and b are connected directly and there can be no arcs separating a a n d b. T h u s there are no crossings of the belt and Dehn twists will have no effect. T h a t is, there are no h o m o t o p y classes with signature nl, n2, nl > 1. It. remains only to show that the only possible system with signature 1 , m , m > 0, is one of the t y p e described above with direct connections across the belt joining a system of k arcs on the left to the corresponding standard system on the left. T h e first integer in the signature being 1 requires that the connections across the belt be direct or direct plus a twist of 4-1 since otherwise there will be more than one arc around b o t h a and b. A twist of - 1 clearly requires that m > 0. A twist of +1 would give a first integer > 1 also, since there would be 1 from the connection to Li and two more from the arc going around the belt to connect L0 to R2k-1. Thus the only feasible case is the set of direct connections and the theorem is proved. •

5. R e m a r k s E v e r y simple curve joining a and b in A\{0, a, b} defines a signature nl, nz and every signature defines a 00. However, there are only a countable n u m b e r of such signatures. Hence, almost all 00 must give rise to a trajectory structure in which the unit disk is a density domain. How is it that density domains do not seem to give rise to problems in the calculations, particularly since numerical methods are of necessity not exact? The answer is that we are looking for information about the m a p p p i n g and most of this information is a continuous function of the p a r a m e t e r 00. It is i m p o r t a n t to verify t h a t a 00 exists defining the solution to the problem. Then nearby 80 will define an approximation to the solution. The exception to this principle occurs when we consider properties t h a t depend on the homotopy class of 7, such as the length of the arc 7. For example, the curves with signature n, n + 2 with n odd and n, n + 2 relatively prime define angles 80 which converge to the 80 corresponding to the signature 1,1, but the lengths of the 3' tend to infinity. T h e above discussion is necessarily somewhat vague since each problem will be different a n d one is faced with m a n y difficult topological problems, each which m u s t be solved on its own merits. However, it is clear that the Schiffer Differential Equation can usually be solved numerically with the help of standard numerical integration and multidimensional zero finding techniques. As a closing remark, we observe that one of the biggest problems in the numerical solution of the Schiffer Differential Equation is in determining the argument of the functions being integrated. For example if we integrate R(w)l/2dw along some path, it is essential t h a t the argument of the integrand be continuous. The complex square root in F O R T R A N or other computer languages has a j u m p if the variable crosses the negative real axis and this can easily lead to erroneous results. The best way of overcoming

Numerical Solutions of the Schiffer Equation

79

this problem is to do the computation of individual factors separately, choosing forms which do not cross the negative real axis along the path of integration. For example, the argument of the factor (b - w) in (3) will change by at most ~r along a line segment, so, if it is multiplied by a constant as necessary so that at least one point along the path makes (b - w) real and positive, there will be no jump in the argument. The alternative is to adjust the argument at each point to make sure that it is continuous.

References

[1]

A. Fathi, F. Laudenbach, V. Poenaru, et al., Travaux de Thurston sur les surfaces, Asterisque, (1979), 66-67.

[2]

P.R. Garabedian, M. Schiffer, The local maximum theorem for the coefficients of univalent functions, Arch. Rational Mech. Anal., 26 (1967), 1-32.

[3]

J. Hummel, B. Pinchuck, Variations for bounded nonvanishin9 univalent functions, J. Analyse Math., 44 (1984/85), 183-199.

[4]

J. Hummel, B. Pinchuk, A minimal distance problem in conformal mapping, Complex Variables, 9 (1987), 211-220.

[5]

J. Krzy~, An extremal length problem and its applications, Proc. of the NRL Conference on Classical Function Theory (1970), Math. Research Center, Naval Research Laboratory, Washington D.C., 143-155.

[8]

L. Liao, Certain extremal problems concerning module and harmonic measure, J. AnMyse Math., 40 (1981), 1-42.

Received: April 5, 1989.

Computational Methods and Function Theory Proceedings, Valparaiso 1989 St. Ruscheweyh, E.B. Sail', L. C. Salinas, R.S. Varga (eds.) Lecture Notes in Mathematics 1435, pp. 81-87 (~) Springer Berlin Heidelberg 1990

Behavior of the Lagrange Interpolants in the Roots of Unity K.G. Ivanov 1 I n s t i t u t e of Mathematics, Bulgarian A c a d e m y of Science Sofia, 1090, Bulgaria and

E.B. S a ~ I n s t i t u t e for Constructive Mathematics, D e p a r t m e n t of M a t h e m a t i c s University of South Florida, T a m p a Florida 33620, USA

Dedicated to R.S. Varga on the occasion of his sixtieth birthday.

A b s t r a c t . Let A0 be the class of functions f analytic in the open unit disk Izl < 1, continuous on Izl < 1, but not analytic on Izl < 1. We investigate the behavior of the Lagrange polynomial interpolants Ln-1 (f, z) to f in the n - t h roots of unity. In contrast with the properties of the partial sums of the Maclaurin expansion, we show that for any w, with lwl > 1, there exists a g E A0 such that L~-l(g,w) = 0 for all n. We also analyze the size of the coefficients of L~-l(f, z) and the asymptotic behavior of the zeros of the L~-l(f, z).

1. Convergence Let f ( z ) = ~,k=Oakz ~ k be continuous on D1 : = {z E C : lz] < 1}. T h e n the L a g r a n g e interpolant to f at the n - t h roots of unity e ( ~ ) , k = O, 1 , . . . , n 1,e(x) : = e 2~i*, can be w r i t t e n as

1The research of this author was conducted while visiting the University of South Florida. 2The research of this author was supported, in part, by the National Science Foundation under grant DMS-881-4026.

K.G. Ivanov and E.B. Saff

82

L,~_~(f, z) - z~ - 1

(1.1)

k

k

n

e(~)f(e(g)) k=o z - - - e ~k-) -

n

j)k

n-1

E c ( j , n ) zj, j=o

where

(12)

c(j,n):=-

1~

e

f

e

k

,j=0,1,...,n-1.

rt k=0

When f is analytic on D1 (that is, f is analytic on Izl _< l + e for some e > 0), several results concerning Walsh's theory of equiconvergence describe the very close behavior of the sequence of Lagrange interpolants {L,~_~(f, z)} and the sequence of partial sums n--1 { s n - l ( f , z)} of its Taylor series, s n - l ( f , z) := ~k=0 akzk" For example, for such f ' s and for any z C C, the sequences {L~-l(f,z)}~ and {s~_~(f, z)}~° either both converge or both diverge (hence the term equiconvergence). But when f belongs to A0 ....... the set of all functions continuous on D1, analytic on o

the interior D], but not analytic on D~ - - , the behavior of the two sequences may be different. Of course, both {L~_~(f, z)}~and {s~_~(f,z)}~°converge (to f(z)) when Iz[ < 1. When Izl = 1 there are several examples of f E A0 such that {Ln-~(f, z)}~converges but {sn-l(f, z)}~diverges (the first goes back to du Bois-Reymond who constructed a function f C A0 with a divergent Maclaurin series at z = 1, but L~_~(f, 1) = f(1)). Conversely, {L~_~(f, z)}~°may diverge at a point on Izl -- 1 where {S~-l(f, z)}~converges (if f is continuous and of bounded variation on Izl = 1, then S~-l(f) converges uniformly to f, but Ln_~(f, z) can diverge for appropriate f and z, e.g. f(z) = (1 - l o g ( 1 + z)) -1/2 at z -- - 1 ) When Izl > 1, then {.%-1(f, z)}~ necessarily diverges (the terms a,~z'~ do not tend to zero). Surprisingly, it is still possible for {L~_~(f, z ) } ~ t o converge for some z with ]z I > 1, as the corollary of the following theorem shows. T h e o r e m 1. Let A be any subset of N and Iet m C N. The foI1owing are equivMent: (a) There exists an f C Ao such that the first m coet~cients c(j, n),j = n - 1,..., n rn of L,~-l(f, z) are zero for every n E A. (b) There exist distinct points wj, lwjt > 1,j -- 1 , 2 , . . . , m , and g E Ao such that L~-I(g, wj) -- 0 for every j = 1, 2 , . . . , m and every n C A.

Prod. (1.3)

(b). From (12) we have ~_e k=0

f(e 7t

(k)) = 0 ,

s=l,2,...,m,

neA.

,e

Let wj,j -- 1 , . . . ,rn, be any rn different points in where p(z) := 1-Ij~__l(Z- wj). Setting

Izl

> 1 and let g(z) := f(z)p(z),

( x v = 0 Or" 1

and using (1.1) and (1.3) we get that for any n C A and j = 1 , 2 , . . . ,rn

Behavior of the Lagrange Interpolants in the Roots o£ Unity

Ln-l(g, wj) -

83

w')-l'~-ln k:o~-'f( e(n))e k k'---P(e--(~))(n)wj-e(k k)

wj

1 Ef(e(k))e(k) ~_~(e~(k ) - w , )

-

n-~

n

k=0

n

n

(_~,

(k)

n

(1.4) w~ - 1 n - 1 12

j-1 92

f(e '-))e

k=0

~

7"~

w ....

s=l

n

,~-lf ( e k e ks k=0

(b) ::~ ( a ) . Keeping the notation from the first part of the proof we again set f(z) :=

g(z)/p(z). T h e n f C A0 because the wj's are outside D1. From (1.4) we have ~W .... jc(n-s,n)=O,

j=l,2,...,m,

for any n e A. But D e t ( W . . . . j)j~=l,~_-i = H,~ 1, there is a g E Ao such that L n - l ( g , w ) = 0 for every n E N.

Proof, According to Theorem 1 it is enough to find f E A0 for which all leading coefficients of the Lagrange interpolants are zero. For the function

k----1

where # is the Mhbius function (of number theory), we know (see [1],[5]) that

~F(e k=0

=0,

heN.

"~

Hence for f(z) := F(z)/z we have £:~-~f(e(~))e(,~) = 0 for any n, which proves the corollary. In this case, g ( z ) = Z~= 1 ( @ - - •U J -~(k+l)~ k+l ] z k -- W. • R e m a r k 1. We do not know whether there exists a g E A0 such that L,~-l(g, wj) = 0, j = 1,2, for every n E N, where [wj] > 1,Wl ~ w~. R e m a r k 2. Any g satisfying Theorem 1 or Corollary 2 will not be smooth. For example, no function with absolutely convergent Maelaurin series on tz] = 1 can satisfy Corollary 2.

K.G. Ivanov and E.B. Saff

84 2. Coefficients

and

the

Distribution

of Zeros

According to a theorem of Jentzsch [6], for any function f E A0, every point on the boundary of O1 is a limit point of the zeros of {S~-l(f, z)}~. One can say even more - the zeros of a special subsequence {s~_~(f, z)} tend weakly to the uniform distribution on the unit circle {z : [z I = 1} (see Szeg5 [7]). The same behavior can be observed for the zeros of the best polynomial approximants to f E A0 (see [2,3]). It is natural to ask whether the sequence of Lagrange interpolants {L~_~(f, z)}~° also possesses this property. A crucial step in establishing the above mentioned facts is the proof that the leading coefficients of the full sequence of polynomials are not geometrically small. For example, for the partial sums of Taylor series, this means limsup [a~l 1/~ = 1, n ---*oo

which is equivalent to f E A0, provided f E C(D1). One cannot expect the same behavior for the leading coefficients of L~-I (f). Indeed, as the example function f E A0 from the proof of Corollary 2 shows, we may have c(n - 1,n) = 0 for every n. But results similar to Jentzsch's and Szegb's theorems still can be established by utilizing the following statement, which is a special case of Theorem 1 in G r o t h m a n n [4]. Let pm be an algebraic polynomial of exact degree n(m). Define the zero-measure v~ associated with pm as vm(A) := # of zeros of pm in A

for any Borel set A C C, where the zeros are counted with their multiplicity. T h e o r e m A, ([4]) Let A be a sequence of positive integers and assume that the following three conditions hold for the sequence {Pm}mEA Of algebraic polynomiMs: (i) l i m s u p ( s u p - ~ l

m~

~zED1I¢(rFt)

loglpm(z)[) < 0 ; o

(ii) for every compact set M CD1, limoo vm( M )

rnEA

= 0;

(iii) there is a compact set K C_ C \ D 1 with liminf sup ~-~ L,~EK

log Ipm(z)l - log Iz

> O.

Then, in the weak-star topology, um tends to the uniform distribution AdO on the unit circle as m --* oc, m E A.

This leads us to investigating

Behavior of the Lagrange Interpolants in the Roots of Unity (r(f,O) := t i m s u p n--*oo

max

(i-O)n<j ~q.

(2.3)

Now our aim is, using (2.1) and (2.2), to obtain an estimate contradicting From (2.1) we get (2.4)

c(l,2s) = c(/,s) - c(l + s,2s),

(2.5)

c(l, 6n) + c(l + 2n, 6n) + c(l + 4n, 6n) = c(l, 2n),

and (2.6)

c(l + 2~, 6~) + 41 + 5~, 6~) -- c(z + 2., 3~).

From (2.5) and (2.6) we get

(2.3).

K.G. Ivanov and E.B. Saf[

86

c(l, 6n) = c(l, 2n) + {c(I + 5n, 6n) - c(1 + 4n, 6n) - c(l + 2n, 3n)}.

(2.7)

F r o m (2.7) with n -- 3 k - ' s , k -- 1, 2 , . . . , m, and (2.4) we o b t a i n

c(I, 2 . 3 m s ) = c(l, s) - c(l + s, 28) + a

(2.8) where m

a = ~{c(l+

- c(l+ 4 . 3 k - ' s , 2 . 3 k s ) - c(1 + 2 . 3 k - l s , 3 k s ) } .

5.3k-'s,2.3ks)

k=l

It is easy to see t h a t all t e r m s on the right-hand side of (2.8) are of the t y p e c(j, n) with ~n < j < n, n > s > I > no. By applying (2.2) in (2.8) we get

Ic(t,2.3m

)t

< qs +

+ ELi{2q (3ks) + q3k,)


1 be fixed. Assume t h a t there exists an e, 0 < e < 1/2, such t h a t limsup

sup

log[L~_~(f,z)l-logr

No a n d for every z, [z[ = r, we have 1 ~(n) log I L ~ - , ( f , z)[ - log r < - 2 e log r, t h a t is,

[L,~-l(f,z)[ < r (~-2~)~(~) -(1-g~)cos¢}

for n -- 0,1, 2, . . .

where 0 _< go < 1 and 0 < gn < 1 for n E N. Then {V~} is a sequence of value regions corresponding to {12~} given by Y2~ :-- En x {1} where (2.10)

E. := {z e ¢; Izl- Re(ze

< 2(1 -g,~-,)gn cos 2 ¢}.

Here, p, in (2.8) is replaced by (1 - g,)cos ¢ to make it easier to recognize the chain sequence (1 - g,~-l)gn in the definition of E,.

Lisa Jacobsen

94 Note that the half planes V~ do not contain e¢.

Example 2.3. (2.11)

Let

V~:={zEC;

Iz-F~[0

satisfies

z+rI_>t

and

->1

F ( z + F)(1 - s 2) = - A .

s

For ~ = ~¢~ E E it therefore follows that

/9 2

-A~=r(z+r)(1

iz+rl2)¢~

where

p=~lz+rl

and

I¢~ - 11 + ,leaf = t¢~ - 11 +

p ~ + r l _ 2 could be modified to give a proof of the g = 0 case. Such a proof would be very different form Andreev's. T h e purpose of the present p a p e r is to present such a proof. We separate the statement of the g = 0 case into two parts: T h e o r e m A below, which deals with standard circle packings, and T h e o r e m B, which allows the circles to intersect at prescribed angles. These theorems have apphcations to conformal mapping [3]. 1. A circle packing on the Riemann sphere or in the plane is a collection of closed disks or the R i e m a n n sphere on in the Euclidean plane with the p r o p e r t y t h a t the interiors of the disks are disjoint. The nerve of such a circle packing is the graph which has a vertex for each disk and an edge connects two vertices if and only if the corresponding closed disks intersect. T h u r s t o n ' s theorem, in the case of circle packings, is the following:

1Research partially supported by the NSF. 2Research partially supported by the NSF and DARPA/ACMP.

A. Marden and B. Rodin

104

T h e o r e m A. Let T be a triangulation of the Riemann sphere P. There exists a c/rc/e packing of P whose nerve is isomorphic to the one dimensional skeleton of T ; any two such circle packings are images of each other under some linear fractional transformation or its complex conjugate. 2. We begin the proof with some formulas for the Euler characteristic. Let V, E , and F denote the number of vertices, edges, and faces in the triangulation T . Then V-E+F=2.

(1)

Since 3 F = 2E, we can eliminate E in (1) and obtain (2)

2V = Y + 4.

It will be convenient to label the vertices of the triangulation T by vl, v 2 , . . . , vv. Let r = ( r l , r 2 , . . . , r v ) be a vector of V positive numbers. Then r determines a polygonal structure on the topological 2-sphere ]T] as follows. Associate to each face of T, with vertices (vl, vj, vk) say, the Euclidean triangle determined by the centers of three mutually (externally) tangent circles of radii rl, rj and rk. Transfer the Euclidean metric on this Euclidean triangle to the associated face of T. Note that the metric is well defined on an edge which is common to two different faces. In this way ITI becomes a locally Euclidean space with cone type singularities at the vertices; we denote this space by T~. The curvature of T-~ at the vertex vi, denoted by ~r(vi), is defined as follows. Consider all faces of ~ which have vi as one of their vertices. Let a(vi) be the sum of each angle at vi in each of these triangles. Then

(3) Let us note that V

~., n~(v,) = 4re.

(4)

i=l

Indeed, the left hand side reduces to 27rV - ~ 8, where ~ 0 denotes the sum of all angles of all triangles of T~. This sum is equal to TrF. An application of Equation (2) now establishes (4). 3. If A > 0 then ~ and ~r~r are similar in the sense that corresponding angles are equal. Therefore nT(vi) = t~aT(vi). It turns out to be advantageous to normalize the map r ~ f ( r ) = (~T(vl), sz~(v2),..., ~ ( v v ) ) by restricting its domain to the simplex (5)

A_-- { ( r l , r 2 , . . . , r v ) ) c R Y : r 1

> 0 , r2 > 0 , . . . r y

>0&rl+r2T...+rv--

It follows form (4) that the range of f can be taken as the hyperplane (6)

Y = { ( y l , y 2 , . . . , y y ) ¢ n V : Yl + Y2 + . . . + yv = 4r}.

1}.

On Thurston's Formulation and Proof of Andreev's Theorem

105

4. For convenience of notation, assume that vl, v2, va are the vertices of a single face r0 of T . We now prove that the existence assertion of Theorem A will follow once it is shown that the point Po = (47r/3, 4~r/3, 47r/3, 0 , . . . , 0), for example, lies in the image of the map f:A-*Y. To see this, suppose f(ro) = po. If we remove that face r0 from T~0 then the remaining triangles can be placed isometrically in the plane, one by one, in an orientation preserving manner, keeping identified edges coincident. Since the curvature is zero at each interior vertex of this complex it can be shown that we obtain in this way an isometric embedding of T~0 less To onto a triangle in the plane. {To prove that this is so, one can first show that the process of placing adjacent faces in the plane yields a well defined isometry once the image of an initial face is fixed. For suppose a sequence of adjacent faces are placed in the plane in this way and suppose the first face in the sequence is the same as the last. Then the placement of the first and of the last face will agree-this is clearly true if the sequence of faces surrounds only one interior vertex of T~0 less r0, and can be shown to be true in general by induction on the number of such vertices. T h e second step in the proof is to use the fact that this placement process provides a locMly isometric embedding of ~0 less r0 into the plane and is an actual embedding of the boundary of T~0 less r0. It is easy to see that a local embedding of a topological disk into the plane which is an actual embedding on the boundary must by a global embedding. One concludes that this placement process is a global isometric embedding of T~0 less r0 onto a triangle in the plane}. We have constructed an isometric embedding, call it ¢, of T~0 less r0 onto a triangle A B C in the plane. It follows from the definition of T~0 that if we center a circle of radius ri at the point ¢(vi) we obtain a circle packing in the plane whose nerve is isomorphic to the one dimensional skeleton of T. Stereographic projection transforms this packing to a packing of the Riemann sphere with the same property. It will be useful when we discuss uniqueness to observe that triangle A B C is necessarily equilateral. To verify this, weld another copy of triangle A B C to this one along corresponding edges. One then obtains an isometric image of all of T~0. We can calculate the curvature at the vertex ¢(vl) which, we may assume, corresponds to the point A, directly from the definition (3) using this isometric image. We see that the curvture at A is 27r less the sum a ( A ) of all angles in this isometric image with this vertex A. This sum a ( A ) is clearly twice the angular measure re(A) of angle A in triangle A B C . On the other hand, we know by the definition of p0 that the curvature must turn out to be 47r/3. Thus 47r/3 = 27r - 2m(A). Hence re(A) = ~r/3. Similarly, m ( B ) = m ( C ) = 7r/3 and so A B C is equilateral. 5. We now show that f : A --, Y is one to one. Let r' = (rtl,r~ . . . . ,r~v) and II r" = (7"1, r 2l ! , . . . , r ~ ) be distinct points in A. Let Y0 be the set of vertices v; of T for which r 'i < r i" . Note that the definition (5) of A implies that V0 is a nonempty proper subset of the set of all vertices of T.

A. Marden and B. Rodin

106

Consider a vertex v E ])0 together with all the faces of T} which have v as vertex. In each such face there is an angle at v, and we classify this angle as type a if it is the only angle in this face which has its vertex in Y0, of type/3 if two vertices in this face are in Y0, and of type 7 if all three vertices of this face are in Y0. Now E x~,,(vi) = E (2~r - a(v)) vEVo vEVo

(7) =

2

lV01 -

of type a) - ~_,(Zs of type fl) - ~_,(Zs of type -y).

Consider three mutually tangent circles in the plane and their triangle of centers. If one of the circles shrinks and the other two either expand or stay the same size, and if the three circles always remain mutually tangent, then the angle in the triangle of centers with vertex at the center of the shrinking circle will (strictly) increase. If two of the circles shrink and the other either expands or stays the same size, then in the triangle of centers the sum of the two angles which have their vertices at the centers of the shrinking circles will increase. These observations show that if r" is replaced by r' in equations (7) then

(8)

E

> E

vEVo

vEVo

Indeed, in passing from r" to # the radii at v E 1)0 shrink and so the first two quantities in of type /~), of type 7) E ( / s of type a),

E(Ls

E(Zs

will each increase, and the third will remain constant. Since ~0 is a n o n e m p t y proper subset of vertices, not all angles are of type 3'. It follows that the inequality in (8) is strict and that f : A ---+y must be one-to-one. 6. We now examine the behaviour of f ( r ) as r tends to a boundary point s = (Sl, s 2 , . . . , sv) of A. It will turn o u t - a n d this seems very remarkable-that f cannot be extended continously to the boundary of zl, yet the set of accumulation points of f ( r ) as r tends to the boundary of A form the boundary of a polyhedron. Let F0 be the set of vertices vl in T for which 8i = 0; F0 is a nonempty proper subset of V. We classify the angles of T~ into types a,/~, 7 as above. Then as r ~ s we have E ( L s of type o~) --+ (9)

Z ( Z s of type/3)

--+ ~I/~I/2,

E ( / s of type v)

--~ ~rl-yl/3,

where Ixl denotes the number of angles of type x. Therefore equation (7) yields (10)

lira ~ ~ ( v ) = 2~r[~2o]- 7rIa ] ~-~ ~eVo

~r]fl] ~r]T] 2 3

On Thurston's Formulation and Proof of Andreev's Theorem

107

= 2rc]ld0I - re- (no. of faces with a vertex in ]do). From (8) and (10) we see that the image f(Zl) of f : A ~ T lies in the bounded convex polyhedron ]I0 formed by intersecting Y with the half spaces (11)

~ ] yi > 2~rlII- rr-(no, of faces with a vertex in ldi - {vi: i E I}) iEI

as I varies over all nonempty proper subsets of { 1 , 2 , . . . , V}. We have also seen that the accumulation points of f ( r ) as r ---* 0A lie on the hyperplanes (12)

~ Yi = 2zr[I[ -- zr. (no. of faces with a vertex in 121 =-- {vi: i E I}). iEI

which form the boundary of Y0. 7. We know that f : A --~ Y0 is a continuous 1-1 mapping. Hence, by Invariance of the Domain, f is a homeomorphism. We also know that f ( r ) -~ OYo as r --* 0A. It follows by elementary topology that f : A ~ Y0 is surjective. Indeed, merely pass to the one point compactifications and apply the simple fact that if X, Y are Hausdorff spaces with X compact and connected and Y connected, and if ¢ : X -* Y is continuous and open, then ¢ is surjective. 8. We complete the proof of the existence part of Theorem A by showing that p0 = ( 4 r r / 3 , 4 r r / 3 , 4 7 r / 3 , 0 , . . . , 0 ) is in the image Y0 of f : A ~ y (see Section 4). According to (11), this can be done by showing that for every nonempty proper subset

Xof {1,2,...,v},

(13)

~Pi

> 2~r[I[- 7r. (no. of faces with a vertex in 12x =- {vi : i E

I}),

iEI

where Po = (Pl, P 2 , . . . , pv) = (47r/3, 4zr/3,4~r/3, 0 , . . . , 0). If [I[ = V - 1 then every face has a vertex in VI - {vi : i E I}. Therefore the right hand side of (13) is, by (2), (14)

27r(V - 1) - 7r. F = 27r.

For subsets I of this cardinality the left hand side of (13) becomes

(15)

Y~Pi = iEI

8¢r/3 or 12~/3.

Thus Po satisfies (13) when IZl = v - 1. If I/t = v - 2 similar reasoning shows that the right hand side of (13) is zero while the left hand side is at least 47r/3. Thus po satisfies (13) in this case also. We shall show that p0 satisfies (13) v/hen 1 < ] I I _< V - 3 by proving that the right hand side of (13) will be negative in these cases. First we rewrite the right hand side in a more invariant form. Let F1, F2, Fa denote, respectively, the number of faces of T which have exactly 1,2,3 vertices in Idl. Then the right side of (13) is ~r(2]I I - F1 - F~ - F3). Let

A. Marden and B. R o d i n

108

: O(e,-j) /

........~.~

F i g u r e 1. T h e angle of intersection of two disks

E2 d e n o t e the n u m b e r of edges of T which have b o t h of their b o u n d a r y vertices in 1;z. T h e n the simplicial complex TI spanned by the vertices of Yl has Euler characteristic X0 = II] - E2 + F3. Since 3F3 + F2 = 2E2, we can eliminate E2 from the expression for X0 a n d o b t a i n 2X0 = 2]II - F3 - F~. Therefore the condition (13) t h a t ( P l , P 2 , . . . , p y ) lies in the image of f can be rewritten as (17)

~-~p, > ~r(2X0 - F1) iEl

for every n o n e m p t y p r o p e r subset I of { 1 , 2 , . . . , V}. We wish to show t h a t the right h a n d side of (17) is negative for 1 k > 0 for ali c E 012, then 12 is spherically k-convex. Proof We begin by showing that 12 is spherically convex. By the preceding lemma, it is sufficient to show that there is a locally supporting great circle at each point c E 012. We m a y assume that c = 0. Then k(0,012) = ks(O, 0Y2) > 0. Hence, k((, 012) > 0 for all ~ C 012 in a neighbourhood of 0, so 12 has a locally supporting euclidean straight line at 0 [S, p. 46]. This straight line through 0 is also a great circle, so f2 has a locally supporting great circle at c.

Hyperbolic Geometry in Spherically k-convex Regions

121

Next, we show that [2 is spherically k-convex:. Fix a, b E /2. Let r be the s u p r e m u m of all t _> 0 such that St[a,b] C/2. Note that r #D(a) with equality if and only if $2 = D. Because ED(a) = Ea(a), this inequality in conjunction with the above formula for #D(Z) completes the proof. • C o r o l l a r y 1. Suppose that $2 is a spherically k-convex region. If f is meromorphic in D and f ( D ) C $2, then for z 6 D (1 - [ z l ~ ) f # ( z ) a/(1 + ~/1 - a(a + k)) with equality if and only if ~ is a spherical disk of radius i arctan(2/k). In the case of equality, f is a conformal mapping of D onto a spherical disk of radius 1 arctan(2/k) that contains the origin and whose boundary is externally tangent to the circle {w : Iwl -- ~/(1 + x/1 - a ( a + k))}. In this case it is straightforward to check that f must have the prescribed form. •

5. T h e spherical B l o c h - L a n d a u c o n s t a n t for t h e family

Ks(k, a)

We derive a sharp lower bound for the spherical density of the hyperbolic metric for a spherically k-convex region in terms of a uniform upper bound on ca. We use an extremal region which has been employed in [M1] and [MM]. Suppose k > 0, 0 E (0, 7r/2) and N = tan0. Let R = ~/N(2 - kN)/(k + 2N); R is selected so that the circle through - R , iN and R has spherical radius ½arctan(2/k) = arctan(v/-~-+ 4 - k)/2, or equivalently, spherical curvature k. Let S = S(N) = i n t S k [ - R , R ] . Note that for N = ( k v / ~ T 4 - k)/2 the set S is actually a spherical disk. In all cases, S contains the disk {z : [z I < N}, but no larger disk centered at the origin, and S is contained in the disk D = {z : Izl < R}. Each of the two circular arcs bounding S makes an angle 2T with the segment [ - R , R], where qp = arctan( N / R). We also introduce a certain collection of triangular spherically k-convex regions. Let T = T ( N ) denote the family of all spherically k-convex regions that contain {z : Izl < N} and are bounded by three distinct circular arcs each of spherical radius ½aretan(2/k) and having the property that the full circles are tangent to Izl = N and contain {z : ]zl < N} in their interior. Each of these circular arcs will meet OD in diametrically opposite points and has euclidean radius k' = (1 + N2)/(k + 2N). Therefore, each region A in T is both spherically k-convex and euclidean k'-convex. From [MM, Lemma 2] we obtain the following result. L e m m a 2. If A E T, then for z E A, #a(z) > (zr/4qO)#D(Z) > (zr/4Tn). T h e o r e m 2. Suppose f2 is a spherically k-convex region. Let 0 = max{¢~(z) : z E g2} and N = tan 0. Then

D. Mej[a and D. Minda

124

k+2N

1

#9(z) >_ ~ V N ~ Z - f f - N )

/ N ( k + 2N)" arctan V 2 : k N

Equality holds at a point a E t71 if and only if there is a rotation T of P such that Y2 = T(S) and a = T(O). Proof. Select a E $2 with Ea(a) = N. From Proposition 3 we see that 1 2 x/~ +42 ¢~(a) O, R = {z : E e ( a , z ) < 0}, F = OR a n d j denote reflection in F. If

(2)

E
O. Then f e Ks(k, a) if and only if 1 + Re ( z f " ( z )

\ if(z)

2zf_(z)f'(z)~ > kf*(z)lzl 1 + If(z)l 2 ] -

for z E D.

Proof. First, suppose f E K~(k, a). Straightforward, but tedious, calculations show that the inequality of Theorem 6 implies the inequality of the corollary. On the other hand, suppose that the above inequality holds. Consider the path 7 : z = z(t) = re ~t, t 6 [0,2~]. Since k~(z,7) = (1 -[z[2)/[z[, we obtain

(zS,,(z) 1 + Re ~ if(z)

k / f ( z ) , f o 7) =

1 + If(z)l ~ ]

izf#(z)[

Therefore, k,(f(z), f o 7) >- k for all z E 7, so f ( { z : lzl < r}) is a spherically k-convex region by Proposition 1. Then f(D) is also spherically k-convex since it is an increasing union of spherically k-convex regions. • C o r o l l a r y 2. Suppose f is holomorphic and univalent in D. Then f(D) is sphericedly k-convex if and on/y if f maps each subdisk o l d onto a sphericalIy k-convex region.

References

[Ba]

K.W. Bauer, Uber die Ab~chdtzung yon Lbsungen gewisser partieller Differentialgleichungen yore elliptischen Typus, Bonner Mathematische Schriften, 10, Bonn, 1960.

Hyperbolic Geometry in Sphericedly k-convex Regions

129

[B] W. Blaschke, Uber den gr~flten Kreis in ether konvcxen Punktmenge, Jahresber. Deutsch. Math. Verein. 23 (1914), 369-374.

[¢1 H.W. Guggenheimer, Differential Geometry, McGraw-Hill, New York, 1963. [MM] D. Mejla, D. Minda, Hyperbolic geometry in k-convex regions, Pacific J. Math. (to appear). [M1] D. Minda, The hyperbolic metric and Bloch constants for spherically convex regions, Complex Variables Theory Appl. 5 (1986), 127-140. [M2] D. Minda, A reflection principle for the hyperbolic metric and applications to geometric function theory, Complex Variables Theory Appl. 8 (1987), 129-144. [M3] D. Minda, Applications of hyperbolic convexity to euclidean and spherical convexity, J. Analyse Math. 49 (1987), 90-105. [S] J.J. Stoker, Differential Geometry, W'iley-Interscience, 1969. Received: May 14, 1989

Computational Methods and Function Theory Proceedings, Valparaiso 1989 St. Ruscheweyh, E.B. Sail', L. C. Salinas, R.S. Varga (eds.) Lecture Notes in Mathematics 1435, pp. 131-142 (~) Springer Berlin Heidelberg 1990

The

Bloch

and

Marden

Constants

D a v i d M i n d a 1' 2 D e p a r t m e n t of Mathematical Sciences, University of Cincinnati Cincinnati, Ohio 45221-0025

1. I n t r o d u c t i o n Suppose F is holomorphic in the open unit disk D. For a E D let r(a, F) denote the largest nonnegative n u m b e r r such that there is a simply connected region 1"2, with a E t2 C D, which is m a p p e d conformally onto D(F(a), r) b y F. (Here we use the notation D(b, r) for the open euclidean disk with center b and radius r. For b = 0 we simplify this to D(r).) Set r(F) = sup{r(a, r ) : a e D}. Clearly, r(a, F ) = 0 if and only if F'(a) = O. T h e Bloch constant B is defined by B = i n f { r ( F ) : F is holomorphic in D and F ' ( 0 ) = 1}. In 1924 Bloch proved that B is positive in spite of the vast collection of functions over which the infimum is taken. Landau ([L1],[L2]) gave various lower bounds for the Bloch constant the best of which is B > 0.396. Landau also showed that B = i n f { r ( F ) : F e B,

IIFII

= 1 and F ' ( 0 ) = 1},

where B is the class of all holomorphic functions F defined on D such t h a t F ( 0 ) = 0 and

IIFll = sup{(1 -Iwl2)lF'(w)l : w • D}
_ f ~/v'5 F ; ( r ) d r = Jo

El(I/v/3) -- v/3/4.

Thus, F I ( D ( 1 / v ~ ) ) D D ( v ~ / 4 ) . Since F l ( 1 / v ~ ) = v ~ / 4 and F~(1/v~) = 0, it follows that r(0, F1) = v/3/4.

The Bloch and Marden Constants

135

It is not difficult to show that r(F) = 3 v ~ / 8 and s(F) = +co. T h e o r e m 1. Suppose that s > O, la[ = s and g is holomorphic and nonconstant on D(s) U {a}. If [g(z)] _< lg(a)l for z • D(s), then c = ag'(a)/g(a) is positive and Re~g(ta)~ > (c+l)t-(c-1) [ g(a) J - (c + l ) - ( c - 1 ) t '

t•(-1,1),

with equedity for some t • (--1, 1) if and only if g(z) = g ( a ) ~c + 1)z -- (c -- 1)a + 1)a -~---1-~" Proof. We start by reducing the general case to the special case in which a = 1 and 9(a) = 1. Set h(z) = g(az)/g(a). Then h is holomorphic in D U {1}, h(D) C D, h(1) = 1 and c = h'(1). Since h(D) C D and h(1) = 1, elementary geometric considerations show that c > 0. Set c-1 c+1 _ (c+l)z-(c-1) V(z) c- 1 (c + 1) - ( c - 1)z" 1z c+l Note that U is a conformal automorphism of D, U(1) = 1 and U'(1) = c. We need to prove that Reh(t) > U(t), t E ( - 1 , 1), with equality if and only if h = U. Set k = U - ' oh. Then k is holomorphic in DU{1}, k(D) C D, k(1) = 1 and 1 = k'(1). Recall that for r e (0, oz) Z

--

-

-

I1 - zl ~ < r} 1 A ( 1 , r ) = {z e D : {zl------1_ 7 =D(l+r,l+r

)

is a horodisk in D based at 1; that is, an open disk in D that is internally tangent to the unit circle at 1. Observe that U(Zl(1,r)) = Zl(1,cr). Julia's Lemma ([A2, pp. 7-9], [C, pp. 23-28]) asserts that for z E D [ 1 - k ( z ) [ ' < ik,(1)11117 ; ; _

II-z] =

1 -IKz)l = -

i -Iz[

I

2'

with equality for some z E D if and only if k = Rb for some b E R, where

Rb(z)

=

(I + z) - (1 + ib)(1 - z) (i + z)+~--ib)(1 z)"

Note that Rb is a conformal automorphism of D that fixes 1 and maps each horodisk A(1, r) onto itself. Julia's Lemma has an elegant geometric interpretation: either k = Rb for some b • R or else k(A(1, r)\{1}) C A(1, r) for all r • (0, oo). Thus, either h = UoRb for some b • R or else h ( A ( i , r ) \ { 1 } ) c U(A(1,r)) for all r • (0,oo). Given t • ( - 1 , 1 ) , select r = (1 - t ) / ( 1 + t ) • (0, oo). Then A(1,r) is the horodisk in D whose boundary meets the real axis in the points t and 1. From the inclusion h(A(1, r)) C U(A(t, r)) we obtain Reh(t) >_ U(t) with equality for some t if and only if

D. Minda

136

I m h ( t ) = 0 a n d h(t) = U(t), t h a t is, k(t) = t. But then equality holds in Julia's L e m m a , so k = Rb for some b • R. But if t = Rb(t) for some t • ( 1 - , 1 ) , then it follows t h a t b = 0. Since/go is the identity function, we deduce h = U. C o r o l l a r y . Suppose s > 0, [a] = s and g is holomorphic and nonconstant on D(s) U {a}. u b(z)t _ b(a)L for z • D(s) and aV(a) = g(a), then

R e { g(ta)~g(a)j >-t'

t•(-1,1),

with equMity for some t • ( - 1 , 1) if and only if g(z) = g(a)z/a. R e m a r k . For the p r o o f of T h e o r e m 1 we only require the weak f o r m of Julia's L e m m a in which g is assumed to be analytic at z = 1, not the general result dealing with the a n g u l a r derivative at z = 1. There is a simple p r o o f of Julia's L e m m a in this special case in [PS, prob. 292, p. 141]. Now, we give a geometric proof of Bonk's distortion t h e o r e m with a careful analysis of the sharpness. T h e o r e m 2. Suppose F • 13, [IFII = 1 and F ' ( 0 ) = 1. Then a e F ' ( w ) _> F;(Iw[) for [w[ < v~/2 with equality at re 'e # 0 if and only if F(w) = e'eFl(e-'ew).

Proo£ T h e r e is no h a r m in assuming t h a t w = u E (0, 1); if not, t h e n consider e-leF(eiew) to treat the case in which the point is re ie. Note t h a t 1

11 + F"(O)w + . . . I = IF'(w)l -< 1 - Iwl ~ - 1 + Iwl = + . . . implies F"(O) = O. Consider any a , 0 < a < 1. Set w = S~(z) = (z + a)/(1 + az) and G~ = F o S~. T h e n IIFII = ]IG=]I and (1 - I z [ 2 ) l G ' ( z ) l = 1 for z = - a since S~(-a) = 0 and F'(O) = 1. Thus, for ]z[ _< a we have 1

1

[G:(z)J (c+l)-(c-1)t = (l+a')+(l-aa2)t '

or

Re{F'(S=(-at))S'=(-at)} >_

(1 -

(1

-

3a 2) + (1 + a2)t

aa)[(1 + a 2) + (1 - 3a=)t] '

with equality for some t • ( - 1 , 1) if and only if

t•(--1,1),

• (-1,1),

The B1och and Marden Constants

137 1

(c + 1 ) z - ( c -

1)(-a)

G'a(z) = 1 - a ~ (c 4- 1 ) ( - a ) - (c - 1)z" Set u = Sa(--at), or t = (a - u)/a(1 --au). Observe that Sa maps the interval ( - a , a ) onto the interval (0, 2a/(1 + aS)) and that -

S'=(-at)

1

au) 2 1 - as

_ (1 -

(S:')'(u) Therefore, (*)

ReF'(u) >

(1

-

2a- (l+3a2)u au)2[2a (1 a2)u] '

u e (0, 2a/(1 + ab).

-

-

This implies that ReF'(w) >

2a - (1 + 3a~)lwl (1 -

alwl)~[2a

-

(1 -

a2)lwl]'

Iwl < 2a/(1 + a2).

Thus, ReF'(w) > 0 for lwl < 2a/(1 + 3ab. The choice a = 1/x/~ produces the largest disk on which a e F ' ( w ) > 0 and also yields ReF'(w) >_ F~(lwl) for lwl < v ~ / 2 . Note that for a = 1/v/3 we have c = 1 and so strict inequality holds in (*) for u E (0, V"3/2) unless a',(z) = - 3 v / 3 z / 2 , or equivalently, G~(z) = a , ( z ) , and so F ( w ) = El(W). Here G1 refers to the function in Example 1. Conversely, if F = F1, then for a = 1/yr3 equality holds in (*) at each point of the interval (0, v~/2). C o r o l l a r y 1. Suppose Y e I3, IlVll = 1 and F'(O) = 1. Then r(O, F) > v/3/4 and s(0, F ) ~ 2 a r t a n h 1/x/~ with strict inequality unless F ( w ) = e~°Fl(e-l°w) for some ~ER.

Proof. Suppose F ( w ) • eieF,(e-'ew) for all 0 e R. Then ReF'(w) > F{(lwl) > 0 when Iwl < v ~ / 2 . In particular, min{aeF'(w) : Iw[ = 1/v~} > 0. Thus, there exists s > 1 / v ~ such that ReF'(w) > 0 on D(s). The Wolff-Warschawski-Noshiro Theorem implies that F is univalent on D(s), so s(0, F) k 2 artanh s > 2 artanh 1/V~. Also, for = d U 4 5 we have iF(w)l > r~e ~ ] -

=

ReF'(re'e)dr > J o

F~(r)dr

4

Hence, min{lF(w)t : Iwl = 1 / 4 5 } > v ~ t 4 , so there exists r > V~14 such that F ( D ( l l v / 3 ) ) contains D(r). Because F is univalent in D(lfv/3), this implies that r(0, F ) > x/~/4. • R e m a r k . For bounded analytic functions there is a sharp analogy of Corollary 1 due to Landau, see ([H2, pp, 36-39]). The extremal functions for Landau's result are two-sheeted branched coverings of O onto itself. This result of Landau is the basis for one elementary proof of a lower bound for the Bloch constant, see ([H2, pp. 46]). C o r o l l a r y 2. B > v ~ / 4 and M > 2 artanh 1/v~.

D. Minda

138

P r o o f Let B1 = { F : F C B, t]F]I = 1 and F'(0) = 1}. The family B1 is a compact normal family. Robinson [R] showed that there exist extremal functions for the Bloch constant, that is, functions F E B1 with r ( F ) = B. Consider any such extremal function F. If F(w) ¢ ei°Fl(e-i°w) for all 0 E R, then r(F) > r(0, F ) > v ~ / 4 by the preceding corollary. On the other hand, if F(w) = e'°Fl(e-i°w) for some 0 e R, then r(F) = 3x/3/8 by Example 1. Thus, in either case, B > x/~/4. Next, we show M > 2 artanh 1/x/~ by a similar argument. Consider any sequence {F~} in B1 with s(F~) --~ M. Because B~ is a compact normal family, we may assume that Fn ~ F E B1, where the convergence is uniform on compact subsets of D. If F is univalent on Dh(a, s), then for each e > 0 there exists N such that for all n _> N, F~ is univalent on Dh(a, s - e). Hence, for n > N, s(F~) >>_s(a, F,~) >_ s(a, F ) - e. By letting n tend to infinity we obtain M > s(a, F) - e. But e > 0 is arbitrary, so M > s(a, F). This yields M >_ s(F), so M = s(F). This demostrates that there are extremal functions for the Marden constant. The remainder of the proof that M > 2 a r t a n h 1/x/3 is now analogous to the proof in the preceding paragraph and is omitted. •

3. G e o m e t r i c

interpretation

Suppose F is holomorphic and nonconstant on D. Let X = X ( F ) be the Riemann image surface of F viewed as spread over the complex plane C. In connection with the Bloch constant problem, it is natural to inquire about the possible location of the largest schlicht disk on X. One plausible guess is that the largest schlicht disk would be centered at a point where the hyperbolic metric attains its minimum value. For a plane region it makes sense to speak of the minimum of the density of the hyperbolic metric, but for a Riemann surface it makes no sense to speak of the value of a metric at a point of the surface. However, for a Riemann surface spread over C there is a. natural way to define a density for the hyperbolic metric. This density will be infinite at all branch points and at all finite boundary points. We will show that a minimum point for this density is always the center of a relatively large schlieht disk, but generally not the largest such disk on the surface. In fact, the work of Bonk yields a sharp lower bound for the radius of a schlicht disk centered at such a minimum point; this is the geometric significance of x/3/4. The result of Bonk also gives the best possible lower bound on the hyperbolic radius of a hyperbolic disk centered at a minimum point which is one-sheeted when viewed as spread over C. This is a geometric interpretation of 2 artanh 1/v/3. This is the same as the lower bound on the hyperbolic distance from a minimum point to the nearest branch point. We begin by making precise the notion of the Riemann image surface and establish some notation. Precisely, X = {(w, F(w)) : w C D} is the graph of F in C x C. Define the two natural projections of X onto the coordinates planes: ~h:X~D

1r2:X~C

~rl(w, F(w)) = w,

7r2(w, F(w)) = F(w).

The surface X is endowed with the unique conformal structure that makes both 7rl and r2 analytic functions. Note that 7h is a conformal mapping of the simply connected

The Bloch and Marden Constants

139

Riemann surface X onto D. The set of branch points of X is b(X) = { ( w , F ( w ) ) : F'(w) = 0}. The function ~r2 can be used as a local coordinate at each point of X\b(X). For convenience, let 7r~-1 = / ~ : 13 ~ X,

~'(w) = (w, F(w)), and observe that 7r2 o ~" = F. There are two natural metrics and associated distance functions on X. First, there is the hyperbolic metric )~x(w)ldwI and the associated distance function dx. Recall that the hyperbolic metric on X is the unique conformal metric on X whose pull-back under any conformal mapping of D onto X is the hyperbolic metric on D; in symbols,

=

D(w)ld

2jdwl , l = 1-7:

where /kD(w)IdwI is the ordinary hyperbolic metric on D with curvature -1. Second, there is the pull-back of the euclidean metric via rr2. Explicitely, ~c(¢)]d~l = 1IdOl is the euclidean metric on C and rr~(~c(~)ld¢[) is a metric on Zkb(X), but just a semi-metric on X itself. In other words, ~r;(~c(~)ld~l) is a continuous, nonnegative linear density on X which vanishes precisely at the branch points. Because the zeros of this semi-metric are isolated points on X, it induces a distance function d on X that is compatible with the topology of X. Note that r(a, F ) is the minimum of the distance from (a, F(a)) to b(X) and the distance from (a,F(a)) to the ideal boundary of X relative to the distance function d. Crudely speaking, it is the radius of the largest disk on X, relative to the distance function d, centered at (a, F(a)) which does not contain a branch point or meet the ideal boundary of X. Similarly, let t(a, F) be the minimum of the distance from (a, F(a)) to b(X) and the distance from (a,F(a)) to the ideal boundary of X relative to the distance function dx. In fact, the latter distance is always infinite, so t(a, F) is just the dx-distance from (a, F(a)) to the set of branch points. The geometric quantity t(a, F) was introduced and studied in [M2]. Also, observe that s(a, F) is the radius (relative to dx) of the largest disk centered at (a,F(a)) in which ~r2 is univalent. In other words, it is the radius of the largest hyperbolic disk centered at (a, F(a)) which is one-sheeted when viewed as spread over the complex plane. Clearly, s(a, F) < t(a, F). The quotient ~x(w)ld~l =

= 7r

(Ac(¢)ld¢l)

of the hyperbolic metric by the pull-back of the euclidean metric defines a positive, continuous function on X which is infinite at each branch point. The function # is the density of the hyperbolic metric when the hyperbolic metric is expressed in terms of the local parameter 7r2. Observe that ~(w, F(w)) = , ( P ( w ) ) =

F*(:~x(~)ld~l) F*(~r~(Ac(C)tdCI))

D. Minda

140

P*(;~x(~)ld~l) F'(.Xc(C)IdCI)

~D(w)ldwI IF'(~)lldwl

Here we have used F* = (~r~ o F)" = / ~ " o 7r] and

2 (X - IwP)lF'(w)l"

P*(Ax(w)ldwl)-- Ao(w)ldwl.

Set

m = re(X) = inf{u(w, F ( w ) ) : (w, F(w)) E X } . From the preceding work it is clear that re(X) = 2/ItFIf. Also, # attains its minimum value rn at the point (a, F(a)) if and only if (1 - la]:)lF'(a)l = I]FII. We can now give a geometric version of Corollary 1 of Theorem 2. T h e o r e m 3. Suppose F is holomorphic in O and (a, F(a)) is a minimum point for #x. Then r(a, F ) m ( X ) >_ v ~ / 4 and t(a, F) > s(a, F ) _> 2 artanh 1 / v ~ . Both of these inequalities are sharp. Proof. We m a y assume that a = 0 and F'(0) > 0. If not, then replace F by AF o S, where A is a unimodular constant and S(z) = (z + a)/(1 + az). Notice that we have m ( X ( F ) ) = m ( X ( A F o S)). Thus, we are assuming F'(0) = ][F]]. Then the function F/I]F]] satisfies the hypotheses of Corollary 1 to Theorem 2, and so the conclusions of Theorem 3 follow immediately. Note that equality holds in all the inequalities for the function/;'1. • E x a m p l e 2. It is interesting to look at this geometric interpretation in the special case of the function F1. The e×tremal function F1 has an intimate connection with the ultrahyperbolie metric used in the proof that B > V~/4 via Ahlfors' method. This sheds light upon why Ahlfors' method and Bonk's distortion theorem yield the same lower bound for the Bloeh constant. Actually, it is simpler to consider the function F0 = FI - v ~ / 4 . Then F0 is a twosheeted branched covering of D onto D(3v/3/4) and we let X0 denote the Riemann image surface of F0. The associated function is Go(z) = Fo o S(z) = - ( 3 v ~ / 4 ) z 2. The function # satisfies 2

2

#(w, Fo(~)) = (1 -f~P)IF,~(w)I = (1 -t=l~)lC~,(z)t If Fo(wx) = Fo(w2), then ao(T(w,)) = Co(T(w2)), so T(w,) = -T(w2). This implies that G'o(T(wl)) = -G'o(T(w2)), so #(w~, Fo(Wl)) = #(w2, Fo(w2)). In brief, the function # is invariant under the sheet-interchange function for X0, so # induces a function a on D(3vf3/4). We explicitely determine a. If ; = Go(z) = Fo(w), then

o(C0(z)) = ~(F0(w)) = o ( ~ ( w , F0(w))) = , ( ~ , Fo(w)) = For ( = Go(z) = - ( 3 v ~ / 4 ) z ~ this gives

o-(C) =

4

i¢1,;~ (3-~-~ - tCI)

(~ - I z P ) l a ~ , ( z ) l

The Bloch and Marden Constants

141

a(C)IdC[ is a conformal metric on the punctured disk D(av~/4)\{0} with constant Gaussian curvature -1. Now, the connection with Ahlfors' method becomes apparent. We employ the notation of [M1, §5]. In Ahlfors' method the appropiate ultrahyperbolic metric is constructed from the conformal metric

v TIdq ,,,,4QId¢l

= i 11/2(3 T

_

I¢1)

(Actually, this metric is twice the metric that appears in [M1, §5]; the reason for this factor of two is that ;~c(~)ld~l was taken to be twice the euclidean metric in [5].) This is a conformal metric on D(3r)\{0} which has constant Gaussian curvature -1. The proof that B > v ~ / 4 via Ahlfors' method uses the value r = v~/4. But for this choice of r, O'1, r =

G.

References [A1] L.V. Ahlfors, An extension of Schwarz's lemma, Trans. Amer. Math. Soc. 43 (1938), 359-364. [A2] L.V. Ahlfors, Conformal invariants: Topics in geometric function theory, McGraw Hill, New York, 1973.

[AG] L.V. Ahlfors, H. Grunsky, ~fber die Blochsche Konstante, Math. Z. 42 (1937), 671-673. [B] M. Bonk, On Bloch's constant, Proc. Amer. Math. Soc. (to appear). [C] C. Carathdodory, Theory of functions of a complex variable, vol. II, 2nd English ed., Chelsea Publishing Co., New York, 1960. [H1] M. Heins, On a class of conforraal metrics, Nagoya Math. J. 21 (1962), 1-60. [H2] M. Heins, Selected topics in the classical theory of functions of a complex variable, Holt, Rinehart and Winston, New York, 1962. ILl] E. Landau, Der Picard-Schottkysche Satz und die Blochsche Konstante, Sitzungsber. Preuss, Akad. Wiss. Berlin, Phys.-Math. K1.32 (1926), 467-474. [L2] E. Landau, Uber die Blochsche Konstante und zwei verwandte Weltkonstanten, Math. Z. 30 (1929), 608-634. [M1] D. Minda, Bloch constants, J. Analyse Math. 41 (1982), 54-84. [M2] D. Minda, Domain Bloch constants, Trans. Amer. Math. Soc. 276 (1983), 645-655. [M3] D. Minda, Marden constants for Bloch and normal functions. J. Analyse Math.

42 (1982/s3), n7-127.

D. Minda

142

[Pel] E. Peschl, Uber die Verwendun9 yon Differentialinvarianten bei gewissen Funktionenfamilien und die Ubertragung einer darauf gegrgndeten Methode auf partielte Differentialgleichungen yore elIiptischen Typus, Ann. Acad. Sci. Fenn. Ser. AI

3a6/6 (1963), 23 pp. [Pe2] E. Peschl, Uber unverzweigte konforme Abbildungen, Osterreich. Akad. Wiss. Math.-Naturwiss. K1. S.-B. II 185 (1976), 55-78. [PS] G. P61ya, G. SzegS, Aufgaben und Lehrs~tze aus der Analysis, Bd. 1, Die Grundlehren der math. Wissenschaften in Einzeldarstellungen, Bd. 19, Springer-Verlag, New York, 1964. [Po] Ch. Pommerenke, On Bloch functions, J. London Math. Soc. (2) 2 (1970), 689-695. [R] R.M. Robinson, Bloch functions, Duke Math. J. 2 (1936), 453-459. Received: March 18, 1989

Computational Methods and Function Theory Proceedings, Valpara/so 1989 St. R,uscheweyh, E.B. SalT, L. C. Salinas, R.S. Varga (eds.) Lecture Notes in Mathematics 1435, pp. 143-154 (~) Springer Berlin Heidelberg 1990

On some analytic and computational aspects of two dimensional vortex sheet evolution O. F. O r e l l a n a 1 D e p a r t a m e n t o de Matemgticas Universidad T6cnica Federico Santa Maria Valpara/so, Chile

A b s t r a c t . This survey paper gives an account of recent analytic and numerical results of the initial value problem:

~(~,t) = Z(7,0)

=

'

~

z(.~,O-z(.~,,t)

,

7 + S(7),

which is the Birkhoff-Rott equation for the evolution of a sligthly perturbed fiat vortex sheet. We will indicate some open problems of current research and propose a new physically desingularized Vortex sheet equation, which agrees with the finite thickness vortex layer equations in the localized approximation.

Introduction T h e r e exist two main motivations to study the problem we are concerned a b o u t in these notes: in three dimensions, it is an important unsolved problem of m a t h e m a t i c a l fluid dynamics to determine whether solutions of the Euler equations develop singularities in finite time, with smooth initial data. This problem is i m p o r t a n t for two reasons: from the physical point of view because the existence of such singularities is connected with the onset of turbulence in a high Reynolds number regime, and, from the numerical point of view because the existence of such singularities will generate instabilities in the calculations. Since, this seems to be a hard problem that has been studied through different numerical methods by several authors it is not a bad idea to try a simpler problem namely: show that initially singular solutions can become more singular after a finite 1The author was economically supported by FONDECYT (grant number 235, 1987-88) and Universidad T$cniea Federico Santa Maria (grants 88.12.08, 1988 and 89.12.08) 1989).

144

O . F . Ore//ana

time. Specifically, suppose the velocity field has a symmetry with respect to one of the axes so that the velocity field is two dimensional and has a discontinuity along a smooth curve (for instance a vortex sheet). Does the initially analytic curve stay analytic for all time or develop a singularity in finite time? A number of authors have shown through analytic a n d / o r numerical techniques the appearance of a singularity in finite time (see [l], [21, [31, [41, [51, [71). The second motivation to study the Birkhoff-Rott equation as a mathematical model for the evolution of a vortex sheet is because of its applications to aerodynamics and consequently to the design of more secure and economic airplanes through the calculation of lift of airfoils, the design of airfoils and knowledge and control of the region of turbulence (see [8]). A mathematically rigorous deduction of the Birkhoff-Rott equation can be found in [9] and [10]. This review article describes mathematical analysis results about the evolution of a slightly perturbed flat vortex sheet, because of its importance to the numerics and computation of vortex sheet evolution. Hence we will also comment on the numerical methods to solve Birkhoff-Rott equation proposed by R. Krasny (see [5] and [6]) and the proof of convergence of such methods given by R. Cafiisch and J. Lowengrub in [17]. We also mention open questions of current research in the appropiate place. In the last section of this paper we comment on a new physically desingularized vortex sheet equation proposed by Caflisch, Orellana and Siegel [14] as an alternative to the desingularized equation proposed by R. Krasny [6].

1. Instability analysis and singularity formation for vortex sheets. We will call a 2-dimensional vortex sheet, a discontinuity curve in a fluid domain, which moves with velocity equal to the average of the velocities on its two sides, and across which the tangential velocity, but not the normal velocity, is discontinuous. The jump in tangential velocity across the sheet in a given point is called vortex sheet strength. A good example is provided by the flow field with velocity components: (1)

(u,v)(x,y):=

/ (-½,0)

if y > 0

t

ify > 1, the singularity is far from the physical plane. As t increases it moves towards the physical plane and reaches it at the critical time tc = 2l log e I + 0(log l log ¢1) At the singularity the curvature of the vortex sheet is infinite, although the slope of the sheet remains finite and continuous. Moore also predicted that the singularity would be of type S = 7 a/2. For t near tc it is not clear whether Moore's approximate solution is asymptotically correct. But his results have been partially confirmed numerically by Krasny [5] and Meiron et al [3]. Moreover, numerical eomputations by Krasny [6] using the vortex blob method show that roll up occurs immediately (or at least very soon) after the first singularity time and experiments indicate that the vortex sheet will roll-up into a tightly wound spiral with two branches. Under the assumption of analyticity, Caflisch and Orellana [4] using a localized approximation method (see Caflisch, Orellana and Siegel [14]) prove existence almost up to the time of expected singularity formation of the solution of (4)-(5) for a small amplitude, odd, periodic perturbation of the flat vortex sheet. Considering Z(% t) = 7 + S(7, t), extending S(% t) analytically to the complex 3~-plane, define S*(7) = S(~))and write S(%t) = S_(% 0 + S+(7,t) where S_(%t)

=

- ~ A , ( t ) e -i~" re=0

S+(%t)

= ~ An(t)e '~" r~----1

because of the oddness of S(~/), and under suitable assumptions on 5'(7, t). We were able to approximately localize the equation (4) as follows:

(6)

OS'.. at (%t)

=

1 L "~ B[S](% t) = -2r---ij_ (~ + s(v + ~) - s(v))-'d{

=

B[S+](% t) + B[S_](% t) + D[S+, S_]

This equation is just a definition of D. It is shown in section 4 of [4] that D is small since it depends essentially on product terms S+S_. The first two terms of (6) can be evaluated explicity by contour integration. Since the integral of {-1 vanishes,

On Some Analytic and Computational Aspects ...

1 ./~

147

s+(r + ¢) - s+(r)

d~

_ 1{ O-yS+ } 2 1 + OrS +

l{a s }

Analogously

B[S_](r) = - ~

1 + OrS_

'

and ~-~{S_~(7)+S*_(7)}=~

1+07S

+

-2

1+07S_

+D[S+'S-]'

which implies that:

(7)

(8)

1{ &s+ }

o,

= ~ 1 +OTS+ +H+D'

~S_(7) OS. ~" +(7) -

1{

2

07S-

1 +OTS_

}

+H-D'

where Hi[S] = S+. Now consider the change of variables y = i7 and the function f(y) = S+(7). From the oddness of S(7, t) it follows that (7) and (8) are equivalent and ,9+(-7) = - S - ( 7 ) . Moreover since ~ = i(-ff) it follows that

f*(Y) = f(Y) = ( S* +)(-5') = -(S:)(7) Hence neglecting H+D in (8) we can rewrite it as:

(9)

O. l{iOyf} -~f (y,t) = --~ l + iOyf

'

which is exactly the equation derived by Moore [1]. Now denoting ¢ = 1 + iOyf and ¢ = ¢* = 1 - iOyf* and applying derivation and • to (9) we obtain:

(10)

Og, _

i 0(1

at

2oy 7 )

which is a system of two conservation laws. Setting

148

O . F . Ore//ana

and asking that g, h be analytic in y and real for y real, i.e. g'(y) = g(y), h*(y) = h(y), we get Moore's conservation laws [2]: 0

0

gh

= Ng,

0 Ng

=

(11) h

h.

As an example of initial data, suppose that S(7, t = O) = ie sin 3' which is Moore's initial data. Then f ( y , O ) = 2 eu

and therefore: ¢(v,0)

= 1+2

¢(y,0)

=

Z~ y l-~-e

is the initial data for (10) and h(y,0)

=

2+

C2

-e

= -2tan- 1

C

is the initial data for (11). Considering the system of conservation laws (10) and its corresponding initial data, we were able to show that the solution developed a singularity after a finite time (using P. Lax's a priory estimates for the time of blow up of a 2 by 2 system of conservation law, see P. Lax [13]). Finally considering (10) with the error terms that came from H + D and, using the abstract Cauchy Kowalewsky theorem established by L. Nieremberg and improved by Nishida [15] we were able to establish a long time existence theorem for the solution of the Birkhoff-Rott equation (see Caflisch and Oretlana [4]). Also under the assumption of analyticity, Sulem, Sulem, Bardos and Frisch [10] were able to prove short time existence for solutions of the initial value problem (4)-(5). Borgers [16] and Caflisch and Lowengrub [17] proved global existence for the solution of a de~ingularized version of the initial value problem (4)-(5) as proposed by R. Krasny

[6]. The need for the analytic function space setting for the vortex sheet problem, has been demostrated by Caflisch and Orellana [7], who showed that the problem is not well-posed in the Sobolev space H ~ for any n > 3/2. We constructed exact nonlinear solutions with small initial norm for which 0~S become infinite in arbitrary short time for any a > 1. Such singular solutions where also constructed by Duchon and Robert [18], and an alternative proof of ill-posedness was given by Ebin [19]. The singular

On Some Analytic and ComputationM Aspects ...

149

solutions produced in [7] are not believed to be typical (i.e. the type of singularity formation at the initial time is still on open question). Moore [20] has derived corrections to the Birkhoff-Rott equation, for a thin vortex layer approximating a vortex sheet. Numerical solutions and asymptotic analysis for thin layers have been carried out by Baker and Shelley [21] and Shelley and Baker [22]. There are other important analytic aspects about the Birkhoff-Rott equation and its solutions, like similarity solutions of the form =t

z

=

which has been analysed by a number of authors. So far, we learn from the analysis that computation of the evolution of vortex sheet in two-dimensional, incompressible inviscid flow is delicate because of two reasons: a) Kelvin-Helmholtz instability, and b) Singularity formation on the interface after finite time, infinite curvature of the interface. Hence any numerical method must take into account this problem, because numerical roundoff error can excite the physical instability to produce irregular results well before the physically correct singularity formation and roll-up of the vortex sheet.

2. T h e point v o r t e x m e t h o d , the v o r t e x blob m e t h o d and c o n v e r g e n c e of this m e t h o d s for v o r t e x s h e e t The difficulties mentioned above to compute the evolution of vortex sheets were overcame by R. Krasny [5] and [6] by two methods. a) A point vortex methods with filtering to eliminate spurious high wavenumber components, and b) A vortex blob method. From a formal point of view both methods are similar. The former consists of replacing the continuous curve representing the interface or vortex sheet Z(7 ,t) at a fixed time by a finite number of point vortex, corresponding to a uniform "),-mesh. This is achieved by discretizing the integral part of Birkhoff-Rott equation by Simpson's rule, and then using a Runge-Kutta 4th order method to solve numerically the system of O.D.E. that was obtained after discretization of the integral part of equation (4). In the process of iteration there is a filter that at each iteration filters the discrete solution to eliminate spurious high wave number components stabilizing the numerical process in time and space which otherwise will produce irregular results (chaotic motion of the point vortices). Using this numerical method R. Krasny simulated singularity formation for the vortex sheet evolution. Moreover, the jump in tangential velocity developed a cusp at the singular points. Krasny's numerical results are in good agreement with Moore's asymptotic results.

150

O . F . Ore//ana

The vortex blob method consists of replacing the continuous curve representing the interface or vortex sheet Z(7, t) at a fixed time by a finite number of circular vortex patches of radius 6 corresponding to a uniform 7-mesh. This is achived by doing exactly the same type of discretization described above, but this time applied to the desingularized integro-differential equation

(12)

0Z

t

1

~

jzZ(% t)

- Z ( e ' , t)d~'

(with no filter). Using this method R. Krasny, was able to simulate roll-up of the vortex sheet, giving numerical evidence of the convergence of the m e t h o d with respect to both discretization parameters (i.e. for the discretization parameters corresponding to the time and space variable respectively) holding 6 equal to const. He also gives numerical evidence of the convergence of the vortex blob method to the vortex sheet computing the solution of equation (12) for several values of the parameter 6 going to zero and holding the other two parameters constant. However, no analytic proof of the convergence had been given until Caflisch and Lowengrub [17] proved convergence of both the point vortex m e t h o d and the vortex blob method with both spatial and temporal discretization and simulated roundoff error. For the vortex point method they proved convergence in the case of a vortex sheet that is initially a small analytic perturbation of a flat, uniform sheet and the perturbation is chosen periodic for simplicity. They proved convergence for a short time. The roundoff error er must satisfy er < max {e -1/h , e-116} This condition on the roundoff error is quite strict, but it is consistent with the numerical results of Krasny. This condition is the numerical interpretation of the requirement of analyticity for vortex sheet solutions. For the spatial discretization size h the m a x i m u m wavenumber is km= 1/h. Analyticity for a function f is (roughly speaking) equivalent to requiring that ] ( k ) < exp(-c[k[). This condition can be verified for all k with Ik[ < k m only if the round off error is sufficiently small; i.e. er < exp(-c/h). First of all Caflisch and Lowengrub prove global existance for analytic solutions of Krasny's desingularized vortex sheet equation (12). This result is established without any reference of closeness to a flat vortex sheet and does not use the Cauchy-Kowalewski Theorem, but it is established for arbitrary analytic initial data. The abstract Cauchy-Kowalewski Theorem as established by Nierenberg and improved by Nishida and a discretized version of it is the basic tool they used to construct the solutions and to prove convergence of the vortex blob method. In the limit 6 -- 0 they get a convergence proof for the point vortex method and for a short time interval, certainly less than the critical time. Hence there is no convergence proof of the point vortex m e t h o d after to. For this result it was necessary to set the analysis in an analytic function space because of the use of Cauchy-Kowalewsky theorem, but more i m p o r t a n t because under this assumption the stability of the point vortex m e t h o d is mantained, otherwise the problem is ill posed as proved in [7]. Hence one of the main contributions of their paper is the clarification of the meaning of analy-ticity for numerical analysis because of its possible application to many other ill-posed problems.

On Some Analytic and Computational Aspects ...

151

3. A n e w p h y s i c a l l y d e s i n g u l a r i z e d v o r t e x s h e e t e q u a t i o n In the previous section we mentioned that R. Krasny [6] formulated a desingularized approximation equation of Birkhoff-Rott equation for the evolution of a vortex sheet and he used it to compute roll-up of a vortex sheet after the critical time (i.e. after singularity formation). Krasny's desingularized equation is:

--

(la)

o,z(%0

= K,[z,-~]

=

~1

T)(z(%~)-- ~!~'' t)d~' ]_'oo00IZ--(~,

After the analytic extension of Z(% t) to the complex 7 plane (13) can be rewritten:

- Z*(~',~)d~' OtZ*(7,*) = K,[Z, Z*] = ~1 f ~oo I z*(%,) Z - - ~ , 7 S 7 # - ( ~ 1 ~ 7 ~ ='

(14)

where z*(%,) = z ( v , 0 .

Even though, from a mathematical point of view, when 6 goes to zero (13) approaches the Birkhoff-Rott equation, the desingularization proposed by R. Krasny seems rather arbitrary. Moreover, from a physical point of view, 6 does not represent the viscosity because the desingularized equation (13) conserves energy, whereas viscosity would dissipate energy. Thus the effect of ~ is dispersive rather dissipative. Here we give an argument that illustrates that ~ does not represent a vortex layer thickness either. If we apply the localization method illustrated in the first section of this paper to equation (14) and the equation of motion of a vortex layer of small thickness derived by Moore [20], namely:

OZ" 1 Ot ( f f ' t ) = M ~ [ Z ' Z * l = ~ i f

dr'

¢ 0 ( OZ -' OZ*~

z(.~,t)-z(~,,t) +-CL-iwiNklO.YI

o'r ]

where the terms of size O(¢ 2) have been ignored; ~ = H/p, H =the dimensional thickness and p the radius of curvature, and w := e~ is the vortex sheet strength with ~ the constant vorticity in the layer; we get:

a,s:

=

. v~[s_,s+]= i5 (

{ a,s+ 17-E~+~j 2 * O~S_

aa~s+

}

+ 8- / (1 + O.rS+)312(1 + O~S*_)31; + (1 + a~s+),/=(1 + a~s:),/= '

1{ 1 +o~s: } O,S_"

o~s; = M,[S_, S;]* = -~

3a{s_ s and

(1 + o,s'_p/~(1 + os+)~n + (1 + ~.,S'_)s/2(1 + oq.~S+)t/2

'

O. F. Orellana

152 o~s:

=

M~[S+,

S:]

1{ 1+c3.yS+ O~S+ } + ~(6wi)_10~{(1+0~8+)-2(1 + 0~S*_)-1 }

2 o,s+

= Mo{S+,S:]"

-

21{ I+0"~S:0~S_*} -e(6wi)-lO~{(l+O'~S*-)-2(l+O'~S+)-'}

respectively. Comparison of these equations shows them to be quite different, hence the desingularization parameter 6 cannot be interpreted as vortex layer thickness. This difference can be understood by noting that Moore's expansion parameter is e/w which has units of (length/velocity). A more physically meaningful desingularization equation is found by replacing 6~ in (13) or (14) by (c/w) 2. Then a factor with units of velocity must be put into the first term of the denominator. Hence we proposed the following physically desingularized vortex sheet equation: --

z

--

z-:

i

t

aZ

2

t

~Z

I

, t

2

p

- ;-:

which agrees up to size O(~): with the vortex layer equation in the localized approximate equation. Analytic and numerical analysis of this last equation is presently being performed.

Conclusion From what has been presented here the importance of taking into account all the aspects involved in the resolution of a given problem is quite obvious. In particular, for this problem, it is useful to notice how the analytic, numerical and physical aspects of the problem play an important role in its resolution and how they relate and complement one other to validate the different solutions and finally the model in question (i.e. Birkhoff-Rott equation as a mathematical model to describe the evolution and dynamics of a vortex sheet).

References [1] D.W. Moore, The spontaneous appearance of a singularity in the shape of an evolving vortex sheet, Proc. Roy. Soc. London, Ser. A, 365 (1979), 105-119.

On Some Analytic and Computational Aspects ...

153

[2] D.W. Moore,

Numerical and Analytical aspects of Helmholtz instability in Theoretical and Applied Mechanics Proe. XVI Internat. Congr. Theoret. Appl. Mech., F.I. Niordson and N. Olhoff, eds., North-Holland, Amsterdam, 1984, 629-633.

[3] D.I. Meiron, R.G. Baker, S.A. Orszag, Analytic structure of vortex sheet dynamics, Part 1, Kelvin-Helmholtz instability, J. Fluid Mech. 114 (1982) 283-298. [4] R.E. Caflisch, O. Orellana. Long Time Existence for a Slightly Perturbed Vortex Sheet, Comm. Pure Appl. Math. 39 (1986) 807-838. [5] R. Krasny, On singularity formation in a vortex sheet and the point vortex approximation, J. Fluid Mech. 167, (1986) 65-93. [6] Krasny, R., Desingularization of periodic vortex sheet roll-up, J. Comp. Phys. 65, (1986) 292-313. [7] R.E. Caflisch, O. Orellana, Singular Solutions and Ill-Posedness for the Evolution of Vortex Sheets, SIAM J. Math. Anal. 20, (1989) 293-307. [8] H.W. Hoeigmakers, W. Vaatstra, W., A higher order panel method applied to vortex sheet roll-up, J. AIAA 21, (1983) 516-523. [9] G. Birkhoff, Helmholtz and Taylor instability in "Hydrodynamic Instability", Proc. Syrup. in Appl. Math. XII, AMS (1962), 55-76

[10] C. Sulem, P.L. Sulem, C. Bardos, U. Frisch, Finite time

analyticity for the two and three dimensional Kelvin-Helmholtz instability, Comm. Math. Phys. 80, (1981) 485-516.

[11] G.K. Batchelor, An introduction to Fluid Dynamics, Cambridge University Press (1967) 511-517. [12] P. Garabedian, Partial differential equations, John Wiley and Sons, 1964. [13] P.D. Lax, Development of singularities of solutions of nonlinear hyperbolic partial differential equations, J. Math. Phys. 5, (1964), 611-613. [14] R. E. Caflisch, O. Orellana, M. Siegel, A Localized Approximation Method for Vortical Flows, SIAM Journal on Applied Mathematics, (to appear). [15] T. Nishida, A note on a theorem of Niremberg, J. Diff. Geom. 12 (1977) 629-633. [16] C. Borges, On the numerical solution of the regularized Birkhoff equation, preprint 1988. [17] R.E. Caflisch, J. Lowengrub, Convergence of the vortex method for vortex sheets, SIAM J. Num., Anal. (to appear). [18] J. Duchon, R. Robert, Solutions globales avec nape tourbillionaire pour les equations d'Euler dans le plan, C.R. Acad. Sci., Paris 302 (1986) 183-186.

154

O.F. Ore//ana

[19] D. Ebin,

Ill-posedness of the Rayleigh-Taylor and Helmholtz problems for incompressible fluids, Comm. P.D.E. 13 (1985) 1265-1295.

[201 D.W. Moore,

The equation of motion of a vortex layer of small thickness, Stud. in Appl. Math. 58, (1978) 119-140.

[21] G.R. Baker, M.J. Shelley, On the connection

between thin vortex layers and vortex

sheets, Part I: J. Fluid Mech. (to appear).

[22] M.J. Shelley, Baker, G.R.,

On the connection between thin vortex layer and vortex sheets, Part II: Numerical Study, J. Fluid Mech. (to appear).

Received: July 30, 1989

Computational Methods and Function Theory Proceedings, Valparaiso 1989 St. Ruscheweyh, E.B. Saff, L. C. Salinaz, R.S. Varga (eds.) Lecture Notes in Mathematics 1435, pp. 155-169 (~) Springer Berlin Heidelberg 1990

On the

Numerical

Decomposition

Performance

Method

N. Papamichael

and

of a Domain

for Conformal

Mapping

N.S. Stylianopoulos

Department of Mathematics and Statistics, Brunel University Uxbridge, Middlesex UB8 3PH, U.K.

1. I n t r o d u c t i o n This paper is a sequel to a recent paper [14], concerning a domain decomposition m e t h o d (hereafter referred to as D D M ) for the conformal mapping of a certain class of quadrilaterals. For the description of the D D M we proceed exactly as in [14:§1], by introducing the following terminology and notations. Let G be a simply-connected Jordan domain in the complex z-plane (z = x + i y ) , and consider a system consisting of G and four distinct points Zl, z2, z3, z4 in counterclockwise order on its boundary OG. Such a system is said to be a quadrilateral Q and is denoted by Q -- {G; zl, z2, zs, z4}. The conformal module re(Q) of Q is defined as follows: Let R be a rectangle of the form (1.1)

n:=

{(~,,):a < ~ < b,c< ~ < d},

in the w-plane (w = ~ + it/) , and let h denote its aspect ratio, i.e. h := (d - c)/(b - a). T h e n rn(Q) is the unique value of h for which Q is conformally equivalent to a rectangle of the form (1.1), in the sense that for h = ra(Q) and for this value only there exists a unique conformal map R -* G which takes the four corners a + ic, b + ic, b + id, and a + id, of R respectively onto the four points zl, z~, z3, z4. In particular, h = r e ( Q ) is the only value of h for which Q is conformally equivalent to a rectangle of the form

(1.2)

Rh{c~} := { ( ( , ~ ) : O < ( < l , a < 7/ < a + h}.

T h e D D M is a method for computing approximations to the conformal modules and associated conformal maps of quadrilaterals of the form illustrated in Figure 1.1(b). T h a t is, the method is concerned with the mapping of quadrilaterals

(1.3a)

Q := {a; z~, z:, z~, z~},

N. Papamichael, N.S. Stylianopoulos

156

where: • The domain G is bounded by the straight lines x = 0 and x = 1 and two Jordan arcs with cartesian equations y = -vl(x) and y = r2(x), where rj; j = 1, 2, are positive in [0, 1], i.e. (1.3b)

a := { ( x , y ) : 0 < x < 1,--TI(X ) < y < T2(X)}.

* The points zl, z2, z3, z4 are the corners where the arcs intersect the straight lines, i.e.

(1.3c)

zl = --irl(O),

z2 = 1 -- it1(1),

z3 = 1 + ir2(1),

z4 = iv:(O).

Let Q be of the form (1.3) and let (1.4a)

Gi := { ( x , y ) : 0 • x .~ 1,--TI(X ) < y < 0},

and (1.45)

G~ := {(z, u ) : 0 < z < 1,0 < u < ~ ( x ) } ,

so that G = G1 U G2. Also, let Q1 and Q2 denote the quadrilaterals (1.4c)

Q, := {G1;z~,z~,l,O}

and

Q~ := {G2;O,l,z3,z4},

and let h := re(Q) and hj := m(Q~); j = 1,2; see Figures 1.2(5) and 1.3(5). Finally, let g and gj; j ---- 1, 2, denote the conformed maps (t.5) (1.6)

g: n h { - h l } -~ G, g l : R h l { - h l } --} G1 and

g2: Rh2{0} ---* G2,

where, with the notation (1.2), R h { - h ~ } := { ( C ~ ) : 0 < ~ < 1 , - h , < ~ < h - hi},

Rhl { - h i }

:= {(~,r]): 0 _ hi + h2 and equality occurs only in the trivial cases where G is a rectangle or rl(X) = 7"2(x), x e [0, 1]; see e.g. [9: p. 437].) The treatment of the DDM contained in [14] is a theoretical investigation leading to estimates of the errors in the approximations (1.7). These error estimates are derived by assuming that the functions rj; j = 1,2, satisfy the following: (i) rj; j = 1,2, are absolutely continuous in [0,11, and (1.8)

dj := ess

[rj(x){ < ce.

sup

0~x__5. The solution conjectured by Krzyz [4] is that An = 2/e and that [an[ < 2/e unless (1)

f(z)

= ce ('Tzn-1)/('rzn+l),

Icl =

= 1.

This conjecture is indeed true for n = 1,2,3,4. A rather natural extension of the above question is to assume

[ 1 [2. f e H p, Mp(f) = sup L2--~Jo r 0 and a~ >__0 (this can be accomplished by replacing f by 3'f(ze i~) for appropriate a and 3'), then

E x t r e m M Problems for Non-vanishing HP Functions

179

(i) al 1 with equaJity if and only if I ( z ) = 2-~/p(1 + z)V,e-Cl-,mci-,)/(,+z) and

(ii) a, ~, p > 1, then result that i f f E N p , la,d < (2/e)1-~ is true, as follows. Under the above assumption, the function

/ ~

~11~

g(z) = [ I I f j ( z ) ] /

, f j ( z ) = f(ze2~i(J-1)/~),

is n-fold symmetric (i.e. g(z) = k(z n) for some k e Np) and satisfies

g(z) = ao + anz ~ + ' " . It therefore follows that a~ for this restricted class has the same bound as al for the entire family. While it is conjectured that the extreme value for la~[ in the entire family is identical to that for lall when p > 1. This is not true for p < 1: As we shall see below, la2] has the sharp bound 2~-2/P(2 + p)2/p-1/p (the square of the sharp bound for Jail) when p < 1.

T.J. Suffridge

180 2. Proof

of main

theorem

Given a non-trivial function f E Np, we may write

f ( z ) = (h(z))2/VI(z),

(6)

where h E H 2, h(z) 7t 0 when z E D and I(z) = e -P(~) is an inner function, h ( 0 ) > O, P(O) >_ O. Lemma

1. Suppose h and k are outer functions in the space H 2 and that h(O) >

o, k(o) > o. zf h" and k" are the corresponding boundary functions and I h * ( e ° ) l

=

[k*(ei°)[ a.e., then h = k. Proof. By hypothesis,

h(z)

=exp

~

~e it-z

,~ e it -- z

= exp = k(z).

L e m m a 2. K h ( z ) = ~ = o ckz k E H ~, h(z) ¢ 0 in D then for 0 < e < 1, the function

he(z) = (1 + eeiZz~)h(z)

[ck12(1+ e2) + 2eRe ~ ckek+~e iz k=O

has the same properties and [[h~[[2 = 1. Proof. Use the fact that for E~=oakz k = f ( z ) C H 2, llfI[~ = Ek=0~ lakl2 and n-1

oo

(1 + c~%")h(z) = F. ~ z~ + ~ ( c ~ + . + c~'ec~)~+". k=0

Lemma

•

k=0

3. I f h and h~ are related as in L e m m a 2, then /" iz ET=o ckek+~

Proof. A straightforward differentiation.

h

•

L e m m a 4. I f L C £ is a non-trivied//near functional and f E Np has the property R e L ( f ) >_ ReL(g) for M1 g C Np, 0 < p < 0% where f = h2/PI, h C H 2 and I is inner, then [[hII2 = 1 and oo

(7)

L ( z ~ f ) = L ( f ) ~_, ck~k+,~ for n = O, 1 , ' " . k=O

Extremal Problems for Non-vanishing H p Functions

181

Proof Clearly [Ifllv = Ilhll2 and flfl[p < 1 implies pf E Np for some p > 1 so that

ReL(pf) = pReL(I)

> a e L ( / ) (because L is non-trivial and the assumed extremal f satisfies R e L ( f ) > 0). Thus, (7) holds for n = 0. Now consider L = (h~)~/~I, h~ as in L e m m a 2. T h e n ReL(f~) _< R e L ( f ) and hence

a e L ~, d~ L o ]

-

We have

df~

= ~(h~)2/p_ll dh~

de ~=0

= 2 f (ei~z~

de ~=0

Reei~?ock~k+~)

P

=

"

Therefore Re

eiZL(z~f) - Re eiz

ck~k+,L

s , c ' > 0 . Thus, Q(z) = Vf~/t-Ijn=l(1 + pjeia'Z) has the property, Q is outer and IO(e'°)t 2 = lh(e~°)12. This means Q = h. This completes the proof of (iii) in the main theorem. To prove that (i), (ii) and (iii) hold for some extremat f in case p = 0% let fp be extremal for L in h r , p _> 2. Since N ~ C - ~ C N2, 2 _< p, the family {fp} = {(hp)2/PIp} is locally uniformly bounded. Thus, as p --* c~, there is a sequence {p~} such that pn ~ c~ and hp, --* h, Ip, -~ I uniformly on compact subsets of the disk D. Clearly, ReL(fp) > maxg~N~ ReL(g) and hence R e L ( f ) _> maxgeN~ ReL(g) by continuity of L. The theorem now follows. •

183

E x t r e m a l P r o b l e m s for Non-vanishing HP Functions

3. Applications

Proposition.

I f f E Np then If(r)l
0) provided the functional L has the property that L( f ) > 0 for extremal functions f with max ReL(g ) = ReL( f ) over the restricted family. Theorem 3. I f f E

when (1 + r ) ( l - r)'/p-'

%, and f(0) > 0, then

2 1, where

and

when (1

+ r ) ( l - ~ ) ~ / p - \ 1, where p is a solution of the equation

Remark 2. Note that the result in Theorem 3 is not the correct bound for the entire class N,. This is easily seen in case p = m. In this case, since Ref(r) 1, if we require f(0) > 0, clearly Re(f ( r ) - f(0)) < 1. However, if f (z) = - e - t P ( z ) where P(0) = 1 and t is small and positive then f(0) can be near -1 but with r near 1, we may choose P so that tP(z) = u vi, u small, v = a. Then f (r) - f (0) will be nearly 2. The problem maxj~,v,Re( f (r) - f (0)) is more difficult.


0 then the function g(z) = (f (z) . fo)1/2 E NP, g(0) = f (0) > 0 and g(r) = If(r)l 2 Ref(r). Thus, by Remark 1, Theorem 1 applies to the functional L(f) = f(r) - f(0). Further, the extremal function can be assumed to have real coefficients. We have rnf (r) = L(znf ) = [f(r) - f (0)](zn, 1 hi2), so that

ExtremM Problems for Non-vanishing HP Functions

where p is real, ]p[ < 1, z = e ~°, and K -

185

rf(O) If [p[ = 1, it is easy to check f(r) - f(O)"

t h a t p # - 1 (i.e., the relation 1 -

r2 -

2r 2

f(O)

2rf(O)

_

f(r) - f(O)

f(r) - f(O)

c a n n o t hold), so Ipl = I implies p = 1. In this case, we find t h a t

h(z)

Since

tlfllp

= 1,

= v~

K-

l+z

and

I - rz

f(z)

~

K 1 / p ( I + z ~ 2/p e - iP-* 'gg. \1 - rz/

1 -2 r , s o t h a t f(r) - f(O) = 1 2r_r f (0),

hence

~1 _+ r~ ~ ] 21pe _ t ( l _ r ) / ( l + r )

_

f(r) -- 11+ r f ( o )

and

-i-r~ -'. 1+

nus, e,+, = ( l + r ) ( 1 - r ) 2/p-a a n d we obtain the value of t given in the t h e o r e m . Because t >_ 0, this can be e x t r e m M only for the values of p for which (1 + r)(1 r ) 2]p-1 > 1 (this clearly includes p >_ 2). In case [p[ < 1, we find as before t h a t p > 0, so t h a t W

~

--

lh(z)l -

1 K 11 - ~ z l ~ 7 I(1 + p z ) ( p + z)i -

z=ei°,andweeoncludeh(z)=,/~SI+pz V p 1-rz' A g a i n u s i n g Ilfllp = 1, w e get K -

f ( r ) - f(O) =

T h u s , since f(O) =

I z l < l . Thus, f ( z ) =

l +p(1 2 p r- +rp2)2

r(1 + 2pr + p2)

~(~-~)

1 K [1 + pzl 2, I1 - ~ z p p

( K ) 1/" ( l + p z ~ 2/" \ ~ ] "

' so

. f(O), f(r) = r +pp ( l + p1r ) f tr02a . , ,

we have

(1 + ? 0 and each factor (1 - zei%) is a factor of h. Now set

h ( z ) = z=h (3) so t h a t [A(eiC~)12 = e-in~[h(ei~)h(ei~)] a n d note t h a t hh is a self-inversive p o l y n o m i a l of degree n + degree h < 2n. Further, each zero of h t h a t lies Oll lzl --- I is also a zero of A a n d therefore is a zero of hh of even multiplicity. Using T h e o r e m I (i), for the extremM, we see t h a t n-1

a. + ~ ( a k e i(~-k)~ + ~-~e-i(~-k)*) = a . e i " ~ h ( e - i ~ ) h ( e -i~) k=O

a n d thus,

ao + •lz + 62z 2 + . . . + a~z ~ + an-1 zn+l "4-"" q- ao z2n = a n h ( z ) h ( 2 ) , when ]z[ = 1. This equality therefore persists for all z and we see t h a t a0 = a,~coc,, so t h a t c,~ > 0 a n d degree h = n. Further,

ao + a l z + . . . + anz ~ + a ~ - l z T M + . . . + aoz 2n = a~h(z)A(z). W e n o w state the next theorem. Theorem

4. I f f E Np satistles f(O) > O, am > 0 and a~ > [g(~)(0)[ for a1I g E Np -

then

n!

(i) f ( z ) = h2/pI, where h is a p o l y n o m i M of degree n, I = e -tP(z), where m 1 q- ze'% t P ( z ) = ~ tj 1 j=l for s o m e m , o f h,

O 1. We believe it is trivial when p < 1. T h e o r e m 6. If f E Np is extremal /'or the problem L ( f ) = a~, n > O, and 1 < p, then the inner part I in the representation f ( z ) = (h)2/PI is non-trivial O.e. 1 7~ 1). Proof. We know that if I is trivial then (h) 2/p ~ a,,h]~. Hence, with h(z) = Co + ' . . + c~z '~, we have cg/p = a,~coc,, However, since h(z) 7~ 0 in D and Ilhll= = 1, 0 < ic~l _< Ic01 < 1. Thus, cg/p < a,~c~ so that c~/p-2 < a,~. However p > 1 implies 2/p - 2 < 0 so a~ > 1. We know this last inequality is false and it follows that I cannot be trivial. • We have a complete solution for n = 2. T h e o r e m 7. If f E Np, then

la~l _< (2/e) 1-V',

(s)

la l < (_L_2

(9)

- \2-p)

1 < p,

_2 p'

p__-(1

l+p 1 + 2 p - p2

(1 - p)3 log(1 + p + / )

p)

(1 - p)~ 2

(1 =3 p)3 o~ ~(1 k=O

- p)k

p)2 (1 - p)3

(1 2

Further

-

3p

l+p

< 1. Thus, it is sufficient to show 1 + p + ( p - p2) _< p [1 - / - 2p(1 - p) - p(1 - p)~ - ~(1 - p)3 + (1 - p)3] _ 4(1 _ p ) 3

3

p.

Finally, 1 + p + p2 < e3!p = 1 + -~p + Sp2 ~ + .-- is sufficient. The last inequality clearly 1 1 2 holds since 0 < ~p - uP , 0 < p < 1. This completes the proof. •

References [1] J. E. Brown, On a Coe~ieient Problem for nonvanishing Hp functions, Complex Variables, Theory and Applications 4 (1985), 253-265. [2] C. Carath4odory, Uber den Variabilitdtsbereich der Fourier3chen Konstanten yon positiven harmonisehen Funktionen, Rend. Circ. Mat. Palermo 32 (1911), 193-217. [3] J. A. Hummel, S. Scheinberg and L. Zaleman, A eoej~cient problem/or bounded nonvanishing functions, J. Analyse Math. 31 (1977), 169-190. [4] J. Krzy~, Coej~cient problems for bounded nonvanishing functions, Ann. Polon. Math. 20 (1968), 314. [5] Delin Tan, Coejficient estimates for bounded nonvanishing functions, Chinese Ann. Math. Ser. A4 (1983), 97-104 (Chinese). [6] O. Toeplitz, (/bet die Fouriersche Entwieklung positiver Funktionen, Rend. Circ. Math. Palermo 32 (1911), 191-192. Received: September 3, 1989.

ComputationalMethods and Function Proceedings, Valpara/so 1989

Theory

St. Ruscheweyh, E.B. SalT, L. C. Salinas, R.S. Varga (eds.) Lecture Notes in Mathematics 1435, pp. 191-200 ~) Springer Berlin Heidelberg 1990

Some

results on separate of continued

convergence

fractions

W.J. Thron 1 Department of Mathematics, University of Colorado Campus Box 426, Boulder, CO 80309, U.S.A.

1. Introduction The term separate convergence was introduced recently to describe the phenomena which result when conditions - - stronger than needed for convergence - - are imposed on continued fractions or similar algorithms. The term separate is motivated by the first result of this type due to Sleszyfiski in 1888. He proved that if the continued fraction K(anz/1) satisfies the condition ~ lant < c~ then not only does the sequence of approximants {A~(z)/Bn(z)} converge but {An(z)} and {Bn(z)} converge separately to entire functions A(z) and B(z). In later investigation the conclusions frequently are less sweeping. One settles for the existence of an "easily described" sequence {Fn(z)} such that {A~(z)/F~(z)} and {Bn(z)/F~(z)} converge separately for z E A. Usually the convergence will be uniform on compact subsets of A, which may be a proper subset of the complex plane C. The restriction that the F,~ can be "easily described" is essential because for convergent continued fractions one can always choose In(z) = Bn(z) and thus the distinction between ordinary and separate convergence would become meaningless. Most, if not all, instances of separate convergence occur for limit periodic continued fractions with elements that are functions of a complex variable. Sometimes separate convergence is a tool in the derivation of results on analytic continuation or behavior on the boundary of the function to which the continued fraction converges. In other cases it may only be a by-product of such investigations. Since orthogonM polynomials can be obtained as denominators of the approximants of certain continued fractions, there is an overlap between our work and known results about the asymptotic behavior of 1This research was supported in part by the U.S. National Science Foundation under grant No. DMS8700498.

W.J. Thron

192

orthogonal polynomials (see, for example [2]). Our results can also be used to obtain information on the asymptotic behavior of orthogonal L-polynomials. Instead of describing at length the results obtained on separate convergence, we give in the References a list of all articles on the subject we know'about. This includes some papers where separate convergence is provable, but is not explicitly established. We shall concentrate here on the proof of a basic theorem from which some, but not all, of the known results on separate convergence can be derived. The theorem also yields new results, in particular on general T-fractions. We shall also discuss briefly the impact of equivalence transformations on separate convergence. An account concerned mainly with separate convergence of PC fractions, Schur fractions and Schur algorithms is being prepared by O. Nj£stad. In a subsequent article we hope to return to the subject and explore some other general approaches which yield results on separate convergence.

2. A s y m p t o t i c behavior of numerators and denominators of approximants of limit periodic continued fractions. In this section we shall prove a quite general and rather involved theorem. In the next section corollaries of a much simpler nature will be given. T h e o r e m 2.1. Let an(z), b~(z), a(z) and b(z) be hotomorphic functions for z E A. Further assume that a,~(z) ¢ 0 and

~im a~(z) = a(z),

j i m b~(z) = b(z)

fo~. z e z~

Then

(ao(z) ~=, \ b~(z) ]

is a limit periodic continued fraction for z E A. Set (2.2)

am(z) = a(z) + 6~(z), b,~(z) = b(z) + ,7,~(z).

Note that ~ ( z ) and ,~(z) are holomorphic and lim 6,~(z) = 0,

?I--+OO

lim rl~(z) = 0

for z e A. Let x f f z ) and x2(z) be the solutions of

~,~ + b( z )w - a( z ) = o and assume that the solutions have been so numbered that (2.3)

xxl(z) -~

Further assume that the series

0 for

z e ,40.

Finally, let ,4(~) be such that I ~ ( z ) - ~(z)l > 2~ for

(2.6)

z e `4(
0

Here A,~(z) and B~(z) are the numerator and denominator, respectively, of the nth approximant of the continued fraction (2.1). Of course the theorem is of interest only if a'¢O. Proof. From now on we shall usually not indicate the dependence of the various functions under consideration on z. We shall simply write an, b~, a, b, 5n, qn, x,, x2, An, B~, C~, Dn. We note that

(2.s)

--a =

XlX2,

-b

=

Xl + x 2 .

As was shown in [16] one then has

B~ + xlBn-1

= (b + xl)Bn-1 + aBn-~ + rl,~Bn-1 + 5,~B~-2 =

(-x2)Bn-, + (-z~x2)Bn-2 + ~,~B,~_, + 6,,B,~_~.

Iterating these equations one arrives at n-1

(2.9)

n-1

Bn + x l m _ , = (-x~) n + E ( - x ~ ) n - ~ - % ÷ , B ~ + F_,(-x~)"-I-%÷~B~-~. k=0

k=0

n-1

n-1

An analogous derivation leads to

(2.1o)

B. + x~m_, = (-~,)~ + ~..(-x,)~-~-%÷,B~ + F , ( - Z l ) " - ~ - % ÷ I B ~ - . k=O

k=O

If Xl ~ x2, one can solve the system of equations (2.9), (2.10) for B,~. The result is

W.J. Thron

194 n-1 - - ( - - X l ) n + l -~- ~-'~.((--Z2) n - k -k=0

(xl - *=)Bn = (_x=)n+,

(--Xl)n-k)rlk+lBk

n-1

+ E ( ( - x ~ ) n-~ _ (-~,)'~-k)6~+~B~_,. k=O

Introduce (2.11)

On

Bn -

(_x~)n+,

In terms of Dn we then have (xl

-

x2)Dn = 1 -(x-'Z'~ n+l+ E

\X2]

1-

77

,k+,Dk

k--0

(2.12) 1 n-'(1 (X__Al~n-kI q--------~2k~=O -- \X2/ ] 5k+lDk-1. Similarly one obtains for (2.13)

Ca

An -

(--Z~)n+l

the formula (Xl

-x2)Cn ~ Xl -

(Xl~n+l n-1 ( ( X l ~ n - - k ~ -x2+~ 1- ~ ~k+lCk \X2/ k=0 \X2/ 1

(2.14) + --X2 k----O

KX2/

]

From (2.12) one can deduce for z E A* the inequality n-1

(2.15)

Ix~ - x211Dnl _< 2 + 2 ~

2

I~k+~l IDkl + ~

k=O

n--1

~

16k+~lIDk-~t .

k=0

We would like to prove that there exists a constant M > 0 such that for z in certain subsets of A0 n A* ] D n I < M for all n > 1. To do this we prove the following lemma. A similar result can be found in [2, p. 455]. Lemma2.2.

Leta>0,%>0,

n>_ 1 and Sn k O, n >__O. If

n-1 (2.16)

S~ < a + ~ %+lSk,

k=O

~hen

n >_ 1, So l.

+7,Q =: P,.,,

k=l

Proof. We have $1 _< a + 71So _< a(1 +71)Set Po = a and assume t h a t Sk 5 Pk for 0 < k < n - 1. T h e n sit

5 ~ + E kn=- 1o ( (

1

. . . . l i p ~+~ - Pk) + 7k) - 1)Pk = a -~ z~,=oV

=a-Po+P.=P~. T h e l e m m a is thus proved by induction. • We apply the l e m m a to (2.15) by setting [Dit I = Sit, a = m a x ( 2 / I z , - a2[, 1/[a2[), %

_

I11

_2 x=l (It/k[ + 17)~k+1 "21/

For K a compact subset of A0 N A* we then have [x2] > dK > 0, [x1/x2[ < rg < 1. In view of (2.4) the sequence {Dit} has a uniform b o u n d M ( K , e) satisfying (2.18)

IDd_< max(1/e, 1 / d g ) f i

(1 + I~kl+ ~ )
0 for z E ¢~.

= D(z)

Here ~ - - is chosen so tha~

Proof. In this ease A = C, A* = A 0 = ¢2 and

[

1 A(~) = Lz : - - ~ a -- z > ~2/4]a[] T h e p r o o f of the convergence of (3.1) under the a s s u m p t i o n (3.2) is also due to Sleszyfiski [13]. O u r conclusion a b o u t s e p a r a t e convergence is not explicitly s t a t e d in his article b u t can easily be deduced. Corollary

3.3 If the g e n e r a / T - f r a c t i o n

F,z +GozJ satist~es E~°°=I [F~[ < cc and E,~=~ [G~ - G[ < cx~, G ~ O, then lira A,~(z) - C*(z), ~-oo (1 + az)T M for all z 6 C ~ [ - l / G ] .

lira

B~(z) - Dr(z) (1 -~- az)T M

The functions C t and D t are holomorphic in C ,~ [ - l / G ] .

Proof. x~ = O, x2 = - 1 - Gz, A = C, A" = C ~ [ - l / G ] = A0, A (~) = [z : [1 + 2c]. Corollary

azl

> •

3.4. In the general T-fraction (3.3) iet

(3.4)

lira F~ = 1,

lim G~ = - 1 .

Further, assume that

~lF,-ll
1. Proof. A = C. Xst case: x I --Z, X2 2 n d c a s e : xl = - l , x2=-z,A*=[z:lz[> :

=

- 1 , A* = [z: [z[ < 1], A(0 = [z: [ z - l [ > 2¢]. 1], A(') = [z : [ 1 - z[ > 2 ¢ ] . "

This corollary overlaps some results of W a a d e l a n d [21] and T h r o n a n d W a a d e l a n d

[151.

W.J. Thron

198 C o r o l l a r y 3.5.

In the J-fraction

(3.5)

K

n~--1

A,,(z)

nli~lTl°°(C -- Z) n+l --

and C* and D* are

C*(z), n---*c¢ lim B,~(z) -D*(z) (C : Z) n-'''-+l

holomorphic in C -~ [c].

P~oor. n = a - = c , z , = 0, x~ = z - c, a 0 = c ~ [cl, a (~) = [ z : Iz - cl > 2 d . This is Theorem 5 in Schwartz [12].

4. C h a n g e s

Let

in separate convergence transformations.

under

equivalence

K~=1(a,~/b,,) be a given continued fraction, let

be defined by (4.2)

5~=7~_:7~a~,

b~=7,~b~,

70=1,

%~0,

n > 1.

Then the two continued fractions are equivalent in the sense that their sequences of approximants are identical. However for the numerators and denominators of the approximants one can show, using the recursion relations, that (4.3)

Ao=AoI]~o

,

B~=B~II%,

v=0

,

n>-:,

v=O

provided one uses the convention FIE = 1, for m < k. We use this idea to prove another result for J-fractions (3.4). The continued fraction (4.4)

K(k'(z-c')-:(z-c'~-:)-l) l

,

Co = z - l ,

is equivalent to (3.4). Under the conditions (4.5)

c,~ ---* co,

< oc,

the numerators Am and denominators/~, of (4.4) satisfy, for z E C ,-, [Cl, c 2 " "] l i m ii. = A,

lim/}. = B.

Some Results on Separate Convergence...

199

It follows from (4.3) that for the J-fraction (3.4) one has

An

Bn

l i m l_i~:o(Z _ c,) - B .

lirn I_t~=o(z _ cv) - A, If one strengthens (4.5) to

one obtains (Theorem 2 in Schwartz [10]) ,~-~ I1~=1(_c. ) - A

1- z

,

lim I]~=~(-c.) - B

v----1

1-

.

v=l

References [1] R. J. Arms, A. Edrei, The Padd tables and continued fractions generated by totally positive sequences, Mathematical Essays dedicated to A. J. Macintyre, Athens, Ohio (1970), 1-21. [2] F. W. Atkinson, Discrete and continuous boundary problems, Academic Press, New York, 1964. [3] A. Auric, Recherches sur les fractions continues algdbriques, J. Math pure et applique, (6) 3 (1907), 105-206. [4] A. Edrei, Sur des suite~ de hombres lides h la thdorie des fractions continues, Bull. Sci. Math., (2) 72 (1948), 45-64. [5] Lisa Jacobsen, A note on separate convergence of continued fractions, preprint. [6.] E. Maillet, Sur tes fractions continues aIgdbriques, J. Ec. poI., (2) 12 (1908), 41-62. [7] O. Perron, Die Lehre yon den Kettenbrgchen, 3. Anti., 2. Band, Teubner, Stuttgart, 1957. [8.] H.-J. Runckel, Bounded analytic functions in the unit disk and the behavior of certain analytic continued fractions near the singular line, J. reine angew. Math., 281 (1976), 97-125. [9]

, Continuity on the boundary and analytic continuation of continued fractions, Math. Zeitschr., 148 (1976), 189-205.

[10]

, Meromorphic extension of analytic continued fractions, Rocky Mtn. J. Math., to appear.

200

W.J. Thron

[11] J. Schur, Uber Potenzreihen die im Innern des Einheitskreises besehrdnkt sind (Fortsetzung), J. reine angew. Math., 148 (1918/19), 122-145. [12] H. M. Schwartz, A class of continued fractions, Duke Math. J., 6 (1940), 48-65. [13] I. V. Sleshinskii (J. Sleszyfiski), Convergence of continued fractions (in Russian), Zapiski matematicheskago otodieleniia Novorossiiskago obshchestvoispytatela, 8 (1888), 97-127. [14] T. J. Stieltjes, Recherches sur les fractions continues, Ann. Fac. Sci. Toulouse, 8 (1894), 1-122. [15] W. J. Thron, Order and type of entire functions arising from separately convergent continued fractions, submitted. [16] W. J. Thron and H. Waadeland, Convergence questions for limit periodic continued fractions, Rocky Mtn. J. Math., 11 (1981), 641-657. [17] Walter van Assche, Asymptotics for orthogonal polynomials and three term recurrences, Proceedings of Columbus conference, to appear. [18] H. von Koch, Quelques thdor~mes concernant Ia theorie gdndraIe des fractions continues, Oversigt av kongl. Vetenskaps-Akademiens FSrhandlingar, 52 (1895). [19]

, Sur la convergence des determinant~ d'ordre infini et des fractions continues, Comptes Rendus, 120 (1895), 145.

[20] . . . . . . . Sur un thdor~me de Stieltjes et sur les fonctions dd~nies par des fractions continues, Bull. Soc. Math. France, 23 (1895), 33-41. [21] H. Waadeland, A convergence property of certain T-fraction expansions, Kgl. norske videnskabers selskabs skrifter 1966, No. 9, 3-22. [22] H. S. Wall, Some recent developments in the theory of continued fractions, Bull. Amer. Math. Soe., 47 (1941), 405-423. [23]

, The behavior of certain Stieltje~ continued fractions near the ~ingutar line, Bull. Amer. Math. Soc., 48 (1942), 427-431.

Received: August 7, 1989, in revised form November 28, 1989.

Computational Methods and Function

Theory

Proceedings, Valparaiso 1989 St. Ruscheweyh, E.B. Saff, L. C. Salinas, R.S. Varga (eds.) Lecture Notes in Mathematics 1435, pp. 201-207 (~) Springer Berlin Heidelberg 1990

A s y m p t o t i c s for t h e Z e r o s o f t h e P a r t i a l S u m s o f e ~'. II R.S. Varga* Institute for Computational Mathematics Kent State University, Kent, OH 44242, USA and

A.J. Carpenter D e p a r t m e n t of Mathematical Sciences Butler University, Indianapolis, IN 46208, USA

1. I n t r o d u c t i o n n

With

s,(z)

:=

~--~zJ/j!

(n = 1 , 2 , . . . ) denoting the familiar partial sums of the

j----0

exponential function e ", we continue our investigation here on the location of the zeros of the normalized partial sums, sn(nz), which are known to lie (of. Anderson, Sail', and Varga [1]) for every n > 1 in the open unit disk A := {z e C : Izl < 1}. For notation, let the Szeg5 curve, D ~ , be defined by

(1.1)

D ~ := {z E C : lzel-Zl = 1 and [z t < 1}.

It is known that D ~ is a simple closed curve in the closed unit disk ~ , and t h a t Do~ is star-shaped with respect to the origin, z = 0. If {zk,n}~=l denotes the zeros of sn(nz) (for n = 1,2,---), then it was shown by Szeg5 [7] in 1924 t h a t each accumulation point of all these zeros, {zk,~}k=l,n=1, m u s t lie on D ~ , and, conversely, that each point of D ~ is an accumulation point of the zeros n~oo n,oo {zk,n}k=l,n=l. Subsequently, it was shown by Buckholtz [2] that the zeros {zk,~}k=l,,=l all lie outside the simple closed curve Doo. As for a measure of the rate at which the zeros, {zk,~}~=l, tend to D ~ , we use the quantity *Research supported by the National Science Foundation

R.S. Varga and A.J. Carpenter

202

dist [{Zk.n}~=l'~D¢01 := max (dist [zk,~;Doo]),

(1.2)

l_0.10890... > 9,

(1.7)

for any fixed ~ with 0 < $ _< 1. In [3], an arc, D~, was defined for each n = 1, 2 , . . . by Dn:=

1-z

zeC:[zd-~l~=r~

~

(1.8)

, tz] _< 1, and I arg zl _> cos-1 ( ~ - ~ )

},

where from Stirling's formula, n! (1.9)

1

vn.-- n~e_~ 2v/~-~ 1 + ~ + 2 8 8 n

1 2

139 51840n 3 + . - . ,

asn-+oc.

n This arc was introduced to provide a much closer approximation to the zeros { Z k,n}~=l of s,(nz), than does the Szeg5 curve. With the notation of (1.5), it was shown in [3] that, for any fixed ~ with 0 < 5 _< 1,

(1.10)

dist [{zk,,}k=1\C6, ON]

O

(n ~ ~ ) ,

and moreover that (1.10) is best possible, as a function of n, since (cf. [3, eq. (3.18)]) (1.11)

limoo{n2 • dist [{zk,,}~=l\c6;n~]} >_ 0.13326... >

0,

Asymptotics for the zeros of the partial sums of e~. H

203

for any fixed 6 with 0 < 6 ~ 1. It turns out (cf. [3, Prop. 3]) that, for each positive integer n, the arc Dn is starshaped with respect to the origin, z = 0, i.e., for each real number 8 in [-Tr, +~r] with [8[ >_ c o s - l ( - ~ ) , there is a unique positive number r = r,~(8) such that z = re i° lies on the arc D,~ of (1.8). Let :D~ be the closed star-shaped (with respect to z = 0) set defined from the arc D~, i.e.,

L A, Izl _< 1, and

D= := {z E C : [zel-~[ ~ _
cos-l(rt : 2 ) } ,

(rt ---- 1,2,'.-).

n Recently, R. Barnard and K. Pierce asked if the zeros, {Z k,~}k=l, of s~(nz) all lie outside ~D~ for every n _> 1. (This would be the natural analogue of the result of Buckholtz [2] which established that all the zeros {zk,~}k=l,~=l lie outside of Doo.) This is not at all 16 and "t f z k,2rIk=l 127 obvious from the graphs of [3], since it appeared that the z e r o s {Zk,16}k=l of s16(16z) and s2r(27z) were, to plotting accuracy, respectively on the curves D16 and D27. It turns out that the zeros, {z~,,}~=l of sn(nz) do not all lie outside :D,, for every n > 1. This follows from our first result below (to be established in §2).

P r o p o s i t i o n 1 If {zk,n}~=l denotes the zeros of s~(nz) with increasing arguments,

i.e,, (1.13)

0 < argzl,, _< argz2,n _< --. ~ arg zn,n < 27r,

then (cf. (1.12)) zl,, is an element of D, for all positive n su~ciently large. As a consequence of Proposition 1, there is a least positive integer, no, such that (cL (1.12)) (1.14) {zk,,~}~=l AD,~ ~ 0 for all positive integers n > no, i.e., at least one zero of s~(nz) lies in ~D~ for every n > no. By direct calculation of the zeros of s~(nz), it appears that (1.15) no = 96, and also that (1.16)

{Zk,n}~=lN~)n

=

$

(n = 1 , 2 , . . . , n 0 ) .

The size of no = 96 is somewhat surprising. Because no is so large, it was necessary to calculate the zeros of s,~(nz) with great precision, and for this, Richard Brent's MP package was used with 120 significant digits. As a consequence of Proposition 1, it is natural to ask if there is a simple modification, say ~ , of the definition of the closed set T~,~of (1.12) which would have all the zeros {zk,,,}~=l outside f),~ for all n > 1. To give an affirmative answer to this question, we define, for each n = 1, 2,-.-, the arc

(1.17)

/ ) . :=

.----I1 -Re z z E C : lzel-~l'~ = T,~.d2~'n 1 - - - ~ 1 ' [zl < 1, and ,~,

~ ~o~-~ ( ~ ) } ,

204

R.S. Varga and A.J. Carpenter

1.0

-

1.0

0.5

0.5

0.0

0.0

-0.5

-0.5

-1.0

-1.0 I

I

I

I

-1.0

-0.5

0.0

0.5

1.0

I

I

l

I

I

-1.0

-0.5

0.0

0.5

1.0

F i g u r e 1. D~7

F i g u r e 2./)27

and its associated closed star-shaped (with respect to z = 0) set

(1.t8)

~n :~_ {Z E C : Izel-zln ~ Tn 2V/'2~ n 1 - R-e z 1, Z

Izl_ 0)

where tl is the zero of erfc(w) :=

which is closest to the origin, w = 0, and it is known numerically (cf. Fettis, Caslin, and C r a m e r [4]) t h a t (2.10) t~ = - 1 . 3 5 4 8 1 0 . . . + i l , 9 9 1 4 6 7 . . . . On evaluating A~(z~,~) from (2.7), (2.9), and (2.10), it can be verified t h a t (2.11)

lim A~(Zl,~) = 0 . 9 8 5 9 6 4 . . - < 1.

n--*OO

Thus, with (2.8), zl,~ is contained in D~ for all positive n sufficiently large, which establishes Proposition 1. • We r e m a r k that it is because the constant, 0.985964.-., of (2.11) is so close to unity, t h a t it is difficult to see, graphically, that there are zeros of s,(nz) which lie interior to :D,~, for all n sufficiently large.

3. P r o o f o f T h e o r e m

2

We consider the integral (cf. (2.3)) (3.1)

L(~¢l-¢)nd~

I,(z) :-~-

(z e C, rt = 0, 1,-.-),

and, with z = re i°, we choose the line segment ~ = pei°(O < p < r) for the p a t h of integration in (3.1). Then,

(3.2)

Ix~(=)l
0. Thus, with (3.2) and (3.3), we have (3.4)

IzJlzel- l II~(z)[ < n'~---~-e z) (0 < [z[ < 1, n = 1,2,..-),

and from (2.3), we further have that (3.5)

r~(z)

= 1 - e-~%(nz).

Thus, if {zk,~}~=l denotes the set of zeros of s~(nz), then ~-~2 I~(zk,~) = 1, which implies from (3.4) that

(3.6)

> 1.

From (1.18), this means that all zeros {z~,~}~=~ of s~(nz) lie outside the set 7~, for all n _> 1, which is the desired result of (1.19) of Theorem 2. The remainder of Theorem 2, to establish (1.20), now similarly follows, as in the proof given in [3, Theorem 4], by expressing a zero, zk,~, of s,(nz), as ~ + ~, where is a suitable boundary point of ~ , and where 6 is assumed small. This argument also shows that the result of (1.20) is best possible. •

References

[1] N. Anderson, E.B. Saff, and R.S. Varga, On the Enestr~m-Kakeya Theorem and its sharpne88, Linear Algebra Appl. 28 (1979), 5-16. [2] J.D. Buckholtz, A characterization of the exponential series, Amer. Math. Monthly 73, Part II (1966), 121-123. [3] A.J. Carpenter, R.S. Varga, and J. WMdvogel, Asymptotics for the zeros of the partial sums of e~. I., Rocky Mount. J. of Math. (to appear). [4] H.E. Fettis, J.C. Caslin, and K.R. Cramer, Complex zeros of the error function and of the complementary error function, Math. Comp. 27 (1973), 401-404. [5] E.B. Saff and R.S. Varga, On the zeros and poles of Padd approximant8 to e~, Numer. Math. 25 (1975), 1-14. [6] E.B. Saff and R.S. Varga, Zero-free parabolic regions for 8equence~ of polynomials, SIAM J. Math. Anal. 7 (1976), 344-357. [7] G. Szeg6, Uber eine Eigen~cha~t der Exponentialreihe, Sitzungsber. Berl. Math. Ges. 23 (1924), 50-64. Received: February 28, 1990

L e c t u r e s presented during the conference R. A. Askey, Madison, USA "Polynomial inequalities". R.W. Barnard, Texas Tech, USA "On two conjectures in geometric function theory".

H.-P. Blatt, Eichst£tt, FRG "Erd6s-Tur£n Theorems on Jordan curves and arcs". P. Borwein, Halifax, Canada "A remarkable cube mean iteration". A. Cdrdova, Wiirzburg, FRG "Maximal range problems for polynomials". C. FitzGerald, San Diego, USA "Slit Mapping problems with no corresponding extremal problem". R. Fournier, Montreal, Canada Starlike univalent functions bounded on a diameter". R. Freund, Wfirzburg, FRG "A constrained Chebyshev approximation problem for ellipses". W.H.J. Fuchs, Cornell, USA "On a conjecture of Fischer and Michelli". W.K. Hayman, York, UK "A functional equation arising from the mortality tables". D. Hough, Z/irich, Switzerland "A Symm-Jacobi collocation method for numerical conformal mapping" J.A. Hummel, Maryland, USA "Numerical solutions of the Schiffer differential equation". L. Jacobsen, Trondheim, Norway "Orthogonal polynomials, chain sequences, three-term recurrence relations and continued fractions". W.B. Jones, Boulder, USA "Zeros of Szeg6 polynomials associated with Wiener-Levinson prediction". A. Lewis, Halifax, Canada "On the convergence of moment problems". D. Mej~a, Medellln, Colombia "Hyperbolic geometry in spherically k-convex regions".

210 D. Minda, Cincinatti, USA "An application of hyperbolic geometry". P.D. Miletta, Santiago, Chile "Approximation of special functions to delay differential equations". P. Nevai, Ohio State, USA "Computational aspects of orthogonal polynomials". O. Orellana, Valparaiso, Chile "On the point vortex method and some applications to problems in a~rodynamics". N. Papamichael, Brunel, UK "A domain decomposition method for conformal mapping onto a rectangle". C. Pommerenke, TU Berlin, FRG "Conformal mapping and the computation of bad curves". B. Rodin, San Diego, USA "Circle packing and conformal mapping". F. R0nning, Trondheim, Norway "A result about the sections of univalent functions". St. Ruscheweyh, Valparaiso, Chile, and W/irzburg, FRG "Convexity preserving operators". A. Ruttan, Kent State, USA "Optimal successive overrelaxation iterative methods for p-cyclic matrices". E.B. Saff, Tampa, USA "Distribution of extreme points on best complex polynomial approximation". L. Salinas, Valparaiso, Chile "On abstract conjugation and some engineering applications". G. Schober, Bloomington, USA "Planar harmonic mappings". D.F. Shea, Madison, USA "An extremal property of entire functions with positive zeros". F. Stenger, Salt Lake City, USA "Explicit exponential and rational approximation of continuous functions on R". H.J. Stetter, TH Vienna, Austria "Numerical inversion of multivariate polynomial systems". T.J. Suffridge, Lexington, USA "On nonvanishing H p functions".

211 W.J. Thron, Boulder, USA "Consequences of separate convergence of continued fractions". R.S. Varga, Kent State, USA "Recent results on the Riemann Hypothesis". R.A. Zalik, Auburn, USA "On the nonlinear Jeffcott equations". D. Zwick, Vermont, USA "Recent progress on best harmonic and subharmonic approximation".

Other Chilean participants J. Almanza, Concepci6n J. Bestagno, Concepci6n H. Burgos, Temuco V. Gruenberg, Valparalso S. Martlnez, Concepci6n W. Moscoso, Temuco F. Novoa, Concepci6n H. Pinto, Valpara/so V. Valderrama, Punta Arenas J.C. Vega, Concepci6n V. Vargas, Temuco

L E C T U R E N O T E S E d i t e d b y A. Dold,

IN I~;~TH EI~;/%T B. E c k m a n n and F. T a k e n s

I C S

Some general remarks on the publication of proceedings of congresses and symposia

Lecture Notes aim to r e p o r t new d e v e l o p m e n t s - quickly, i n f o r m a l l y and at a high level. The following describes c r i t e r i a and procedures which apply to proceedings volumes. The editors of a volume are strongly advised to inform contributors about these points at an early stage. §i.

One (or more) e x p e r t participant(s) of the m e e t i n g should act as the r e s p o n s i b l e editor(s) of the proceedings. They select the papers which are suitable (cf. §§ 2, 3) for i n c l u s i o n in the proceedings, and have t h e m i n d i v i d u a l l y r e f e r e e d (as for a journal). It should not be a s s u m e d that the p u b l i s h e d p r o c e e d i n g s must reflect c o n f e r e n c e events faithfully and in their entirety. Contributions to the m e e t i n g w h i c h are not i n c l u d e d in the proceedings can be l i s t e d by title. The series e d i t o r s will normally not interfere w i t h the editing of a particular proceedings volume - except in fairly obvious cases, or on technical m a t ters, such as d e s c r i b e d in §§ 2, 3. The names of the r e s p o n s i b l e editors appear on the title page of the volume.

§2.

The p r o c e e d i n g s s h o u l d be r e a s o n a b l y h o m o g e n e o u s (concerned w i t h a limited area). For instance, the p r o c e e d i n g s of a congress on "Analysis" or "Mathematics in Wonderland" w o u l d n o r m a l l y not be sufficiently homogeneous. One or two longer survey articles on r e c e n t d e v e l o p m e n t s in the field are often v e r y useful additions to such p r o c e e d i n g s - even if they do not c o r r e s p o n d to actual lectures at the congress. A n extensive i n t r o d u c t i o n on the subject of the c o n g r e s s would be desirable.

§3.

The c o n t r i b u t i o n s should be of a high m a t h e m a t i c a l standard and of current interest. R e s e a r c h articles s h o u l d p r e s e n t new m a t e rial and not d u p l i c a t e other papers a l r e a d y p u b l i s h e d or due to be published. T h e y s h o u l d contain s u f f i c i e n t i n f o r m a t i o n and m o tivation and they should p r e s e n t proofs, or at least outlines of such, in s u f f i c i e n t detail to enable an e x p e r t to complete them. Thus resumes and mere announcements of p a p e r s a p p e a r i n g elsewhere cannot be included, although m o r e d e t a i l e d versions of a contribution m a y well be p u b l i s h e d in other p l a c e s later. Contributions in n u m e r i c a l m a t h e m a t i c s m a y be a c c e p t a b l e w i t h o u t formal theorems resp. proofs if they p r e s e n t n e w algorithms solving problems (previously u n s o l v e d or less w e l l solved) or develop innovative q u a l i t a t i v e methods, not yet a m e n a b l e to a more formal t r e a t m e n t . . Surveys, if included, should cover a s u f f i c i e n t l y b r o a d topic, and should in g e n e r a l not simply r e v i e w the author's own recent research. In the case of such surveys, exceptionally, proofs of results may not be necessary.

§4.

"Mathematical Reviews" and "Zentralblatt f~r Mathematik" recommend that p a p e r s in proceedings volumes carry an e x p l i c i t statement that they are in final form and that no similar paper has been or is b e i n g s u b m i t t e d elsewhere, if these papers are to be c o n s i d e r e d for a review. Normally, papers that satisfy the criteria of the L e c t u r e Notes in M a t h e m a t i c s series also satisfy

this requirement, ting authors be b e g i n n i n g or end cases where this the paper is still

but we strongly r e c o m m e n d that the c o n t r i b u asked to give this g u a r a n t e e e x p l i c i t l y at the of their paper. There will occasionally be does not apply but where, for special reasons, a c c e p t a b l e for LNM.

55.

Proceedings should appear soon after the meeeting. The p u b l i s h e r should, therefore, receive the complete m a n u s c r i p t (preferably in duplicate) w i t h i n nine m o n t h s of the date of the m e e t i n g at the latest.

§6.

Plans or proposals for p r o c e e d i n g s volumes s h o u l d be sent to one of the editors of the series or to Springer-Verlag Heidelberg. They should give sufficient i n f o r m a t i o n on the conference or symposium, and on the p r o p o s e d proceedings. In particular, they should contain a list of the e x p e c t e d c o n t r i b u t i o n s with their p r o s p e c t i v e length. A b s t r a c t s or early v e r s i o n s (drafts) of some of the c o n t r i b u t i o n s are helpful.

57.

Lecture Notes are printed by p h o t o - o f f s e t from c a m e r a - r e a d y typed copy p r o v i d e d by the editors. For this p u r p o s e S p r i n g e r Verlag provides editors w i t h technical i n s t r u c t i o n s for the preparation of m a n u s c r i p t s and these should be d i s t r i b u t e d to all contributing authors. Springer-Verlag can also, on request, supply stationery on w h i c h the prescribed t y p i n g area is outlined. Some h o m o g e n e i t y in the p r e s e n t a t i o n of the c o n t r i b u t i o n s is desirable. Careful preparation of manuscripts will h e l p k e e p p r o d u c t i o n time short and ensure a s a t i s f a c t o r y a p p e a r a n c e of the f i n i s h e d book. The actual p r o d u c t i o n of a Lecture Notes v o l u m e n o r m a l l y takes 6 -8 weeks. M a n u s c r i p t s should be at least I00 pages should include a table of contents.

§8.

long.

The

final v e r s i o n

Editors receive a total of 50 free copies of their v o l u m e for d i s t r i b u t i o n to the c o n t r i b u t i n g authors, but no royalties. (Unfortunately, no reprints of individual contributions can be supplied.) They are e n t i t l e d to purchase further copies of their book for their personal use at a d i s c o u n t of 33.3 %, other Springer m a t h e m a t i c s books at a discount of 20 % d i r e c t l y from Springer-Verlag. C o n t r i b u t i n g authors may p u r c h a s e the v o l u m e in which their article appears at a discount of 33.3 %. C o m m i t m e n t to publish is made by letter of intent rather than by signing a formal contract. S p r i n g e r - V e r l a g secures the c o p y r i g h t for each volume.

Addresses: Professor A. Dold, Mathematisches Institut, Universit~t Heidelberg, Im Neuenheimer Feld 288, 6900 Heidelberg, Federal Republic of Germany Professor B. Eckmann, Mathematik, ETH-Zentrum 8092 ZUrich, Switzerland Prof. F. Takens, Mathematisch Instituut, Rijksuniversiteit Groningen, Postbus 800, 9700 AV Groningen, The Netherlands Springer-Verlag, Mathematics Editorial, Tiergartenstr. 17, 6900 Heidelberg, Federal Republic of Germany, Tel.: (06221) 487-410 Springer-Verlag, Mathematics Editorial, 175, Fifth Avenue, New York, New York i0010, USA, Tel.: (212) 460-1596

L E C T U R E

N O T E S

ESSENTIALS FOR THE PREPARATION OF C A M E R A - R E A D Y M A N U S C R I P T S

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Symbols.

Anything

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ONLY

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AND

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BLACK

ink.

(A

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Differential equations. : u s e

output

Differential equations. : u s e

Differential equations.

size

output

: use

13.5

size

output

16

size

x

20 x 18

cm.

23.5 x

cm.

26.5

cm.

I n t e r l i n e spacing: 5.5 m m b a s e - t o - b a s e for 14 p o i n t c h a r a c t e r s stand a r d f o r m a t of 18 x 26.5 cm). If in any doubt, p l e a s e s e n d us 1 - 2 s a m p l e p a g e s for e x a m i n a t i o n . We w i l l be g l a d to g i v e advice.