Geometric Function Theory in One and Higher Dimensions

GEOMETRIC FUNCTION THEORY IN ONE AND HIGHER DIMENSIONS IAN GRAHAM University of Toronto Toronto, Ontario, Canada

MARCEL DEKKER, INC.

GABRIELA KOHR Babe§-Bolyai University Cluj-Napoca, Romania

NEW YORK • BASEL

Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress. ISBN: 0-8247-0976-4 This book is printed on acid-free paper. Headquarters Marcel Dekker, Inc. 270 Madison Avenue, New York, NY 10016 tel: 212-696-9000; fax: 212-685-4540 Eastern Hemisphere Distribution Marcel Dekker AG Hutgasse 4, Postfach 812, CH-4001 Basel, Switzerland tel: 41-61-260-6300; fax: 41-61-260-6333 World Wide Web http://www.dekker.com The publisher offers discounts on this book when ordered in bulk quantities. For more information, write to Special Sales/Professional Marketing at the headquarters address above.

Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved. Neither this book nor any part may be reproduced or transmitted in any form or by an> means, electronic or mechanical, including photocopying, microfilming, and recording, oi by any information storage and retrieval system, without permission in writing from th( publisher. Current printing (last digit): 10 9 8 7 6 5 4 3 2 1 PRINTED IN THE UNITED STATES OF AMERICA

PURE AND APPLIED MATHEMATICS A Program of Monographs, Textbooks, and Lecture Notes

EXECUTIVE EDITORS Earl J. Taft Rutgers University New Brunswick, New Jersey

Zuhair Nashed University of Central Florida Orlando, Florida

EDITORIAL BOARD M. S. Baouendi University of California, San Diego Jane Cronin Rutgers University Jack K. Hale Georgia Institute of Technology

Anil Nerode Cornell University Donald Passman University of Wisconsin, Madison Fred S. Roberts Rutgers University

S. Kobayashi University of California, Berkeley

David L. Russell Virginia Polytechnic Institute and State University

Marvin Marcus University of California, Santa Barbara

Walter Schempp Universitat Siegen

W. S. Massey Yale University

Mark Teply University of Wisconsin, Milwaukee

MONOGRAPHS AND TEXTBOOKS IN PURE AND APPLIED MATHEMATICS 1. K. Yano, Integral Formulas in Riemannian Geometry (1970) 2. S. Kobayashi, Hyperbolic Manifolds and Hplomorphic Mappings (1970) 3. V. S. Vladimirov, Equations of Mathematical Physics (A. Jeffrey, ed.; A. LitHewood, trans.) (1970) 4. 6. N. Pshenichnyi, Necessary Conditions for an Extremum (L. Neustadt, translation ed.; K. Makowski, trans.) (1971) 5. L. Narici et a/., Functional Analysis and Valuation Theory (1971) 6. S. S. Passman, Infinite Group Rings (1971) 7. L. Domhoff, Group Representation Theory. Part A: Ordinary Representation Theory. Part B: Modular Representation Theory (1971,1972) 8. W. Boothby and G. L Weiss, eds., Symmetric Spaces (1972) 9. Y. Matsushima, Differentiable Manifolds (E. T. Kobayashi, trans.) (1972) 10. L £ Ward, Jr., Topology (1972) 11. A. Babakhanian, Cohomological Methods in Group Theory (1972) 12. R. Gilmer, Multiplicative Ideal Theory (1972) 13. J. Yeh, Stochastic Processes and the Wiener Integral (1973) 14. J. Barros-Neto, Introduction to the Theory of Distributions (1973) 15. R. Larsen, Functional Analysis (1973) 16. K. Yano and S. Ishihara, Tangent and Cotangent Bundles (1973) 17. C. Procesi, Rings with Polynomial Identities (1973) 18. R. Hermann, Geometry, Physics, and Systems (1973) 19. N. R. Wallach, Harmonic Analysis on Homogeneous Spaces (1973) 20. J. Dieudonne, Introduction to the Theory of Formal Groups (1973) 21. /. Vaisman, Cohomology and Differential Forms (1973) 22. B.-Y. Chen, Geometry of Submanifolds (1973) 23. M. Marcus, Finite Dimensional Multilinear Algebra (in two parts) (1973,1975) 24. R. Larsen, Banach Algebras (1973) 25. R. O. Kujala and A. L Vitter, eds., Value Distribution Theory: Part A; Part B: Deficit and Bezout Estimates by Wilhelm Stoll (1973) 26. K. B. Stolarsky, Algebraic Numbers and Diophantine Approximation (1974) 27. A. R. Magid, The Separable Galois Theory of Commutative Rings (1974) 28. B. R. McDonald, Finite Rings with Identity (1974) 29. J. Satake, Linear Algebra (S. Koh et al., trans.) (1975) 30. J. S. Golan, Localization of Noncommutative Rings (1975) 31. G. Klambauer, Mathematical Analysis (1975) 32. M. K. Agoston, Algebraic Topology (1976) 33. K. R. Goodearl, Ring Theory (1976) 34. L E. Mansfield, Linear Algebra with Geometric Applications (1976) 35. N. J. Pullman, Matrix Theory and Its Applications (1976) 36. B. R. McDonald, Geometric Algebra Over Local Rings (1976) 37. C. W. Groetsch, Generalized Inverses of Linear Operators (1977) 38. J. E. Kuczkowski and J. L Gersting, Abstract Algebra (1977) 39. C. O. Christenson and W. L Voxman, Aspects of Topology (1977) 40. M. Nagata, Field Theory (1977) 41. R. L. Long, Algebraic Number Theory (1977) 42. W. F. Pfeffer, Integrals and Measures (1977) 43. R. L. Wheeden and A. Zygmund, Measure and Integral (1977) 44. J. H. Curtiss, Introduction to Functions of a Complex Variable (1978) 45. K. Hrbacek and T. Jech, Introduction to Set Theory (1978) 46. W. S. Massey, Homology and Cohomology Theory (1978) 47. M. Marcus, Introduction to Modem Algebra (1978) 48. E. C. Young, Vector and Tensor Analysis (1978) 49. S. B. Nadler, Jr., Hyperspaces of Sets (1978) 50. S. K. Segal, Topics in Group Kings (1978) 51. A. C. M. van Rooij, Non-Archimedean Functional Analysis (1978) 52. L. Corwin and R. Szczarba, Calculus in Vector Spaces (1979) 53. C. Sadosky, Interpolation of Operators and Singular Integrals (1979) 54. J. Cronin, Differential Equations (1980) 55. C. W. Groetsch, Elements of Applicable Functional Analysis (1980)

56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. 79. 80. 81. 82. 83. 84. 85. 86. 87. 88. 89. 90. 91. 92. 93. 94. 95. 96. 97. 98. 99. 100. 101. 102. 103. 104. 105. 106. 107. 108. 109. 110. 111. 112.

/. Vaisman, Foundations of Three-Dimensional Euclidean Geometry (1980) H. I. Freedan, Deterministic Mathematical Models in Population Ecology (1980) S. 8. Chae, Lebesgue Integration (1980) C. S. Rees et a/., Theory and Applications of Fourier Analysis (1981) L Nachbin, Introduction to Functional Analysis (R. M. Aron, trans.) (1981) G. Orzech and M. Orzech, Plane Algebraic Curves (1981) R. Johnsonbaugh and W. E. Pfaffenberger, Foundations of Mathematical Analysis (1981) W. L Voxman and R. H. Goetschel, Advanced Calculus (1981) L. J. Corwin and R. H. Szczarba, Multivariable Calculus (1982) V. I. Istratescu, Introduction to Linear Operator Theory (1981) R. D. Jarvinen, Finite and Infinite Dimensional Linear Spaces (1981) J. K. Beem and P. E. Ehrtich, Global Lorentzian Geometry (1981) D. L Armacost, The Structure of Locally Compact Abelian Groups (1981) J. W. Brewer and M. K. Smith, eds., Emmy Noether: A Tribute (1981) K. H. Kim, Boolean Matrix Theory and Applications (1982) T. W. Wieting, The Mathematical Theory of Chromatic Plane Ornaments (1982) D. B.Gauld, Differential Topology (1982) R. L. Faber, Foundations of Euclidean and Non-Euclidean Geometry (1983) M. Carmeli, Statistical Theory and Random Matrices (1983) J. H. Camith et a/., The Theory of Topological Semigroups (1983) R. L Faber, Differential Geometry and Relativity Theory (1983) S. Bamett, Polynomials and Linear Control Systems (1983) G. Karpilovsky, Commutative Group Algebras (1983) F. Van Oystaeyen and A. Verschoren, Relative Invariants of Rings (1983) /. Vaisman, A First Course in Differential Geometry (1984) G. W. Swan, Applications of Optimal Control Theory in Biomedicine (1984) T. Petrie and J. D. Randall, Transformation Groups on Manifolds (1984) K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings (1984) T. Albu and C. Nastasescu, Relative Finiteness in Module Theory (1984) K. Hrbacek and T. Jech, Introduction to Set Theory: Second Edition (1984) F. Van Oystaeyen and A. Verschoren, Relative Invariants of Rings (1984) B. R. McDonald, Linear Algebra Over Commutative Rings (1984) M. Namba, Geometry of Projective Algebraic Curves (1984) G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics (1985) M. R. Bremner et a/.. Tables of Dominant Weight Multiplicities for Representations of Simple Lie Algebras (1985) A. E. Fekete, Real Linear Algebra (1985) S. B. Chae, Holomorphy and Calculus in Normed Spaces (1985) A. J. Jerri, Introduction to Integral Equations with Applications (1985) G. Karpilovsky, Projective Representations of Finite Groups (1985) L. Narici and E. Beckenstein, Topological Vector Spaces (1985) J. Weeks, The Shape of Space (1985) P. R. Gribik and K. O. Kortanek, Extremal Methods of Operations Research (1985) J.-A. Chao and W. A. Woyczynski, eds., Probability Theory and Harmonic Analysis (1986) G. D. Crown et a/., Abstract Algebra (1986) J. H. Carruth et a/., The Theory of Topological Semigroups, Volume 2 (1986) R. S. Doran and V. A. Belfi, Characterizations of C*-Algebras (1986) M. W. Jeter, Mathematical Programming (1986) M. Altman, A Unified Theory of Nonlinear Operator and Evolution Equations with Applications (1986) A. Verschoren, Relative Invariants of Sheaves (1987) R. A. Usmani, Applied Linear Algebra (1987) P. 8/ass and J. Lang, Zariski Surfaces and Differential Equations in Characteristic p > 0(1987) J. A. Reneke et a/., Structured Hereditary Systems (1987) H. Busemann and B. B. Phadke, Spaces with Distinguished Geodesies (1987) R. Harte, Invertibility and Singularity for Bounded Linear Operators (1988) G. S. Ladde et a/., Oscillation Theory of Differential Equations with Deviating Arguments (1987) L. Dudkin et a/., Iterative Aggregation Theory (1987) T. Okubo, Differential Geometry (1987)

113. 114. 115. 116. 117. 118. 119. 120. 121. 122. 123. 124. 125. 126. 127. 128. 129. 130. 131. 132. 133. 134. 135. 136. 137. 138. 139. 140. 141. 142. 143. 144. 145. 146. 147. 148. 149. 150. 151. 152. 153. 154. 155. 156. 157. 158. 159. 160. 161. 162. 163. 164. 165. 166. 167. 168. 169. 170. 171.

D. L Stand and M. L Stand, Real Analysis with Point-Set Topology (1987) T. C. Gard, Introduction to Stochastic Differential Equations (1988) S. S. Abhyankar, Enumerative Combinatorics of Young Tableaux (1988) H. Sfrade and R. Famsteiner, Modular Lie Algebras and Their Representations (1988) J. A. Huckaba, Commutative Rings with Zero Divisors (1988) W. D. Wallis, Combinatorial Designs (1988) W. Wiestaw, Topological Fields (1988) G. Karpilovsky, Field Theory (1988) S. Caenepeel and F. Van Oystaeyen, Brauer Groups and the Cohomology of Graded Rings (1989) W. Kozlowski, Modular Function Spaces (1988) E. Lowen-Colebunders, Function Classes of Cauchy Continuous Maps (1989) M. Pavel, Fundamentals of Pattern Recognition (1989) V. Lakshmikantham et a/.. Stability Analysis of Nonlinear Systems (1989) R. Sivaramakrishnan, The Classical Theory of Arithmetic Functions (1989) N. A. Watson, Parabolic Equations on an Infinite Strip (1989) K. J. Hastings, Introduction to the Mathematics of Operations Research (1989) B. Fine, Algebraic Theory of the Bianchi Groups (1989) D. N. Dikranjan et a/., Topological Groups (1989) J. C. Morgan II, Point Set Theory (1990) P. BilerandA. Wrtkowski, Problems in Mathematical Analysis (1990) H. J. Sussmann, Nonlinear Controllability and Optimal Control (1990) J.-P. Florens et a/., Elements of Bayesian Statistics (1990) N. Shell, Topological Fields and Near Valuations (1990) 8. F. Doolin and C. F. Martin, Introduction to Differential Geometry for Engineers (1990) S. S. Holland, Jr., Applied Analysis by the Hilbert Space Method (1990) J. Oknfnski, Semigroup Algebras (1990) K. Zhu, Operator Theory in Function Spaces (1990) G. B. Price, An Introduction to Multicomplex Spaces and Functions (1991) R. B. Darst, Introduction to Linear Programming (1991) P. L Sachdev, Nonlinear Ordinary Differential Equations and Their Applications (1991) T. Husain, Orthogonal Schauder Bases (1991) J. Foran, Fundamentals of Real Analysis (1991) W. C. Brown, Matrices and Vector Spaces (1991) M. M. Rao and Z. D. Ren, Theory of Oriicz Spaces (1991) J. S. Golan and T. Head, Modules and the Structures of Rings (1991) C. Small, Arithmetic of Finite Fields (1991) K. Yang, Complex Algebraic Geometry (1991) D.G. Hoffman et a/.. Coding Theory (1991) M. O. Gonzalez, Classical Complex Analysis (1992) M. O. Gonzalez, Complex Analysis (1992) L. W. Baggett, Functional Analysis (1992) M. Sniedovich, Dynamic Programming (1992) R. P. Agarwal, Difference Equations and Inequalities (1992) C. Brezinski, Biorthogonality and Its Applications to Numerical Analysis (1992) C. Swartz, An Introduction to Functional Analysis (1992) S. 8. Nadler, Jr., Continuum Theory (1992) M. A. AI-Gwaiz, Theory of Distributions (1992) E. Perry, Geometry: Axiomatic Developments with Problem Solving (1992) E. Castillo and M. R. Ruiz-Cobo, Functional Equations and Modelling in Science and Engineering (1992) A. J. Jerri, Integral and Discrete Transforms with Applications and Error Analysis (1992) A. Charlier et al., Tensors and the Clifford Algebra (1992) P. Bilerand T. Nadzieja, Problems and Examples in Differential Equations (1992) E. Hansen, Global Optimization Using Interval Analysis (1992) S. Guerre-Delabriere, Classical Sequences in Banach Spaces (1992) V. C. Wong, Introductory Theory of Topological Vector Spaces (1992) S. H. KulkamiandB. V. Limaye, Real Function Algebras (1992) W. C. Brown, Matrices Over Commutative Rings (1993) J. Loustau and M. Dillon, Linear Geometry with Computer Graphics (1993) W. V. Petryshyn, Approximation-Solvability of Nonlinear Functional and Differential Equations (1993)

172. 173. 174. 175. 176. 177. 178. 179. 180. 181. 182. 183. 184. 185. 186. 187. 188. 189. 190. 191. 192. 193. 194. 195. 196. 197. 198. 199. 200. 201. 202. 203. 204. 205. 206. 207. 208. 209. 210. 211. 212. 213. 214. 215. 216. 217. 218. 219. 220. 221. 222. 223. 224. 225.

E. C. Young, Vector and Tensor Analysis: Second Edition (1993) T. A. Bick, Elementary Boundary Value Problems (1993) M. Pavel, Fundamentals of Pattern Recognition: Second Edition (1993) S. A. Albeverio et a/., Noncommutative Distributions (1993) W. Fulks, Complex Variables (1993) M. M. Rao, Conditional Measures and Applications (1993) A. Janicki and A. Weron, Simulation and Chaotic Behavior of a-Stable Stochastic Processes (1994) P. Neittaanmaki and D. Tiba, Optimal Control of Nonlinear Parabolic Systems (1994) J. Cronin, Differential Equations: Introduction and Qualitative Theory, Second Edition (1994) S. HeikkilS and V. Lakshmikantham, Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations (1994) X. Mao, Exponential Stability of Stochastic Differential Equations (1994) B. S. Thomson, Symmetric Properties of Real Functions (1994) J. E. Rubio, Optimization and Nonstandard Analysis (1994) J. L Bueso et a/., Compatibility, Stability, and Sheaves (1995) A. N. Michel and K. Wang, Qualitative Theory of Dynamical Systems (1995) M. R. Darnel, Theory of Lattice-Ordered Groups (1995) Z. Naniewicz and P. D. Panagiotopoulos, Mathematical Theory of Hemivariational Inequalities and Applications (1995) L J. Corwin and R. H. Szczarba, Calculus in Vector Spaces: Second Edition (1995) L H. Erbe et a/., Oscillation Theory for Functional Differential Equations (1995) S. Agaian era/., Binary Polynomial Transforms and Nonlinear Digital Filters (1995) M. I. Gil', Norm Estimations for Operation-Valued Functions and Applications (1995) P. A. Grillet, Semigroups: An Introduction to the Structure Theory (1995) S. Kichenassamy, Nonlinear Wave Equations (1996) V. F. Krotov, Global Methods in Optimal Control Theory (1996) K. I. Beidaretal., Rings with Generalized Identities (1996) V. I. Amautov et a/., Introduction to the Theory of Topological Rings and Modules (1996) G. Sierksma, Linear and Integer Programming (1996) R. Lasser, Introduction to Fourier Series (1996) V. Sima, Algorithms for Linear-Quadratic Optimization (1996) D. Redmond, Number Theory (1996) J. K. Beem et a/., Global Lorentzian Geometry: Second Edition (1996) M. Fontana et a/., Prufer Domains (1997) H. Tanabe, Functional Analytic Methods for Partial Differential Equations (1997) C. Q. Zhang, Integer Flows and Cycle Covers of Graphs (1997) E. Spiegel and C. J. O'Donnell, Incidence Algebras (1997) B. Jakubczyk and W. Respondek, Geometry of Feedback and Optimal Control (1998) T. W. Haynes et a/., Fundamentals of Domination in Graphs (1998) T. W. Haynes et a/., ecfs., Domination in Graphs: Advanced Topics (1998) L. A. D'Alotto et a/., A Unified Signal Algebra Approach to Two-Dimensional Parallel Digital Signal Processing (1998) F. Halter-Koch, Ideal Systems (1998) N. K. Govil et a/., eds., Approximation Theory (1998) R. Cross, Multivalued Linear Operators (1998) A. A. Martynyuk, Stability by Liapunov's Matrix Function Method with Applications (1998) A. Favini and A. Yagi, Degenerate Differential Equations in Banach Spaces (1999) A. Illanes and S. Nadler, Jr., Hyperspaces: Fundamentals and Recent Advances (1999) G. Kato and D. Struppa, Fundamentals of Algebraic Microlocal Analysis (1999) G. X.-Z. Yuan, KKM Theory and Applications in Nonlinear Analysis (1999) D. Motreanu and N. H. Pavel, Tangency, Flow Invariance for Differential Equations, and Optimization Problems (1999) K. Hrbacek and T. Jech, Introduction to Set Theory, Third Edition (1999) G. E. Kolosov, Optimal Design of Control Systems (1999) N. L Johnson, Subplane Covered Nets (2000) B. Fine and G. Rosenberger, Algebraic Generalizations of Discrete Groups (1999) M. Vath, Volterra and Integral Equations of Vector Functions (2000) S. S. Miller and P. T. Mocanu, Differential Subordinations (2000)

226. R. Li et a/., Generalized Difference Methods for Differential Equations: Numerical Analysis of Finite Volume Methods (2000) 227. H. Li and F. Van Oystaeyen, A Primer of Algebraic Geometry (2000) 228. R. P. Agarwal, Difference Equations and Inequalities: Theory, Methods, and Applications, Second Edition (2000) 229. A. B. Kharazishvili, Strange Functions in Real Analysis (2000) 230. J. M. Appell et a/., Partial Integral Operators and Integra-Differential Equations (2000) 231. A. I. Prilepko et a/., Methods for Solving Inverse Problems in Mathematical Physics (2000) 232. F. Van Oystaeyen, Algebraic Geometry for Associative Algebras (2000) 233. D. L. Jagerman, Difference Equations with Applications to Queues (2000) 234. D. R. Hankerson et a/., Coding Theory and Cryptography: The Essentials, Second Edition, Revised and Expanded (2000) 235. S. Dascalescu et a/., Hopf Algebras: An Introduction (2001) 236. R. Hagen et a/., C*-Algebras and Numerical Analysis (2001) 237. V. Talpaert, Differential Geometry: With Applications to Mechanics and Physics (2001) 238. R. H. Villameal, Monomial Algebras (2001) 239. A. N. Michel et a/., Qualitative Theory of Dynamical Systems: Second Edition (2001) 240. A. A. Samarskii, The Theory of Difference Schemes (2001) 241. J. Knopfmacher and W.-B. Zhang, Number Theory Arising from Finite Fields (2001) 242. S. Leader, The Kurzweil-Henstock Integral and Its Differentials (2001) 243. M. Biliotti et a/., Foundations of Translation Planes (2001) 244. A. N. Kochubei, Pseudo-Differential Equations and Stochastics over Non-Archimedean Fields (2001) 245. G. Sierksma, Linear and Integer Programming: Second Edition (2002) 246. A. A. Martynyuk, Qualitative Methods in Nonlinear Dynamics: Novel Approaches to Liapunov's Matrix Functions (2002) 247. B. G. Pachpatte, Inequalities for Finite Difference Equations (2002) 248. A. N. Michel and D. Liu, Qualitative Analysis and Synthesis of Recurrent Neural Networks (2002) 249. J. R. Weeks, The Shape of Space: Second Edition (2002) 250. M. M. Rao and Z. D. Ren, Applications of Oriicz Spaces (2002) 251. V. Lakshmikantham and D. Trigiante, Theory of Difference Equations: Numerical Methods and Applications, Second Edition (2002) 252. T. Albu, Cogalois Theory (2003) 253. A. Bezdek, Discrete Geometry (2003) 254. M. J. Cortess and A. E. Frazho, Linear Systems and Control: An Operator Perspective (2003) 255. /. Graham and G. Kohr, Geometric Function Theory in One and Higher Dimensions (2003) Additional Volumes in Preparation

To my twin sister, Mirela Gabriela Kohr

To Norberto Kerzman Ian Graham

Preface In this book we give a combined treatment of classical results in univalent function theory (as well as newer results in geometric function theory in one variable) and generalizations of these results to higher dimensions, in which there has been much recent progress. The one-variable topics treated include the class 5 of normalized univalent functions on the unit disc and various subclasses, the theory of Loewner chains and applications, Bloch functions and the Bloch constant, and linear-invariant families. Our treatment of these topics is designed to prepare the ground for the several-variables material. The second part of the book begins with a concise introduction to those aspects of the theory of several complex variables and complex analysis in infinite dimensions which are needed. We then study the class S(B) of normalized biholomorphic mappings from the unit ball B of C™ into C71. We consider growth, covering, and distortion theorems and coefficient estimates for various subclasses of S(B), some of which are direct generalizations of familiar subclasses of S, and some of which are not. We give a detailed exposition of the theory of Loewner chains in several variables with applications. We also consider Bloch mappings and analogs of the Bloch constant problem, and the theory of linear-invariant families in several variables. Finally we study extension operators such as the Roper-Suffridge operator which can be used to construct biholomorphic mappings of the unit ball with certain geometric properties using univalent functions of the unit disc with related properties. The book is intended for both graduate students and research mathematicians. The prerequisites are a good first course in complex analysis, including

vi

Preface

the Riemann mapping theorem, a course in measure theory, and some basic notions of functional analysis. A course in several complex variables is not a prerequisite (though we hope that one-variable readers will be led to explore other aspects of this subject); the necessary background is given in the first section of Chapter 6. In fact, the book can be used as an introduction to several complex variables. Numerous exercises are given throughout. A more detailed description of the contents appears in the Introduction. We would like to acknowledge a number of people. Gabriela Kohr wishes to express her gratitude to Professor Petru T. Mocanu for his help and encouragement and for all that she learned from him over many years. She particularly wishes to thank Hidetaka Hamada for his great help throughout a long and valuable collaboration. Professor Ted Suffridge has provided much useful advice over the years. Professor John Pfaltzgraff has given some much appreciated encouragement and ideas. Ian Graham wishes to thank David Minda for discussions about geometric function theory in one variable, and Dror Varolin for discussions about covering theorems. Among earlier mathematical influences, he would like to mention the advice and enthusiasm of Norberto Kerzman and his collaboration with H. Wu. We also thank Professor Sheng Gong for discussions about geometric function theory of several complex variables. We would like to thank all those who assisted with the preparation of the manuscript, especially Georgeta Bonda of Babe§-Bolyai University and Ida Bulat of the University of Toronto. The figures were made with the help of Nadia Villani and Miranda Tang of the University of Toronto and Radu Trimbit;a§ of Babe§-Bolyai University. We would also like to acknowledge the hospitality of each other's university and the support of the Natural Sciences and Engineering Research Council of Canada. Finally we would like to thank the staff at Marcel Dekker Inc., including Maria Allegra, for their help with the publication of this book. Ian Graham and Gabriela Kohr

Contents Preface

v

Introduction

I

xiii

Univalent functions

1

1 Elementary properties of univalent functions 3 1.1 Univalence in the complex plane 3 1.1.1 Elementary results in the theory of univalent functions. Examples of univalent functions 3 1.1.2 The area theorem 9 1.1.3 Growth, covering and distortion results in the class S . 13 1.1.4 The maximum modulus of univalent functions 18 1.1.5 Two-point distortion results for the class S 21 2 Subclasses of univalent functions in the unit disc 2.1 Functions with positive real part. Subordination and the Herglotz formula 2.1.1 The Caratheodory class. Subordination 2.1.2 Applications of the subordination principle 2.2 Starlike and convex functions 2.3 Starlikeness and convexity of order a. Alpha convexity 2.3.1 Starlikeness and convexity of order a 2.3.2 Alpha convexity vii

27 27 27 32 36 54 54 58

viii

Contents 2.4

Close-to-convexity, spirallikeness and ^-likeness in the unit disc 2.4.1 Close-to-convexity in the unit disc 2.4.2 Spirallike functions in the unit disc 2.4.3 $-like functions on the unit disc

63 63 73 79

3 The Loewner theory 3.1 Loewner chains and the Loewner differential equation 3.1.1 Kernel convergence 3.1.2 Subordination chains and kernel convergence 3.1.3 Loewner's differential equation 3.1.4 Remarks on Bieberbach's conjecture 3.2 Applications of Loewner's differential equation to the study of univalent functions 3.2.1 The radius of starlikeness for the class S and the rotation theorem 3.2.2 Applications of the method of Loewner chains to characterize some subclasses of S 3.3 Univalence criteria 3.3.1 Becker's univalence criteria 3.3.2 Univalence criteria involving the Schwarzian derivative . 3.3.3 A generalization of Becker's and Nehari's univalence criteria

87 87 87 94 100 112 117 118 126 130 130 132 140

4 Bloch functions and the Bloch constant 145 4.1 Preliminaries concerning Bloch functions 145 4.2 The Bloch constant problem and Bonk's distortion theorem . . 151 4.3 Locally univalent Bloch functions 157 4.3.1 Distortion results for locally univalent Bloch functions . 157 4.3.2 The case of convex functions 163 5 Linear invariance in the unit disc 5.1 General ideas concerning linear-invariant families 5.2 Extremal problems and radius of univalence

165 165 172

Contents

ix

5.2.1 5.2.2

II

Bounds for coefficients of functions in linear-invariant families

172

Radius problems for linear-invariant families

174

Univalent mappings in several complex variables and complex Banach spaces

181

6 Univalence in several complex variables 6.1

Preliminaries concerning holomorphic mappings in C™ and complex Banach spaces 184 Holomorphic functions in C71

184

6.1.2

Classes of domains in C". Pseudoconvexity

188

6.1.3

Holomorphic mappings

191

6.1.4

Automorphisms of the Euclidean unit ball and the unit polydisc

195

6.1.1

6.2

6.1.5

Holomorphic mappings in complex Banach spaces

6.1.6

Generalizations of functions with positive real part . . . 202

6.4

. . . 197

6.1.7 Examples and counterexamples Criteria for starlikeness 6.2.1

6.3

183

210 213

Criteria for starlikeness on the unit ball in C1 or in a complex Banach space

213

6.2.2

Starlikeness criteria on more general domains in C"

6.2.3

Sufficient conditions for starlikeness for mappings of class C1 219

6.2.4

Starlikeness of order 7^0*

Criteria for convexity

. . 217

221 223

6.3.1

Criteria for convexity on the unit polydisc and the Euclidean unit ball 223

6.3.2

Necessary and sufficient conditions for convexity in complex Banach spaces 230

6.3.3

Quasi-convex mappings on the unit ball of C"

Spirallikeness and ^-likeness in several complex variables

238 . . . 244

x

Contents

7 Growth, covering and distortion results for starlike and convex mappings in C" and complex Banach spaces 7.1 Growth, covering and distortion results for starlike mappings in several complex variables and complex Banach spaces 7.1.1 Growth and covering results for starlike mappings on the unit ball and some pseudoconvex domains in C™. Extensions to complex Banach spaces 7.1.2 Bounds for coefficients of normalized starlike mappings inC n 7.1.3 A distortion result for a subclass of starlike mappings in Cn 7.2 Growth, covering and distortion results for convex mappings in several complex variables and complex Banach spaces 7.2.1 Growth and covering results for convex mappings . . . . 7.2.2 Covering theorem and the translation theorem in the case of nonunivalent convex mappings in several complex variables 7.2.3 Bounds for coefficients of convex mappings in Cn and complex Hilbert spaces 7.2.4 Distortion results for convex mappings in Cn and complex Hilbert spaces 8 Loewner chains in several complex variables 8.1 Loewner chains and the Loewner differential equation in several complex variables 8.1.1 The Loewner differential equation in Cn 8.1.2 Transition mappings associated to Loewner chains on the unit ball of C1 8.2 Close-to-starlike and spirallike mappings of type alpha on the unit ball of C1 8.2.1 An alternative characterization of spirallikeness of type alpha in terms of Loewner chains 8.2.2 Close-to-starlike mappings on the unit ball of Cn . . . .

255 256

256 262 268

271 271

278 281 286 295 295 295 312 322 322 324

Contents

xi

8.3 Univalent mappings which admit a parametric representation . 8.3.1 Examples of mappings which admit parametric representation on the unit ball of Cn 8.3.2 Growth results and coefficient bounds for mappings in Sj(B) 8.4 Applications of the method of Loewner chains to univalence criteria on the unit ball of C™ 8.5 Loewner chains and quasiconformal extensions of holomorphic mappings in several complex variables 8.5.1 Construction of quasiconformal extensions by means of Loewner chains 8.5.2 Strongly starlike and strongly spirallike mappings of type a on the unit ball of C™

330

9 Bloch constant problems in several complex variables 9.1 Preliminaries and a generalization of Bonk's distortion theorem 9.2 Bloch constants for bounded and quasiregular holomorphic mappings 9.3 Bloch constants for starlike and convex mappings in several complex variables

377 377

10 Linear invariance in several complex variables 10.1 Preliminaries concerning the notion of linear invariance in several complex variables 10.1.1 L.I.F.'s and trace order in several complex variables . . 10.1.2 Examples of L.I.F.'s on the Euclidean unit ball of C" . 10.2 Distortion results for linear-invariant families in several complex variables 10.2.1 Distortion results for L.I.F.'s on the Euclidean unit ball ofC71 10.2.2 Distortion results for L.I.F.'s on the unit polydisc of Cn 10.3 Examples of L.I.F.'s of minimum order on the Euclidean unit ball and the unit polydisc of C"

395

330 334

348 353 353 370

384 390

396 396 399 401 401 410 414

xii

Contents 10.3.1 Examples of L. I. F. 's of minimum order on the Euclidean unit ball of C1 10.3.2 Examples of L.I.F.'s of minimum order on the unit polydisc of C1 10.4 Norm order of linear-invariant families in several complex variables 10.5 Norm order and univalence on the Euclidean unit ball of C™ . . 10.6 Linear-invariant families in complex Hilbert spaces

11 Univalent mappings and the Roper-Suffridge extension operator 11.1 Convex, starlike and Bloch mappings and the Roper-Suffridge extension operator 11.2 Growth and covering theorems associated with the RoperSuffridge extension operator 11.3 Loewner chains and the operator 3>n,a 11.4 Radius problems and the operator 3>nja 11.5 Linear-invariant families and the operator $n>a

414 426 429 434 440

443 444 456 461 466 469

Bibliography

477

List of Symbols

521

Index

527

Introduction The theory of univalent functions is one of the most beautiful topics in one complex variable. There are many remarkable theorems dealing with extremal problems for the class S of normalized univalent functions on the unit disc, from the Bieberbach conjecture which was solved by de Branges in 1985, to others of a purely geometrical nature. A great variety of methods was developed to study these problems. The study of the direct analog of the class S in several variables, i.e. the class S(B) of normalized biholomorphic mappings of the unit ball B in C™, was comparatively slow to develop, although it was suggested by H. Cartan in 1933 [Cart2]. Perhaps this was because of the failure of the Riemann mapping theorem in higher dimensions, and its replacement by many new types of mapping questions. Moreover, some of the most obvious questions that one can formulate about the class S(B) lead to counterexamples rather than generalizations of one-variable theorems. However, in recent years there have been many developments in univalent mappings in higher dimensions, and this subject now includes a significant body of results. It is our belief that a book which combines both classical results in univalent function theory and analogous recent results in higher dimensions will be useful at this time. Indeed, it is our hope that the book will lead to increased interaction between specialists in one and in several complex variables. The book begins with the classical growth, covering and distortion theorems for the class S. In Chapter 2 we consider various subclasses of S, including not only starlike and convex functions but also functions which are spirallike, close-to-convex,

Xlll

xiv

Introduction

starlike of order a, or a-convex. (Part of the reason for doing so is that in several variables it is necessary to consider proper subclasses of the normalized univalent mappings on the unit ball in C™ in order to obtain nontrivial theorems.) These subclasses are denned by geometric conditions which can be reformulated as analytic conditions, which in turn lead to interesting theorems. Our intention is not to give an exhaustive treatment of subclasses of S, but to give a number of applications which are typical of the results which can be found in the one-variable literature. The study of Loewner chains (Chapter 3) will be of special interest, partly by way of comparison with recent results in this area in several variables. We shall give some well-known and beautiful applications of this method in one variable, including the radius of starlikeness, the rotation theorem, the bound for the third coefficient of functions in 5, alternative characterizations of starlikeness, convexity, spirallikeness, and close-to-convexity, and univalence criteria. We have omitted the proof of de Branges' theorem, since some of the methods do not generalize to several variables. (Proofs can be found in the books of Conway, Hayman, Henrici, or Rosenblum and Rovnyak.) Some of the ideas from univalent function theory extend naturally to the study of certain classes of non-univalent functions. In Chapter 4 we study Bloch functions, including Bonk's distortion theorem and estimates for the Bloch constant. In Chapter 5 we consider linear-invariant families, introduced by Pommerenke, in which the study of estimates for the second coefficient is extended to families of locally univalent functions. The second part of the book begins with a summary of results from the general theory of several complex variables which will be needed, and some examples which show that not all of the results of classical univalent function theory can be expected to carry over to higher dimensions. We then treat particular subclasses of normalized univalent mappings on the unit ball (and in some cases on more general domains and in infinite dimensions). The convex and starlike mappings are of course analogs of well-known subclasses of 5, but other new classes in several variables are introduced. As in one variable, the focus is on growth, distortion, and covering theorems and on coefficient estimates. Among the recent results treated here, we mention the compactness of

Introduction

xv

the class Ai which plays the role of the Caratheodory class in several variables. This is the subject of Chapters 6 and 7. The theory of Loewner chains in several variables (Chapter 8) is one of the main themes of the second part of the book. There are many recent results in this area, including improvements in the existence theorems resulting from the compactness of the class A4, and new applications. Of particular importance is the subclass 5° (B) of S(B) consisting of mappings which have parametric representation, because many of the results for the class 5 in one variable can be generalized to this class, and many useful subclasses of S(B) are also subclasses of S°(B). Surprisingly, in higher dimensions S°(B) turns out to be a proper subclass of the class of normalized holomorphic mappings of B which can be embedded as the first element of a Loewner chain. We also consider Bloch mappings in higher dimensions (Chapter 9), and we give a detailed exposition of the theory of linear-invariant families on the Euclidean unit ball and the polydisc in Chapter 10. The book concludes with a study of the Roper-Suffridge extension operator (Chapter 11), a particularly interesting way of constructing mappings of the unit ball in Cn which extend univalent functions on the disc, preserving certain properties. Many of the results and methods of previous chapters are tied together in this chapter.

GEOMETRIC FUNCTION THEORY IN ONE AND HIGHER DIMENSIONS

Part I

Univalent functions

Chapter 1

Elementary properties of univalent functions The theory of univalent functions is one of the most beautiful subjects in geometric function theory. Its origins (apart from the Riemann mapping theorem) can be traced to the 1907 paper of Koebe [Koe], to Gronwall's proof of the area theorem in 1914 [Gro], and to Bieberbach's estimate for the second coefficient of a normalized univalent function in 1916 and its consequences [Biel]. By then, univalent function theory was a subject in its own right. We begin the one-variable part of the book with the study of basic notions about the class 5 of normalized univalent functions on the unit disc, including growth, covering, and distortion theorems. Most of the results in the theory of univalent functions that we present here are classical, but there are some which are relatively new and provide a slightly different viewpoint of older results.

1.1 1.1.1

Univalence in the complex plane Elementary results in the theory of univalent functions. Examples of univalent functions

Let C be the complex plane. If ZQ €. C and r > 0, we let U(ZQ, r) = {z € C : \z — ZQ\ < r} be the open disc of radius r centered at ZQ. The closure of

4

Elementary properties of univalent functions

U(zQ,r) will be denoted by U(zo,r) and its boundary by dU(zo,r). The open disc (7(0, r) will be denoted by Z7r, and the unit disc U\ will be denoted by U. If G is an open subset of C, let H(G) denote the set of holomorphic functions on G with values in C. With the topology of local uniform convergence (or uniform convergence on compact subsets), H(G) becomes a topological space. Let D be a domain in C. A function / : £ > — > C is called univalent if / G H(D] and / is one-to-one on D. We shall be interested in the study of the class HU(D} of univalent functions on D. It is well known that the class HU(C) contains only functions of the form f ( z ) = az + b, z G C, where a, b € C, a ^ 0. However for a general domain D, HU(D) contains many other functions. A function / 6 H(D} is called locally univalent if each point z £ D has a neighbourhood V such that f\v is univalent. Since / is holomorphic, local univalence is equivalent to the condition that f ' ( z ) ^ 0, z € D. If / G H(D) is locally univalent and z G D, the derivative $'(z) determines the local geometric behaviour of / at z. The quantity |/'(^)| gives the local magnification factor for lengths, and argf'(z) is the local rotation factor. Moreover, if / is viewed as a transformation from a domain D C M2 to R 2 , the Jacobian determinant of this transformation is given by |/'(^)|2. A locally univalent function therefore preserves angles and orientation. For this reason it is customary to refer to a univalent function as a conformal mapping or a conformal equivalence. The condition that f ' ( z ) ^ 0 on a domain D C C is necessary but of course not sufficient for the univalence of / on D. For example, /(z) = ekz is locally univalent on U for all k G C, but is not globally univalent on U if \k\ > TT. It is more difficult to give conditions for global univalence than for local univalence, but as we shall see there are many such conditions, some of them quite remarkable. One of the most easily stated and proved is the following criterion of Noshiro [Nos], Warschawski [War], and Wolff [Wol] (see Lemma 2.4.1): If / is holomorphic on a convex domain D C C and Re f ' ( z ) ^ 0, z G D, then / is univalent on D. One of the most basic results in the theory of univalent functions in one variable is the Riemann mapping theorem. Its failure in several variables is one

1.1. Univalence in the complex plane

5

of the key differences between complex analysis in one variable and in higher dimensions. Riemann mapping theorem. Every simply connected domain D, which is a proper subset of C, can be mapped conformally onto the unit disc. Moreover, if ZQ G D, there is a unique conformal map of D onto U such that f ( z 0 ) = 0 and /'(ZQ) > 0. The entire complex plane cannot be conformally equivalent to the unit disc U by Liouville's theorem, although these domains are homeomorphic. We note that there is a stronger version of the Riemann mapping theorem in the case when the boundary of D is a closed Jordan curve, due to Caratheodory: Caratheodory's theorem. Let D C C be a simply connected domain bounded by a closed Jordan curve. Then any conformal map of D onto U extends to a homeomorphism of D onto U. In view of the Riemann mapping theorem, it suffices to study many questions involving univalence on the unit disc U rather than on a general simply connected domain. For this purpose, we introduce the class 5 of functions / e HU(U) which are normalized by the condition /(O) = /'(O) — 1 = 0. (Any holomorphic function / on U which satisfies /(O) = /'(O) — 1 = 0 will be said to be normalized.) If g is any univalent function on U and h = (