![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 devel-
oped 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 map-
ping 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 univa-
lent 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 theo-
rems for the class S.
In Chapter 2 weconsider various subclasses of S, including not only starlike
and convex functions but also functions which are spirallike, close-to-convex,
Xlll
Copyright © 2003 Marcel Dekker, Inc.](https://image.slidesharecdn.com/70105-250416075808-d330138c/85/Geometric-Function-Theory-in-One-and-Higher-Dimensions-1st-Edition-Ian-Graham-22-320.jpg)
![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 1916and 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 Univalence in the complex plane
1.1.1 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
Copyright © 2003 Marcel Dekker, Inc.](https://image.slidesharecdn.com/70105-250416075808-d330138c/85/Geometric-Function-Theory-in-One-and-Higher-Dimensions-1st-Edition-Ian-Graham-27-320.jpg)
![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 func-
tions 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
/ GH(D] and / is one-to-one on D. Weshall 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 GC, 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 fv 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) deter-
mines 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 R2
,
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 GD, 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
Copyright © 2003 Marcel Dekker, Inc.](https://image.slidesharecdn.com/70105-250416075808-d330138c/85/Geometric-Function-Theory-in-One-and-Higher-Dimensions-1st-Edition-Ian-Graham-28-320.jpg)
Geometric Function Theory in One and Higher Dimensions 1st Edition Ian Graham
- 1. Geometric Function Theory in One and Higher Dimensions 1st Edition Ian Graham pdf download https://ebookname.com/product/geometric-function-theory-in-one- and-higher-dimensions-1st-edition-ian-graham/ Get Instant Ebook Downloads – Browse at https://ebookname.com
- 2. Instant digital products (PDF, ePub, MOBI) available Download now and explore formats that suit you... New Developments in Geometric Function Theory First Edition Georgia Irina Oros https://ebookname.com/product/new-developments-in-geometric- function-theory-first-edition-georgia-irina-oros/ Geometric Function Theory Explorations in Complex Analysis Cornerstones 1st Edition Steven G. Krantz https://ebookname.com/product/geometric-function-theory- explorations-in-complex-analysis-cornerstones-1st-edition-steven- g-krantz/ Black Holes in Higher Dimensions 1st Edition Gary T. Horowitz https://ebookname.com/product/black-holes-in-higher- dimensions-1st-edition-gary-t-horowitz/ Wiley Not for Profit GAAP 2011 Interpretation and Application of Generally Accepted Accounting Principles 8th Edition Richard F. Larkin https://ebookname.com/product/wiley-not-for-profit- gaap-2011-interpretation-and-application-of-generally-accepted- accounting-principles-8th-edition-richard-f-larkin/
- 3. Boundaries of Toleration Alfred Stepan (Editor) https://ebookname.com/product/boundaries-of-toleration-alfred- stepan-editor/ Achieving Cultural Competency A Case Based Approach to Training Health Professionals 1st Edition Lisa Hark Phd Rd https://ebookname.com/product/achieving-cultural-competency-a- case-based-approach-to-training-health-professionals-1st-edition- lisa-hark-phd-rd/ Breaking Cardinal Rules Katina Powell & Dick Cady [Powell https://ebookname.com/product/breaking-cardinal-rules-katina- powell-dick-cady-powell/ Handbook of Clinical Drug Data 10th Edition Philip Anderson https://ebookname.com/product/handbook-of-clinical-drug- data-10th-edition-philip-anderson/ Simply Rational Decision Making in the Real World 1st Edition Gerd Gigerenzer https://ebookname.com/product/simply-rational-decision-making-in- the-real-world-1st-edition-gerd-gigerenzer/
- 4. The Brooklyn Bridge Building America Then and Now 1st Edition G. S. Prentzas https://ebookname.com/product/the-brooklyn-bridge-building- america-then-and-now-1st-edition-g-s-prentzas/
- 5. GEOMETRIC FUNCTION THEORY IN ONE AND HIGHER DIMENSIONS IAN GRAHAM GABRIELA KOHR University of Toronto Babe§-Bolyai University Toronto, Ontario, Canada Cluj-Napoca, Romania MARCEL DEKKER, INC. NEW YORK • BASEL Copyright © 2003 Marcel Dekker, Inc.
- 6. 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 infor- mation, write to Special Sales/Professional Marketingat 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): 1 0 9 8 7 6 5 4 3 2 1 PRINTED IN THE UNITEDSTATES OF AMERICA Copyright © 2003 Marcel Dekker, Inc.
- 7. PURE AND APPLIEDMATHEMATICS A Program of Monographs, Textbooks, and Lecture Notes EXECUTIVE EDITORS Earl J. Taft Zuhair Nashed Rutgers University University of Central Florida New Brunswick, New Jersey Orlando, Florida EDITORIAL BOARD M. S. Baouendi Anil Nerode University of California, Cornell University San Diego Donald Passman Jane Cronin University of Wisconsin, Rutgers University Madison Jack K. Hale Fred S. Roberts Georgia Institute of Technology Rutgers University S. Kobayashi David L. Russell University of California, Virginia Polytechnic Institute Berkeley and State University Marvin Marcus Walter Schempp University of California, Universitat Siegen Santa Barbara Mark Teply W. S. Massey University of Wisconsin, Yale University Milwaukee Copyright © 2003 Marcel Dekker, Inc.
- 8. MONOGRAPHS AND TEXTBOOKSIN 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) Copyright © 2003 Marcel Dekker, Inc.
- 9. 56. /. Vaisman, Foundations of Three-Dimensional Euclidean Geometry (1980) 57. H. I. Freedan, Deterministic Mathematical Models in Population Ecology (1980) 58. S. 8. Chae, Lebesgue Integration (1980) 59. C. S. Rees et a/., Theory and Applications of Fourier Analysis (1981) 60. L Nachbin, Introduction to Functional Analysis (R. M.Aron, trans.) (1981) 61. G. Orzech and M. Orzech, Plane Algebraic Curves (1981) 62. R. Johnsonbaugh and W. E. Pfaffenberger, Foundations of Mathematical Analysis (1981) 63. W. L Voxman and R. H. Goetschel, Advanced Calculus (1981) 64. L. J. Corwin and R. H. Szczarba, Multivariable Calculus (1982) 65. V. I. Istratescu, Introduction to Linear Operator Theory (1981) 66. R. D. Jarvinen, Finite and Infinite Dimensional Linear Spaces (1981) 67. J. K. Beem and P. E. Ehrtich, Global Lorentzian Geometry (1981) 68. D. L Armacost, The Structure of Locally Compact Abelian Groups (1981) 69. J. W. Brewer and M. K. Smith, eds.,Emmy Noether: A Tribute (1981) 70. K. H. Kim, Boolean Matrix Theory and Applications (1982) 71. T. W. Wieting, The Mathematical Theory of Chromatic Plane Ornaments (1982) 72. D. B.Gauld, Differential Topology (1982) 73. R. L. Faber, Foundations of Euclidean and Non-Euclidean Geometry (1983) 74. M. Carmeli, Statistical Theory and Random Matrices (1983) 75. J. H. Camith et a/., The Theory of Topological Semigroups (1983) 76. R. L Faber, Differential Geometry and Relativity Theory (1983) 77. S. Bamett, Polynomials and Linear Control Systems (1983) 78. G. Karpilovsky, Commutative Group Algebras (1983) 79. F. Van Oystaeyen and A. Verschoren, Relative Invariants of Rings (1983) 80. /. Vaisman, A First Course in Differential Geometry (1984) 81. G. W. Swan, Applications of Optimal Control Theory in Biomedicine (1984) 82. T. Petrie and J. D. Randall, Transformation Groups on Manifolds (1984) 83. K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings (1984) 84. T. Albu and C. Nastasescu, Relative Finiteness in Module Theory (1984) 85. K. Hrbacek and T. Jech, Introduction to Set Theory: Second Edition (1984) 86. F. Van Oystaeyen and A. Verschoren, Relative Invariants of Rings (1984) 87. B. R. McDonald, Linear Algebra Over Commutative Rings (1984) 88. M. Namba, Geometry of Projective Algebraic Curves (1984) 89. G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics (1985) 90. M. R. Bremner et a/.. Tables of Dominant Weight Multiplicities for Representations of Simple Lie Algebras (1985) 91. A. E. Fekete, Real Linear Algebra (1985) 92. S. B. Chae, Holomorphy and Calculus in Normed Spaces (1985) 93. A. J. Jerri, Introduction to Integral Equations with Applications (1985) 94. G. Karpilovsky, Projective Representations of Finite Groups (1985) 95. L. Narici and E. Beckenstein, Topological Vector Spaces (1985) 96. J. Weeks, The Shape of Space (1985) 97. P. R. Gribik and K. O. Kortanek, Extremal Methods of Operations Research (1985) 98. J.-A. Chao and W. A. Woyczynski, eds., Probability Theory and Harmonic Analysis (1986) 99. G. D. Crown et a/., Abstract Algebra (1986) 100. J. H. Carruth et a/., The Theory of Topological Semigroups, Volume 2 (1986) 101. R. S. Doran and V. A. Belfi, Characterizations of C*-Algebras (1986) 102. M. W. Jeter, Mathematical Programming (1986) 103. M. Altman, A Unified Theory of Nonlinear Operator and Evolution Equations with Applications (1986) 104. A. Verschoren, Relative Invariants of Sheaves (1987) 105. R. A. Usmani, Applied Linear Algebra (1987) 106. P. 8/ass and J. Lang, Zariski Surfaces and Differential Equations in Characteristic p > 0(1987) 107. J. A. Reneke et a/., Structured Hereditary Systems (1987) 108. H. Busemann and B. B. Phadke, Spaces with Distinguished Geodesies (1987) 109. R. Harte, Invertibility and Singularity for Bounded Linear Operators(1988) 110. G. S. Ladde et a/., Oscillation Theory of Differential Equations with Deviating Argu- ments (1987) 111. L. Dudkin et a/., Iterative Aggregation Theory (1987) 112. T. Okubo, Differential Geometry (1987) Copyright © 2003 Marcel Dekker, Inc.
- 10. 113. D. L Stand and M. L Stand, Real Analysis with Point-Set Topology (1987) 114. T. C. Gard, Introduction to Stochastic Differential Equations (1988) 115. S. S. Abhyankar, Enumerative Combinatorics of Young Tableaux (1988) 116. H. Sfrade and R. Famsteiner, Modular Lie Algebras and Their Representations (1988) 117. J. A. Huckaba, Commutative Rings with Zero Divisors (1988) 118. W. D. Wallis, Combinatorial Designs (1988) 119. W. Wiestaw, Topological Fields (1988) 120. G. Karpilovsky, Field Theory (1988) 121. S. Caenepeel and F. Van Oystaeyen, Brauer Groups and the Cohomology of Graded Rings (1989) 122. W. Kozlowski, Modular Function Spaces (1988) 123. E. Lowen-Colebunders, Function Classes of Cauchy Continuous Maps (1989) 124. M. Pavel, Fundamentals of Pattern Recognition (1989) 125. V. Lakshmikantham et a/.. Stability Analysis of Nonlinear Systems (1989) 126. R. Sivaramakrishnan, The Classical Theory of Arithmetic Functions (1989) 127. N. A. Watson, Parabolic Equations on an Infinite Strip (1989) 128. K. J. Hastings, Introduction to the Mathematics of Operations Research (1989) 129. B. Fine, Algebraic Theory of the Bianchi Groups (1989) 130. D. N. Dikranjan et a/., Topological Groups (1989) 131. J. C. Morgan II, Point Set Theory (1990) 132. P. BilerandA. Wrtkowski, Problems in Mathematical Analysis (1990) 133. H. J. Sussmann, Nonlinear Controllability and Optimal Control (1990) 134. J.-P.Florens et a/., Elements of Bayesian Statistics (1990) 135. N. Shell, Topological Fields and Near Valuations (1990) 136. 8. F. Doolin and C. F. Martin, Introduction to Differential Geometry for Engineers (1990) 137. S. S. Holland, Jr.,Applied Analysis by the Hilbert Space Method (1990) 138. J. Oknfnski, Semigroup Algebras (1990) 139. K. Zhu, Operator Theory in Function Spaces (1990) 140. G. B. Price, An Introduction to Multicomplex Spacesand Functions (1991) 141. R. B. Darst, Introduction to Linear Programming (1991) 142. P. L Sachdev, Nonlinear Ordinary Differential Equations andTheir Applications (1991) 143. T. Husain, Orthogonal Schauder Bases (1991) 144. J. Foran, Fundamentals of Real Analysis (1991) 145. W. C. Brown, Matrices and Vector Spaces (1991) 146. M. M. Rao and Z. D. Ren, Theory of Oriicz Spaces (1991) 147. J. S. Golan and T. Head, Modules and the Structures of Rings (1991) 148. C. Small, Arithmetic of Finite Fields (1991) 149. K. Yang, Complex Algebraic Geometry (1991) 150. D.G. Hoffman et a/.. Coding Theory (1991) 151. M. O. Gonzalez, Classical Complex Analysis (1992) 152. M. O. Gonzalez, Complex Analysis (1992) 153. L. W. Baggett, Functional Analysis (1992) 154. M. Sniedovich, Dynamic Programming (1992) 155. R. P. Agarwal, Difference Equations and Inequalities (1992) 156. C. Brezinski, Biorthogonality and Its Applications to Numerical Analysis (1992) 157. C. Swartz, An Introduction to Functional Analysis (1992) 158. S. 8. Nadler, Jr., Continuum Theory (1992) 159. M. A. AI-Gwaiz, Theory of Distributions (1992) 160. E. Perry, Geometry: Axiomatic Developments with Problem Solving (1992) 161. E. Castillo and M. R. Ruiz-Cobo, Functional Equations and Modelling in Science and Engineering (1992) 162. A. J. Jerri, Integral and Discrete Transforms with Applications and Error Analysis (1992) 163. A. Charlier et al., Tensors and the Clifford Algebra (1992) 164. P. Bilerand T. Nadzieja, Problems and Examples in Differential Equations (1992) 165. E. Hansen, Global Optimization Using Interval Analysis (1992) 166. S. Guerre-Delabriere, Classical Sequences in Banach Spaces (1992) 167. V. C. Wong, Introductory Theory of Topological Vector Spaces (1992) 168. S. H. KulkamiandB. V. Limaye, Real Function Algebras (1992) 169. W. C. Brown, Matrices Over Commutative Rings (1993) 170. J. Loustau and M. Dillon, Linear Geometry with Computer Graphics (1993) 171. W. V. Petryshyn, Approximation-Solvability of Nonlinear Functional and Differential Equations (1993) Copyright © 2003 Marcel Dekker, Inc.
- 11. 172. E. C. Young, Vector and Tensor Analysis: Second Edition (1993) 173. T.A. Bick, Elementary Boundary Value Problems (1993) 174. M. Pavel, Fundamentals of Pattern Recognition: Second Edition (1993) 175. S. A. Albeverio et a/., Noncommutative Distributions (1993) 176. W. Fulks, Complex Variables (1993) 177. M. M. Rao,Conditional Measures and Applications (1993) 178. A. Janicki and A. Weron, Simulation and Chaotic Behavior of a-Stable Stochastic Processes (1994) 179. P. Neittaanmaki and D. Tiba, Optimal Control of Nonlinear Parabolic Systems (1994) 180. J. Cronin, Differential Equations: Introduction and Qualitative Theory, Second Edition (1994) 181. S. HeikkilS and V. Lakshmikantham, Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations (1994) 182. X. Mao, Exponential Stability of Stochastic Differential Equations (1994) 183. B. S. Thomson, Symmetric Properties of Real Functions (1994) 184. J. E. Rubio, Optimization and Nonstandard Analysis (1994) 185. J. L Bueso et a/., Compatibility, Stability, and Sheaves (1995) 186. A. N. Michel and K. Wang, Qualitative Theory of Dynamical Systems (1995) 187. M. R. Darnel, Theory of Lattice-Ordered Groups (1995) 188. Z. Naniewicz and P. D. Panagiotopoulos, Mathematical Theory of Hemivariational Inequalities and Applications (1995) 189. L J. Corwin and R. H. Szczarba, Calculus in Vector Spaces: Second Edition (1995) 190. L H. Erbe et a/., Oscillation Theory for Functional Differential Equations (1995) 191. S. Agaian era/., Binary Polynomial Transforms and Nonlinear Digital Filters (1995) 192. M. I. Gil', Norm Estimations for Operation-Valued Functions and Applications (1995) 193. P. A. Grillet, Semigroups: An Introduction to the Structure Theory (1995) 194. S. Kichenassamy, Nonlinear Wave Equations (1996) 195. V. F. Krotov, Global Methods in Optimal Control Theory (1996) 196. K. I. Beidaretal., Rings with Generalized Identities (1996) 197. V. I. Amautov et a/., Introduction to the Theory of Topological Rings and Modules (1996) 198. G. Sierksma, Linear and Integer Programming (1996) 199. R. Lasser, Introduction to Fourier Series (1996) 200. V. Sima, Algorithms for Linear-Quadratic Optimization (1996) 201. D. Redmond, Number Theory (1996) 202. J. K. Beem et a/., Global Lorentzian Geometry: Second Edition (1996) 203. M. Fontana et a/., Prufer Domains (1997) 204. H. Tanabe, Functional Analytic Methods for Partial Differential Equations (1997) 205. C. Q. Zhang, Integer Flows and Cycle Covers of Graphs (1997) 206. E. Spiegel and C. J. O'Donnell, Incidence Algebras (1997) 207. B. Jakubczyk and W. Respondek, Geometry of Feedback and Optimal Control (1998) 208. T. W. Haynes et a/., Fundamentals of Domination in Graphs (1998) 209. T. W. Haynes et a/., ecfs., Domination in Graphs: Advanced Topics (1998) 210. L. A. D'Alotto et a/., A Unified Signal Algebra Approach to Two-Dimensional Parallel Digital Signal Processing (1998) 211. F. Halter-Koch, Ideal Systems (1998) 212. N. K. Govil et a/., eds., Approximation Theory (1998) 213. R. Cross, Multivalued Linear Operators (1998) 214. A. A. Martynyuk, Stability by Liapunov's Matrix Function Method with Applications (1998) 215. A. Favini and A. Yagi, Degenerate Differential Equations in Banach Spaces (1999) 216. A. Illanes and S. Nadler, Jr., Hyperspaces: Fundamentals and Recent Advances (1999) 217. G. Kato and D. Struppa, Fundamentals of Algebraic Microlocal Analysis (1999) 218. G. X.-Z. Yuan, KKM Theory and Applications in Nonlinear Analysis (1999) 219. D. Motreanu and N. H. Pavel, Tangency, Flow Invariance for Differential Equations, and Optimization Problems (1999) 220. K. Hrbacek and T. Jech, Introduction to Set Theory, Third Edition (1999) 221. G. E. Kolosov, Optimal Design of Control Systems (1999) 222. N. L Johnson, Subplane Covered Nets (2000) 223. B. Fine and G. Rosenberger, Algebraic Generalizations of Discrete Groups (1999) 224. M. Vath, Volterra and Integral Equations of Vector Functions (2000) 225. S. S. Miller and P. T. Mocanu, Differential Subordinations (2000) Copyright © 2003 Marcel Dekker, Inc.
- 12. 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 Applica- tions, 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 Copyright © 2003 Marcel Dekker, Inc.
- 13. To my twin sister, Mirela Gabriela Kohr To Norberto Kerzman Ian Graham Copyright © 2003 Marcel Dekker, Inc.
- 14. 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 nor- malized 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 exposi- tion 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 mathemati- cians. The prerequisites are a good first course in complex analysis, including Copyright © 2003 Marcel Dekker, Inc.
- 15. 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 Chapte r 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. Mo- canu 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 func- tion 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 Copyright © 2003 Marcel Dekker, Inc.
- 16. Contents Preface v Introduction xiii I 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 27 2.1 Functions with positive real part. Subordination and the Her- glotz formula 27 2.1.1 The Caratheodory class. Subordination 27 2.1.2 Applications of the subordination principle 32 2.2 Starlike and convex functions 36 2.3 Starlikeness and convexity of order a. Alpha convexity 54 2.3.1 Starlikeness and convexity of order a 54 2.3.2 Alpha convexity 58 vii Copyright © 2003 Marcel Dekker, Inc.
- 17. viii Contents 2.4 Close-to-convexity, spirallikeness and ^-likeness in the unit disc 63 2.4.1 Close-to-convexity in the unit disc 63 2.4.2 Spirallike functions in the unit disc 73 2.4.3 $-like functions on the unit disc 79 3 The Loewner theory 87 3.1 Loewner chains and the Loewner differential equation 87 3.1.1 Kernel convergence 87 3.1.2 Subordination chains and kernel convergence 94 3.1.3 Loewner's differential equation 100 3.1.4 Remarks on Bieberbach's conjecture 112 3.2 Applications of Loewner's differential equation to the study of univalent functions 117 3.2.1 The radius ofstarlikeness for the class S and the rotation theorem 118 3.2.2 Applications of the method of Loewner chains to characterize some subclasses of S 126 3.3 Univalence criteria 130 3.3.1 Becker's univalence criteria 130 3.3.2 Univalence criteria involving the Schwarzian derivative . 132 3.3.3 A generalization of Becker's and Nehari's univalence cri- teria 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 165 5.1 General ideas concerning linear-invariant families 165 5.2 Extremal problems and radius of univalence 172 Copyright © 2003 Marcel Dekker, Inc.
- 18. Contents ix 5.2.1 Bounds for coefficients of functions in linear-invariant families 172 5.2.2 Radius problems for linear-invariant families 174 II Univalent mappings in several complexvariables and complex Banach spaces 181 6 Univalence in several complex variables 183 6.1 Preliminaries concerning holomorphic mappings in C™ and com- plex Banach spaces 184 6.1.1 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 6.1.5 Holomorphic mappings in complex Banach spaces . . . 197 6.1.6 Generalizations of functions with positive real part . . . 202 6.1.7 Examples and counterexamples 210 6.2 Criteria for starlikeness 213 6.2.1 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" . . 217 6.2.3 Sufficient conditions for starlikeness for mappings of class C1 219 6.2.4 Starlikeness of order 7^0* 221 6.3 Criteria for convexity 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 com- plex Banach spaces 230 6.3.3 Quasi-convex mappings on the unit ball of C" 238 6.4 Spirallikeness and ^-likeness in several complex variables . . . 244 Copyright © 2003 Marcel Dekker, Inc.
- 19. x Contents 7 Growth, covering and distortion results for starlike and convex mappings in C" and complex Banach spaces 255 7.1 Growth, covering and distortion results for starlike mappings in several complex variables and complex Banach spaces 256 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 256 7.1.2 Bounds for coefficients of normalized starlike mappings inCn 262 7.1.3 A distortion result for a subclass of starlike mappings in Cn 268 7.2 Growth, covering and distortion results for convex mappings in several complex variables and complex Banach spaces 271 7.2.1 Growth and covering results for convex mappings . . . . 271 7.2.2 Covering theorem and the translation theorem in the case of nonunivalent convexmappings in severalcomplex variables 278 7.2.3 Bounds for coefficients of convex mappings in Cn and complex Hilbert spaces 281 7.2.4 Distortion results for convex mappings in Cn and com- plex Hilbert spaces 286 8 Loewner chains in several complex variables 295 8.1 Loewnerchains and the Loewner differential equation in several complex variables 295 8.1.1 The Loewner differential equation in Cn 295 8.1.2 Transition mappings associated to Loewner chains on the unit ball of C1 312 8.2 Close-to-starlike and spirallike mappings of type alpha on the unit ball of C1 322 8.2.1 An alternative characterization of spirallikeness of type alpha in terms of Loewner chains 322 8.2.2 Close-to-starlike mappings on the unit ball of Cn . . . . 324 Copyright © 2003 Marcel Dekker, Inc.
- 20. Contents xi 8.3 Univalent mappings which admit a parametric representation . 330 8.3.1 Examples of mappings which admit parametric repre- sentation on the unit ball of Cn 330 8.3.2 Growth results and coefficient bounds for mappings in Sj(B) 334 8.4 Applications of the method of Loewner chains to univalence criteria on the unit ball of C™ 348 8.5 Loewner chains and quasiconformal extensions of holomorphic mappings in several complex variables 353 8.5.1 Construction of quasiconformal extensions by means of Loewner chains 353 8.5.2 Strongly starlike and strongly spirallike mappings of type a on the unit ball of C™ 370 9 Bloch constant problems in several complex variables 377 9.1 Preliminaries and a generalization of Bonk's distortion theorem 377 9.2 Bloch constants for bounded and quasiregular holomorphic mappings 384 9.3 Bloch constants for starlike and convex mappings in several complex variables 390 10 Linear invariance in several complex variables 395 10.1 Preliminaries concerning the notion of linear invariance in sev- eral complex variables 396 10.1.1 L.I.F.'s and trace order in several complex variables . . 396 10.1.2 Examples of L.I.F.'s on the Euclidean unit ball of C" . 399 10.2 Distortion results for linear-invariant families in several complex variables 401 10.2.1 Distortion results for L.I.F.'s on the Euclidean unit ball ofC71 401 10.2.2 Distortion results for L.I.F.'s on the unit polydisc of Cn 410 10.3 Examples of L.I.F.'s of minimum order on the Euclidean unit ball and the unit polydisc of C" 414 Copyright © 2003 Marcel Dekker, Inc.
- 21. xii Contents 10.3.1 Examples of L.I.F.'s of minimum order on the Euclidean unit ball of C1 414 10.3.2 Examples of L.I.F.'s of minimum order on the unit poly- disc of C1 426 10.4 Norm order of linear-invariant families in several complex vari- ables 429 10.5 Norm order and univalence on the Euclidean unit ball of C™ . . 434 10.6 Linear-invariant families in complex Hilbert spaces 440 11 Univalent mappings and the Roper-Suffridge extension operator 443 11.1 Convex, starlike and Bloch mappings and the Roper-Suffridge extension operator 444 11.2 Growth and covering theorems associated with the Roper- Suffridge extension operator 456 11.3 Loewner chains and the operator 3>n,a 461 11.4 Radius problems and the operator 3>nja 466 11.5 Linear-invariant families and the operator $n>a 469 Bibliography 477 List of Symbols 521 Index Copyright © 2003 Marcel Dekker, Inc.
- 22. 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 devel- oped 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 map- ping 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 univa- lent 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 theo- rems for the class S. In Chapter 2 weconsider various subclasses of S, including not only starlike and convex functions but also functions which are spirallike, close-to-convex, Xlll Copyright © 2003 Marcel Dekker, Inc.
- 23. 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 normal- ized 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 theo- rems. 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 par- ticular 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-knownsubclasses 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 esti- mates. Among the recent results treated here, we mention the compactness of Copyright © 2003 Marcel Dekker, Inc.
- 24. Introduction xv the class Ai which plays the role ofthe 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 concludeswith 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. Copyright © 2003 Marcel Dekker, Inc.
- 25. GEOMETRIC FUNCTION THEORY IN ONE AND HIGHER DIMENSIONS Copyright © 2003 Marcel Dekker, Inc.
- 26. Part I Univalent functions Copyright © 2003 Marcel Dekker, Inc.
- 27. 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 1916and 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 Univalence in the complex plane 1.1.1 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 Copyright © 2003 Marcel Dekker, Inc.
- 28. 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 func- tions 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 / GH(D] and / is one-to-one on D. Weshall 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 GC, 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 fv 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) deter- mines 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 R2 , 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 GD, 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 Copyright © 2003 Marcel Dekker, Inc.
- 29. 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. More- over, if ZQ G D, there is a unique conformal map of D onto U such that f(z0 ) = 0 and /'(ZQ) > 0. The entire complex plane cannot be conformallyequivalent 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 theo- rem 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 ques- tions 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 = (<7 —<7(0))/</(0), then h G5, so the study of the class S provides information about any univa- lent function on U. Similarly, for 0 < r < 1 we shall sometimes consider the class S(Ur) of functions / E Hu(Ur) which are normalized. A function / in the class S has a Taylor series expansion of the form (1.1.1) f ( z ) = z +a2z2 + ... +anzn + ..., zeU. We also introduce the class S of functions y> which are univalent on A = {C GC : |£| > 1}, with a simple pole at oo, and which are normalized so that the Laurent series expansion of <p at oo has the form (1.1.2) y,(C) = C + *o + ^ + ... + ^ + ..., |C|>1. Any such function <p € S maps A onto the complement of a connected compact set. Copyright © 2003 Marcel Dekker, Inc.
- 30. Another Random Document on Scribd Without Any Related Topics
- 31. with active links or immediate access to the full terms of the Project Gutenberg™ License. 1.E.6. You may convert to and distribute this work in any binary, compressed, marked up, nonproprietary or proprietary form, including any word processing or hypertext form. However, if you provide access to or distribute copies of a Project Gutenberg™ work in a format other than “Plain Vanilla ASCII” or other format used in the official version posted on the official Project Gutenberg™ website (www.gutenberg.org), you must, at no additional cost, fee or expense to the user, provide a copy, a means of exporting a copy, or a means of obtaining a copy upon request, of the work in its original “Plain Vanilla ASCII” or other form. Any alternate format must include the full Project Gutenberg™ License as specified in paragraph 1.E.1. 1.E.7. Do not charge a fee for access to, viewing, displaying, performing, copying or distributing any Project Gutenberg™ works unless you comply with paragraph 1.E.8 or 1.E.9. 1.E.8. You may charge a reasonable fee for copies of or providing access to or distributing Project Gutenberg™ electronic works provided that: • You pay a royalty fee of 20% of the gross profits you derive from the use of Project Gutenberg™ works calculated using the method you already use to calculate your applicable taxes. The fee is owed to the owner of the Project Gutenberg™ trademark, but he has agreed to donate royalties under this paragraph to the Project Gutenberg Literary Archive Foundation. Royalty payments must be paid within 60 days following each date on which you prepare (or are legally required to prepare) your periodic tax returns. Royalty payments should be clearly marked as such and sent to the Project Gutenberg Literary Archive Foundation at the address specified in Section 4, “Information
- 32. about donations to the Project Gutenberg Literary Archive Foundation.” • You provide a full refund of any money paid by a user who notifies you in writing (or by e-mail) within 30 days of receipt that s/he does not agree to the terms of the full Project Gutenberg™ License. You must require such a user to return or destroy all copies of the works possessed in a physical medium and discontinue all use of and all access to other copies of Project Gutenberg™ works. • You provide, in accordance with paragraph 1.F.3, a full refund of any money paid for a work or a replacement copy, if a defect in the electronic work is discovered and reported to you within 90 days of receipt of the work. • You comply with all other terms of this agreement for free distribution of Project Gutenberg™ works. 1.E.9. If you wish to charge a fee or distribute a Project Gutenberg™ electronic work or group of works on different terms than are set forth in this agreement, you must obtain permission in writing from the Project Gutenberg Literary Archive Foundation, the manager of the Project Gutenberg™ trademark. Contact the Foundation as set forth in Section 3 below. 1.F. 1.F.1. Project Gutenberg volunteers and employees expend considerable effort to identify, do copyright research on, transcribe and proofread works not protected by U.S. copyright law in creating the Project Gutenberg™ collection. Despite these efforts, Project Gutenberg™ electronic works, and the medium on which they may be stored, may contain “Defects,” such as, but not limited to, incomplete, inaccurate or corrupt data, transcription errors, a copyright or other intellectual property infringement, a defective or
- 33. damaged disk or other medium, a computer virus, or computer codes that damage or cannot be read by your equipment. 1.F.2. LIMITED WARRANTY, DISCLAIMER OF DAMAGES - Except for the “Right of Replacement or Refund” described in paragraph 1.F.3, the Project Gutenberg Literary Archive Foundation, the owner of the Project Gutenberg™ trademark, and any other party distributing a Project Gutenberg™ electronic work under this agreement, disclaim all liability to you for damages, costs and expenses, including legal fees. YOU AGREE THAT YOU HAVE NO REMEDIES FOR NEGLIGENCE, STRICT LIABILITY, BREACH OF WARRANTY OR BREACH OF CONTRACT EXCEPT THOSE PROVIDED IN PARAGRAPH 1.F.3. YOU AGREE THAT THE FOUNDATION, THE TRADEMARK OWNER, AND ANY DISTRIBUTOR UNDER THIS AGREEMENT WILL NOT BE LIABLE TO YOU FOR ACTUAL, DIRECT, INDIRECT, CONSEQUENTIAL, PUNITIVE OR INCIDENTAL DAMAGES EVEN IF YOU GIVE NOTICE OF THE POSSIBILITY OF SUCH DAMAGE. 1.F.3. LIMITED RIGHT OF REPLACEMENT OR REFUND - If you discover a defect in this electronic work within 90 days of receiving it, you can receive a refund of the money (if any) you paid for it by sending a written explanation to the person you received the work from. If you received the work on a physical medium, you must return the medium with your written explanation. The person or entity that provided you with the defective work may elect to provide a replacement copy in lieu of a refund. If you received the work electronically, the person or entity providing it to you may choose to give you a second opportunity to receive the work electronically in lieu of a refund. If the second copy is also defective, you may demand a refund in writing without further opportunities to fix the problem. 1.F.4. Except for the limited right of replacement or refund set forth in paragraph 1.F.3, this work is provided to you ‘AS-IS’, WITH NO OTHER WARRANTIES OF ANY KIND, EXPRESS OR IMPLIED,
- 34. INCLUDING BUT NOT LIMITED TO WARRANTIES OF MERCHANTABILITY OR FITNESS FOR ANY PURPOSE. 1.F.5. Some states do not allow disclaimers of certain implied warranties or the exclusion or limitation of certain types of damages. If any disclaimer or limitation set forth in this agreement violates the law of the state applicable to this agreement, the agreement shall be interpreted to make the maximum disclaimer or limitation permitted by the applicable state law. The invalidity or unenforceability of any provision of this agreement shall not void the remaining provisions. 1.F.6. INDEMNITY - You agree to indemnify and hold the Foundation, the trademark owner, any agent or employee of the Foundation, anyone providing copies of Project Gutenberg™ electronic works in accordance with this agreement, and any volunteers associated with the production, promotion and distribution of Project Gutenberg™ electronic works, harmless from all liability, costs and expenses, including legal fees, that arise directly or indirectly from any of the following which you do or cause to occur: (a) distribution of this or any Project Gutenberg™ work, (b) alteration, modification, or additions or deletions to any Project Gutenberg™ work, and (c) any Defect you cause. Section 2. Information about the Mission of Project Gutenberg™ Project Gutenberg™ is synonymous with the free distribution of electronic works in formats readable by the widest variety of computers including obsolete, old, middle-aged and new computers. It exists because of the efforts of hundreds of volunteers and donations from people in all walks of life. Volunteers and financial support to provide volunteers with the assistance they need are critical to reaching Project Gutenberg™’s goals and ensuring that the Project Gutenberg™ collection will
- 35. remain freely available for generations to come. In 2001, the Project Gutenberg Literary Archive Foundation was created to provide a secure and permanent future for Project Gutenberg™ and future generations. To learn more about the Project Gutenberg Literary Archive Foundation and how your efforts and donations can help, see Sections 3 and 4 and the Foundation information page at www.gutenberg.org. Section 3. Information about the Project Gutenberg Literary Archive Foundation The Project Gutenberg Literary Archive Foundation is a non-profit 501(c)(3) educational corporation organized under the laws of the state of Mississippi and granted tax exempt status by the Internal Revenue Service. The Foundation’s EIN or federal tax identification number is 64-6221541. Contributions to the Project Gutenberg Literary Archive Foundation are tax deductible to the full extent permitted by U.S. federal laws and your state’s laws. The Foundation’s business office is located at 809 North 1500 West, Salt Lake City, UT 84116, (801) 596-1887. Email contact links and up to date contact information can be found at the Foundation’s website and official page at www.gutenberg.org/contact Section 4. Information about Donations to the Project Gutenberg Literary Archive Foundation Project Gutenberg™ depends upon and cannot survive without widespread public support and donations to carry out its mission of increasing the number of public domain and licensed works that can be freely distributed in machine-readable form accessible by the widest array of equipment including outdated equipment. Many
- 36. small donations ($1 to $5,000) are particularly important to maintaining tax exempt status with the IRS. The Foundation is committed to complying with the laws regulating charities and charitable donations in all 50 states of the United States. Compliance requirements are not uniform and it takes a considerable effort, much paperwork and many fees to meet and keep up with these requirements. We do not solicit donations in locations where we have not received written confirmation of compliance. To SEND DONATIONS or determine the status of compliance for any particular state visit www.gutenberg.org/donate. While we cannot and do not solicit contributions from states where we have not met the solicitation requirements, we know of no prohibition against accepting unsolicited donations from donors in such states who approach us with offers to donate. International donations are gratefully accepted, but we cannot make any statements concerning tax treatment of donations received from outside the United States. U.S. laws alone swamp our small staff. Please check the Project Gutenberg web pages for current donation methods and addresses. Donations are accepted in a number of other ways including checks, online payments and credit card donations. To donate, please visit: www.gutenberg.org/donate. Section 5. General Information About Project Gutenberg™ electronic works Professor Michael S. Hart was the originator of the Project Gutenberg™ concept of a library of electronic works that could be freely shared with anyone. For forty years, he produced and distributed Project Gutenberg™ eBooks with only a loose network of volunteer support.
- 37. Project Gutenberg™ eBooks are often created from several printed editions, all of which are confirmed as not protected by copyright in the U.S. unless a copyright notice is included. Thus, we do not necessarily keep eBooks in compliance with any particular paper edition. Most people start at our website which has the main PG search facility: www.gutenberg.org. This website includes information about Project Gutenberg™, including how to make donations to the Project Gutenberg Literary Archive Foundation, how to help produce our new eBooks, and how to subscribe to our email newsletter to hear about new eBooks.
- 38. Welcome to our website – the ideal destination for book lovers and knowledge seekers. With a mission to inspire endlessly, we offer a vast collection of books, ranging from classic literary works to specialized publications, self-development books, and children's literature. Each book is a new journey of discovery, expanding knowledge and enriching the soul of the reade Our website is not just a platform for buying books, but a bridge connecting readers to the timeless values of culture and wisdom. With an elegant, user-friendly interface and an intelligent search system, we are committed to providing a quick and convenient shopping experience. Additionally, our special promotions and home delivery services ensure that you save time and fully enjoy the joy of reading. Let us accompany you on the journey of exploring knowledge and personal growth! ebookname.com