Geometric function theory in one and higher dimensions

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Geometric function theory in one and higher dimensions / Ian Graham, Gabriela Kohr

資料種別:

図書

出版情報:

New York : Marcel Dekker, c2003

形態:

xv, 530 p. ; 24 cm

シリーズ名:

Monographs and textbooks in pure and applied mathematics ; 255 <BA00046446>

著者名:

ISBN:

9780824709761 [0824709764]

書誌ID:

BA6208601X

子書誌情報

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書誌詳細

注記:

Bibliography: p. 477-520
Includes index

主題:

Functions of several complex variables; Geometric function theory; Univalent functions

言語:

英語

目次情報:

Preface

Introduction

Univalent functions / I：

Elementary properties of univalent functions / 1：

Univalence in the complex plane / 1.1：

Elementary results in the theory of univalent functions. Examples of univalent functions / 1.1.1：

The area theorem / 1.1.2：

Growth, covering and distortion results in the class S / 1.1.3：

The maximum modulus of univalent functions / 1.1.4：

Two-point distortion results for the class S / 1.1.5：

Subclasses of univalent functions in the unit disc / 2：

Functions with positive real part. Subordination and the Herglotz formula / 2.1：

The Caratheodory class. Subordination / 2.1.1：

Applications of the subordination principle / 2.1.2：

Starlike and convex functions / 2.2：

Starlikeness and convexity of order [alpha]. Alpha convexity / 2.3：

Starlikeness and convexity of order [alpha] / 2.3.1：

Alpha convexity / 2.3.2：

Close-to-convexity, spirallikeness and [Phi]-likeness in the unit disc / 2.4：

Close-to-convexity in the unit disc / 2.4.1：

Spirallike functions in the unit disc / 2.4.2：

[Phi]-like functions on the unit disc / 2.4.3：

The Loewner theory / 3：

Loewner chains and the Loewner differential equation / 3.1：

Kernel convergence / 3.1.1：

Subordination chains and kernel convergence / 3.1.2：

Loewner's differential equation / 3.1.3：

Remarks on Bieberbach's conjecture / 3.1.4：

Applications of Loewner's differential equation to the study of univalent functions / 3.2：

The radius of starlikeness for the class S and the rotation theorem / 3.2.1：

Applications of the method of Loewner chains to characterize some subclasses of S / 3.2.2：

Univalence criteria / 3.3：

Becker's univalence criteria / 3.3.1：

Univalence criteria involving the Schwarzian derivative / 3.3.2：

A generalization of Becker's and Nehari's univalence criteria / 3.3.3：

Bloch functions and the Bloch constant / 4：

Preliminaries concerning Bloch functions / 4.1：

The Bloch constant problem and Bonk's distortion theorem / 4.2：

Locally univalent Bloch functions / 4.3：

Distortion results for locally univalent Bloch functions / 4.3.1：

The case of convex functions / 4.3.2：

Linear invariance in the unit disc / 5：

General ideas concerning linear-invariant families / 5.1：

Extremal problems and radius of univalence / 5.2：

Bounds for coefficients of functions in linear-invariant families / 5.2.1：

Radius problems for linear-invariant families / 5.2.2：

Univalent mappings in several complex variables and complex Banach spaces / II：

Univalence in several complex variables / 6：

Preliminaries concerning holomorphic mappings in C[superscript n] and complex Banach spaces / 6.1：

Holomorphic functions in C[superscript n] / 6.1.1：

Classess of domains in C[superscript n]. Pseudoconvexity / 6.1.2：

Holomorphic mappings / 6.1.3：

Automorphisms of the Euclidean unit ball and the unit polydisc / 6.1.4：

Holomorphic mappings in complex Banach spaces / 6.1.5：

Generalizations of functions with positive real part / 6.1.6：

Examples and counterexamples / 6.1.7：

Criteria for starlikeness / 6.2：

Criteria for starlikeness on the unit ball in C[superscript n] or in a complex Banach space / 6.2.1：

Starlikeness criteria on more general domains in C[superscript n] / 6.2.2：

Sufficient conditions for starlikeness for mappings of class C[superscript 1] / 6.2.3：

Starlikeness of order [gamma] in C[superscript n] / 6.2.4：

Criteria for convexity / 6.3：

Criteria for convexity on the unit polydisc and the Euclidean unit ball / 6.3.1：

Necessary and sufficient conditions for convexity in complex Banach spaces / 6.3.2：

Quasi-convex mappings on the unit ball of C[superscript n] / 6.3.3：

Spirallikeness and [Phi]-likeness in several complex variables / 6.4：

Growth, covering and distortion results for starlike and convex mappings in C[superscript n] and complex Banach spaces / 7：

Growth, covering and distortion results for starlike mappings in several complex variables and complex Banach spaces / 7.1：

Growth and covering results for starlike mappings on the unit ball and some pseudoconvex domains in C[superscript n]. Extensions to complex Banach spaces / 7.1.1：

Bounds for coefficients of normalized starlike mappings in C[superscript n] / 7.1.2：

A distortion result for a subclass of starlike mappings in C[superscript n] / 7.1.3：

Growth, covering and distortion results for convex mappings in several complex variables and complex Banach spaces / 7.2：

Growth and covering results for convex mappings / 7.2.1：

Covering theorem and the translation theorem in the case of nonunivalent convex mappings in several complex variables / 7.2.2：

Bounds for coefficients of convex mappings in C[superscript n] and complex Hilbert spaces / 7.2.3：

Distortion results for convex mappings in C[superscript n] and complex Hilbert spaces / 7.2.4：

Loewner chains in several complex variables / 8：

Loewner chains and the Loewner differential equation in several complex variables / 8.1：

The Loewner differential equation in C[superscript n] / 8.1.1：

Transition mappings associated to Loewner chains on the unit ball of C[superscript n] / 8.1.2：

Close-to-starlike and spirallike mappings of type alpha on the unit ball of C[superscript n] / 8.2：

An alternative characterization of spirallikeness of type alpha in terms of Loewner chains / 8.2.1：

Close-to-starlike mappings on the unit ball of C[superscript n] / 8.2.2：

Univalent mappings which admit a parametric representation / 8.3：

Examples of mappings which admit parametric representation on the unit ball of C[superscript n] / 8.3.1：

Growth results and coefficient bounds for mappings in S[superscript 0 subscript g](B) / 8.3.2：

Applications of the method of Loewner chains to univalence criteria on the unit ball of C[superscript n] / 8.4：

Loewner chains and quasiconformal extensions of holomorphic mappings in several complex variables / 8.5：

Construction of quasiconformal extensions by means of Loewner chains / 8.5.1：

Strongly starlike and strongly spirallike mappings of type [alpha] on the unit ball of C[superscript n] / 8.5.2：

Bloch constant problems in several complex variables / 9：

Preliminaries and a generalization of Bonk's distortion theorem / 9.1：

Bloch constants for bounded and quasiregular holomorphic mappings / 9.2：

Bloch constants for starlike and convex mappings in several complex variables / 9.3：

Linear invariance in several complex variables / 10：

Preliminaries concerning the notion of linear invariance in several complex variables / 10.1：

L.I.F.'s and trace order in several complex variables / 10.1.1：

Examples of L.I.F.'s on the Euclidean unit ball of C[superscript n] / 10.1.2：

Distortion results for linear-invariant families in several complex variables / 10.2：

Distortion results for L.I.F.'s on the Euclidean unit ball of C[superscript n] / 10.2.1：

Distortion results for L.I.F.'s on the unit polydisc of C[superscript n] / 10.2.2：

Examples of L.I.F.'s of minimum order on the Euclidean unit ball and the unit polydisc of C[superscript n] / 10.3：

Examples of L.I.F.'s of minimum order on the Euclidean unit ball of C[superscript n] / 10.3.1：

Examples of L.I.F.'s of minimum order on the unit polydisc of C[superscript n] / 10.3.2：

Norm order of linear-invariant families in several complex variables / 10.4：

Norm order and univalence on the Euclidean unit ball of C[superscript n] / 10.5：

Linear-invariant families in complex Hilbert spaces / 10.6：

Univalent mappings and the Roper-Suffridge extension operator / 11：

Convex, starlike and Bloch mappings and the Roper-Suffridge extension operator / 11.1：

Growth and covering theorems associated with the Roper-Suffridge extension operator / 11.2：

Loewner chains and the operator [Phi subscript n, alpha] / 11.3：

Radius problems and the operator [Phi subscript n, alpha] / 11.4：

Linear-invariant families and the operator [Phi subscript n, alpha] / 11.5：

Bibliography

List of Symbols

Index

Preface

Introduction

Univalent functions / I：

Elementary properties of univalent functions / 1：

Univalence in the complex plane / 1.1：

Elementary results in the theory of univalent functions. Examples of univalent functions / 1.1.1：

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