Multivalent Functions

W. K. Hayman

doi:10.1017/CBO9780511526268

The class of multivalent functions is an important one in complex analysis. They occur for example in the proof of De Branges' theorem which, in 1985, settled the long-standing Bieberbach conjecture. The second edition of Professor Hayman's celebrated book contains a full and self-contained proof of this result, with a chapter devoted to it. Another chapter deals with coefficient differences. It has been updated in several other ways, with theorems of Baernstein and Pommerenke on univalent functions of restricted growth, and an account of the theory of mean p-valent functions. In addition, many of the original proofs have been simplified. Each chapter contains examples and exercises of varying degrees of difficulty designed both to test understanding and illustrate the material. Consequently it will be useful for graduate students, and essential for specialists in complex function theory.

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Multivalent Functions

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Contents

Frontmatter
pp i-iv

Contents
pp v-viii

Preface
pp ix-x

Preface to the second edition
pp xi-xii

1 - Elementary bounds for univalent functions
pp 1-27

2 - The growth of finitely mean valent functions
pp 28-65

3 - Means and coefficients
pp 66-102

4 - Symmetrization
pp 103-143

5 - Circumferentially mean p–valent functions
pp 144-164

6 - Differences of successive coefficients
pp 165-196

7 - The Löwner theory
pp 197-229

8 - De Branges' Theorem
pp 230-254

Bibliography
pp 255-260

Index
pp 261-263

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Multivalent Functions

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