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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