Topics in Hardy Classes and Univalent Functions (Birkhäuser Advanced Texts Basler Lehrbücher)

Rosenblum, Marvin; Rovnyak, James

ISBN 10: 376435111X / ISBN 13: 9783764351113

Published by Birkhäuser, 1994

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Bibliographic Details

• Title: Topics in Hardy Classes and Univalent Functions (Birkhäuser Advanced Texts Basler Lehrbücher)
• Publisher: Birkhäuser
• Publication Date: 1994
• Language: English
• ISBN 10: 376435111X
• ISBN 13: 9783764351113
• Binding: Hardcover
• Condition: New

About this title

Synopsis

These notes are based on lectures given at the University of Virginia over the past twenty years. They may be viewed as a course in function theory for nonspecialists. Chapters 1-6 give the function-theoretic background to Hardy Classes and Operator Theory, Oxford Mathematical Monographs, Oxford University Press, New York, 1985. These chapters were written first, and they were origi­ nally intended to be a part of that book. Half-plane function theory continues to be useful for applications and is a focal point in our account (Chapters 5 and 6). The theory of Hardy and Nevanlinna classes is derived from proper­ ties of harmonic majorants of subharmonic functions (Chapters 3 and 4). A selfcontained treatment of harmonic and subharmonic functions is included (Chapters 1 and 2). Chapters 7-9 present concepts from the theory of univalent functions and Loewner families leading to proofs of the Bieberbach, Robertson, and Milin conjectures. Their purpose is to make the work of de Branges accessible to students of operator theory. These chapters are by the second author. There is a high degree of independence in the chapters, allowing the material to be used in a variety of ways. For example, Chapters 5-6 can be studied alone by readers familiar with function theory on the unit disk. Chapters 7-9 have been used as the basis for a one-semester topics course.

Synopsis

This text deals with classical and contemporary topics in function theory and is designed to be used after a one-year course in real and complex analysis. It can be used as a text for topics courses or courses in function theory, operator theory and applied areas. The first six chapters supplement the authors' book "Hardy Classes and Operator Theory". The theory of harmonic majorants for subharmonic functions is used to introduce Hardy-Orlicz classes, which are specialized to standard Hardy classes on the unit disk. The theorem of Szegoe-Solomentsev characterizes boundary behaviour. Half-plane function theory receives equal treatment and features the theorem of Flett and Kuran on existence of harmonic majorants and applications of the Phragmen-Lindeloef principle. The last three chapters contain an introduction to univalent functions, leading to a self-contained account of Loewner's differential equation and de Branges' proof of the Milin conjecture.