Derivatives of Inner Functions was inspired by a conference held at the Fields Institute in 2011 entitled "Blaschke Products and Their Applications." Inner functions form an important subclass of bounded analytic functions. Since they have unimodular boundary values, they appear in many extremal problems of complex analysis. They have been extensively studied since the early twentieth century and the literature on this topic is vast. This book is devoted to a concise study of derivatives of inner functions and is confined to treating the integral means of derivatives and presenting a comprehensive list of results on Hardy and Bergman means. This self-contained monograph allows researchers to get acquainted with the essentials of inner functions, rendering this theory accessible to graduate students while providing the reader with rapid access to the frontiers of research in this field.
Notes
Description based upon print version of record.
Bibliographic references
Includes bibliographical references and index.
Language note
English
Contents
Derivatives of Inner Functions; Preface; Contents; Chapter 1: Inner Functions; 1.1 The Poisson Integral of a Measure; 1.2 The Hardy Space Hp(D); 1.3 Two Classes of Inner Functions; 1.4 The Canonical Factorization; 1.5 A Characterization of Blaschke Products; 1.6 The Nevanlinna Class N and Its Subclass N+; 1.7 Bergman Spaces; Chapter 2: The Exceptional Set of an Inner Function; 2.1 Frostman Shifts and the Exceptional Set ε; 2.2 Capacity; 2.3 Hausdorff Dimension; 2.4 ε Has Logarithmic Capacity Zero; 2.5 The Cluster Set at a Boundary Point; Chapter 3: The Derivative of Finite Blaschke Products
3.1 Elementary Formulas for B'3.2 The Cardinality of the Zeros of B'; 3.3 A Formula for |B'|; 3.4 The Locus of the Zeros of B' in D; 3.5 B Has a Nonzero Residue; Chapter 4: Angular Derivative; 4.1 Elementary Formulas for B' and S'; 4.2 Some Estimations for Hp-Means; 4.3 Some Estimations for Ap-Means; 4.4 The Angular Derivative; 4.5 The Carathéodory Derivative; 4.6 Another Characterization of the Carathéodory Derivative; Chapter 5: Hp-Means of S'; 5.1 The Effect of Singular Factors; 5.2 A Characterization of Φ' Hp(D); 5.3 We Never Have S' H12(D); 5.4 The Distance Function
10.2 Hp-Means of the First Derivative10.3 Hp-Means of Higher Derivatives; 10.4 Ap-Means of the First Derivative; References; Index
ISBN
1-283-90980-4
1-4614-5611-8
OCLC
821031019
Doi
10.1007/978-1-4614-5611-7
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