ABSTRACT

Presenting new results along with research spanning five decades. Fractional Cauchy Transforms provides a full treatment of the topic, from its roots in classical complex analysis to its current state. Self-contained, it includes introductory material and classical results, such as those associated with complex-valued measures on the unit circle, that form the basis of the developments that follow. The authors focus on concrete analytic questions, with functional analysis providing the general framework., After examining basic properties, the authors study integral means and relationships between the fractional Cauchy transforms and the Hardy and Dirichlet spaces. They then study radial and nontangential limits, followed by chapters devoted to multipliers, composition operators, and univalent functions. The final chapter gives an analytic characterization of the family of Cauchy transforms when considered as functions defined in the complement of the unit circle.

chapter Chapter 1|15 pages

Introduction

chapter Chapter 2|30 pages

Basic Properties of Fα

chapter Chapter 3|19 pages

Integral Means and the Hardy and Dirichlet Spaces

chapter Chapter 4|24 pages

Radial Limits

chapter Chapter 5|16 pages

Zeros

chapter Chapter 6|24 pages

Multipliers: Basic Results

chapter Chapter 7|53 pages

Multipliers: Further Results

chapter Chapter 8|18 pages

Composition

chapter Chapter 9|14 pages

Univalent Functions

chapter Chapter 10|19 pages

A Characterization of Cauchy Transforms