Univalent Functions

Derek K. Thomas, Nikola Tuneski, Allu Vasudevarao

• Publisher: De Gruyter
• Year: 2018
• ISBN:
  9783110560121
• Edition: 1
• Topic: Mathematics
• Subtopic: Complex Analysis
• Index: Mathematics

The study of univalent functions dates back to the early years of the 20th century, and is one of the most popular research areas in complex analysis. This book is directed at introducing and bringing up to date current research in the area of univalent functions, with an emphasis on the important subclasses, thus providing an accessible resource suitable for both beginning and experienced researchers.

Contents
Univalent Functions – the Elementary Theory
Definitions of Major Subclasses
Fundamental Lemmas
Starlike and Convex Functions
Starlike and Convex Functions of Order ?
Strongly Starlike and Convex Functions
Alpha-Convex Functions
Gamma-Starlike Functions
Close-to-Convex Functions
Bazilevi? Functions
B1(?) Bazilevi? Functions
The Class U(?)
Convolutions
Meromorphic Univalent Functions
Loewner Theory
Other Topics
Open Problems

1Univalent Functions – the Elementary Theory

1.1Definitions and Basic Properties

A domain is an open connected subset of the complex plane ℂ.

A complex-valued function f of a complex variable is called univalent in a domain D if it does not take the same value twice, so that for z1, z2 ∈ D,

$f\left(z_1\right)\ne f\left(z_2\right)\text{for}{z}_1\ne z_2.\left(\mathrm{1.1.1}\right)$

Let D be a simply connected domain, which is a proper subset of the complex plane and z be a given point in D. Then the Riemann mapping theorem guarantees that there is a unique function f which maps D conformally onto the unit disk, with the properties f(z) = 0 and f'(z) > 0.

A necessary condition for a function f analytic in D to be univalent in D is that its derivative does not vanish on D, i.e.,

$f\text{is}\text{univalent}\text{in}D\Rightarrow f^{\prime }\left(z\right)\ne 0\text{for}\text{all}z\in D.$

A proof of this can be obtained using Rouche’s the...