On the radius constants for classes of analytic functions

TL;DR

This paper determines sharp radius constants for various classes of analytic functions on the unit disk, including starlikeness, parabolic starlikeness, Bernoulli lemniscate starlikeness, and uniform convexity, and discusses conjectures on non-sharp constants.

Contribution

It provides the first sharp radius constants for multiple classes of analytic functions and proposes conjectures for non-sharp constants.

Findings

Sharp radius constants for several classes of analytic functions.

Conjectures on non-sharp radius constants.

Results applicable to geometric function theory.

Abstract

Radius constants for several classes of analytic functions on the unit disk are obtained. These include the radius of starlikeness of a positive order, radius of parabolic starlikeness, radius of Bernoulli lemniscate starlikeness, and radius of uniform convexity. In the main, the radius constants obtained are sharp. Conjectures on the non-sharp constants are given.