Ruscheweyh's univalence criterion and quasiconformal extensions

TL;DR

This paper refines Ruscheweyh's univalence criterion to include quasiconformal extensions using Becker’s method, providing solutions to open problems in the theory of univalent functions.

Contribution

It introduces a refined quasiconformal extension criterion based on Ruscheweyh's work, advancing the understanding of univalent functions and their extensions.

Findings

Established a new quasiconformal extension criterion

Solved an open problem posed by Ruscheweyh

Enhanced the theoretical framework for univalent functions

Abstract

Ruscheweyh extended the work of Becker and Ahlfors on sufficient conditions for a normalized analytic function on the unit disk to be univalent there. In this paper we refine the result to a quasiconformal extension criterion with the help of Becker's method. As an application, a positive answer is given to an open problem proposed by Ruscheweyh.