Loewner chain and quasiconformal extension of some classes of univalent functions

TL;DR

This paper develops methods to extend certain classes of univalent functions to quasiconformal maps on the entire complex plane, using Loewner chains and established theorems.

Contribution

It introduces new quasiconformal extension techniques for specific classes of univalent functions via Loewner chains and classical results.

Findings

Constructed quasiconformal extensions for functions on the unit disk and exterior.

Applied Loewner chains to derive extensions.

Utilized known theorems to obtain additional extensions.

Abstract

In this article, we obtain quasiconformal extensions of some classes of conformal maps defined either on the unit disc or on the exterior of it onto the extended complex plane. Some of these extensions have been obtained by constructing suitable Loewner chains and others have been obtained by applying a well-known result.