Ahlfors's quasiconformal extension condition and $\Phi$-likeness

TL;DR

This paper explores conditions under which analytic functions on the unit disk are univalent and admit quasiconformal extensions, generalizing classical criteria including Ahlfors's condition, through the concept of $\

Contribution

It introduces new necessary and sufficient conditions for quasiconformal extendability based on $\\Phi$-like functions, extending classical univalence and extension criteria.

Findings

Derived generalized criteria for quasiconformal extension

Connected $\\Phi$-likeness with Ahlfors's condition

Provided new necessary and sufficient conditions

Abstract

The notion of $\Phi$-like functions is known to be a necessary and sufficient condition for univalence. By applying the idea, we derive several necessary conditions and sufficient conditions for that an analytic function defined on the unit disk is not only univalent but also has a quasiconformal extension to the Riemann sphere, as generalizations of well-known univalence and quasiconformal extension criteria, in particular, Ahlfors's quasiconformal extension condition.