On quasiconformal extensions of harmonic mappings associated with pre-Schwarzian derivative

TL;DR

This paper extends classical univalence and quasiconformal extension criteria from analytic functions to harmonic mappings in the unit disk, providing a general extension for harmonic Teichmüller mappings with sharp dilatation estimates.

Contribution

It introduces new quasiconformal extension criteria for harmonic mappings and generalizes extensions for harmonic Teichmüller mappings with optimal dilatation bounds.

Findings

Extended Ahlfors's univalent criteria to harmonic mappings

Provided a general quasiconformal extension for harmonic Teichmüller mappings

Achieved asymptotically sharp maximal dilatation estimates

Abstract

In this paper, we extend Ahlfors's univalent criteria and Ahlfors's quasiconformal extension for analytic functions to harmonic mappings defined in the unit disk. Moreover, we give a general quasiconformal extension of harmonic Teichm\"{u}ller mappings, whose maximal dilatation estimate is asymptotically sharp.