Criteria for univalence and quasiconformal extension of harmonic mappings in terms of the Schwarzian DERIVATIVE

TL;DR

This paper establishes criteria based on the Schwarzian derivative norm that ensure harmonic mappings in the unit disk are globally univalent and extendable as quasiconformal mappings.

Contribution

It provides new conditions involving the Schwarzian derivative norm that guarantee univalence and quasiconformal extension of harmonic mappings.

Findings

Small Schwarzian norm implies global univalence.

Harmonic mappings with small Schwarzian norm extend quasiconformally.

Criteria improve understanding of harmonic mapping extensions.

Abstract

We prove that if the Schwarzian norm of a given complex-valued locally univalent harmonic mapping $f$ in the unit disk is small enough, then $f$ is, indeed, globally univalent and can be extended to a quasiconformal mapping in the extended complex plane.