On boundary correspondence of q.c. harmonic mappings between smooth Jordan domains

TL;DR

This paper provides explicit Lipschitz constants and characterizations for quasiconformal harmonic mappings between the unit disk and Jordan domains with smooth boundaries, enhancing understanding of boundary correspondence in complex analysis.

Contribution

It introduces a quantitative inequality for quasiconformal harmonic mappings and characterizes such mappings using boundary functions and Hilbert transforms.

Findings

Explicit Lipschitz constants depending on domain structure and quasiconformality

Characterization of mappings via boundary functions and Hilbert transforms

Sharp quasiconformal constants in terms of boundary data

Abstract

A quantitative version of an inequality obtained in \cite[Theorem~2.1]{mathz} is given. More precisely, for normalized $K$ quasiconformal harmonic mappings of the unit disk onto a Jordan domain $\Omega\in C^{1,\mu} $ ($0<\mu\le 1$) we give an explicit Lipschitz constant depending on the structure of $\Omega$ and on $K$. In addition we give a characterization of q.c. harmonic mappings of the unit disk onto an arbitrary Jordan domain with $C^{2,\alpha}$ boundary in terms of boundary function using the Hilbert transformations. Moreover it is given a sharp explicit quasiconformal constant in terms of the boundary function.