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

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

2009-10-26

arxiv.org/abs/0910.4950v1

MATH. NACHR., 285, Issue 2-3, (2012) 283--294

Mathematics

Complex Variables

Scientific paper

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.

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