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

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Kalaj, David
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Abstract
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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.
Keywords
Mathematics - Complex Variables, Mathematics - Analysis of PDEs
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