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

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 $Ω\in C^{1,μ}$ ($0<μ\le 1$) we give an explicit Lipschitz constant depending on the structure of $Ω$ and on $K$. In addition we give a characterization of q.c. harmonic mappings of the unit disk onto an arbitrary Jordan domain with $C^{2,α}$ 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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