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 $\mathrm{\Omega }\in C^{1,\mu }$ ($0<\mu \le 1$) we give an explicit Lipschitz constant depending on the structure of $\mathrm{\Omega }$ 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.