  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.
