  We prove the following theorem: every quasiconformal harmonic mapping between
two plane domains with $C^{1,\alpha}$ ($\alpha<1$), respectively $C^{1,1}$
compact boundary is bi-Lipschitz. The distance function with respect to the
boundary of the image domain is used. This in turn extends a similar result of
the author in \cite{kalajan} for Jordan domains, where stronger boundary
conditions for the image domain were needed.
