Harmonic mappings and distance function

Abstract: 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.