Quasiconformality of Harmonic Mappings Between Jordan Domains

Miodrag Mateljević

Suppose that $h$ is a harmonic mapping of the unit disc onto a $C^{1,\alpha}$ domain $D$. Then $h$ is q.c. if and only if it is bi-Lipschitz. In particular, we consider sufficient and necessary conditions in terms of boundary function that $h$ is q.c. We give a review of recent related results including the case when domain is the upper half plane. We also consider harmonic mapping with respect to $\rho$ metric on codomain.