A Schwarz–Pick inequality for harmonic quasiconformal mappings and its applications

The main result of this paper is the sharp generalized Schwarz–Pick inequality for euclidean harmonic quasiconformal mappings with convex ranges, which generalizes a result given by Mateljević. As its applications, we obtain the property of quasi-isometry with respect to the Poincaré distance and an analogue of the Koebe theorem for this class of mappings.