Quasiconformal and Lipschitz harmonic mappings of the unit disk onto bounded convex domains

Abstract

We study the Lipschitz property of a harmonic injective and sense-preserving mapping F of the unit disk D onto a bounded convex domain in the complex plane C. In particular we show that F is bi-Lipschitz iff F is quasiconformal and Lipschitz. To this end we establish some auxiliary properties of harmonic mappings dealing with the boundary radial limiting values of the formal derivatives @F and ¯ @F.