Quasiconformal and harmonic mappings between Jordan domains

Abstract

Let Ω and Ω1 be Jordan domains, let μ ∈ (0, 1], and let \(f: \Omega \mapsto \Omega_1\) be a harmonic homeomorphism. The object of the paper is to prove the following results: (a) If f is q.c. and ∂Ω, ∂Ω1 ∈ C 1,μ , then f is Lipschitz; (b) if f is q.c., ∂Ω, ∂Ω1 ∈ C 1,μ and Ω1 is convex, then f is bi-Lipschitz; and (c) if Ω is the unit disk, Ω1 is convex, and ∂Ω1 ∈ C 1,μ , then f is quasiconformal if and only if its boundary function is bi-Lipschitz and the Hilbert transform of its derivative is in L ∞. These extend the results of Pavlovic (Ann. Acad. Sci. Fenn. 27:365–372, 2002).