On boundary correspondences under quasiconformal harmonic mappings between smooth Jordan domains

Abstract A quantitative version of an inequality obtained in [ 8 , Theorem 2.1] is given. More precisely, for normalized K quasiconformal (q.c.) harmonic mappings of the unit disk onto a Jordan domain Ω ∈ C1, μ (0 < μ ≤ 1), we give an explicit Lipschitz constant depending on the structure of Ω and on K. In addition, we give a characterization of q.c. harmonic mappings of the unit disk onto an arbitrary Jordan domain with C2, α boundary in terms of the boundary function using the Hilbert transform. Moreover, a sharp explicit quasiconformal constant is given in terms of the boundary function.