Some estimates for harmonic mappings with given boundary function

Let w(z) = P[f](z) = h(z) + g(z) over bar be a bounded harmonic mapping defined in the unit disk D with the boundary function f, where h(z) = Sigma(infinity)(n=1) a(n)z(n) and g(z) = Sigma(infinity)(n=1) b(n)z(n) are analytic in D. In this paper, using the boundary condition of f, we improve the estimate for vertical bar a(n)vertical bar + vertical bar b(n)vertical bar. In addition if f is a sense-preserving homeomorphism of the unit circle onto the boundary of a bounded convex domain ohm, then we obtain the sufficient and necessary conditions on f such that w(z) = P[f](z) is a bi-Lipschitz harmonic mapping which in particular is a quasiconformal mapping. (C) 2013 Elsevier Inc. All rights reserved.