Some estimates for harmonic mappings with given boundary function

Let w ( z ) = P [ f ] ( z ) = h ( z ) + g ( z ) ¯ be a bounded harmonic mapping defined in the unit disk D with the boundary function f, where h ( z ) = ∑ n = 1 ∞ a n z n and g ( z ) = ∑ n = 1 ∞ b n z n are analytic in D . In this paper, using the boundary condition of f, we improve the estimate for | a n | + | b n | . In addition if f is a sense-preserving homeomorphism of the unit circle onto the boundary of a bounded convex domain Ω, 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.