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 DD with the boundary function f  , where h(z)=∑n=1∞anzn and g(z)=∑n=1∞bnzn are analytic in DD. In this paper, using the boundary condition of f  , we improve the estimate for |an|+|bn||an|+|bn|. 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)w(z)=P[f](z) is a bi-Lipschitz harmonic mapping which in particular is a quasiconformal mapping.