On The Properties Of Some Univalent Harmonic Close-To-Convex Mappings

In 2002, D. Bshouty and A. Lyzzaik presented a question in the paper " On problems and conjectures in planar harmonic mappings". It says that if What is the maximum valency of of if g’=z2h’? This problem and some interesting related results they obtained attract many mathematicians.In this paper, we discuss the univalent property, Landau Theorem and Bloch constant for the harmonic mapping f(z)=h(z)+g(z) with its analytic part h(z)First, the analytic representing formula of the analytic part h(z) is given for some stable close-to-convex harmonic mappings. And then the stable close-to-convex criterion of f(z) is proved when the dilatation function w(z)is a linear function, which gernerlizes the results made by Bshouty and Nagpal. We also obtain the coefficient estimates for these mappings. Next, we estimate the stable close-to-convex radius of f=h+g when the dilation function is w(z)=z2, as well as w(z)=zn, If the dilation just satisfies |w(z)|< 1, the univalent radius is also estimated. At last, we consider the Landau Theorem for harmonic functions with some other conditions, our result improves the one made by Huaihui Chen and P. M. Gauthier.