Some results about log-harmonic mappings

Let A(α,β) be a subclass of certain analytic functions and H (D) is to be a linear space of all analytic functions defined on the open unit disc D = {z| |z| < 1}. A sense-preserving log-harmonic function is the solution of the non-linear elliptic partial differential equation;where w(z) is analytic, satisfies the condition |w(z)| < 1 for every z ε D and is called the second dilatation of f. It has been shown that if f is a non-vanishing log-harmonic mapping, then f can be represented by; f (z) = h(z)g(z), where h(z) and g(z) are analytic in D with h(0) = 0, g(0) = 1([1]). If f vanishes at z = 0, but it is not identically zero, then f admits the representation; f (z) = z |z|^2β h(z)g(z), where Reβ >-1/2, h(z) and g(z) are analytic in D with g(0) = 1 and h(0) = 0. The class of sense-preserving log-harmonic mappins is denoted by SLH. The aim of this paper is to give some distortion theorems of these classes.