Abstract

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 ∈ 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;   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;  where Reβ  >-½, 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.

Key words: Starlike, subordination, distortion.