Some results on a starlike log-harmonic mapping of order alpha

Let H(D)H(D) be the linear space of all analytic functions defined on the open unit disc D=z∈C:|z|<1D=z∈C:|z|<1. A sense preserving log-harmonic mapping is the solution of the non-linear elliptic partial differential equation fz=w(z)fz(fz/f)fz=w(z)fz(fz/f) where w(z)∈H(D)w(z)∈H(D) is the second dilatation of ff such that |w(z)|<1|w(z)|<1 for all z∈Dz∈D.A sense preserving log-harmonic mapping is a solution of the non-linear elliptic partial differential equation equation(0.1)fz¯f¯=w(z).fzf where w(z)w(z) the second dilatation of ff and w(z)∈H(D)w(z)∈H(D), |w(z)|<1|w(z)|<1 for every z∈Dz∈D. It has been shown that if ff is a non-vanishing log-harmonic mapping, then ff can be expressed as equation(0.2)f(z)=h(z)g(z)¯ where h(z)h(z) and g(z)g(z) are analytic in DD with the normalization h(0)≠0h(0)≠0, g(0)=1g(0)=1. On the other hand if ff vanishes at z=0z=0, but it is not identically zero, then ff admits the following representation equation(0.3)f(z)=z.z2βh(z)g(z)¯ where Reβ>−12, h(z)h(z) and g(z)g(z) are analytic in the open disc DD with the normalization h(0)≠0h(0)≠0, g(0)=1g(0)=1  (Abdulhadi and Bshouty, 1988) [2], (Abdulhadi and Hengartner, 1996) [3].In the present paper, we will give the extent of the idea, which was introduced by Abdulhadi and Bshouty (1988) [2]. One of the interesting applications of this extent idea is an investigation of the subclass of log-harmonic mappings for starlike log-harmonic mappings of order alpha.