Notes on the starlike log--harmonic mappings of order alpha

Abstract

Let $h$ and $g$ be two analytic functions in the unit disc $\Delta$ that $g(0)=1$. Also let $\beta$ be a complex number with $\mathrm{Re}\{\beta \}>-1/2$. A function $f$ is said to be log--harmonic mapping if it has the following representation \begin{equation*} f(z)=z |z|^{2β} h(z)\overline{g(z)}\quad (z\in Δ). \end{equation*} A log--harmonic mapping $f$ is said to be starlike log--harmonic mapping of order $\alpha$, where $0\le \alpha <1$, if \begin{equation*} {\rm Re}\left\{\frac{zf_z -\overline{z}f_{\overline{z}}}{f}\right\}>α\quad(z\in Δ). \end{equation*} In this paper, by use of the subordination principle, we study some geometric properties of the starlike log--harmonic mappings of order $\alpha$. Also, we estimate the Jacobian of log--harmonic mappings.

8 pages, 4 figures