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

Let $h$ and $g$ be two analytic functions in the unit disc $\mathrm{\Delta }$ that $g(0)=1$. Also let $\beta$ be a complex number with $\mathrm{R}\mathrm{e}\{\beta \}>-1\mathrm{/}2$. A function $f$ is said to be log–harmonic mapping if it has the following representation \begin{equation*} f(z)=z |z|^{2\beta} h(z)\overline{g(z)}\quad (z\in \Delta). \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}>\alpha\quad(z\in \Delta). \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.