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

TL;DR

This paper investigates the geometric properties and Jacobian estimates of starlike log-harmonic mappings of order alpha in the unit disk, using subordination principles to extend understanding of their structure.

Contribution

It introduces new results on the geometric behavior and Jacobian bounds of starlike log-harmonic mappings of order alpha, expanding the theoretical framework in this area.

Findings

Derived conditions for starlikeness of log-harmonic mappings

Established bounds on the Jacobian of these mappings

Applied subordination principles to analyze geometric properties

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 ${\rm Re}\{\beta\}>-1/2$. A function $f$ is said to be log--harmonic mapping if it has the following representation \begin{equation*} f(z)=z |z|^{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 $0\leq \alpha<1$, 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.