Improved Bohr radius for the class of starlike log-harmonic mappings

Abstract

Let $ \mathcal{H}(\mathbb{D}) $ be the linear space of analytic functions on the unit disk $ \mathbb{D}=\{z\in\mathbb{C}: |z|<1\} $ and let $ \mathcal{B}=\{w\in \mathcal{H}(\mathbb{D}: |w(z)|<1)\} $. The classical Bohr's inequality states that if a power series $ f(z)=\sum_{n=0}^{\infty}a_nz^n $ converges in $ \mathbb{D} $ and $ |f(z)|<1 $ for $ z\in\mathbb{D} $, then \begin{equation*} \sum_{n=0}^{\infty}|a_n|r^n\leq 1\;\;\mbox{for}\;\; r\leq \frac{1}{3} \end{equation*} and the constant $ 1/3 $ is the best possible. The constant $ 1/3 $ is known as Bohr radius. A function $ f : \mathbb{D}\rightarrow\mathbb{C} $ is said to be log-harmonic if there is a $ w\in\mathcal{B} $ such that $ f $ is a non-constant solution of the non-linear elliptic partial differential equation \begin{equation*} \bar{f}_{\bar{z}}(z)/\bar{f}(z)=w(z)f_{z}(z)/f(z). \end{equation*} The class of log-harmonic mappings is denoted by $ \mathcal{S}_{LH} $. The set of all starlike log-harmonic mapping is defined by \begin{equation*} \mathcal{ST}_{LH}=\bigg\{f\in\mathcal{S}_{LH}:\frac{\partial}{\partial\theta}{\rm Arg}(f(e^{i\theta}))={\rm Re}\left(\frac{zf_{z}-\bar{z}f_{\bar{z}}}{f}\right)>0\;\; \mbox{in}\;\; \mathbb{D}\bigg\}. \end{equation*} In this paper, we study several improved Bohr radius for the class $ \mathcal{ST}^{0}_{LH} $, a subclass of $ \mathcal{ST}_{LH} $, consisting of functions $ f\in\mathcal{ST}_{LH} $ which map the unit disk $ \mathbb{D} $ onto a starlike domain (with respect to the origin).