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 $f(z)={\sum }_{n=0}^{\infty }{a}_n{z}^n$ 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 $f:\mathbb{D}\to \mathbb{C}$ 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}{\partialθ}{\rm Arg}(f(e^{iθ}))={\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 ${\mathcal{S}\mathcal{T}}_{LH}^0$, a subclass of ${\mathcal{S}\mathcal{T}}_{LH}$, consisting of functions $f\in {\mathcal{S}\mathcal{T}}_{LH}$ which map the unit disk $\mathbb{D}$ onto a starlike domain (with respect to the origin).

Cited by in corpus (1)

• Bohr-Rogosinski and improved Bohr type inequalities for certain fully starlike harmonic mappings