Sharp Bohr Radius Constants For Certain Analytic Functions

Abstract

The Bohr radius for a class $\mathcal{G}$ consisting of analytic functions $f(z)={\sum }_{n=0}^{\infty }{a}_n{z}^n$ in unit disc $\mathbb{D}=\{z\in \mathbb{C}:|z|<1\}$ is the largest $r^{\ast }$ such that every function $f$ in the class $\mathcal{G}$ satisfies the inequality \begin{equation*} d\left(\sum_{n=0}^{\infty}|a_nz^n|, |f(0)|\right) = \sum_{n=1}^{\infty}|a_nz^n|\leq d(f(0), \partial f(\mathbb{D})) \end{equation*} for all $|z|=r\le r^{\ast }$, where $d$ is the Euclidean distance. In this paper, our aim is to determine the Bohr radius for the classes of analytic functions $f$ satisfying differential subordination relations $z{f}^{\prime }(z)/f(z)\prec h(z)$ and $f(z)+{\hat{I}}^{2}z{f}^{\prime }(z)+{\hat{I}}^3{z}^2{f}^{\prime \prime }(z)\prec h(z)$, where $h$ is the Janowski function. Analogous results are obtained for the classes of $\hat{I}\pm$-convex functions and typically real functions, respectively. All obtained results are sharp.