Sharp Bohr Radius Constants for Certain Analytic Functions

Abstract

The Bohr radius for a class ${\mathcal {G}}$ consisting of analytic functions $f(z)=\sum _{n=0}^{\infty }a_nz^n$ in unit disc ${\mathbb {D}}=\{z\in {\mathbb {C}}:|z|<1\}$ is the largest $r^*$ such that every function f in the class ${\mathcal {G}}$ satisfies the inequality

$$\begin{aligned} d\left( \sum _{n=0}^{\infty }|a_nz^n|, |f(0)|\right) = \sum _{n=1}^{\infty }|a_nz^n|\le d(f(0), \partial f({\mathbb {D}})) \end{aligned}$$

for all $|z|=r \le r^*$, 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 $zf'(z)/f(z) \prec h(z)$ and $f(z)+\beta z f'(z)+\gamma z^2 f''(z)\prec h(z)$, where h is the Janowski function. Analogous results are obtained for the classes of $\alpha$-convex functions and typically real functions, respectively. All obtained results are sharp.