Bohr radius for certain classes of starlike and convex univalent functions

Abstract

We say that a class $\mathcal{F}$ consisting of analytic functions $f(z)={\sum }_{n=0}^{\infty }{a}_n{z}^n$ in the unit disk $\mathbb{D}:=\{z\in \mathbb{C}:|z|<1\}$ satisfies a Bohr phenomenon if there exists $r_f\in (0,1)$ such that $$\sum _{n=1}^{\infty }|a_n{z}^n|\le d(f(0),\partial f(\mathbb{D}))$$ for every function $f\in \mathcal{F}$ and $|z|=r\le r_f$, where $d$ is the Euclidean distance. The largest radius $r_f$ is the Bohr radius for the class $\mathcal{F}$. In this paper, we establish the Bohr phenomenon for the classes consisting of Ma-Minda type starlike functions and Ma-Minda type convex functions as well as for the class of starlike functions with respect to a boundary point.

16