Bohr radius for certain classes of starlike and convex univalent functions

TL;DR

This paper investigates the Bohr phenomenon for specific classes of univalent functions, determining their Bohr radii and extending the concept to Ma-Minda starlike and convex functions, as well as boundary point starlike functions.

Contribution

It establishes the Bohr phenomenon and calculates the Bohr radii for Ma-Minda type starlike and convex functions, and for boundary point starlike functions.

Findings

Bohr radii are determined for Ma-Minda starlike functions.

Bohr phenomenon is verified for Ma-Minda convex functions.

Results extend the Bohr phenomenon to boundary point starlike 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 $$ \sum_{n=1}^{\infty} |a_{n}z^{n}|\leq d(f(0),\partial f(\mathbb{D})) $$ for every function $f \in \mathcal{F}$ and $|z|=r\leq r_{f}$, 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.