A generalization of the Bohr inequality for bounded analytic functions on simply connected domains and its applications

TL;DR

This paper extends Bohr's inequality to bounded analytic functions on specific simply connected domains and explores its applications, including calculating Bohr-type radii for certain integral operators.

Contribution

It introduces a generalized Bohr inequality for functions on domains parameterized by gamma and applies it to determine Bohr radii for integral operators.

Findings

Derived a generalized Bohr inequality for the domain mma.

Calculated Bohr-type radii for specific integral operators.

Extended classical results to new domain classes.

Abstract

Bohr's classical theorem and its generalizations are now active areas of research and have been the source of investigations in numerous function spaces. In this article, we study a generalized Bohr's inequality for the class of bounded analytic functions defined on the simply connected domain $$ \Omega_{\gamma}:=\bigg\{z\in \mathbb{C}: \bigg|z+\frac{\gamma}{1-\gamma}\bigg|<\frac{1}{1-\gamma}\bigg\}, \,\ \text{for } 0\leq \gamma<1. $$ Part of its applications, we calculate the Bohr-type radii for some known integral operators.