Bohr inequalities for unimodular bounded functions on simply connected domains

TL;DR

This paper investigates Bohr inequalities for analytic and harmonic functions on simply connected domains, establishing sharp bounds and improved Bohr radii beyond the classical unit disk case.

Contribution

It extends Bohr inequalities to more general simply connected domains and provides new sharp bounds for both analytic and harmonic mappings.

Findings

Established sharp Bohr radius for analytic functions on simply connected domains.

Derived improved Bohr radii for harmonic mappings.

Extended classical Bohr inequality beyond the unit disk.

Abstract

Let $ \mathcal{H}(\mathbb{D}) $ be the class of analytic functions in the unit disk $ \mathbb{D} : =\{z\in\mathbb{C} : |z|<1\} $. The classical Bohr's inequality states that if a power series $ f(z)=\sum_{n=0}^{\infty}a_nz^n $ converges in $ \mathbb{D} $ and $ |f(z)|<1 $ for $ z\in\mathbb{D} $, then \begin{equation*} \sum_{n=0}^{\infty}|a_n|r^n\leq 1\;\;\mbox{for}\;\; r\leq \frac{1}{3} \end{equation*} and the constant $ 1/3 $ cannot be improved. The constant $ 1/3 $ is known as Bohr radius. In this paper, we study Bohr phenomenon for analytic as well as harmonic mappings on simply connected domains. We prove several sharp results on improved Bohr radius for analytic functions as well as for harmonic mappings on simply connected domains.