On the Bohr inequality

TL;DR

This paper surveys recent developments and generalizations of the Bohr inequality, exploring its applications to various classes of functions, domains, and metrics, including harmonic, starlike, and higher-dimensional cases.

Contribution

It provides a comprehensive overview of recent advances in Bohr inequality research, extending classical results to new function classes and geometric settings.

Findings

Determined the Bohr radius for various domains and functions.

Extended Bohr inequality concepts to harmonic and starlike logharmonic mappings.

Analyzed the Bohr phenomenon using different metrics and in higher dimensions.

Abstract

The Bohr inequality, first introduced by Harald Bohr in 1914, deals with finding the largest radius $r$, $0<r<1$, such that $\sum_{n=0}^\infty |a_n|r^n \leq 1$ holds whenever $|\sum_{n=0}^\infty a_nz^n|\leq 1$ in the unit disk $\mathbb{D}$ of the complex plane. The exact value of this largest radius, known as the \emph{Bohr radius}, has been established to be $1/3.$ This paper surveys recent advances and generalizations on the Bohr inequality. It discusses the Bohr radius for certain power series in $\mathbb{D},$ as well as for analytic functions from $\mathbb{D}$ into particular domains. These domains include the punctured unit disk, the exterior of the closed unit disk, and concave wedge-domains. The analogous Bohr radius is also studied for harmonic and starlike logharmonic mappings in $\mathbb{D}.$ The Bohr phenomenon which is described in terms of the Euclidean distance is further investigated using the spherical chordal metric and the hyperbolic metric. The exposition concludes with a discussion on the $n$-dimensional Bohr radius.