Refinement of the Classical Bohr Inequality

TL;DR

This paper refines the classical Bohr inequality, providing improved bounds and related versions, advancing the understanding of power series behavior within the unit disk.

Contribution

The paper introduces a refined version of Bohr's inequality and presents several improved related results, enhancing previous bounds and understanding.

Findings

New refined version of Bohr's inequality

Improved bounds for power series in the unit disk

Extended results related to classical inequalities

Abstract

The classical inequality of Bohr asserts that if a power series converges in the unit disk and its sum has modulus less than or equal to $1$, then the sum of absolute values of its terms is less than or equal to $1$ for the subdisk $|z|<1/3$ and $1/3$ is the best possible constant. Recently, there has been a number of investigations on this topic. In this article, we present a refined version of Bohr's inequality along with few other related improved versions of previously known results.