Some Refinements and Generalizations of Bohr’s Inequality

Abstract

In this article, we delve into the classic Bohr inequality for complex numbers, a fundamentalresult in complex analysis with broad mathematical applications. We offer refinements and generalizations of Bohr’s inequality, expanding on the established inequalities of N. G. de Bruijn and Radon, as well as leveraging the class of functions defined by the Daykin–Eliezer–Carlitz inequality. Our novel contribution lies in demonstrating that Bohr’s and Bergström’s inequalities can be derived from one another, revealing a deeper interconnectedness between these results. Furthermore, we present several new generalizations of Bohr’s inequality, along with other notable inequalities from theliterature, and discuss their various implications. By providing more comprehensive and verifiableconditions, our work extends previous research and enhances the understanding and applicability ofBohr’s inequality in mathematical studies.