Note on Weighted Bohr's Inequality

TL;DR

This paper introduces new generalizations of Bohr's inequality applicable to bounded analytic functions, sense-preserving quasiconformal harmonic mappings, and subordinate functions, expanding its scope in complex analysis.

Contribution

It provides novel generalizations of Bohr's inequality for specific classes of harmonic and subordinate functions, broadening its theoretical framework.

Findings

New bounds for bounded analytic functions.

Generalized Bohr's inequality for quasiconformal harmonic mappings.

Extensions to subordinate function classes.

Abstract

In this paper, first we give a new generalization of the Bohr's inequality for the class of bounded analytic functions $\mathcal{B'}$ and for the class of sense-preserving $K$-quasiconformal harmonic mappings of the form $f=h+\overline{g},$ where $h \in \mathcal{B'}.$ Finally we give a new generalization of the Bohr's inequality for the class of analytic functions subordinate to univalent functions and for the class of sense-preserving $K$-quasiconformal harmonic mappings of the form $f=h+\overline{g},$ where $h$ is subordinated to some analytic function.