Improved Bohr's inequality for locally univalent harmonic mappings

TL;DR

This paper presents sharper versions of Bohr's inequality for certain harmonic mappings, incorporating area-related terms and achieving sharp bounds, with specific improvements for special classes of mappings.

Contribution

The authors introduce improved Bohr's inequalities for harmonic mappings, including area terms, and establish sharp bounds with enhancements for particular classes.

Findings

Derived sharper Bohr's inequalities for harmonic mappings.

Incorporated area of the image into the inequalities.

Results are proven to be sharp for the considered classes.

Abstract

We prove several improved versions of Bohr's inequality for the harmonic mappings of the form $f=h+\overline{g}$, where $h$ is bounded by 1 and $|g'(z)|\le|h'(z)|$. The improvements are obtained along the lines of an earlier work of Kayumov and Ponnusamy, i.e. \cite{KayPon2}, for example a term related to the area of the image of the disk $D(0,r)$ under the mapping $f$ is considered. Our results are sharp. In addition, further improvements of the main results for certain special classes of harmonic mappings are provided.