A Unified Study of Bohr's Inequality for analytic and harmonic mappings on the Unit Disk

TL;DR

This paper extends Bohr's inequality to analytic and harmonic mappings on the unit disk, introducing new bounds and generalizations using area measures and special functions, with all results proven to be sharp.

Contribution

It provides a unified framework for Bohr's inequality for both analytic and harmonic mappings, incorporating new functions and sharp bounds.

Findings

Improved Bohr inequalities for analytic selfmaps using area measures.

Generalization of Bohr inequalities for harmonic mappings with new sequences.

Application of Hurwitz Lerch Zeta function to derive consequences.

Abstract

We investigate improved forms of the Bohr inequality, using the quantity $S_r/\pi$, for analytic selfmaps in class $\mathcal{B}$ of $\mathbb{D}$, where $S_r$ is the area measure of $\mathbb{D}_r$. We then generalize the inequality for harmonic mappings ($\mathcal{P}^0_{\mathcal{H}}(M)$ and $\mathcal{W}^0_{\mathcal{H}}(\alpha)$ of the form $f = h + \overline{g}$) by introducing a sequence $\{\varphi_n(r)\}_{n=0}^\infty$ of differentiable, increasing functions on $[0, 1)$. The Hurwitz Lerch Zeta function is utilized for some consequences, and all results are shown to be sharp.