Improved Bohr inequalities for certain class of harmonic univalent functions

TL;DR

This paper establishes improved and sharp Bohr-type inequalities for a specific class of harmonic univalent functions, extending classical results and providing new bounds within the unit disk.

Contribution

It derives the first sharp Bohr-Rogosinski, improved, refined, and Bohr-type inequalities for the harmonic univalent class \( \mathcal{P}_{\mathcal{H}}^{0}(M) \), advancing the understanding of harmonic function bounds.

Findings

Established sharp Bohr-Rogosinski inequality for the class.

Derived improved Bohr inequality with tighter bounds.

Proved refined and Bohr-type inequalities for harmonic univalent functions.

Abstract

Let $ \mathcal{H} $ be the class of complex-valued harmonic mappings $ f=h+\bar{g}$ defined in the unit disk $ \mathbb{D} : =\{z\in\mathbb{C} : |z|<1\} $, where $ h $ and $ g $ are analytic functions in $ \mathbb{D} $ with the normalization $ h(0)=0=h^{\prime}(0)-1 $ and $ g(0)=0 $. Let $ \mathcal{H}_{0}=\{f=h+\bar{g}\in\mathcal{H} : g^{\prime}(0)=0\}. $ Ghosh and Vasudevrao \cite{Ghosh-Vasudevarao-BAMS-2020} have studied the following interesting harmonic univalent class $ \mathcal{P}^{0}_{\mathcal{H}}(M) $ which is defined by $$\mathcal{P}^{0}_{\mathcal{H}}(M) :=\{f=h+\overline{g} \in \mathcal{H}_{0}: \mathrm{Re} (zh^{\prime\prime}(z))> -M+|zg^{\prime\prime}(z)|,\; z \in \mathbb{D}\; \mbox{and}\;\; M>0\}. $$ In this paper, we obtain the sharp Bohr-Rogosinski inequality, improved Bohr inequality, refined Bohr inequality and Bohr-type inequality for the class $ \mathcal{P}_{\mathcal{H}}^{0}(M) $.