Improved Bohr inequalities for certain classes of harmonic mappings

TL;DR

This paper establishes improved Bohr inequalities for harmonic functions in the unit disk, identifying sharp radii where these inequalities hold, with applications to specific classes of harmonic and univalent functions.

Contribution

The paper introduces two new types of improved Bohr inequalities for harmonic mappings, including univalent and stable harmonic functions, with precise sharp radius values.

Findings

Derived sharp Bohr radii for harmonic functions.

Extended Bohr inequalities to classes of univalent and stable harmonic mappings.

Provided corollaries with exact sharp bounds for the Bohr radius.

Abstract

The Bohr radius for the class of harmonic functions of the form $ f(z)=h+\overline{g} $ in the unit disk $ \mathbb{D}:=\{z\in\mathbb{C} : |z|<1\} $, where $ h(z)=\sum_{n=0}^{\infty}a_nz^n $ and $ g(z)=\sum_{n=1}^{\infty}b_nz^n $ is to find the largest radius $ r_f $, $ 0<r_f<1 $ such that \begin{equation*} \sum_{n=1}^{\infty}\left(|a_n|+|b_n|\right)|z|^n\leq d(f(0),\partial f(\mathbb{D})) \end{equation*} holds for $ |z|=r\leq r_f $, where $ d(f(0),\partial f(\mathbb{D})) $ is the Euclidean distance between $ f(0) $ and the boundary of $ f(\mathbb{D}) $. In this paper, we prove two-type of improved versions of the Bohr inequalities, one for a certain class of harmonic and univalent functions and the other for stable harmonic mappings. It is observed in the paper that to obtain sharp Bohr inequalities it is enough to consider any non-negative real coefficients of the quantity $ S_r/\pi $. As a consequence of the main result, we prove corollaries showing the precise value of the sharp Bohr radius.