Bohr phenomenon for certain classes of harmonic mappings

TL;DR

This paper investigates the Bohr phenomenon for harmonic and univalent functions, establishing improved sharp Bohr radii and deriving related corollaries to advance understanding of these classes.

Contribution

The paper introduces new, sharper bounds for the Bohr radius specific to harmonic and univalent functions, extending classical results.

Findings

Established improved sharp Bohr radii for harmonic functions.

Derived several corollaries from the main theorems.

Extended Bohr phenomenon results to broader classes of functions.

Abstract

Bohr phenomenon for analytic functions $ f $ where $ f(z)=\sum_{n=0}^{\infty}a_nz^n $, first introduced by Harald Bohr in $ 1914 $, deals with finding the largest radius $ r_f $, $ 0<r_f<1 $, such that the inequality $ \sum_{n=0}^{\infty}|a_nz^n|<1 $ holds whenever $ |f(z)|<1 $ holds in the unit disk $ \mathbb{D}=\{z\in\mathbb{C} : |z|<1\} $. The Bohr phenomenon for the harmonic functions of the form $ f(z)=h+\overline{g} $, 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|\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 several improved versions of the sharp Bohr radius for the classes of harmonic and univalent functions. Further, we prove several corollaries as a consequence of the main results.