Bohr phenomenon for certain Subclasses of Harmonic Mappings

TL;DR

This paper explores the Bohr phenomenon for harmonic functions, determining the largest radius where certain inequalities hold for subclasses of harmonic mappings in the unit disk.

Contribution

It extends the Bohr phenomenon to various subclasses of harmonic functions, providing new radius estimates and insights into harmonic mappings.

Findings

Established Bohr radius for harmonic subclasses.

Derived bounds for harmonic functions with specific properties.

Extended classical Bohr radius results to harmonic mappings.

Abstract

The Bohr phenomenon for analytic functions of the form $f(z)=\sum_{n=0}^{\infty} a_{n}z^{n}$, first introduced by Harald Bohr in 1914, deals with finding the largest radius $r_{f}$, $0<r_{f}<1$, such that the inequality $\sum_{n=0}^{\infty} |a_{n}z^{n}| \leq 1$ holds whenever the inequality $|f(z)|\leq 1 $ holds in the unit disk $\mathbb{D}=\{z \in \mathbb{C}: |z|<1\}$. The exact value of this largest radius known as Bohr radius, which has been established to be $r_{f}=1/3$. The Bohr phenomenon \cite{Abu-2010} for harmonic functions $f$ of the form $f(z)=h(z)+\overline {g(z)}$, where $h(z)=\sum_{n=0}^{\infty} a_{n}z^{n}$ and $g(z)=\sum_{n=1}^{\infty} b_{n}z^{n}$ is to find the largest radius $r_{f}$, $0<r_{f}<1$ such that $$\sum\limits_{n=1}^{\infty} (|a_{n}|+|b_{n}|) |z|^{n}\leq d(f(0),\partial f(\mathbb{D})) %\quad\mbox { for } |z|\leq r_{f}. $$ holds for $|z|\leq r_{f}$, here $d(f(0),\partial f(\mathbb{D})) $ denotes the Euclidean distance between $f(0)$ and the boundary of $f(\mathbb{D})$. In this paper, we investigate the Bohr radius for several classes of harmonic functions in the unit disk $\mathbb{D}.$