Bohr radius for some classes of Harmonic mappings

TL;DR

This paper introduces a broad class of harmonic mappings, establishes fundamental theorems including the Bohr radius, and applies these results to various classes of harmonic and univalent functions.

Contribution

It defines a new class of harmonic mappings with coefficient bounds and derives the Bohr radius, growth, and covering theorems for this class and related functions.

Findings

Established the Bohr radius for the new harmonic class

Proved growth and covering theorems for the class

Applied results to various harmonic and univalent function classes

Abstract

We introduce a general class of sense-preserving harmonic mappings defined as follows: \begin{equation*} \mathcal{S}^0_{h+\bar{g}}(M):= \{f=h+\bar{g}: \sum_{m=2}^{\infty}(\gamma_m|a_m|+\delta_m|b_m|)\leq M, \; M>0 \}, \end{equation*} where $h(z)=z+\sum_{m=2}^{\infty}a_mz^m$, $g(z)=\sum_{m=2}^{\infty}b_m z^m$ are analytic functions in $\mathbb{D}:=\{z\in\mathbb{C}: |z|\leq1 \}$ and \begin{equation*} \gamma_m,\; \delta_m \geq \alpha_2:=\min \{\gamma_2, \delta_2\}>0, \end{equation*} for all $m\geq2$. We obtain Growth Theorem, Covering Theorem and derive the Bohr radius for the class $\mathcal{S}^0_{h+\bar{g}}(M)$. As an application of our results, we obtain the Bohr radius for many classes of harmonic univalent functions and some classes of univalent functions.