Bohr radius for some classes of Harmonic mappings

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}(γ_m|a_m|+δ_m|b_m|)\leq M, \; M>0 \}, \end{equation*} where $h(z)=z+{\sum }_{m=2}^{\infty }{a}_m{z}^m$, $g(z)={\sum }_{m=2}^{\infty }{b}_m{z}^m$ are analytic functions in $\mathbb{D}:=\{z\in \mathbb{C}:|z|\le 1\}$ and \begin{equation*} γ_m,\; δ_m \geq α_2:=\min \{γ_2, δ_2\}>0, \end{equation*} for all $m\ge 2$. We obtain Growth Theorem, Covering Theorem and derive the Bohr radius for the class ${\mathcal{S}}_{h+\bar{g}}^0(M)$. As an application of our results, we obtain the Bohr radius for many classes of harmonic univalent functions and some classes of univalent functions.