Bohr radius for some classes of Harmonic mappings (2010.01304v1)

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.