The sharp refined Bohr-Rogosinski inequalities for certain classes of harmonic mappings

Abstract

A class $ \mathcal{F} $ consisting of analytic functions $ f(z)=\sum_{n=0}^{\infty}a_nz^n $ in the unit disc $ \mathbb{D}=\{z\in\mathbb{C}:|z|<1\} $ satisfies a Bohr phenomenon if there exists an $ r_f>0 $ such that \begin{equation*} I_f(r):=\sum_{n=1}^{\infty}|a_n|r^n\leq{d}\left(f(0),\partial \mathbb{D}\right) \end{equation*} for every function $ f\in\mathcal{F} $, and $ |z|=r\leq r_f $. The largest radius $ r_f $ is the Bohr radius and the inequality $ I_f(r)\leq{d}\left(f(0),\partial \mathbb{D}\right) $ is Bohr inequality for the class $ \mathcal{F} $, where `$ d $' is the Euclidean distance. If there exists a positive real number $ r_0 $ such that $ I_f(r)\leq {d}\left(f(0),\partial \mathbb{D}\right) $ holds for every element of the class $ \mathcal{F} $ for $ 0\leq r<r_0 $ and fails when $ r>r_0 $, then we say that $ r_0 $ is sharp bound for the inequality w.r.t. the class $ \mathcal{F} $. In this paper, we prove sharp refinement of the Bohr-Rogosinski inequality for certain classes of harmonic mappings.