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

A class $\mathcal{F}$ consisting of analytic functions $f(z)={\sum }_{n=0}^{\mathrm{\infty }}{a}_n{z}^n$ in the unit disc $\mathbb{D}=\{z\in \mathbb{C}:\mathrm{\mid }z\mathrm{\mid }<1\}$ satisfies a Bohr phenomenon if there exists an $r_f>0$ such that \begin{equation*} I_f®:=\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 $\mathrm{\mid }z\mathrm{\mid }=r\le r_f$. The largest radius $r_f$ is the Bohr radius and the inequality $I_f(r)\le d(f(0),\mathrm{\partial }\mathbb{D})$ 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 $I_f(r)\le d(f(0),\mathrm{\partial }\mathbb{D})$ holds for every element of the class $\mathcal{F}$ for $0\le r<r_0$ 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.