Bohr–Rogosinski radius for a certain class of close-to-convex harmonic mappings

Abstract

AbstractLet$ \mathcal {B} $be the class of analytic functions$ f $in the unit disk$ \mathbb {D}=\{z\in \mathbb {C} : |z|<1\} $such that$ |f(z)|<1 $for all$ z\in \mathbb {D} $. If$ f\in \mathcal {B} $of the form$ f(z)=\sum _{n=0}^{\infty }a_nz^n $, then$ \sum _{n=0}^{\infty }|a_nz^n|\leq 1 $for$ |z|=r\leq 1/3 $and$ 1/3 $cannot be improved. This inequality is called Bohr inequality and the quantity$ 1/3 $is called Bohr radius. If$ f\in \mathcal {B} $of the form$ f(z)=\sum _{n=0}^{\infty }a_nz^n $, then$ |\sum _{n=0}^{N}a_nz^n|<1\;\; \mbox {for}\;\; |z|<{1}/{2} $and the radius$ 1/2 $is the best possible for the class$ \mathcal {B} $. This inequality is called Bohr–Rogosinski inequality and the corresponding radius is called Bohr–Rogosinski radius. Let$ \mathcal {H} $be the class of all complex-valued harmonic functions$ f=h+\bar {g} $defined on the unit disk$ \mathbb {D} $, where$ h $and$ g $are analytic in$ \mathbb {D} $with the normalization$ h(0)=h^{\prime }(0)-1=0 $and$ g(0)=0 $. Let$ \mathcal {H}_0=\{f=h+\bar {g}\in \mathcal {H} : g^{\prime }(0)=0\}. $For$ \alpha \geq 0 $and$ 0\leq \beta <1 $, let$$ \begin{align*} \mathcal{W}^{0}_{\mathcal{H}}(\alpha, \beta)=\{f=h+\overline{g}\in\mathcal{H}_{0} : \mathrm{Re}\left(h^{\prime}(z)+\alpha zh^{\prime\prime}(z)-\beta\right)>|g^{\prime}(z)+\alpha zg^{\prime\prime}(z)|,\;\; z\in\mathbb{D}\} \end{align*} $$be a class of close-to-convex harmonic mappings in$ \mathbb {D} $. In this paper, we prove the sharp Bohr–Rogosinski radius for the class$ \mathcal {W}^{0}_{\mathcal {H}}(\alpha , \beta ) $.