Bohr phenomenon for certain classes of harmonic mappings

Abstract

Bohr phenomenon for analytic functions $f$ where $f(z)={\sum }_{n=0}^{\infty }{a}_n{z}^n$, first introduced by Harald Bohr in $1914$, deals with finding the largest radius $r_f$, $0<r_f<1$, such that the inequality ${\sum }_{n=0}^{\infty }|a_n{z}^n|<1$ holds whenever $|f(z)|<1$ holds in the unit disk $\mathbb{D}=\{z\in \mathbb{C}:|z|<1\}$. The Bohr phenomenon for the harmonic functions of the form $f(z)=h+\overline{g}$, where $h(z)={\sum }_{n=0}^{\infty }{a}_n{z}^n$ and $g(z)={\sum }_{n=1}^{\infty }{b}_n{z}^n$ is to find the largest radius $r_f$, $0<r_f<1$ such that \begin{equation*} \sum_{n=1}^{\infty}\left(|a_n|+|b_n|\right)|z|^n\leq d(f(0),\partial f(\mathbb{D})) \end{equation*} holds for $|z|\le r_f$, where $d(f(0),\partial f(\mathbb{D}))$ is the Euclidean distance between $f(0)$ and the boundary of $f(\mathbb{D})$. In this paper, we prove several improved versions of the sharp Bohr radius for the classes of harmonic and univalent functions. Further, we prove several corollaries as a consequence of the main results.