Bohr phenomenon for certain Subclasses of Harmonic Mappings

Abstract

The Bohr phenomenon for analytic functions of the form $f(z)=\sum_{n=0}^{\infty} a_{n}z^{n}$, first introduced by Harald Bohr in 1914, deals with finding the largest radius $r_{f}$, $0<r_{f}<1$, such that the inequality $\sum_{n=0}^{\infty} |a_{n}z^{n}| \leq 1$ holds whenever the inequality $|f(z)|\leq 1 $ holds in the unit disk $\mathbb{D}=\{z \in \mathbb{C}: |z|<1\}$. The exact value of this largest radius known as Bohr radius, which has been established to be $r_{f}=1/3$. The Bohr phenomenon \cite{Abu-2010} for harmonic functions $f$ of the form $f(z)=h(z)+\overline {g(z)}$, 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 $$\sum\limits_{n=1}^{\infty} (|a_{n}|+|b_{n}|) |z|^{n}\leq d(f(0),\partial f(\mathbb{D})) %\quad\mbox { for } |z|\leq r_{f}. $$ holds for $|z|\leq r_{f}$, here $d(f(0),\partial f(\mathbb{D})) $ denotes the Euclidean distance between $f(0)$ and the boundary of $f(\mathbb{D})$. In this paper, we investigate the Bohr radius for several classes of harmonic functions in the unit disk $\mathbb{D}.$