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 ${\sum }_{n=0}^{\infty }|a_n{z}^n|\le 1$ holds whenever the inequality $|f(z)|\le 1$ 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 $$\sum _{n=1}^{\infty }(|a_n|+|b_n|)|z{|}^n\le d(f(0),\partial f(\mathbb{D}))$$ holds for $|z|\le r_f$, 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 $\mathbb{D}.$

24 pages