Bohr phenomenon for certain classes of harmonic mappings

Abstract

Bohr phenomenon for analytic functions $f$, where $f(z)={\sum }_{n=0}^{\infty }{a}_n{z}^n$, was first introduced by Harald Bohr in $1914$ and deals with finding the largest radius $r_f$, , such that the inequality holds for $\left|z\right|=r\le r_f$ whenever holds in the unit disk . The Bohr phenomenon for harmonic functions of the form $f(z)=h+ḡ$, 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 $r_f$, such that

$$\sum _{n=1}^{\infty }(|a_n|+|b_n|)|z{|}^n\le d(f(0),\partial f(\mathbb{𝔻}))$$

holds for $\left|z\right|\le r_f$, where $d(f(0),\partial f(\mathbb{𝔻}))$ is the Euclidean distance between $f(0)$ and the boundary of $f(\mathbb{𝔻})$. 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.