Bohr inequalities for the class of unimodular bounded functions on shifted disks

Abstract

Let $H\left(D\right)$ denote the class of analytic functions in the unit disk $D:=\{z\in C:|z|<1\}$. The classical Bohr's inequality [21] states that if $f\in H\left(D\right)$ is given by $f\left(z\right)={\sum }_{n=0}^{\infty }{a}_n{z}^n$ such that $\left|f\right(z\left)\right|<1$ for $z\in D$, then$\sum _{n=0}^{\infty }\left|a_n\right|r^n\le 1\text{for}r\le \frac{1}{3}$ and the constant 1/3 cannot be improved. The constant 1/3 is known as Bohr radius. In this paper, we study Bohr phenomenon for classes of analytic as well as harmonic mappings on shifted disks. We prove several sharp results on improved Bohr radius for the classes of analytic functions as well as for the class of harmonic mappings on certain shifted disks.