The Bohr phenomenon for analytic functions on shifted disks

Abstract

In this paper, we investigate the Bohr phenomenon for the class of analytic functions defined on the simply connected domain ${\mathrm{\Omega }}_{\gamma }=\{z\in \mathbb{C}:|z+\frac{\gamma }{1-\gamma }|<\frac{1}{1-\gamma }\}$ for $0\le \gamma <1.$ We study improved Bohr radius, Bohr-Rogosinski radius and refined Bohr radius for the class of analytic functions defined in ${\mathrm{\Omega }}_{\gamma }$, and obtain several sharp results.