  In this paper, we investigate the Bohr phenomenon for the class of analytic
functions defined on the simply connected domain
  \begin{equation*}
  \Omega_{\gamma}=\bigg\{z\in\mathbb{C} :
\bigg|z+\frac{\gamma}{1-\gamma}\bigg|<\frac{1}{1-\gamma}\bigg\}\;\;
\text{for}\;\; 0\leq \gamma<1.
  \end{equation*} We study improved Bohr radius, Bohr-Rogosinski radius and
refined Bohr radius for the class of analytic functions defined in $
\Omega_{\gamma} $, and obtain several sharp results.
