  Bohr's classical theorem and its generalizations are now active areas of
research and have been the source of investigations in numerous function
spaces. In this article, we study a generalized Bohr's inequality for the class
of bounded analytic functions defined on the simply connected domain $$
\Omega_{\gamma}:=\bigg\{z\in \mathbb{C}:
\bigg|z+\frac{\gamma}{1-\gamma}\bigg|<\frac{1}{1-\gamma}\bigg\}, \,\ \text{for
} 0\leq \gamma<1. $$ Part of its applications, we calculate the Bohr-type radii
for some known integral operators.
