  In this article, we first establish a generalized Bohr inequality and examine
its sharpness for a class of analytic functions $f$ in a simply connected
domain $\Omega_\gamma,$ where $0\leq \gamma<1$ with a sequence $\{\varphi_n(r)
\}^{\infty}_{n=0}$ of non-negative continuous functions defined on $[0,1)$ such
that the series $\sum_{n=0}^{\infty}\varphi_n(r)$ converges locally uniformly
on $[0,1)$. Our results represent twofold generalizations corresponding to
those obtained for the classes $\mathcal{B}(\mathbb{D})$ and
$\mathcal{B}(\Omega_{\gamma})$, where \begin{align*}
  \Omega_{\gamma}:=\biggl\{z\in \mathbb{C}:
\bigg|z+\dfrac{\gamma}{1-\gamma}\bigg|<\dfrac{1}{1-\gamma}\biggr\}.
\end{align*} As a convolution counterpart, we determine the Bohr radius for
hypergeometric function on $ \Omega_{\gamma} $. Lastly, we establish a
generalized Bohr inequality and its sharpness for the class of $ K
$-quasiconformal, sense-preserving harmonic maps of the form $f=h+\overline{g}$
in $\Omega_{\gamma}.$
