The Bohr inequality on a simply connected domain and its applications

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 ${\mathrm{\Omega }}_{\gamma },$ where $0\le \gamma <1$ with a sequence $\{{\phi }_n(r){\}}_{n=0}^{\mathrm{\infty }}$ of non-negative continuous functions defined on $[0,1)$ such that the series ${\sum }_{n=0}^{\mathrm{\infty }}{\phi }_n(r)$ converges locally uniformly on $[0,1)$. Our results represent twofold generalizations corresponding to those obtained for the classes $\mathcal{B}(\mathbb{D})$ and $\mathcal{B}({\mathrm{\Omega }}_{\gamma })$, where \begin{align*} Ω_γ:=\biggl{z\in $\mathbb{C}$: \bigg|z+\dfracγ{1-γ}\bigg|<\dfrac{1}{1-γ}\biggr}. \end{align*} As a convolution counterpart, we determine the Bohr radius for hypergeometric function on ${\mathrm{\Omega }}_{\gamma }$. Lastly, we establish a generalized Bohr inequality and its sharpness for the class of $K$-quasiconformal, sense-preserving harmonic maps of the form $f=h+\bar{g}$ in ${\mathrm{\Omega }}_{\gamma }\mathrm{.}$