The Bohr inequality on a simply connected domain and its applications
Ahammed, Sabir, Molla Basir Ahamed, Partha Pratim Roy · arXiv (Cornell University) · 2024
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 $Ω_γ,$ where $0\leq γ<1$ with a sequence $\{φ_n(r) \}^{\infty}_{n=0}$ of non-negative continuous functions defined on $[0,1)$ such that the series $\sum_{n=0}^{\infty}φ_n(r)$ converges locally uniformly on $[0,1)$. Our results represent twofold generalizations corresponding to those obtained for the classes $\mathcal{B}(\mathbb{D})$ and $\mathcal{B}(Ω_γ)$, 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 $ Ω_γ $. 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 $Ω_γ.$