On Bloch norm and Bohr phenomenon for harmonic Bloch functions on simply connected domains

In this article, we introduce the class $\mathcal{B}^{*}_{\mathcal{H},Ω}(α)$ of harmonic $α$-Bloch-type mappings on $Ω$ as a generalization of the class $\mathcal{B}_{\mathcal{H},Ω}(α)$ of harmonic $α$-Bloch mappings on $Ω$, where $Ω$ is arbitrary proper simply connected domain in the complex plane. We study several interesting properties of the classes $\mathcal{B}_{\mathcal{H},Ω}(α)$ and $\mathcal{B}^{*}_{\mathcal{H},Ω}(α)$ on arbitrary proper simply connected domain $Ω$ and on the shifted disk $Ω_γ$ containing $\mathbb{D}$, where $$ Ω_γ:=\bigg\{z\in\mathbb{C} : \bigg|z+\fracγ{1-γ}\bigg|<\frac{1}{1-γ}\bigg\ $$ and $0 \leq γ<1$. We establish the Landau's theorem for the harmonic Bloch space $\mathcal{B}_{\mathcal{H},Ω_γ}(α)$ on the shifted disk $Ω_γ$. For $f \in \mathcal{B}_{\mathcal{H},Ω}(α)$ (respectively $\mathcal{B}^{*}_{\mathcal{H},Ω}(α)$) of the form $f(z)=h(z) + \overline{g(z)}=\sum_{n=0}^{\infty}a_nz^n + \overline{\sum_{n=1}^{\infty}b_nz^n}$ in $\mathbb{D}$ with Bloch norm $||f||_{\mathcal{H},Ω, α} \leq 1$ (respectively $||f||^{*}_{\mathcal{H},Ω, α} \leq 1$), we define the Bloch-Bohr radius for the space $\mathcal{B}_{\mathcal{H},Ω}(α)$ (respectively $\mathcal{B}^{*}_{\mathcal{H},Ω}(α)$) to be the largest radius $r_{Ω,f} \in (0,1)$ such that $\sum_{n=0}^{\infty}(|a_n|+|b_{n}|) r^n\leq 1$ for $r \leq r_{Ω, α}$ and for all $f \in \mathcal{B}_{\mathcal{H},Ω}(α)$ (respectively $\mathcal{B}^{*}_{\mathcal{H},Ω}(α)$). We investigate Bloch-Bohr radius for the classes $\mathcal{B}_{\mathcal{H},Ω}(α)$ and $\mathcal{B}^{*}_{\mathcal{H},Ω}(α)$ on simply connected domain $Ω$ containing $\mathbb{D}$.

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