Bohr phenomenon for harmonic Bloch functions

Abstract

For \(\alpha \in (0,\infty)\), let \(\mathcal{B}_{\mathcal{H},\Omega}(\alpha)\) denote the class of \(\alpha\)-Bloch mappings on a proper simply connected domain \(\Omega \subseteq \mathbb{C}\). In this article, we introduce the class \(\mathcal{B}^{*}_{\mathcal{H},\Omega}(\alpha)\) of harmonic \(\alpha\)-Bloch-type mappings on a proper simply connected domain \(\Omega \subseteq \mathbb{C}\) and study several interesting properties of the classes \(\mathcal{B}_{\mathcal{H},\Omega}(\alpha)\) and \(\mathcal{B}^{*}_{\mathcal{H},\Omega}(\alpha)\) when \(\Omega\) is proper simply connected domain and the shifted disk \(\Omega_{\gamma}\) containing \(\mathbb{D}\), where

$$\Omega_{\gamma}:=\big\{z\in\mathbb{C} : \big|z+\frac{\gamma}{1-\gamma}\big|<\frac{1}{1-\gamma}\big\}$$

and \(0 \leq \gamma <1\). For \(f \in \mathcal{B}_{\mathcal{H},\Omega}(\alpha)\) (respectively \(\mathcal{B}^{*}_{\mathcal{H},\Omega}(\alpha))\) 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 \( \lVert f \rVert _{\mathcal{H},\Omega, \alpha} \leq 1\) (respectively \( \lVert f \rVert ^{*}_{\mathcal{H},\Omega, \alpha} \leq 1\)), we define the Bloch–Bohr radius for the class \(\mathcal{B}_{\mathcal{H},\Omega}(\alpha)\) (respectively \(\mathcal{B}^{*}_{\mathcal{H},\Omega}(\alpha))\) to be the largest radius \(r_{\Omega,\alpha} \in (0,1)\) such that \(\sum_{n=0}^{\infty}(|a_n|+|b_{n}|) r^n\leq 1\) for \(r \leq r_{\Omega, \alpha}\) and for all \(f \in \mathcal{B}_{\mathcal{H},\Omega}(\alpha)\) (respectively \(\mathcal{B}^{*}_{\mathcal{H},\Omega}(\alpha))\). We also investigate Bloch–Bohr radius for the classes \(\mathcal{B}_{\mathcal{H},\Omega}(\alpha)\) and \(\mathcal{B}^{*}_{\mathcal{H},\Omega}(\alpha)\) on simply connected domain \(\Omega\) containing \(\mathbb{D}\).