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

Abstract

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

arXiv commentWe missed to cite the paper of Liu and Ponnusamy [49] at the proper places. We have cited the article [49] at the proper places. 36 pages, 14 figures