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

Abstract

In this article, we introduce the class B * H,Ω (α) of harmonic α-Bloch-type mappings on Ω as a generalization of the class B 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 B H,Ω (α) and B * H,Ω (α) on arbitrary proper simply connected domain Ω and on the shifted disk Ω γ containing D, whereWe establish the Landau's theorem for the harmonic Bloch space B H,Ωγ (α), we define the Bloch-Bohr radius for the space B H,Ω (α) (respectively B * H,Ω (α)) to be the largest radius r Ω,f ∈ (0, 1) such thatand for all f ∈ B H,Ω (α) (respectively B * H,Ω (α)). We investigate Bloch-Bohr radius for the classes B H,Ω (α) and B * H,Ω (α) on simply connected domain Ω containing D.