On a logarithmic Hardy–Bloch type space, II

Abstract

Suppose and . Let $\mathcal{ℬ}{\mathcal{\mathscr{H}}}_{p,\alpha }$ denote the logarithmic Hardy–Bloch type space of those functions $f$ which are analytic in the unit disk $\mathbb{𝔻}$ such that

In this paper, we mainly obtain the relation between the logarithmic Hardy–Bloch type space $\mathcal{ℬ}{\mathcal{\mathscr{H}}}_{p,\alpha }$ and the Hardy space $H^p$ (or the Dirichlet space ${\mathcal{𝒟}}_{p-1}^p$). We also give the characterization of lacunary series on $\mathcal{ℬ}{\mathcal{\mathscr{H}}}_{p,\alpha }$ when .