On a logarithmic Hardy-Bloch type space

Abstract

In this paper, given $0\lt p\lt \infty$, we define a logarithmic Hardy-Bloch type space \begin{multline*} BH_{p,L}=\left\{f(z)\in H(D):||f||_{p,L}\right.\\ \left.=\sup_{z\in D}(1-|z|)\log\frac{e}{1-|z|} M_p(|z|,f')\lt \infty\right\}. \end{multline*} Then we mainly study the relation between $BH_{p,L}$ and three classical spaces: Hardy space, Dirichlet type space and VMOA. We also obtain some estimates on the growth of $f\in BH_{p,L}$.