Suppose 0 < p <= infinity and -infinity < alpha < infinity. Let BHp,alpha denote the logarithmic Hardy-Bloch type space of those functions f which are analytic in the unit disk D such that parallel to f parallel to(p,alpha) = sup(vertical bar z vertical bar<1)(1 - vertical bar z vertical bar)(log e/1 - vertical bar z vertical bar)M-alpha(p)(vertical bar z vertical bar, f') < infinity. In this paper, we mainly obtain the relation between the logarithmic Hardy-Bloch type space BHp,alpha and the Hardy space H-p (or the Dirichlet space a D-p-1(p)). We also give the characterization of lacunary series on BHp,alpha when 1 < p <= infinity.
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