Spaces of analytic functions of Hardy-Bloch type

For 0<p≤∞ and 0<q≤∞, the space of Hardy-Bloch type ℬ(p,q) consists of those functionsf which are analytic in the unit diskD such that (1−r)M _p (r,f′)⊂L ^q (dr/(1−r)). We note that ℬ(∞,∞) coincides with the Bloch space ℬ and that ℬ⊂ℬ(p,∞) for allp. Also, the space ℬ(p,p) is the Dirichlet spaceD _p−1 ^p . We prove a number of results on decomposition of spaces with logarithmic weights which allow us to obtain sharp results about the mean growth of the ℬ(p,q). In particular, we prove that iff is an analytic function inD and 2≤p<∞, then the conditionM _p (r,f′)=O((1−r)⁻¹), asr→1, implies that. This result is an improvement of the well-known estimate of Clunie and MacGregor and Makarov about the integral means of Bloch functions, and it also improves the main result in a recent paper by Girela and Peláez. We also consider the question of characterizing the univalent functions in the spaces ℬ(p,2), 0<p<∞, and in some other related spaces and give some applications of our estimates to study the Carleson measures for the spaces ℬ(p,2) andD _p−1 ^p . The first and third authors were supported by grants from “E1 Ministerio de Educación y Ciencia”, Spain (MTN2004-00078 and MTN2004-21420-E) and by a grant from “La Junta de Andalucía” (FQM-210). The second author was supported in part by MNZŽS Grant, No. ON144010, Serbia.