On new Bloch-type spaces

We introduce a new Bloch-type space, so called, the logarithmic Bloch-type space B-log beta(alpha)(D) on the unit disc D, as the space of all holomorphic functions f on D such that sup(z is an element of D)(1-|z|)(alpha) (ln e(beta/alpha)/1-|z|)(beta) |f'(z)| < infinity, where alpha > 0 and beta >= 0, and present some basic properties of the space. A necessary and a sufficient condition for a function with Hadamard gaps to belong to the logarithmic Bloch-type space are given, as well as some applications of these results to a composition operator. (C) 2009 Elsevier Inc. All rights reserved.