Composition operators from logarithmic Bloch spaces to weighted Bloch spaces

We characterize the analytic self-maps phi of the unit disk D in C that induce continuous composition operators C-phi from the log-Bloch space B-log(D) to mu-Bloch spaces B-mu(D) in terms of the sequence of quotients of the mu-Bloch semi-norm of the nth power of phi and the log-Bloch semi-norm of the nth power F-n of the identity function on D, where mu : D -> (0, infinity) is continuous and bounded. We also obtain an expression that is equivalent to the essential norm of C-phi between these spaces, thus characterizing phi such that C-phi is compact. After finding a pairwise norm equivalent family of log-Bloch type spaces that are defined on the unit ball B-n of C-n and include the log-Bloch space, we obtain an extension of our boundedness/compactness/essential norm results for C-phi acting on B-log to the case when C-phi acts on these more general log-Bloch-type spaces. Published by Elsevier Inc.