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a b s t r a c tWe characterize the analytic self-maps / of the unit disk D in C that induce continuous composition operators C / from the log-Bloch space B log ðDÞ to l-Bloch spaces B l ðDÞ in terms of the sequence of quotients of the l-Bloch semi-norm of the nth power of / and the log-Bloch semi-norm of the nth power F n of the identity function on D, where l : D ! ð0; 1Þ is continuous and bounded. We also obtain an expression that is equivalent to the essential norm of C / between these spaces, thus characterizing / such that C / 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 / acting on B log to the case when C / acts on these more general log-Bloch-type spaces.Published by Elsevier Inc.

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