Composition operators from logarithmic Bloch spaces to weighted Bloch spaces

We characterize the analytic self-maps ϕ   of the unit disk DD in CC that induce continuous composition operators CϕCϕ from the log-Bloch space Blog(D)Blog(D) to μ  -Bloch spaces Bμ(D)Bμ(D) in terms of the sequence of quotients of the μ-Bloch semi-norm of the nth power of ϕ and the log-Bloch semi-norm of the n  th power FnFn of the identity function on DD, where μ:D→(0,∞)μ:D→(0,∞) is continuous and bounded. We also obtain an expression that is equivalent to the essential norm of CϕCϕ between these spaces, thus characterizing ϕ   such that CϕCϕ is compact. After finding a pairwise norm equivalent family of log-Bloch type spaces that are defined on the unit ball BnBn of CnCn and include the log-Bloch space, we obtain an extension of our boundedness/compactness/essential norm results for CϕCϕ acting on BlogBlog to the case when CϕCϕ acts on these more general log-Bloch-type spaces.