Composition operators from logarithmic Bloch spaces to Bloch-type spaces

Abstract

In this paper, we characterize the analytic self-maps ϕ of the unit disk 𝔻 in ℂ that induce continuous and compact composition operators from the logarithmic Bloch spaces, introduced by Stević [Appl. Math. Comput. 215 (2009), 841–849], into the μ-Bloch space, where μ is any weight defined on 𝔻. We give characterizations in terms of quotients of the logarithmic Bloch norm of the n-th power of ϕ and the μ-Bloch norm of the n-th power of the identity function I on 𝔻. This solves partially the problem proposed by Zhao [Proc. Amer. Math. Soc. 138 (2010), 2537–2546].