We give a new characterization for the boundedness of composition operator followed by differentiation operator acting on Bloch-type spaces and calculate its essential norm in terms of the $n$-th power of the induced analytic self-map on the unit disk. From which some sufficient and necessary conditions of compactness of the operator follow immediately.
Yu-Xia Liang. Ze-Hua Zhou. "New estimate of essential norm of composition followed by differentiation between Bloch-type spaces." Banach J. Math. Anal. 8 (1) 118 - 137, 2014. https://doi.org/10.15352/bjma/1381782092