Weighted composition followed by differentiation between Bloch-type spaces

We investigate the behavior of D u C φ : B α → B β , that is, the product of a weighted composition operator u C φ and the differentiation operator D , between Bloch-type spaces with standard weights. For all 0 < α , β < ∞ , we characterize the boundedness and estimate the essential norm of D u C φ in terms of the analytic function u : D → C and the symbol function φ : D → D . As a corollary, we characterize the compactness of D u C φ .