Weighted composition followed by differentiation between weighted Banach spaces of holomorphic functions

Let phi be an analytic self-map of the open unit disk D in the complex plane and psi be an analytic map on D. Such maps induce a weighted composition operator followed by differentiation DC phi,psi acting between the weighted Banach spaces of holomorphic functions. We characterize the boundedness and compactness of such operators in terms of the involved weights as well as the functions phi and psi.