Composition followed by differentiation between weighted Bergman spaces and weighted Banach spaces of holomorphic functions

Let Φ be an analytic self-map of the open unit disk D in the complex plane. Such a map induces through composition a linear composition operator C_Φ : f ↦ f◦Φ.We are interested in the combination of C_Φ with the differentiation operator D, that is in the operator DC_Φ : f ↦ Φ` · (f ◦ Φ) acting between weighted Bergman spaces and weighted Banach spaces of holomorphic functions