Products of Composition and Differentiation between the Fractional Cauchy Spaces and the Bloch-Type Spaces

The operators DCΦ and CΦD are defined by DCΦf=f∘Φ′ and CΦDf=f′∘Φ where Φ is an analytic self-map of the unit disc and f is analytic in the disc. A characterization is provided for boundedness and compactness of the products of composition and differentiation from the spaces of fractional Cauchy transforms Fα to the Bloch-type spaces Bβ, where α>0 and β>0. In the case β<2, the operator DCΦ:Fα⟶Bβ is compact ⇔DCΦ:Fα⟶Bβ is bounded ⇔Φ′∈Bβ,ΦΦ′∈Bβ and Φ∞<1. For β<1, CΦD:Fα⟶Bβ is compact ⇔CΦD:Fα⟶Bβ is bounded ⇔Φ∈Bβ and Φ∞<1