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The operators  D C Φ and  C Φ D are defined by  D C Φ f = f ∘ Φ ′ and  C Φ D f = 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  F α to the Bloch-type spaces  B β , where  α &gt; 0 and  β &gt; 0 . In the case  β &lt; 2 , the operator  D C Φ : F α ⟶ B β is compact  ⇔ D C Φ : F α ⟶ B β is bounded  ⇔ Φ ′ ∈ B β , Φ Φ ′ ∈ B β and  Φ ∞ &lt; 1 . For  β &lt; 1 ,  C Φ D : F α ⟶ B β is compact  ⇔ C Φ D : F α ⟶ B β is bounded  ⇔ Φ ∈ B β and  Φ ∞ &lt; 1 .

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