Sharp results on the integrability of the derivative of an interpolating Blaschke product.

The Schwarz-Pick lemma readily implies that the derivative of any Blaschke product belongs to all the Bergman spaces Ap with 0 < p < 1. It is also well known that this result is sharp: there exist a Blaschke product whose derivative does not belong to A1. However, the question of whether there exists an interpolating Blaschke product B with B′ ∉ A1 remained open. In this paper we give an explicit construction of such an interpolating Blaschke product B. A result of W. S. Cohn asserts that if and B is an interpolating Blaschke product with sequence of zeros of , then B′ ∈ Hp if and only if (1 – | ak|)1– p < ∞. We prove that Cohn's result is no longer true for . Indeed, we construct: (a) an interpolating Blaschke product B whose sequence of zeros of satisfies (1 – | ak|)1/2 < ∞ but B′ ∉ H1/2, and (b) an interpolating Blaschke products B whose sequence of zeros of satisfies (1 – | ak|)1– p < ∞, for all p ∈ (0, 1/2), whose derivative B′ does not belong to the Nevanlinna class. [ABSTRACT FROM AUTHOR]