Mean growth of the derivative of a Blaschke product

Abstract

If $B$ is a Blaschke product with zeros $\{a_n\}$ and if $\sum_n(1-|a_n|)^{\alpha}$ is finite for some $\alpha \in (1/2,1]$, then limits are found on the rate of growth of $\int_0^{2\pi} |B'(re^{it}|^p\, dt$ in agreement with a known result for $\alpha \in (0,1/2)$. Also, a converse is established in the case of an interpolating Blaschke product, whenever $0<\alpha<1$.