Blaschke products with derivative in function spaces

Abstract

Let $B$ be a Blaschke product with zeros {$a_n$}. If $B′ \in A^p_α$ for certain $p$ and $α$, it is shown that $Σ_n (1 - |a_n|)^β < ∞$ for appropriate values of β. Also, if {$a_n$} is uniformly discrete and if $B′ \in H^p$ or $B′ \in A^{1+p}$ for any $p \in (0,1)$, it is shown that $Σ_n (1 - |a_n|)^1-p < ∞$.