Blaschke products with derivative in function spaces

TL;DR

This paper investigates the relationship between the derivatives of Blaschke products and the distribution of their zeros within specific function spaces, establishing conditions for zero sequences based on derivative integrability.

Contribution

It provides new criteria linking the integrability of a Blaschke product's derivative to the summability of its zeros' distances from the boundary.

Findings

If B' in A^p_{α}, then sum of (1 - |a_n|)^β converges.

If zeros are uniformly discrete and B' in H^p or A^{1+p}, then sum of (1 - |a_n|)^{1-p} converges.

Derived conditions connect derivative space membership to zero distribution.

Abstract

Let $B$ be a Blaschke product with zeros $\{a_n\}$. If $B' \in A^p_{\alpha}$ for certain $p$ and $\alpha$, it is shown that $\sum_n (1 - |a_n|)^{\beta} < \infty$ for appropriate values of $\beta$. 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 $\sum_n (1 - |a_n|)^{1-p} < \infty$.