Mean growth of the derivative of a Blaschke product (Q1772514)

In this article it is proved that if \(B\) is a Blaschke product with zeros \(\{\alpha_n\}\) such that \(\sum_n(1 -| \alpha_n| )^\alpha < \infty\) for some \(\alpha\in (\frac{1}{2},1]\), then for each \( p \geq \alpha\), \[ \int_0^{2\pi} | B'(re^{it})| ^p \,dt = o\left(\frac{1}{(1-r)^{p +\alpha - 1}}\right). \] This agrees with a known result for \(\alpha\in (0,\frac{1}{2})\) obtained by \textit{M. Kutbi} [Kodai Math. J. 24, 86--97 (2001; Zbl 0980.30027)]. Also, the author establish a converse in the case of an interpolating Blaschke product whenever \(0 < \alpha < 1.\)