Integral means for the \(n\)'th derivative of Blaschke products. (Q1872579)

Let \(n\) be a positive integer and \(B\) be a Blaschke product in the unit disk whose zeros satisfy \(\sum_{k=1}^{\infty} (1-| a_k| )^{\alpha} < \infty,\) where \(0<\alpha < 1/(n+1).\) Protas, for \(n=1\) in 1973, and Linden, for general \(n\) in 1976, proved that the n'th derivative of \(B\) belongs to the Hardy space \(H^p\) with \(p = \frac{1-\alpha}{n},\) but need not belong to \(H^p\) for larger \(p.\) In the article under review the author shows that for the larger p's the mean of order \(p\) of the n'th derivative on the circle \(| z| = r\) is \(o[(1- r)^{1+\alpha - np}]\) as \(r\rightarrow 1,\) and that this order of magnitude is sharp.