Integral means for the first derivative of Blaschke products

Abstract

Suppose that B(z, {a_k}) is a Blaschke product associated with the sequence {a_k}. If ∑_k=1^∞(1−|a_k|)^α<∞ for some α in (0, 1/2), it is known that B'(z, {a_k})∈H^p for all p in (0, 1−α]. We extend this result by considering the situation when p>1−α. In these cases we obtain that B'(z, {a_k})∉H^p. We then give some counterexamples to indicate to what extent our results may be regarded as being best possible.