Suppose that BzY fa k g is a Blaschke product associated with the sequence fa k g. If y k1 1 À ja k j `y for some in 0Y 1a2, it is known that B H zY fa k g e H p for all p in 0Y 1 À . We extend this result by considering the situation when p b 1 À . In these cases we obtain that B H zY fa k g f H p . We then give some counterexamples to indicate to what extent our results may be regarded as being best possible.
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