Abstract
A result of D.J. Newman asserts that a uniformly separated sequence contained in a Stolz angle is a finite union of exponential sequences. We extend this by obtaining several equivalent characterizations of such sequences. If the zeros of a Blaschke product B lie in a Stolz angle, then \(B^\prime\in A^p\) for all \(p<\frac 32\) and it has recently been shown that this result cannot be improved. Also, Newman's result can be used to prove that if B is an interpolating Blaschke product whose zeros lie in a Stolz angle, then \(B^\prime\in\bigcap_{0<p<1}H^p\subset \bigcap_{0<p<2}A^p\). In this paper we prove that if the zeros of an interpolating Blaschke product lie in a disk internally tangent to the unit circle, then \(B^\prime\in \bigcap_{0<p<3/2}A^p\) and we show that this cannot be improved. We also obtain sharp results concerning the membership in Bergman spaces of the derivative of an interpolating Blaschke product whose sequence of zeros lies in other regions internally tangent to the unit circle.
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Girela, D., Pelaez, J. & Vukotic, D. Interpolating Blaschke Products: Stolz and Tangential Approach Regions. Constr Approx 27, 203–216 (2008). https://doi.org/10.1007/s00365-006-0651-6
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DOI: https://doi.org/10.1007/s00365-006-0651-6