Interpolating Blaschke products: Stolz and tangential approach regions

Abstract. A result of D.J. Newman asserts that a uniformly sepa-rated sequence contained in a Stolz angle is a finite union of exponential sequences. We extend this by obtaining several equivalent characteriza-tions of such sequences. If the zeros of a Blaschke product B lie in a Stolz angle then B ′ ∈ Ap for all p < 3/2 and it has been recently 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 ′ ∈ ∩0<p<1Hp ⊂ ∩0<p<2Ap. In this paper we prove that if the zeros of an interpolating Blaschke product lie in a disc internally tangent to the unit circle then B ′ ∈ ∩0<p<3/2Ap and 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. 1. Introduction an