The infinite range of infinite Blaschke product

Abstract

For an infinite Blaschke product B B , does there necessarily exist δ > 0 \delta >0 such that each w w satisfying | w | > δ |w|>\delta is assumed infinitely often by B B ? Stephenson raised this question in 1979 and then constructed a counterexample in 1988 to prove that the answer to his problem is negative. In this paper, we find two sufficient conditions under which the answer to the problem is positive.