On the boundedness of the total variation of the logarithm of a Blaschke product

We establish that, for a Blaschke product B(z) convergent in the unit disk, the condition - ∞ < \smallint ₀¹ log(1 - t)n(t,B)dt\smallint _0^1 \log (1 - t)n(t,B)dt is sufficient for the total variation of logB to be bounded on a circle of radiusr, 0 <r < 1. For products B(z) with zeros concentrated on a single ray, this condition is also necessary. Here, n(t, B) denotes the number of zeros of the functionB (z) in a disk of radiust.