If p(z) be a polynomial of degree n, which does not vanish in |z| < k, k < 1, then it was conjectured by Aziz [Bull. Austral. Math. Soc. 35 (1987), 245-256] that [...] In this paper, we consider the case k < r < 1 and present a generalization as well as improvement of the above inequality.

We estimate the maximum of ${\prod }_{j=1}^n|1-x^{a_j}|$ on the unit circle where 1 ≤ a₁ ≤ a₂ ≤ ... is a sequence of integers. We show that when $a_j$ is $j^k$ or when $a_j$ is a quadratic in j that takes on positive integer values, the maximum grows as exp(cn), where c is a positive constant. This complements results of Sudler and Wright that show exponential growth when $a_j$ is j.    In contrast we show, under fairly general conditions, that the maximum is less than $2^n/n^r$, where r is an arbitrary positive number. One consequence is that the...