Zeros of extremal functions in weighted Bergman spaces

For -1 < α ≤ 0 and 0 < p < ∞, the solutions of certain extremal problems are known to act as contractive zero-divisors in the weighted Bergman space A(α)(p). We show that for 0 < α ≤ 1 and 0 < p < ∞, the analogous extremal functions do not have any extra zeros in the unit disk and, hence, have the potential to act as zero-divisors. As a corollary, we find that certain families of hypergeometric functions either have no zeros in the unit disk or have no zeros in a half-plane.