Monotonicity of zeros of Jacobi polynomials

Denote by xn,k(α, β), k = 1, . . . , n, the zeros of the Jacobi polynomial P (α,β) n (x). It is well known that xn,k(α, β) are increasing functions of β and decreasing functions of α. In this paper we investigate the question of how fast the functions 1 − xn,k(α, β) decrease as β increases. We prove that the products tn,k(α, β) := fn(α, β) (1− xn,k(α, β)), where fn(α, β) = 2n2 + 2n(α + β + 1) + (α + 1)(β + 1), are already increasing functions of β and that, for any fixed α > −1, fn(α, β) is the asymptotically extremal, with respect to n, function of β that forces the products tn,k(α, β) to increase.