Monotonicity of zeros of Jacobi polynomials

Denote by xnk(α,β), k=1,…,n, the zeros of the Jacobi polynomial . It is well known that xnk(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-xnk(α,β) decrease as β increases. We prove that the products tnk(α,β)≔fn(α,β)(1-xnk(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 tnk(α,β) to increase.