Convexity of the Extreme Zeros of Classical Orthogonal Polynomials

Let Cλ n(x), n = 0, 1, . . . , λ > −1/2, be the ultraspherical (Gegenbauer) polynomials, orthogonal in (−1, 1) with respect to the weight function (1−x2)λ−1/2. Denote by xnk(λ), k = 1, . . . , n, the zeros of Cλ n(x) enumerated in decreasing order. In this short note we prove that, for any n ∈ IN , the product (λ+1)xn1(λ) is a convex function of λ if λ ≥ 0. The result is applied to obtain some inequalities for the largest zeros of Cλ n(x). If xnk(α), k = 1, . . . , n, are the zeros of Laguerre polynomial Ln(x), also enumerated in decreasing order, we prove that xn1(λ)/(α+ 1) is a convex function of α for α > −1.