Convexity of the extreme zeros of Gegenbauer and Laguerre polynomials

摘要

Let Cnλ(x),n=0,1,…,λ>−12, 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 Cnλ(x) enumerated in decreasing order. In this short note, we prove that, for any n∈N, the product (λ+1)3/2xn1(λ) is a convex function of λ if λ⩾0. The result is applied to obtain some inequalities for the largest zeros of Cnλ(x). If xnk(α),k=1,…,n, are the zeros of Laguerre polynomial Lnα(x), also enumerated in decreasing order, we prove that xn1(λ)/(α+1) is a convex function of α for α>−1.