Convexity of the extreme zeros of Gegenbauer and Laguerre polynomials (Q1874154)

In this paper it is proved that for any \(n\in\mathbb{N}\), the product \((\lambda+1)^{3/2} x_{n1}(\lambda)\) is a convex function of \(\lambda\) if \(\lambda\geq 0\), where \(x_{nk}(\lambda)\), \(k= 1,\dots, n\) are denoting zeros of ultraspherical polynomials \(C^\lambda_n(x)\) enumerated in decreasing order. This result is applied to obtain some inequalities for the largest zeros of \(C^\lambda_n(x)\). If \(x_{nk}(\alpha)\), \(k= 1,\dots, n\) are zeros of Laguerre polynomial \(L^\alpha_n(x)\) (also enumerated in decreasing order) it is proved that \(x_{n1}(\lambda)/(\alpha+ 1)\) is a convex function of \(\alpha\) for \(\alpha> -1\).