On a Theorem by Bojanov and Naidenov Applied to Families of Gegenbauer-Sobolev Polynomials

Abstract

Let $\{Q_{n,\lambda}^{(\alpha)}\}_{n\ge 0}$ be the sequence of monic orthogonal polynomials with respect the Gegenbauer-Sobolev inner product $$\langle f,g\rangle s:=\int_{-1}^1 f(x)g(x)(1-x^2)^{\alpha-\frac{1}{2}} dx+\lambda \int_{-1}^1 f'(x)g'(x)(1-x^2)^{\alpha-\frac{1}{2}}dx,$$ where $\alpha \gt -\frac{1}{2}$ and $\lambda \ge 0$. In this paper we use a recent result due to B.D. Bojanov and N. Naidenov [3], in order to study the maximization of a local extremum of the $k$th derivative $\frac{d^k}{dx^k}$ in $[-M_{n,\lambda},M_{n,\lambda}]$, where $M_{n,\lambda}$ is a suitable value such that all zeros of the polynomial $Q_{n,\lambda}^{(\alpha)}$ are contained in $[-M_{n,\lambda},M_{n,\lambda}]$ and the function $\left|Q_{n,\lambda}^{(\alpha)}\right|$ attains its maximal value at the end-points of such interval. Also, some illustrative numerical examples are presented.