On a Theorem by Bojanov and Naidenov applied to families of Gegenbauer-Sobolev polynomials

Let $\{Q^{(\alpha)}_{n,\lambda}\}_{n\geq 0}$ 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 $\alpha>-\frac{1}{2}$ and $\lambda\geq 0$. In this paper we use a recent result due to B.D. Bojanov and N. Naidenov \cite{BN2010}, in order to study the maximization of a local extremum of the $k$th derivative $\frac{d^k}{dx^k}Q^{(\alpha)}_{n,\lambda}$ in $[-M_{n,\lambda}, M_{n,\lambda}]$, where $M_{n,\lambda}$ is a suitable value such that all zeros of the polynomial $Q^{(\alpha)}_{n,\lambda}$ are contained in $[-M_{n,\lambda}, M_{n,\lambda}]$ and the function $\left|Q^{(\alpha)}_{n,\lambda}\right|$ attains its maximal value at the end-points of such interval. Also, some illustrative numerical examples are presented.