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

Let \(\α)n,\λ\\≥ 0 be the sequence of monic polynomials with respect the Gegenbauer-Sobolev inner product\,gS:=\∫₋₁¹f(x)g(x)(1-x²)\α-\1/2dx+\λ\∫₋₁¹f'(x)g'(x)(1-x²)\α-\1/2 dx, where\α>-\1/2 and \λ\≥ 0. In this paper we use a recent result to B.D. Bojanov and N. Naidenov \{BN2010}, in order to study the maximization of a local of the kth derivative ᵏ/dxᵏQ(\α)n,\λ in[-Mn,\λ, Mn,\λ], where Mn,\λ is a suitable value such that all zeros of the polynomial(\α)n,\λ are contained in [-Mn,\λ, Mn,\λ] the function \|Q(\α)n,\λ\| attains its maximal at the end-points of such interval. Also, some illustrative numerical are presented.