On the Markov inequality in the $L_2$-norm with Gegenbauer weight

Abstract

Let $w_{\lambda}(t)=(1-t^2)^{\lambda-1/2}$, $\lambda>-1/2$, be the Gegenbauer weight function, and $\Vert\cdot\Vert$ denote the associated $L_2$-norm, i.e., $$ \Vert f\Vert:=\Big(\int_{-1}^{1}w_{\lambda}(t)\vert f(t)\vert^2\,dt\Big)^{1/2}. $$ Denote by $\mathcal{P}_n$ the set of algebraic polynomials of degree not exceeding $n$. We study the best (i.e., the smallest) constant $c_{n,\lambda}$ in the Markov inequality $$ \Vert p^{\prime}\Vert\leq c_{n,\lambda}\,\Vert p\Vert,\qquad p\in \mathcal{P}_n, $$ and prove that $$ c_{n,\lambda}< \frac{(n+1)(n+2\lambda+1)}{2\sqrt{2\lambda+1}},\qquad \lambda>-1/2\,. $$ Moreover, we prove that the extremal polynomial in this inequality is even or odd depending on whether $n$ is even or odd.