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

Abstract

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