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

Abstract: Let $w_{\lambda}(t) := (1-t^2)^{\lambda-1/2}$, where $\lambda > -\frac{1}{2}$, be the Gegenbauer weight function, let $|\cdot|{w{\lambda}}$ be the associated $L_2$-norm, $$ |f|{w{\lambda}} = \left{\int_{-1}^1 |f(x)|^2 w_{\lambda}(x)\,dx\right}^{1/2}\,, $$ and denote by $\mathcal{P}n$ the space of algebraic polynomials of degree $\le n$. We study the best constant $c_n(\lambda)$ in the Markov inequality in this norm $$ |p_n'|{w_{\lambda}} \le c_n(\lambda) |p_n|{w{\lambda}}\,,\qquad p_n \in \mathcal{P}n\,, $$ namely the constant $$ c_n(\lambda) := \sup{p_n \in \mathcal{P}n} \frac{|p_n'|{w_{\lambda}}}{|p_n|{w{\lambda}}}\,. $$ We derive explicit lower and upper bounds for the Markov constant $c_n(\lambda)$, which are valid for all $n$ and $\lambda$.