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

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