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

Let $w_{\lambda }(t)=(1-t^2{)}^{\lambda -1\mathrm{/}2}$, $\lambda >-1\mathrm{/}2$, be the Gegenbauer weight function, and $\mathrm{\parallel }\cdot \mathrm{\parallel }$ denote the associated $L_2$-norm, i.e., $$\mathrm{\parallel }f\mathrm{\parallel }:=({\int }_{-1}^1{w}_{\lambda }(t)\mathrm{\mid }f(t){\mathrm{\mid }}^{2}dt{)}^{1\mathrm{/}2}\mathrm{.}$$ 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 $$\mathrm{\parallel }{p}^{\mathrm{\prime }}\mathrm{\parallel }\le c_{n,\lambda }\mathrm{\parallel }p\mathrm{\parallel },p\in {\mathcal{P}}_n,$$ and prove that $$c_{n,\lambda }<\frac{(n+1)(n+2\lambda +1)}{2\sqrt{2\lambda +1}},\lambda >-1\mathrm{/}2\mathrm{.}$$ Moreover, we prove that the extremal polynomial in this inequality is even or odd depending on whether $n$ is even or odd.