Markov $L_2$ inequality with the Gegenbauer weight

For the Gegenbauer weight function $w_{\lambda}(t)=(1-t^2)^{\lambda-1/2}$, $\lambda>-1/2$, we denote by $\Vert\cdot\Vert_{w_{\lambda}}$ the associated $L_2$-norm, $$\Vert f\Vert_{w_{\lambda}}:=\Big(\int_{-1}^{1}w_{\lambda}(t)f^2(t)\,dt\Big)^{1/2}.$$ We study the Markov inequality $$\Vert p^{\prime}\Vert_{w_{\lambda}}\leq c_{n}(\lambda)\,\Vert p\Vert_{w_{\lambda}},\qquad p\in \mathcal{P}_n,$$ where $\mathcal{P}_n$ is the class of algebraic polynomials of degree not exceeding $n$. Upper and lower bounds for the best Markov constant $c_{n}(\lambda)$ are obtained, which are valid for all $n\in \mathbb{N}$ and $\lambda>-\frac{1}{2}$.