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