Title: Markov $L_2$-inequality with the Laguerre weight,
Abstract: Let $w_\alpha(t) := t^{\alpha}\,e^{-t}$, where $\alpha > -1$, be the Laguerre
weight function, and let $\|\cdot\|_{w_\alpha}$ be the associated $L_2$-norm,
$$ \|f\|_{w_\alpha} = \left\{\int_{0}^{\infty} |f(x)|^2
w_\alpha(x)\,dx\right\}^{1/2}\,. $$ By $\mathcal{P}_n$ we denote the set of
algebraic polynomials of degree $\le n$.
We study the best constant $c_n(\alpha)$ in the Markov inequality in this
norm $$ \|p_n'\|_{w_\alpha} \le c_n(\alpha) \|p_n\|_{w_\alpha}\,,\qquad p_n \in
\mathcal{P}_n\,, $$ namely the constant $$ c_n(\alpha) := \sup_{p_n \in
\mathcal{P}_n} \frac{\|p_n'\|_{w_\alpha}}{\|p_n\|_{w_\alpha}}\,. $$ We derive
explicit lower and upper bounds for the Markov constant $c_n(\alpha)$, as well
as for the asymptotic Markov constant $$
c(\alpha)=\lim_{n\rightarrow\infty}\frac{c_n(\alpha)}{n}\,. $$