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 $$ c(\alpha)=\lim_{ n\rightarrow\infty}\frac{ c_n(\alpha)}{ n}\,. $$
Categories: math.CA