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