On the $L_2$ Markov Inequality with Laguerre Weight

Let $w_{\alpha}(t)=t^{\alpha}\,e^{-t}$, $\alpha>-1$, be the Laguerre weight function, and $|\cdot|_{w_\alpha}$ denote the associated $L_2$-norm, i.e., $$| f|_{w_\alpha}:=\Big(\int_{0}^{\infty}w_{\alpha}(t)| f(t)|^2\,dt\Big)^{1/2}.$$ Denote by ${\cal P}_n$ the set of algebraic polynomials of degree not exceeding $n$. We study the best constant $c_n(\alpha)$ in the Markov inequality in this norm, $$| p^{\prime}|_{w_\alpha}\leq c_n(\alpha)\,| p|_{w_\alpha}\,,\quad p\in {\cal P}_n\,,$$ namely the constant $$c_{n}(\alpha)=\sup_{\mathop{}^{p\in {\cal P}_n}_{p\ne 0}}\frac{| p^{\prime}|_{w_\alpha}}{| p|_{w_\alpha}}\,,$$ and we are also interested in its asymptotic value $$c(\alpha)=\lim_{n\rightarrow\infty}\frac{c_{n}(\alpha)}{n}\,.$$ In this paper we obtain lower and upper bounds for both $c_{n}(\alpha)$ and $c(\alpha)$. % Note that according to a result of P. D\"{o}rfler from 2002, $c(\alpha)=[j_{(\alpha-1)/2,1}]^{-1}$, with $j_{\nu,1}$ being the first positive zero of the Bessel function $J_{\nu}(z)$, hence our bounds for $c(\alpha)$ imply bounds for $j_{(\alpha-1)/2,1}$ as well.