On the $L_2$ Markov Inequality with Laguerre Weight

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

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