On the L2 markov inequality with laguerre weight

Abstract

Let wα(t) = tα e-t, α > -1, be the Laguerre weight function, and ‖·‖wα denote the associated L2-norm, i.e., ‖f‖wα: (Formula Presented) Denote by Pn the set of algebraic polynomials of degree not exceeding n. We study the best constant cn(α) in the Markov inequality in this norm, ‖p′‖ wα ≤ cn.(α) ‖p‖wα, p ϵ Pn, namely the constant cn.(α) (Formula Presented) and we are also interested in its asymptotic value cn(α) (Formula Presented) In this paper we obtain lower and upper bounds for both cn(α) and c (α). Note that according to a result of P. Dörfler from 2002, c. (α) = [j(α-1)/ 2,1]-1, with jv,1 being the first positive zero of the Bessel function Jv(z), hence our bounds for c(α) imply bounds for j(α-1)/2,1 as well.