ESTIMATES FOR THE BEST CONSTANT IN A MARKOV L2–INEQUALITY WITH THE ASSISTANCE OF COMPUTER ALGEBRA

Abstract

We prove two-sided estimates for the best (i.e., the smallest possible) constant $c_n(\alpha )$ in the Markov inequality $${\Vert p{\prime }_n\Vert }_{w_{\alpha }}\le c_n(\alpha ){\Vert p_n\Vert }_{w_{\alpha }},p_n\in {\mathcal{P}}_n.$$

Here, ${\mathcal{P}}_n$ stands for the set of algebraic polynomials of degree $\le n,w_{\alpha }(x):=x^{\alpha }{e}^{-x},\alpha >-1$, is the Laguerre weight function, and ${\Vert \cdot \Vert }_{w_{\alpha }}$ is the associated $L_2$-norm, $${\Vert f\Vert }_{w_{\alpha }}=({\int }_0^{\mathrm{\infty }}|f(x){|}^2{w}_{\alpha }(x)dx{)}^{1/2}.$$

Our approach is based on the fact that $c_n^{-2}(\alpha )$ equals the smallest zero of a polynomial $Q_n$, orthogonal with respect to a measure supported on the positive axis and defined by an explicit three-term recurrence relation. We employ computer algebra to evaluate the seven lowest degree coefficients of $Q_n$ and to obtain thereby bounds for $c_n(\alpha )$. This work is a continuation of a recent paper [5], where estimates for $c_n(\alpha )$ were proven on the basis of the four lowest degree coefficients of $Q_n$.