Estimates for the best constant in a Markov $L_2$-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 $$\Vert p_n^{\prime }{\Vert }_{w_{\alpha }}\le c_n(\alpha )\Vert p_n{\Vert }_{w_{\alpha }},p_n\in {\mathcal{P}}_n.$$ 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, $$\Vert f{\Vert }_{w_{\alpha }}={\left({\int }_0^{\infty }|f(x){|}^2{w}_{\alpha }(x)dx\right)}^{1/2}.$$ 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 $Q_n$.

References in corpus (1)

• Markov $L_2$-inequality with the Laguerre weight