Estimates for the best constant in a Markov $L_2$-inequality with the assistance of computer algebra

TL;DR

This paper derives two-sided estimates for the optimal constant in a Markov inequality involving Laguerre weights, using orthogonal polynomials and computer algebra to improve bounds based on polynomial coefficients.

Contribution

It introduces a novel approach employing computer algebra to evaluate polynomial coefficients for tighter bounds on the Markov inequality constant.

Findings

Derived two-sided bounds for the Markov constant c_n(α).

Used computer algebra to evaluate polynomial coefficients for improved estimates.

Extended previous work by considering higher-degree polynomial coefficients.

Abstract

We prove two-sided estimates for the best (i.e., the smallest possible) constant $\,c_n(\alpha)\,$ in the Markov inequality $$ \|p_n'\|_{w_\alpha} \le c_n(\alpha) \|p_n\|_{w_\alpha}\,, \qquad p_n \in {\cal P}_n\,. $$ Here, ${\cal P}_n$ 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 $\|\cdot\|_{w_\alpha}$ is the associated $L_2$-norm, $$ \|f\|_{w_\alpha} = \left(\int_{0}^{\infty} |f(x)|^2 w_\alpha(x)\,dx\right)^{1/2}\,. $$ 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 $\,Q_n$.