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

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 \(\,Q_n\).