Estimates for the best constant in a Markov $L_2$-inequality with the assistance of computer algebra (1711.07398v1)

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 paper [5], where estimates for $\,c_n(\alpha)\,$ were proven on the basis of the four lowest degree coefficients of $\,Q_n$.