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 \[ {\Vert p\prime_n \Vert}_{w_\alpha} \leq c_n(\alpha){\Vert p_n \Vert}_{w_\alpha}, p_n \in \mathcal{P}_n . \]

Here, $\mathcal{P}_n$ stands for the set of algebraic polynomials of degree $ \leq 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, \[ {\Vert f \Vert}_{w_\alpha} = \Bigg(\int_{0}^{\infty}\,|f(x)|^2w_\alpha(x)\,dx\Bigg)^{1/2}\quad. \]

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 $Q_n$.