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 $\,Q_n$.
Categories: math.CA