{
  "id": "1711.07398",
  "version": "v1",
  "published": "2017-11-20T16:41:08.000Z",
  "updated": "2017-11-20T16:41:08.000Z",
  "title": "Estimates for the best constant in a Markov $L_2$-inequality with the assistance of computer algebra",
  "authors": [
    "Geno Nikolov",
    "Rumen Uluchev"
  ],
  "categories": [
    "math.CA"
  ],
  "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 $\\,Q_n$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-11-20T16:41:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "41A17"
    ],
    "keywords": [
      "best constant",
      "inequality",
      "seven lowest degree coefficients",
      "assistance",
      "explicit three-term recurrence relation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}