{
  "id": "2212.05692",
  "version": "v1",
  "published": "2022-12-12T04:13:15.000Z",
  "updated": "2022-12-12T04:13:15.000Z",
  "title": "Hutchinson's intervals and entire functions from the Laguerre-Pólya class",
  "authors": [
    "Thu Hien Nguyen",
    "Anna Vishnyakova"
  ],
  "comment": "10 pages",
  "categories": [
    "math.CV"
  ],
  "abstract": "We find the intervals $[\\alpha, \\beta (\\alpha)]$ such that if a univariate real polynomial or entire function $f(z) = a_0 + a_1 z + a_2 z^2 + \\cdots $ with positive coefficients satisfy the conditions $ \\frac{a_{k-1}^2}{a_{k-2}a_{k}} \\in [\\alpha, \\beta(\\alpha)]$ for all $k \\geq 2,$ then $f$ belongs to the Laguerre--P\\'olya class. For instance, from J.I.~Hutchinson's theorem, one can observe that $f$ belongs to the Laguerre--P\\'olya class (has only real zeros) when $q_k(f) \\in [4, + \\infty).$ We are interested in finding those intervals which are not subsets of $[4, + \\infty).$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-12-12T04:13:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "30C15",
      "30D15",
      "30D35",
      "26C10"
    ],
    "keywords": [
      "entire function",
      "laguerre-pólya class",
      "hutchinsons intervals",
      "laguerre-polya class",
      "univariate real polynomial"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}