{
  "id": "2001.06302",
  "version": "v1",
  "published": "2020-01-16T00:18:26.000Z",
  "updated": "2020-01-16T00:18:26.000Z",
  "title": "On the closest to zero roots and the second quotients of Taylor coefficients of entire functions from the Laguerre-Pólya I class",
  "authors": [
    "Thu Hien Nguyen",
    "Anna Vishnyakova"
  ],
  "comment": "18 pages. Submitted to Results in Mathematics. Partially supported by the Akhiezer Foundation. arXiv admin note: substantial text overlap with arXiv:1903.09070, arXiv:1912.10035",
  "categories": [
    "math.CV",
    "math.FA"
  ],
  "abstract": "For an entire function $f(z) = \\sum_{k=0}^\\infty a_k z^k, a_k>0,$ we show that if $f$ belongs to the Laguerre-P\\'olya class, and the quotients $q_k := \\frac{a_{k-1}^2}{a_{k-2}a_k}, k=2, 3, \\ldots $ satisfy the condition $q_2 \\leq q_3,$ then $f$ has at least one zero in the segment $[-\\frac{a_1}{a_2},0].$ We also give necessary conditions and sufficient conditions of the existence of such a zero in terms of the quotients $q_k$ for $k=2,3, 4.$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-01-16T00:18:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "30C15",
      "30D15",
      "30D35",
      "26C10"
    ],
    "keywords": [
      "entire function",
      "zero roots",
      "second quotients",
      "taylor coefficients",
      "laguerre-pólya"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}