{
  "id": "2008.04754",
  "version": "v1",
  "published": "2020-08-08T19:42:23.000Z",
  "updated": "2020-08-08T19:42:23.000Z",
  "title": "On the entire functions from the Laguerre-Pólya I class having the increasing second quotients of Taylor coefficients",
  "authors": [
    "Thu Hien Nguyen",
    "Anna Vishnyakova"
  ],
  "comment": "16 pages. arXiv admin note: text overlap with arXiv:2001.06302, arXiv:1912.10035, arXiv:1903.09070",
  "categories": [
    "math.CV",
    "math.FA"
  ],
  "abstract": "We prove that if $f(x) = \\sum_{k=0}^\\infty a_k x^k,$ $a_k >0, $ is an entire function such that the sequence $Q := \\left( \\frac{a_k^2}{a_{k-1}a_{k+1}} \\right)_{k=1}^\\infty$ is non-decreasing and $\\frac{a_1^2}{a_{0}a_{2}} \\geq 2\\sqrt[3]{2},$ then all but a finite number of zeros of $f$ are real and simple. We also present a criterion in terms of the closest to zero roots for such a function to have only real zeros (in other words, for belonging to the Laguerre--P\\'olya class of type I) under additional assumption on the sequence $Q.$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-08-08T19:42:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "30C15",
      "30D15",
      "30D35",
      "26C10"
    ],
    "keywords": [
      "increasing second quotients",
      "entire function",
      "taylor coefficients",
      "laguerre-pólya",
      "finite number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}