{
  "id": "0903.1984",
  "version": "v1",
  "published": "2009-03-11T14:36:24.000Z",
  "updated": "2009-03-11T14:36:24.000Z",
  "title": "Sharp inequalities for polygamma functions",
  "authors": [
    "Feng Qi",
    "Bai-Ni Guo"
  ],
  "comment": "11 pages",
  "categories": [
    "math.CA"
  ],
  "abstract": "The main aim of this paper is to prove that the double inequality \\frac{(k-1)!}{\\Bigl\\{x+\\Bigl[\\frac{(k-1)!}{|\\psi^{(k)}(1)|}\\Bigr]^{1/k}\\Bigr\\}^k} +\\frac{k!}{x^{k+1}}<\\bigl|\\psi^{(k)}(x)\\bigr|<\\frac{(k-1)!}{\\bigl(x+\\frac12\\bigr)^k}+\\frac{k!}{x^{k+1}} holds for $x>0$ and $k\\in\\mathbb{N}$ and that the constants $\\Bigl[\\frac{(k-1)!}{|\\psi^{(k)}(1)|}\\Bigr]^{1/k}$ and $\\frac12$ are the best possible. In passing, some related inequalities and (logarithmically) complete monotonicity results concerning the gamma, psi and polygamma functions are surveyed.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-03-11T14:36:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "26A48",
      "26D07",
      "26D15",
      "33B15"
    ],
    "keywords": [
      "polygamma functions",
      "sharp inequalities",
      "main aim",
      "complete monotonicity results concerning",
      "related inequalities"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0903.1984Q"
    }
  }
}