{
  "id": "1705.06547",
  "version": "v1",
  "published": "2017-05-18T12:19:55.000Z",
  "updated": "2017-05-18T12:19:55.000Z",
  "title": "Inequalities for the inverses of the polygamma functions",
  "authors": [
    "Necdet Batir"
  ],
  "comment": "submitted",
  "categories": [
    "math.CA"
  ],
  "abstract": "We provide an elementary proof of the left side inequality and improve the right inequality in \\bigg[\\frac{n!}{x-(x^{-1/n}+\\alpha)^{-n}}\\bigg]^{\\frac{1}{n+1}}&<((-1)^{n-1}\\psi^{(n)})^{-1}(x) &<\\bigg[\\frac{n!}{x-(x^{-1/n}+\\beta)^{-n}}\\bigg]^{\\frac{1}{n+1}}, where $\\alpha=[(n-1)!]^{-1/n}$ and $\\beta=[n!\\zeta(n+1)]^{-1/n}$, which was proved in \\cite{6}, and we prove the following inequalities for the inverse of the digamma function $\\psi$. \\frac{1}{\\log(1+e^{-x})}<\\psi^{-1}(x)< e^{x}+\\frac{1}{2}, \\quad x\\in\\mathbb{R}. The proofs are based on nice applications of the mean value theorem for differentiation and elementary properties of the polygamma functions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-05-18T12:19:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "33B15",
      "26D07"
    ],
    "keywords": [
      "polygamma functions",
      "left side inequality",
      "mean value theorem",
      "right inequality",
      "elementary proof"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}