{
  "id": "0902.2513",
  "version": "v3",
  "published": "2009-02-15T01:52:21.000Z",
  "updated": "2011-01-25T00:53:44.000Z",
  "title": "An extension of an inequality for ratios of gamma functions",
  "authors": [
    "Feng Qi",
    "Bai-Ni Guo"
  ],
  "comment": "8 pages",
  "journal": "Bai-Ni Guo and Feng Qi, An extension of an inequality for ratios of gamma functions, Journal of Approximation Theory 163 (2011), no. 9, 1208--1216",
  "doi": "10.1016/j.jat.2011.04.003",
  "categories": [
    "math.CA"
  ],
  "abstract": "In this paper, we prove that for $x+y>0$ and $y+1>0$ the inequality {equation*} \\frac{[\\Gamma(x+y+1)/\\Gamma(y+1)]^{1/x}}{[\\Gamma(x+y+2)/\\Gamma(y+1)]^{1/(x+1)}} <\\biggl(\\frac{x+y}{x+y+1}\\biggr)^{1/2} {equation*} is valid if $x>1$ and reversed if $x<1$ and that the power $\\frac12$ is the best possible, where $\\Gamma(x)$ is the Euler gamma function. This extends the result in [Y. Yu, \\textit{An inequality for ratios of gamma functions}, J. Math. Anal. Appl. \\textbf{352} (2009), no.~2, 967\\nobreakdash--970.] and resolves an open problem posed in [B.-N. Guo and F. Qi, \\emph{Inequalities and monotonicity for the ratio of gamma functions}, Taiwanese J. Math. \\textbf{7} (2003), no.~2, 239\\nobreakdash--247.].",
  "revisions": [
    {
      "version": "v3",
      "updated": "2011-01-25T00:53:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "26D07",
      "33B15"
    ],
    "keywords": [
      "inequality",
      "euler gamma function",
      "open problem"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0902.2513Q"
    }
  }
}