{
  "id": "math/0703084",
  "version": "v2",
  "published": "2007-03-03T12:14:34.000Z",
  "updated": "2008-11-11T03:42:27.000Z",
  "title": "Inequalities and monotonicity of ratios for generalized hypergeometric function",
  "authors": [
    "D. Karp",
    "S. M. Sitnik"
  ],
  "comment": "15 pages, this is the form accepted by Journal of Approximation Theory. Many typos corrected, important references added, several remarks added thoughout the paper. Principal results did not change",
  "categories": [
    "math.CA"
  ],
  "abstract": "We find two-sided inequalities for the generalized hypergeometric function of the form ${_{q+1}}F_{q}(-x)$ with positive parameters restricted by certain additional conditions. Both lower and upper bounds agree with the value of ${_{q+1}}F_{q}(-x)$ at the endpoints of positive semi-axis and are asymptotically precise at one of the endpoints. The inequalities are derived from a theorem asserting the monotony of the quotient of two generalized hypergeometric functions with shifted parameters. The proofs hinge on a generalized Stieltjes representation of the generalized hypergeometric function. This representation also provides yet another method to deduce the second Thomae relation for ${_{3}F_{2}}(1)$ and leads to an integral representations of ${_{4}F_{3}}(x)$ in terms of the Appell function $F_3$. In the last section of the paper we list some open questions and conjectures.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2008-11-11T03:42:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "33C20"
    ],
    "keywords": [
      "generalized hypergeometric function",
      "inequalities",
      "monotonicity",
      "upper bounds agree",
      "second thomae relation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007math......3084K"
    }
  }
}