{
  "id": "1206.4911",
  "version": "v2",
  "published": "2012-06-21T15:19:40.000Z",
  "updated": "2012-06-24T15:21:19.000Z",
  "title": "Refinements of Mitrinović-Cusa inequality",
  "authors": [
    "Zhen-Hang Yang"
  ],
  "comment": "13 pages",
  "categories": [
    "math.CA"
  ],
  "abstract": "The Mitrinovi\\'c-Cusa inequality states that for x\\in(0,{\\pi}/2) (cos x)^{1/3}<((sin x)/x)<((2+cos x)/3) hold. In this paper, we prove that (cos x)^{1/3}<(cos px)^{1/(3p^{2})}<((sin x)/x)<(cos qx)^{1/(3q^{2})}<((2+cos x)/3) hold for x\\in(0,{\\pi}/2) if and only if p\\in[p_{1},1) and q\\in(0,1/\\surd5], where p_{1}=0.45346830977067.... And the function p\\mapsto(cos px)^{1/(3p^{2})} is decreasing on (0,1]. Our results greatly refine the Mitrinovi\\'c-Cusa inequality.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-06-24T15:21:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "26D05",
      "26D15",
      "26A48",
      "33F05"
    ],
    "keywords": [
      "mitrinović-cusa inequality",
      "refinements",
      "mitrinovic-cusa inequality states",
      "results greatly refine"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1206.4911Y"
    }
  }
}