{
  "id": "1301.6430",
  "version": "v1",
  "published": "2013-01-28T02:57:25.000Z",
  "updated": "2013-01-28T02:57:25.000Z",
  "title": "Some Bernstein functions and integral representations concerning harmonic and geometric means",
  "authors": [
    "Feng Qi",
    "Xiao-Jing Zhang",
    "Wen-Hui Li"
  ],
  "comment": "19 pages",
  "categories": [
    "math.CA",
    "math.CV"
  ],
  "abstract": "It is general knowledge that the harmonic mean $H(x,y)=\\frac2{\\frac1x+\\frac1y}$ and that the geometric mean $G(x,y)=\\sqrt{xy}\\,$, where $x$ and $y$ are two positive numbers. In the paper, the authors show by several approaches that the harmonic mean $H_{x,y}(t)=H(x+t,y+t)$ and the geometric mean $G_{x,y}(t)=G(x+t,y+t)$ are all Bernstein functions of $t\\in(-\\min\\{x,y\\},\\infty)$ and establish integral representations of the means $H_{x,y}(t)$ and $G_{x,y}(t)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-01-28T02:57:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "26E60",
      "26A48",
      "30E20",
      "44A10"
    ],
    "keywords": [
      "integral representations concerning harmonic",
      "geometric mean",
      "bernstein functions",
      "harmonic mean",
      "general knowledge"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1301.6430Q"
    }
  }
}