{
  "id": "1503.00345",
  "version": "v1",
  "published": "2015-03-01T20:29:19.000Z",
  "updated": "2015-03-01T20:29:19.000Z",
  "title": "Exact upper and lower bounds on the difference between the arithmetic and geometric means",
  "authors": [
    "Iosif Pinelis"
  ],
  "comment": "8 pages; to appear in the Bulletin of the Australian Mathematical Society",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $X$ denote a random variable with $\\mathsf{E} X<\\infty$. Upper and lower bounds on $\\mathsf{E} X-\\exp\\mathsf{E}\\ln X$ are obtained, which are exact, in terms of $V_X$ and $E_X$ for the upper bound and in terms of $V_X$ and $F_X$ for the lower bound, where $V_X:=\\mathsf{Var}\\sqrt X$, $E_X:=\\mathsf{E}\\big(\\sqrt X-\\sqrt{m_X}\\,\\big)^2$, $F_X:=\\mathsf{E}\\big(\\sqrt{M_X}-\\sqrt X\\,\\big)^2$, $m_X:=\\inf S_X$, $M_X:=\\sup S_X$, and $S_X$ is the support set of the distribution of $X$. Note that, if $X$ takes each of distinct real values $x_1,\\dots,x_n$ with probability $1/n$, then $\\mathsf{E} X$ and $\\exp\\mathsf{E}\\ln X$ are, respectively, the arithmetic and geometric means of $x_1,\\dots,x_n$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-03-01T20:29:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60E15",
      "26D15",
      "90C46"
    ],
    "keywords": [
      "lower bound",
      "geometric means",
      "exact upper",
      "arithmetic",
      "difference"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150300345P"
    }
  }
}