{
  "id": "1301.6432",
  "version": "v1",
  "published": "2013-01-28T03:01:37.000Z",
  "updated": "2013-01-28T03:01:37.000Z",
  "title": "A new proof of the geometric-arithmetic mean inequality by Cauchy's integral formula",
  "authors": [
    "Feng Qi",
    "Xiao-Jing Zhang",
    "Wen-Hui Li"
  ],
  "comment": "5 pages",
  "journal": "Feng Qi, Xiao-Jing Zhang, and Wen-Hui Li, L\\'evy-Khintchine representation of the geometric mean of many positive numbers and applications, Mathematical Inequalities & Applications 17 (2014), no. 2, 719--729",
  "doi": "10.7153/mia-17-53",
  "categories": [
    "math.CA",
    "math.CV"
  ],
  "abstract": "Let $a=(a_1,a_2,...c,a_n)$ for $n\\in\\mathbb{N}$ be a given sequence of positive numbers. In the paper, the authors establish, by using Cauchy's integral formula in the theory of complex functions, an integral representation of the principal branch of the geometric mean {equation*} G_n(a+z)=\\Biggl[\\prod_{k=1}^n(a_k+z)\\Biggr]^{1/n} {equation*} for $z\\in\\mathbb{C}\\setminus(-\\infty,-\\min\\{a_k,1\\le k\\le n\\}]$, and then provide a new proof of the well known GA mean inequality.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-01-28T03:01:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "26E60",
      "30E20",
      "26A48",
      "44A20"
    ],
    "keywords": [
      "cauchys integral formula",
      "geometric-arithmetic mean inequality",
      "ga mean inequality",
      "complex functions",
      "integral representation"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 5,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1301.6432Q"
    }
  }
}