{
  "id": "1405.4384",
  "version": "v1",
  "published": "2014-05-17T11:56:10.000Z",
  "updated": "2014-05-17T11:56:10.000Z",
  "title": "Sharp Bounds for Neuman Means in Terms of Geometric, Arithemtic and Quadratic Means",
  "authors": [
    "Zhi-Jun Guo",
    "Yan Zhang",
    "Yu-Ming Chu",
    "Ying-Qing Song"
  ],
  "comment": "11 pages",
  "categories": [
    "math.CA",
    "math.OC"
  ],
  "abstract": "In this paper, we find the greatest values $\\alpha_{1}$, $\\alpha_{2}$, $\\alpha_{3}$, $\\alpha_{4}$, $\\alpha_{5}$, $\\alpha_{6}$, $\\alpha_{7}$, $\\alpha_{8}$ and the least values $\\beta_{1}$, $\\beta_{2}$, $\\beta_{3}$, $\\beta_{4}$, $\\beta_{5}$, $\\beta_{6}$, $\\beta_{7}$, $\\beta_{8}$ such that the double inequalities $$A^{\\alpha_{1}}(a,b)G^{1-\\alpha_{1}}(a,b)<N_{GA}(a,b)<A^{\\beta_{1}}(a,b)G^{1-\\beta_{1}}(a,b),$$ $$\\frac{\\alpha_{2}}{G(a,b)}+\\frac{1-\\alpha_{2}}{A(a,b)}<\\frac{1}{N_{GA}(a,b)}<\\frac{\\beta_{2}}{G(a,b)}+\\frac{1-\\beta_{2}}{A(a,b)},$$ $$A^{\\alpha_{3}}(a,b)G^{1-\\alpha_{3}}(a,b)<N_{AG}(a,b)<A^{\\beta_{3}}(a,b)G^{1-\\beta_{3}}(a,b),$$ $$\\frac{\\alpha_{4}}{G(a,b)}+\\frac{1-\\alpha_{4}}{A(a,b)}<\\frac{1}{N_{AG}(a,b)}<\\frac{\\beta_{4}}{G(a,b)}+\\frac{1-\\beta_{4}}{A(a,b)},$$ $$Q^{\\alpha_{5}}(a,b)A^{1-\\alpha_{5}}(a,b)<N_{AQ}(a,b)<Q^{\\beta_{5}}(a,b)A^{1-\\beta_{5}}(a,b),$$ $$\\frac{\\alpha_{6}}{A(a,b)}+\\frac{1-\\alpha_{6}}{Q(a,b)}<\\frac{1}{N_{AQ}(a,b)}<\\frac{\\beta_{6}}{A(a,b)}+\\frac{1-\\beta_{6}}{Q(a,b)},$$ $$Q^{\\alpha_{7}}(a,b)A^{1-\\alpha_{7}}(a,b)<N_{QA}(a,b)<Q^{\\beta_{7}}(a,b)A^{1-\\beta_{7}}(a,b),$$ $$\\frac{\\alpha_{8}}{A(a,b)}+\\frac{1-\\alpha_{8}}{Q(a,b)}<\\frac{1}{N_{QA}(a,b)}<\\frac{\\beta_{8}}{A(a,b)}+\\frac{1-\\beta_{8}}{Q(a,b)}$$ hold for all $a, b>0$ with $a\\neq b$, where $G$, $A$ and $Q$ are respectively the geometric, arithmetic and quadratic means, and $N_{GA}$, $N_{AG}$, $N_{AQ}$ and $N_{QA}$ are the Neuman means derived from the Schwab-Borchardt mean.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-05-17T11:56:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "26E60",
      "G.1.2"
    ],
    "keywords": [
      "quadratic means",
      "sharp bounds",
      "arithemtic",
      "greatest values",
      "schwab-borchardt mean"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1405.4384G"
    }
  }
}