{
  "id": "1409.1968",
  "version": "v1",
  "published": "2014-09-06T00:02:38.000Z",
  "updated": "2014-09-06T00:02:38.000Z",
  "title": "The proof of three power-exponential inequalities",
  "authors": [
    "Anibal Coronel",
    "Fernando huancas"
  ],
  "comment": "9 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "In this paper we prove three power-exponential inequalities for positive real numbers. In particular, we conclude that this proofs give affirmatively answers to three, until now, open problems (conjectures~4.4, 2.1 and 2.2) posed by C{\\^i}rtoaje in the following two works: \"{\\it J. Inequal. Pure Appl. Math.} 10, Article 21, 2009\" and \"{\\it J. Nonlinear Sci. Appl.} 4:2:130-137, 2011\". Moreover, we present a new proof of the inequality $a^{ra}+b^{rb}\\ge a^{rb}+b^{ra}$ for all positive real numbers $a$ and $b$ and $r\\in [0,e]$. In addition, three new conjectures are presented.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-09-06T00:02:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "26D10",
      "26D07"
    ],
    "keywords": [
      "inequality",
      "power-exponential inequalities",
      "positive real numbers",
      "open problems",
      "pure appl"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1409.1968C"
    }
  }
}