{
  "id": "2004.07567",
  "version": "v1",
  "published": "2020-04-16T10:08:47.000Z",
  "updated": "2020-04-16T10:08:47.000Z",
  "title": "A tight Hermite-Hadamard's inequality and a generic method for comparison between residuals of inequalities with convex functions",
  "authors": [
    "Milan Merkle",
    "Zoran D. Mitrović"
  ],
  "comment": "11 pages, 3 figures",
  "categories": [
    "math.CA"
  ],
  "abstract": "We present a tight parametrical Hermite-Hadamard type inequality with probability measure, which yields a considerably closer upper bound for the mean value of convex function than the classical one. Our inequality becomes equality not only with affine functions, but also with a family of V-shaped curves determined by the parameter. The residual (error) of this inequality is strictly smaller than in the classical Hermite-Hadamard inequality under any probability measure and with all non-affine convex functions. In the framework of Karamata's theorem on the inequalities with convex functions, we propose a method of measuring a global performance of inequalities in terms of average residuals over functions of the type $x\\mapsto |x-u|$. Using average residuals enables comparing two or more inequalities as themselves, with same or different measures and without referring to a particular function. Our method is applicable to all Karamata's type inequalities, with integrals or sums. A numerical experiment with three different measures indicates that the average residual in our inequality is about 4 times smaller than in classical right Hermite-Hadamard, and also is smaller than in Jensen's inequality, with all three measures.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-04-16T10:08:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "26A51",
      "60E15",
      "26D15"
    ],
    "keywords": [
      "convex function",
      "tight hermite-hadamards inequality",
      "generic method",
      "average residual",
      "tight parametrical hermite-hadamard type inequality"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}