{
  "id": "1411.7708",
  "version": "v1",
  "published": "2014-11-27T21:21:03.000Z",
  "updated": "2014-11-27T21:21:03.000Z",
  "title": "Levin Steckin theorem and inequalities of the Hermite-Hadamard type",
  "authors": [
    "Tomasz Szostok"
  ],
  "categories": [
    "math.CA"
  ],
  "abstract": "Recently Ohlin lemma on convex stochastic ordering was used to obtain some inequalities of Hermite-Hadamard type. Continuing this idea, we use Levin-Ste\\v{c}kin result to determine all inequalities of the forms: $$\\sum_{i=1}^3a_if(\\alpha_ix+(1-\\alpha_i)y)\\leq \\frac{1}{y-x}\\int_{x}^yf(t),$$ $$a_1f(x)+\\sum_{i=2}^3a_if(\\alpha_ix+(1-\\alpha_i)y)+a_4f(y)\\geq \\frac{1}{y-x}\\int_{x}^yf(t)$$ and $$af(\\alpha_1x+(1-\\alpha_1)y)+(1-a)f(\\alpha_2x+(1-\\alpha_2)y)\\leq b_1f(x)+b_2f(\\beta x+(1-\\beta)y)+b_3f(y)$$ which are satisfied by all convex functions $f:[x,y]\\to{\\mathbb R}.$ As it is easy to see, the same methods may be applied to deal with longer expressions of the forms considered. As particular cases of our results we obtain some known inequalities.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-11-27T21:21:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "26A51",
      "26D10",
      "39B62"
    ],
    "keywords": [
      "levin steckin theorem",
      "hermite-hadamard type",
      "inequalities",
      "longer expressions",
      "ohlin lemma"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1411.7708S"
    }
  }
}