{
  "id": "1604.01857",
  "version": "v1",
  "published": "2016-04-07T03:08:36.000Z",
  "updated": "2016-04-07T03:08:36.000Z",
  "title": "The Hermite-Hadamard inequality on hypercuboid",
  "authors": [
    "Mohammad W. Alomari"
  ],
  "comment": "12 pages",
  "categories": [
    "math.CA"
  ],
  "abstract": "Given any ${\\bf{a}}: = \\left( {a_1 ,a_2 , \\ldots ,a_n } \\right)$ and ${\\bf{b}}: = \\left( {b_1 ,b_2 , \\ldots ,b_n } \\right)$ in $\\mathbb{R}^n$. The $\\textbf{n}$-fold convex function defined on $\\left[ {{\\bf{a}},{\\bf{b}}} \\right]$, ${\\bf{a}},{\\bf{b}} \\in \\mathbb{R}^n$ with ${\\bf{a}}<{\\bf{b}}$ is a convex function in each variable separately. In this work we prove an inequality of Hermite-Hadamard type for $\\textbf{n}$-fold convex functions. Namely, we establish the inequality \\begin{align*} f\\left( {\\frac{{{\\bf{a}} + {\\bf{b}}}}{2}} \\right) \\le \\frac{1}{{{\\bf{b}} - {\\bf{a}}}}\\int_{\\bf{a}}^{\\bf{b}} {f\\left( {\\bf{x}} \\right)d{\\bf{x}}} \\le \\frac{1}{{2^n }}\\sum\\limits_{\\bf{c}} {f\\left( {\\bf{c}} \\right)}, \\end{align*} where $\\sum\\limits_{\\bf{c}} {f\\left( {\\bf{c}} \\right)} : = \\sum\\limits_{\\mathop {c_i \\in \\left\\{ {a_i ,b_i } \\right\\}}\\limits_{1 \\le i \\le n} } {f\\left( {c_1, c_2, \\ldots ,c_n } \\right)}$. Some other related result are given.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-04-07T03:08:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "26B25",
      "26B35",
      "52A20",
      "52A41",
      "26D07"
    ],
    "keywords": [
      "hermite-hadamard inequality",
      "fold convex function",
      "hypercuboid"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160401857A"
    }
  }
}