{
  "id": "1512.07272",
  "version": "v1",
  "published": "2015-12-22T21:45:15.000Z",
  "updated": "2015-12-22T21:45:15.000Z",
  "title": "Convexity with respect to families of means",
  "authors": [
    "Gyula Maksa",
    "Zsolt Páles"
  ],
  "journal": "Aequationes Mathematicae 89(1) (2015), 161-167",
  "doi": "10.1007/s00010-014-0281-7",
  "categories": [
    "math.CA"
  ],
  "abstract": "In this paper we investigate continuity properties of functions $f:\\mathbb{R}_+\\to\\mathbb{R}_+$ that satisfy the $(p,q)$-Jensen convexity inequality $$ f\\big(H_p(x,y)\\big)\\leq H_q(f(x),f(y)) \\qquad(x,y>0), $$ where $H_p$ stands for the $p$th power (or H\\\"older) mean. One of the main results shows that there exist discontinuous multiplicative functions that are $(p,p)$-Jensen convex for all positive rational number $p$. A counterpart of this result states that if $f$ is $(p,p)$-Jensen convex for all $p\\in P\\subseteq\\mathbb{R}_+$, where $P$ is a set of positive Lebesgue measure, then $f$ must be continuous.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-12-22T21:45:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "39B62",
      "26D07",
      "26D15"
    ],
    "keywords": [
      "jensen convexity inequality",
      "main results",
      "positive rational number",
      "th power",
      "positive lebesgue measure"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Springer"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151207272M"
    }
  }
}