{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $I$ be an [[interval|0TK8Y3J5RW]] of $\\R$ and let $f \\colon I \\to \\R$ be a [[convex function|0VV0HSW1ZS]].\nLet $x_1 \\ge x_2 \\ge \\dots \\ge x_n$ and $y_1 \\ge y_2 \\ge \\dots \\ge y_n$ be numbers in $I$ such that $(x_1, \\dots, x_n)$ [[majorizes|5K7VT2VKZT]] $(y_1, \\dots, y_n)$, that is,\n$$\\sum_{i=1}^{k} x_i \\ge \\sum_{i=1}^{k} y_i \\quad \\text{for } k = 1, \\dots, n - 1, \\qquad \\sum_{i=1}^{n} x_i = \\sum_{i=1}^{n} y_i.$$\nThen\n$$\\sum_{i=1}^{n} f(x_i) \\ge \\sum_{i=1}^{n} f(y_i).$$\n\n## Remarks\n\nProof idea: let $c_i$ be the slope of the chord of $f$ between $x_i$ and $y_i$, or a [[subderivative|9C5WXYYQEK]] at $x_i$ when $x_i = y_i$.\nSince [[the slopes of a convex function increase|0BY5FTHEEE]] and both sequences decrease, $c_1 \\ge c_2 \\ge \\dots \\ge c_n$.\nThen $\\sum_i \\big(f(x_i) - f(y_i)\\big) = \\sum_i c_i (x_i - y_i)$, and summation by parts writes this as a sum of products of the nonnegative numbers $c_k - c_{k+1}$ with the nonnegative partial sums $\\sum_{i \\le k} (x_i - y_i)$.\n\nExample: $(2, 0)$ majorizes $(1, 1)$, so $f(2) + f(0) \\ge 2 f(1)$, which is midpoint convexity.\nMore generally $(x_1, \\dots, x_n)$ majorizes the constant vector $(\\bar x, \\dots, \\bar x)$ with $\\bar x$ the mean, which gives [[Jensen's inequality|0RWMRTPC87]] with equal weights.\n\nThe equality of the total sums is needed.\nWith $n = 1$, $f(x) = -x$, $x_1 = 1$ and $y_1 = 0$, one has $x_1 \\ge y_1$ but $f(x_1) = -1 \u003c 0 = f(y_1)$.\n\n## Explanations\n\n### L1 human\n\nKaramata's inequality compares two lists of numbers with the same total.\nIf one list is more spread out than the other, then summing a convex function over it gives a larger result.\nA convex function rewards spreading out, because its graph bends upward.\n\n### L2 human\n\nThe inequality says that $x \\mapsto \\sum_i f(x_i)$ is a [[Schur-convex function|A72XTVF2GK]] whenever $f$ is convex.\nIt contains Jensen's inequality with equal weights and implies many symmetric inequalities, for instance by comparing a vector with a less spread out one.\nA converse also holds: if the conclusion is true for every convex $f$, then $x$ majorizes $y$.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "about": [
      "07Q7CNM5WF",
      "5K7VT2VKZT"
    ],
    "kind": "statement",
    "msc": [
      "26D15",
      "26A51"
    ],
    "name": "Karamata's inequality",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q3656857",
      "wikipedia": "Karamata's inequality"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Karamata%27s_inequality"
      }
    ],
    "tag": "FWKSHEJ9RP",
    "uses": [
      "0TK8Y3J5RW",
      "0VV0HSW1ZS",
      "5K7VT2VKZT"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/FWKSHEJ9RP/",
  "links": [
    {
      "field": "about",
      "target": "07Q7CNM5WF"
    },
    {
      "field": "about",
      "target": "5K7VT2VKZT"
    },
    {
      "field": "uses",
      "target": "0TK8Y3J5RW"
    },
    {
      "field": "uses",
      "target": "0VV0HSW1ZS"
    },
    {
      "field": "uses",
      "target": "5K7VT2VKZT"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/FWKSHEJ9RP.md",
  "msc": [
    "26D15",
    "26A51"
  ],
  "name": "Karamata's inequality",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/FWK/FWKSHEJ9RP.md",
  "status": null,
  "tag": "FWKSHEJ9RP",
  "uri": "m2://FWKSHEJ9RP",
  "url": "https://math2.tamnd.com/wiki/karamatas-inequality/",
  "writing": {
    "tag": "FWKSHEJ9RP",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}