{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $\\mathbb{D}$ be the [[unit disc|8G5ABXRK85]], and let $f \\colon \\mathbb{D} \\to \\C$ be a [[univalent function|R91E83SDBD]].\nThen the image $f(\\mathbb{D})$ contains the open disc with centre $f(0)$ and radius $\\abs{f'(0)}/4$.\n\n## Remarks\n\nAfter replacing $f$ by $(f - f(0))/f'(0)$, one may assume $f(z) = z + a_2 z^2 + \\cdots$ is in the class $S$.\nIf $w \\notin f(\\mathbb{D})$, then $g = wf/(w - f)$ is again in $S$, and its second coefficient is $a_2 + 1/w$.\nBieberbach's inequality $\\abs{a_2} \\le 2$ for functions in $S$, which follows from the [[Grönwall area theorem|BYZDAZNHP4]], gives $\\abs{a_2 + 1/w} \\le 2$ and $\\abs{a_2} \\le 2$, hence $\\abs{1/w} \\le 4$, that is, $\\abs{w} \\ge 1/4$.\nThe constant $1/4$ is sharp: the [[Koebe function|EZQFWKF03Q]] $k(z) = z/(1-z)^2$ is in $S$ and omits the value $-1/4$.\nInjectivity is necessary: $f(z) = (e^{Nz} - 1)/N$ has $f(0) = 0$ and $f'(0) = 1$, but omits the value $-1/N$, which is close to $0$ for large $N$.\n\n## Explanations\n\n### L1 human\n\nA univalent function on the unit disc cannot squeeze its image too much near the image of the centre.\nAfter normalising so that $f(0) = 0$ and $f'(0) = 1$, the image always contains the disc of radius $1/4$ around $0$.\nThe Koebe function shows that a quarter is the best one can say.\n\n### L2 human\n\nThe theorem is the basic covering theorem of univalent function theory.\nIt gives the distance estimate $\\frac{1}{4}\\abs{f'(z)}(1 - \\abs{z}^2) \\le \\operatorname{dist}(f(z), \\partial f(\\mathbb{D})) \\le \\abs{f'(z)}(1 - \\abs{z}^2)$ for univalent $f$, where the upper bound comes from the Schwarz lemma.\nThis compares the hyperbolic metric of a simply connected domain with the reciprocal of the distance to its boundary, up to a factor $4$.\nTogether with the Koebe distortion theorem, it gives the compactness of the class $S$ in the topology of locally uniform convergence.\nFor maps that need not be injective, [[Bloch's theorem|KW31KDX7TB]] gives a much weaker covering result.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "about": [
      "R91E83SDBD"
    ],
    "kind": "statement",
    "msc": [
      "30C25",
      "30C55"
    ],
    "name": "Koebe quarter theorem",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q588218",
      "wikipedia": "Koebe quarter theorem"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Koebe_quarter_theorem"
      }
    ],
    "tag": "WX0HMW4GT7",
    "uses": [
      "R91E83SDBD",
      "8G5ABXRK85",
      "EZQFWKF03Q",
      "BYZDAZNHP4"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/WX0HMW4GT7/",
  "links": [
    {
      "field": "about",
      "target": "R91E83SDBD"
    },
    {
      "field": "uses",
      "target": "R91E83SDBD"
    },
    {
      "field": "uses",
      "target": "8G5ABXRK85"
    },
    {
      "field": "uses",
      "target": "EZQFWKF03Q"
    },
    {
      "field": "uses",
      "target": "BYZDAZNHP4"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/WX0HMW4GT7.md",
  "msc": [
    "30C25",
    "30C55"
  ],
  "name": "Koebe quarter theorem",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/WX0/WX0HMW4GT7.md",
  "status": null,
  "tag": "WX0HMW4GT7",
  "uri": "m2://WX0HMW4GT7",
  "url": "https://math2.tamnd.com/wiki/koebe-quarter-theorem/",
  "writing": {
    "tag": "WX0HMW4GT7",
    "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
      }
    ]
  }
}