{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $f$ be a [[univalent function|R91E83SDBD]] on the unit disc $\\mathbb{D}$ with $f(0) = 0$ and $f'(0) = 1$, and write\n$$f(z) = z + \\sum_{n=2}^{\\infty} a_n z^n, \\qquad \\abs{z} \u003c 1.$$\nThen $\\abs{a_n} \\le n$ for every $n \\ge 2$.\nIf $\\abs{a_n} = n$ for some $n \\ge 2$, then $f(z) = e^{-i\\theta} k(e^{i\\theta} z)$ for some $\\theta \\in \\R$, where $k(z) = z/(1-z)^2$ is the [[Koebe function|EZQFWKF03Q]].\n\n## Remarks\n\nBieberbach proved the case $n = 2$ and stated the general bound as a conjecture.\nLoewner proved the case $n = 3$ with his [[differential equation|51G4QJZ345]].\nLouis de Branges proved the full statement, which is therefore also called de Branges's theorem.\nThe conjecture was known to follow from the Robertson conjecture on odd univalent functions, which in turn follows from the Milin conjecture on the logarithmic coefficients of $f$.\nDe Branges proved the Milin conjecture, using the Loewner equation and an inequality on sums of Jacobi polynomials that had been proved by Askey and Gasper.\nThe bound is sharp for every $n$ at once, because the Koebe function has $a_n = n$.\n\n## Explanations\n\n### L1 human\n\nNormalise a univalent function on the unit disc so that it starts like $z$.\nThe theorem says that its $n$-th Taylor coefficient has absolute value at most $n$.\nThe Koebe function, which maps the disc onto the plane minus a slit, is the extreme case with coefficients exactly $n$.\n\n### L2 human\n\nThe case $n = 2$, Bieberbach's inequality $\\abs{a_2} \\le 2$, is a short consequence of the [[Grönwall area theorem|BYZDAZNHP4]] and implies the [[Koebe quarter theorem|WX0HMW4GT7]] and the distortion theorems.\nFor larger $n$ the problem resisted for a long time, and partial results such as $\\abs{a_n} \u003c e\\,n$ and the proofs for small $n$ produced many methods of univalent function theory.\nThe final proof goes through the stronger Milin inequality and the [[Loewner equation|51G4QJZ345]].\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "about": [
      "R91E83SDBD"
    ],
    "kind": "statement",
    "msc": [
      "30C50",
      "30C55"
    ],
    "name": "Bieberbach conjecture",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q857089",
      "wikipedia": "De Branges's theorem"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/De_Branges%27s_theorem"
      }
    ],
    "tag": "ZMRFSYVWQT",
    "uses": [
      "R91E83SDBD",
      "EZQFWKF03Q",
      "51G4QJZ345"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/ZMRFSYVWQT/",
  "links": [
    {
      "field": "about",
      "target": "R91E83SDBD"
    },
    {
      "field": "uses",
      "target": "R91E83SDBD"
    },
    {
      "field": "uses",
      "target": "EZQFWKF03Q"
    },
    {
      "field": "uses",
      "target": "51G4QJZ345"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/ZMRFSYVWQT.md",
  "msc": [
    "30C50",
    "30C55"
  ],
  "name": "Bieberbach conjecture",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/ZMR/ZMRFSYVWQT.md",
  "status": null,
  "tag": "ZMRFSYVWQT",
  "uri": "m2://ZMRFSYVWQT",
  "url": "https://math2.tamnd.com/wiki/bieberbach-conjecture/",
  "writing": {
    "tag": "ZMRFSYVWQT",
    "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
      }
    ]
  }
}