{
  "aliases": [
    "schwarzs-lemma"
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  "body": "\n## Statement\n\nLet $\\mathbb{D} = \\{z \\in \\mathbb{C} : |z| \u003c 1\\}$ be the open unit disc.\nLet $f : \\mathbb{D} \\to \\mathbb{D}$ be [[holomorphic|0DG0Z72CQD]] on $\\mathbb{D}$ with $f(0) = 0$.\nThen $|f(z)| \\le |z|$ for each $z \\in \\mathbb{D}$, and $|f'(0)| \\le 1$.\nIf $|f(z_0)| = |z_0|$ for some $z_0 \\neq 0$ in $\\mathbb{D}$, or if $|f'(0)| = 1$, then there is $a \\in \\mathbb{C}$ with $|a| = 1$ such that $f(z) = a z$ for each $z \\in \\mathbb{D}$.\n\n## Remarks\n\nProof idea: $g(z) = f(z)/z$, with $g(0) = f'(0)$, is holomorphic on $\\mathbb{D}$. The [[maximum modulus principle|048F2P90WR]] on the discs $|z| \\le r$ gives $|g| \\le 1/r$ for each $r \u003c 1$, so $|g| \\le 1$.\nIt follows that the bijective holomorphic maps $\\mathbb{D} \\to \\mathbb{D}$ with $f(0) = 0$ are the rotations $z \\mapsto az$ with $|a| = 1$.\nThe Schwarz–Pick theorem is the version without the condition $f(0) = 0$.\n\n## Explanations\n\n### L1 human\n\nA holomorphic map from the unit disc to itself that keeps $0$ fixed cannot move any point farther away from $0$.\nThe only maps that keep the distance to $0$ of some point are the rotations.\n",
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    "aliases": [
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    "name": "Schwarz lemma",
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        "locator": "Statement of the classical Schwarz Lemma",
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        "locator": "Proof of the classical Schwarz lemma",
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  "msc": [
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  "name": "Schwarz lemma",
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