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  "body": "\n## Statement\n\nLet $\\mathbb{D} = \\set{z \\in \\C : \\abs{z} \u003c 1}$ be the open [[unit disc|8G5ABXRK85]], and let $f \\colon \\mathbb{D} \\to \\mathbb{D}$ be [[holomorphic|0DG0Z72CQD]].\nThen for all $z, w \\in \\mathbb{D}$,\n$$\\abs{\\frac{f(z) - f(w)}{1 - \\overline{f(w)} f(z)}} \\le \\abs{\\frac{z - w}{1 - \\bar w z}},$$\nand for all $z \\in \\mathbb{D}$,\n$$\\frac{\\abs{f'(z)}}{1 - \\abs{f(z)}^2} \\le \\frac{1}{1 - \\abs{z}^2}.$$\nAssume equality holds in the first inequality for some $z \\neq w$, or in the second for some $z$.\nThen $f$ is a bijective [[Möbius transformation|0B346DDAM5]] of $\\mathbb{D}$ onto itself.\n\n## Remarks\n\nProof idea: for $a \\in \\mathbb{D}$ let $\\varphi_a(z) = \\frac{z - a}{1 - \\bar a z}$, a Möbius transformation of $\\mathbb{D}$ onto itself with $\\varphi_a(a) = 0$.\nThe map $g = \\varphi_{f(w)} \\circ f \\circ \\varphi_w^{-1}$ sends $\\mathbb{D}$ into $\\mathbb{D}$ and fixes $0$.\nThe [[Schwarz lemma|0MB48XDBY3]] gives $\\abs{g(\\zeta)} \\le \\abs{\\zeta}$, which is the first inequality with $\\zeta = \\varphi_w(z)$; letting $z \\to w$ gives the second.\nIn terms of the hyperbolic metric $\\frac{2 \\abs{dz}}{1 - \\abs{z}^2}$ of the [[Poincaré disk model|85MM2EA0TK]], the theorem says that $f$ does not increase hyperbolic distances.\nThe Möbius transformations of $\\mathbb{D}$ onto itself are exactly the hyperbolic isometries that preserve orientation.\nExample: $f(z) = z^2$ gives $\\abs{2z}/(1 - \\abs{z}^4) \\le 1/(1 - \\abs{z}^2)$, that is $2\\abs{z} \\le 1 + \\abs{z}^2$, which is $(1 - \\abs{z})^2 \\ge 0$.\n\n## Explanations\n\n### L1 human\n\nThe Schwarz lemma is about maps of the disc into itself that fix the center.\nThe Schwarz–Pick theorem drops the condition at the center.\nIt says that such a map can only bring points closer together, when distance is measured in the right way.\nThat right way is the hyperbolic distance, for which points near the edge of the disc are very far apart.\n\n### L2 human\n\nThe theorem shows that the hyperbolic metric is the natural metric for holomorphic maps of the disc.\nEvery holomorphic self-map is a contraction for it, and the automorphisms are the isometries.\nThis point of view leads to the Ahlfors–Schwarz lemma for metrics of negative curvature and to the Kobayashi metric in several complex variables.\nIt also gives quick proofs of distortion estimates, for example $\\abs{f'(0)} \\le 1 - \\abs{f(0)}^2$.\n",
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