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  "body": "\n## Statement\n\nLet $\\mathbb{D}$ be the [[unit disc|8G5ABXRK85]], and let $f \\colon \\mathbb{D} \\to \\C$ be [[holomorphic|0DG0Z72CQD]] with $f'(z) \\ne 0$ for all $z \\in \\mathbb{D}$.\nLet $S(f)$ be its [[Schwarzian derivative|6ZPKWE0VAZ]].\nIf\n$$\\abs{S(f)(z)} \\le \\frac{2}{(1 - \\abs{z}^2)^2} \\qquad \\text{for all } z \\in \\mathbb{D},$$\nthen $f$ is [[univalent|R91E83SDBD]] on $\\mathbb{D}$.\n\n## Remarks\n\nLocally, $f = u_1/u_2$ for two independent solutions of the linear equation $u'' + \\frac{1}{2} S(f)\\, u = 0$.\nIf $f(a) = f(b)$ with $a \\ne b$, then some solution of this equation vanishes at both $a$ and $b$.\nA disc automorphism moves $a$ and $b$ onto a diameter.\nThe bound on $S(f)$ then allows a comparison of Sturm type with the equation $u'' + (1 - x^2)^{-2} u = 0$ on $(-1, 1)$.\nThat equation has the solution $\\sqrt{1 - x^2}$ without zeros, which leads to a contradiction.\nThe constant $2$ cannot be replaced by any larger constant.\nIn the other direction, the Kraus–Nehari theorem says that every univalent $f$ on $\\mathbb{D}$ satisfies $\\abs{S(f)(z)} \\le 6/(1 - \\abs{z}^2)^2$, with equality at $z = 0$ for the [[Koebe function|EZQFWKF03Q]].\nExample: $f(z) = e^{cz}$ has $S(f) = -c^2/2$, so for $\\abs{c} \\le 2$ the bound holds, since $(1 - \\abs{z}^2)^{-2} \\ge 1$; such $f$ is univalent on $\\mathbb{D}$, as two points of $\\mathbb{D}$ cannot differ by $2\\pi i k / c$ with $k \\ne 0$.\n\n## Explanations\n\n### L1 human\n\nThe Schwarzian derivative measures how far a map is from being a Möbius map, and Möbius maps are injective.\nNehari's theorem says that if the Schwarzian derivative is small enough, measured against the size of the disc near the boundary, then the map is injective.\nIt turns a global property, injectivity, into a pointwise bound that can be checked.\n\n### L2 human\n\nThe weight $(1 - \\abs{z}^2)^{-2}$ is the square of the density of the hyperbolic metric, so the condition is invariant under automorphisms of the disc.\nTogether with the necessary bound $6$, it shows that the Schwarzian derivatives of univalent functions all have bounded hyperbolic size, and that all small enough ones come from univalent functions.\nThis point of view leads to the Bers embedding of the universal Teichmüller space.\nIt also leads to the theorem of Ahlfors and Weill that a slightly stronger bound gives a quasiconformal extension of $f$ to the plane.\n",
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