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  "body": "\n## Definition\n\nLet $U \\subseteq \\C$ be an [[open set|01ZQQDCERP]].\nA function $f \\colon U \\to \\C$ is *univalent* on $U$ if it is [[holomorphic|0DG0Z72CQD]] on $U$ and [[injective|0PVG1CRAJN]] on $U$.\n\nThe class $S$ is the set of univalent functions $f$ on the [[unit disc|8G5ABXRK85]] $\\mathbb{D}$ with $f(0) = 0$ and $f'(0) = 1$, so that $f(z) = z + \\sum_{n \\ge 2} a_n z^n$.\n\n## Remarks\n\nA univalent function has $f'(z) \\ne 0$ for all $z \\in U$, and its inverse $f^{-1} \\colon f(U) \\to U$ is holomorphic.\nExample: $f(z) = e^z$ on the strip $\\set{z : \\abs{\\operatorname{Im} z} \u003c \\pi}$; if $e^z = e^w$ then $z - w \\in 2\\pi i\\Z$, and two points of the strip differ in imaginary part by less than $2\\pi$, so $z = w$.\nNon-example: $e^z$ on all of $\\C$ is holomorphic but not injective, since $e^{0} = e^{2\\pi i}$.\nNon-example: $z \\mapsto z^2$ on $\\mathbb{D}$ is not univalent, though it is injective near every point except $0$.\nThe converse of the first remark fails: $f' \\ne 0$ only gives local injectivity, as $e^z$ on $\\C$ shows.\nSome authors also call injective meromorphic functions univalent; then the class $\\Sigma$ of univalent functions $g(z) = z + b_0 + b_1 z^{-1} + \\cdots$ on $\\set{\\abs{z} \u003e 1}$ is studied together with $S$.\nThe German term *schlicht* is used with the same meaning.\n",
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