Univalent function

Remarks

A univalent function has $f'(z) \ne 0$ for all $z \in U$, and its inverse $f^{-1} \colon f(U) \to U$ is holomorphic. Example: $f(z) = e^z$ on the strip $\set{z : \abs{\operatorname{Im} z} < \pi}$; 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 $z = w$. Non-example: $e^z$ on all of $\C$ is holomorphic but not injective, since $e^{0} = e^{2\pi i}$. Non-example: $z \mapsto z^2$ on $\mathbb{D}$ is not univalent, though it is injective near every point except $0$. The converse of the first remark fails: $f' \ne 0$ only gives local injectivity, as $e^z$ on $\C$ shows. Some 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 $\set{\abs{z} > 1}$ is studied together with $S$. The German term schlicht is used with the same meaning.