---
tag: 199C7G1B36
kind: definition
name: Univalent function
msc: [30C55, 30C45]
defines: R91E83SDBD
uses: [0DG0Z72CQD, 01ZQQDCERP, 0PVG1CRAJN, 8G5ABXRK85]
sources:
  - url: https://en.wikipedia.org/wiki/Univalent_function
---

## Definition

Let $U \subseteq \C$ be an [[open set|01ZQQDCERP]].
A function $f \colon U \to \C$ is *univalent* on $U$ if it is [[holomorphic|0DG0Z72CQD]] on $U$ and [[injective|0PVG1CRAJN]] on $U$.

The 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 $f(z) = z + \sum_{n \ge 2} a_n z^n$.

## 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.