---
tag: R91E83SDBD
kind: concept
name: Univalent function
msc: [30C55, 30C45]
specializes: [0NAHXR1FRH]
sources:
  - url: https://en.wikipedia.org/wiki/Univalent_function
same_as:
  wikidata: Q2608535
  wikipedia: Univalent function
---

## Summary

A univalent function is a holomorphic function that is injective on its domain.
It is a conformal map of its domain onto its image, which is again an open set.
The theory of univalent functions studies the class $S$ of univalent functions on the unit disc with $f(0) = 0$ and $f'(0) = 1$.
Its main results are distortion, covering and coefficient theorems, such as the [[Koebe quarter theorem|WX0HMW4GT7]] and the [[Bieberbach conjecture|ZMRFSYVWQT]].

## Explanations

### L1 human

A univalent function never takes the same value twice.
So it maps its domain onto a region without folding or overlapping, like a sheet of rubber stretched without tearing or doubling.
The map $z \mapsto z^2$ is not univalent on the unit disc, since $z$ and $-z$ have the same square.
On the right half-plane, $z \mapsto z^2$ is univalent, and its image is the plane minus the negative real axis and $0$.