# Univalent function

In mathematics, in the branch of complex analysis, a holomorphic function on an open subset of the complex plane is called univalent if it is injective.[1]

## Examples

Consider the application ${\displaystyle \phi _{a}}$ mapping the open unit disc to itself such that

${\displaystyle \phi _{a}(z)={\frac {z-a}{1-{\bar {a}}z}}.}$

We have that ${\displaystyle \phi _{a}}$ is univalent when ${\displaystyle |a|<1}$.

## Basic properties

One can prove that if ${\displaystyle G}$ and ${\displaystyle \Omega }$ are two open connected sets in the complex plane, and

${\displaystyle f:G\to \Omega }$

is a univalent function such that ${\displaystyle f(G)=\Omega }$ (that is, ${\displaystyle f}$ is surjective), then the derivative of ${\displaystyle f}$ is never zero, ${\displaystyle f}$ is invertible, and its inverse ${\displaystyle f^{-1}}$ is also holomorphic. More, one has by the chain rule

${\displaystyle (f^{-1})'(f(z))={\frac {1}{f'(z)}}}$

for all ${\displaystyle z}$ in ${\displaystyle G.}$

## Comparison with real functions

For real analytic functions, unlike for complex analytic (that is, holomorphic) functions, these statements fail to hold. For example, consider the function

${\displaystyle f:(-1,1)\to (-1,1)\,}$

given by ƒ(x) = x3. This function is clearly injective, but its derivative is 0 at x = 0, and its inverse is not analytic, or even differentiable, on the whole interval (−1, 1). Consequently, if we enlarge the domain to an open subset G of the complex plane, it must fail to be injective; and this is the case, since (for example) f(εω) = f(ε) (where ω is a primitive cube root of unity and ε is a positive real number smaller than the radius of G as a neighbourhood of 0).