Univalent function

For other uses, see Univalent.

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][2]

Examples

The function ${\displaystyle f\colon z\mapsto 2z+z^{2}}$ is univalent in the open unit disc, as ${\displaystyle f(z)=f(w)}$ implies that ${\displaystyle f(z)-f(w)=(z-w)(z+w+2)=0}$. As the second factor is non-zero in the open unit disc, ${\displaystyle z=w}$ so ${\displaystyle f}$ is injective.

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 ${\displaystyle f(x)=x^{3}}$. This function is clearly injective, but its derivative is 0 at ${\displaystyle x=0}$, and its inverse is not analytic, or even differentiable, on the whole interval ${\displaystyle (-1,1)}$. Consequently, if we enlarge the domain to an open subset ${\displaystyle G}$ of the complex plane, it must fail to be injective; and this is the case, since (for example) ${\displaystyle f(\varepsilon \omega )=f(\varepsilon )}$ (where ${\displaystyle \omega }$ is a primitive cube root of unity and ${\displaystyle \varepsilon }$ is a positive real number smaller than the radius of ${\displaystyle G}$ as a neighbourhood of ${\displaystyle 0}$).

See also

• Biholomorphic mapping – Bijective holomorphic function with a holomorphic inversePages displaying short descriptions of redirect targets
• De Branges's theorem – Statement in complex analysis; formerly the Bieberbach conjecture
• Koebe quarter theorem – Statement in complex analysis
• Riemann mapping theorem – Mathematical theorem
• Nevanlinna's criterion – Characterization of starlike univalent holomorphic functions