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.

Examples

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

Basic properties

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

:$f: G \to \Omega$

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

:$(f^)'(f(z)) = \frac$

for all $z$ in $G.$

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

:$f: (-1, 1) \to (-1, 1) \,$

given by $f(x)=x^3$. 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) $f(\varepsilon \omega) = f(\varepsilon)$ (where $\omega$ is a primitive cube root of unity and $\varepsilon$ is a positive real number smaller than the radius of $G$ as a neighbourhood of $0$).

See also

*
*
*
*
*

Note

References

*
*
*
*
*
*

{{Authority control
Analytic functions

is:Eintæk vörpun