Univalent function

Template:Short description 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 $f:z\mapsto 2z+z^2$ 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 $\mathrm{\Omega }$ are two open connected sets in the complex plane, and

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

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

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

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

See also

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References

This article incorporates material from univalent analytic function on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.