Univalent Functions

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, $f$ must be 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 ''ƒ''(''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''(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).

See also

* Biholomorphic mapping
* De Branges's theorem
* Koebe quarter theorem
* Riemann mapping theorem
* Schlicht function

Note

References

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{{Authority control
Analytic functions

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