Univalent function

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title: "Univalent function" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["analytic-functions"] description: "Mathematical concept" topic_path: "general/analytic-functions" source: "https://en.wikipedia.org/wiki/Univalent_function" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Mathematical concept ::

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^{-1} is also holomorphic. More, one has by the chain rule

:(f^{-1})'(f(z)) = \frac{1}{f'(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(\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).

Note

References

is:Eintæk vörpun

References

1. {{harv. Conway. 1995
2. {{harv. Nehari. 1975

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