Univalent function

Last updated September 01, 2024

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 $f\colon z\mapsto 2z+z^{2}$ 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

↑ ( Conway 1995 , p. 32, chapter 14: Conformal equivalence for simply connected regions, Definition 1.12: "A function on an open set is univalent if it is analytic and one-to-one.")
↑ ( Nehari 1975 )

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References

Conway, John B. (1995). "Conformal Equivalence for Simply Connected Regions". Functions of One Complex Variable II. Graduate Texts in Mathematics. Vol. 159. doi:10.1007/978-1-4612-0817-4. ISBN 978-1-4612-6911-3.
"Univalent Functions". Sources in the Development of Mathematics. 2011. pp. 907–928. doi:10.1017/CBO9780511844195.041. ISBN 9780521114707.
Duren, P. L. (1983). Univalent Functions. Springer New York, NY. p. XIV, 384. ISBN 978-1-4419-2816-0.
Gong, Sheng (1998). Convex and Starlike Mappings in Several Complex Variables. doi:10.1007/978-94-011-5206-8. ISBN 978-94-010-6191-9.
Jarnicki, Marek; Pflug, Peter (2006). "A remark on separate holomorphy". Studia Mathematica. 174 (3): 309–317. arXiv: math/0507305 . doi: 10.4064/SM174-3-5 . S2CID 15660985.
Nehari, Zeev (1975). Conformal mapping. New York: Dover Publications. p. 146. ISBN 0-486-61137-X. OCLC 1504503.

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[1] ( Conway 1995 , p. 32, chapter 14: Conformal equivalence for simply connected regions, Definition 1.12: "A function on an open set is univalent if it is analytic and one-to-one.")

[2] ( Nehari 1975 )

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