Univalent function

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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#edit

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

Basic properties#edit

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

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

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

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

for all ${\displaystyle z}$ in ${\displaystyle G.}$

Comparison with real functions#edit

For real analytic functions, unlike for complex analytic (that is, holomorphic) functions, these statements fail to hold. For example, consider the function

${\displaystyle f:(-1,1)\to (-1,1)\,}$

given by ${\displaystyle f(x)=x^{3$ . This function is clearly injective, but its derivative is 0 at ${\displaystyle x=0}$ , and its inverse is not analytic, or even differentiable, on the whole interval ${\displaystyle (-1,1)}$ . Consequently, if we enlarge the domain to an open subset ${\displaystyle G}$ of the complex plane, it must fail to be injective; and this is the case, since (for example) ${\displaystyle f(\varepsilon \omega )=f(\varepsilon )}$ (where ${\displaystyle \omega }$ is a primitive cube root of unity and ${\displaystyle \varepsilon }$ is a positive real number smaller than the radius of ${\displaystyle G}$ as a neighbourhood of ${\displaystyle 0}$ ).

Note#edit

^ (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)

References#edit

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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