Inmathematics, in the branch ofcomplex analysis, aholomorphic function on anopen subset of thecomplex plane is calledunivalent if it isinjective.^[1][2]

The function${\displaystyle f:z\mapsto 2z+z^2}$ 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.

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

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

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

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

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

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

• Biholomorphic mapping – Bijective holomorphic function with a holomorphic inversePages displaying short descriptions of redirect targets
• De Branges's theorem – Statement in complex analysis; formerly the Bieberbach conjecture
• Koebe quarter theorem – Statement in complex analysis
• Riemann mapping theorem – Mathematical theorem
• Nevanlinna's criterion – Characterization of starlike univalent holomorphic functions

• 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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