View source for Univalent function

{{Short description|Mathematical concept}}
{{Other uses|Univalent (disambiguation){{!}}Univalent}}
In [[Mathematics|mathematics]], in the branch of [[Complex analysis|complex analysis]], a [[Holomorphic function|holomorphic function]] on an open subset of the [[Complex plane|complex plane]]  is called '''univalent''' if it is [[Injective function|injective]].<ref>{{harv|Conway|1995|page=32|loc=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."}}</ref><ref>{{harv|Nehari|1975}}</ref>

== Examples==
The function <math>f \colon z \mapsto 2z + z^2</math> is univalent in the open unit disc, as <math>f(z) = f(w)</math> implies that <math>f(z) - f(w) = (z-w)(z+w+2) = 0</math>. As the second factor is non-zero in the open unit disc, <math>z = w</math> so <math>f</math> is injective.

==Basic properties==
One can prove that if <math>G</math> and <math>\Omega</math> are two open [[Connected space|connected]] sets in the complex plane, and

:<math>f: G \to \Omega</math>

is a univalent function such that <math>f(G) = \Omega</math> (that is, <math>f</math> is [[Surjective function|surjective]]), then the derivative of <math>f</math> is never zero, <math>f</math> is invertible, and its inverse <math>f^{-1}</math> is also holomorphic. More, one has by the [[Chain rule|chain rule]]

:<math>(f^{-1})'(f(z)) = \frac{1}{f'(z)}</math>

for all <math>z</math> in <math>G.</math>

== Comparison with real functions ==

For [[Real number|real]] [[Analytic function|analytic function]]s, unlike for complex analytic (that is, holomorphic) functions, these statements fail to hold. For example, consider the function

:<math>f: (-1, 1) \to (-1, 1) \, </math>

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

== See also ==
* {{annotated link|De Branges's theorem}}
* {{annotated link|Koebe quarter theorem}}
* {{annotated link|Riemann mapping theorem}}
* {{annotated link|Nevanlinna's criterion}}

== Note ==
{{Reflist}}
== References ==
*{{cite book |first1=John B. |last1=Conway|doi=10.1007/978-1-4612-0817-4|title=Functions of One Complex Variable II |series=Graduate Texts in Mathematics |year=1995 |volume=159 |isbn=978-1-4612-6911-3|chapter=Conformal Equivalence for Simply Connected Regions|chapter-url={{Google books|yV74BwAAQBAJ|page=32|plainurl=yes}}}}
*{{cite book |chapter-url=https://doi.org/10.1017/CBO9780511844195.041|doi=10.1017/CBO9780511844195.041 |chapter=Univalent Functions |title=Sources in the Development of Mathematics |year=2011 |pages=907–928 |isbn=9780521114707 }}
*{{cite book |last1=Duren |first1=P. L. |title=Univalent Functions |date=1983 |publisher=Springer New York, NY |isbn=978-1-4419-2816-0 |page=XIV, 384}}
*{{cite book |doi=10.1007/978-94-011-5206-8|title=Convex and Starlike Mappings in Several Complex Variables |year=1998 |last1=Gong |first1=Sheng |isbn=978-94-010-6191-9 }}
*{{cite journal |doi=10.4064/SM174-3-5|title=A remark on separate holomorphy |year=2006 |last1=Jarnicki |first1=Marek |last2=Pflug |first2=Peter |journal=Studia Mathematica |volume=174 |issue=3 |pages=309–317 |s2cid=15660985 |doi-access=free |arxiv=math/0507305 }}
*{{Cite book |last=Nehari |first=Zeev |title=Conformal mapping |date=1975 |publisher=Dover Publications |isbn=0-486-61137-X |location=New York |oclc=1504503|page=146}}
{{PlanetMath attribution|title=univalent analytic function|urlname=UnivalentAnalyticFunction}}

[[Category:Analytic functions]]

is:Eintæk vörpun

{{Sourceattribution|Univalent function}}