Univalent function: Difference between revisions

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{{Short description|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 function|injective]].

{{Other uses|Univalent (disambiguation){{!}}Univalent}}

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 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==

AnyThe mappingfunction <math>f \phi_acolon z \mapsto 2z + z^2</math> ofis univalent in the open [[unit disc]], toas itself,<math>f(z) = :f(w)</math>\phi_a implies that <math>f(z) - f(w) =\frac{ (z-a}{1w)(z+w+2) = 0</math>. As the second factor is non-zero \bar{a}z}in the open unit disc, <math>z = w</math> whereso <math>|a|<1,f</math> is univalentinjective.

==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_functionSurjective 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]]

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

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== Comparison with real functions ==

For [[real number|real]] [[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 ''&fnof;''<math>f(''x'')&nbsp;=&nbsp;''x''<sup>^3</supmath>. This function is clearly one-to-oneinjective, however,but its derivative is 0 at ''<math>x''&nbsp;=&nbsp;0</math>, and its inverse is not analytic, or even differentiable, on the whole interval&nbsp; <math>(&minus;-1,&nbsp;1)</math>. Consequently, if we enlarge the ___domain to an open subset ''<math>G''</math> of the complex plane, it must fail to be one-to-oneinjective; and this is the case, since (for example) ''<math>f''(&epsilon;&\varepsilon \omega;)&nbsp; = ''f''(&epsilon;\varepsilon) </math> (where &<math>\omega; </math> is a [[primitive root of unity|primitive cube root of unity]] and &epsilon;<math>\varepsilon</math> is a positive real number smaller than the radius of ''<math>G''</math> as a neighbourhood of <math>0</math>).

== ReferencesSee also ==

* {{annotated link|Biholomorphic mapping}}

* John B. Conway. ''Functions of One Complex Variable I''. Springer-Verlag, New York, 1978. ISBN 0-387-90328-3.

* {{annotated link|De Branges's theorem}}

* John B. Conway. ''Functions of One Complex Variable II''. Springer-Verlag, New York, 1996. ISBN 0-387-94460-5.

* {{annotated link|Koebe quarter theorem}}

* {{annotated link|Riemann mapping theorem}}

* {{annotated link|Nevanlinna's criterion}}

== Note ==

{{PlanetMath attribution|title=univalent analytic function|id=5633}}

{{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|idurlname=5633UnivalentAnalyticFunction}}

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[[Category:Analytic functions]]