Univalent function: Difference between revisions

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{{Short description|Mathematical concept}}

{{other uses|Univalent}}

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

{{Merge to|Injective function|discuss=Talk:Injective function#Proposed merge of Univalent function into Injective function|date=August 2022}}

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>[[John B. {{harv|Conway]] (1996) ''Functions of One Complex Variable II'', |1995|page=32|loc=chapter 14: Conformal equivalence for simply connected regions, page 32, Springer-Verlag, New York, {{ISBN|0-387-94460-5}}. Definition 1.12: "A function on an open set is ''univalent'' if it is analytic and one-to-one."}}</ref><ref>{{Cite book harv|last=Nehari |first=Zeev |url=https://www.worldcat.org/oclc/1504503 |title=Conformal mapping |date=1975 |publisher=Dover Publications |isbn=0-486-61137-X |___location=New York |oclc=1504503|p=146}}</ref>

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>[[John B. Conway]] (1996) ''Functions of One Complex Variable II'', chapter 14: Conformal equivalence for simply connected regions, page 32, Springer-Verlag, New York, {{ISBN|0-387-94460-5}}. Definition 1.12: "A function on an open set is ''univalent'' if it is analytic and one-to-one."</ref><ref>{{Cite book |last=Nehari |first=Zeev |url=https://www.worldcat.org/oclc/1504503 |title=Conformal mapping |date=1975 |publisher=Dover Publications |isbn=0-486-61137-X |___location=New York |oclc=1504503|p=146}}</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>fz = w</math> mustso <math>f</math> beis injective.

==Basic properties==

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

given by ''&fnof;''<math>f(''x'')&nbsp;=&nbsp;''x''<sup>^3</supmath>. This function is clearly injective, 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 injective; 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>).

== See also ==

* {{annotated link|Biholomorphic mapping}}

* [[Schlicht function]]

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

* {{annotated link|Koebe quarter theorem}}

* {{annotated link|Riemann mapping theorem}}

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

== ReferencesNote ==

{{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}}}}

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

*{{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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