Univalent function: Difference between revisions

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{{Short description|Mathematical concept}}

{{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>{{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>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]]}}

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

* [[{{annotated link|Koebe quarter theorem]]}}

* [[{{annotated link|Riemann mapping theorem]]}}

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

* [[Schlicht function]]

== Note ==

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*{{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 journalbook |doi=10.40641007/SM174978-394-5011-5206-8|title=AConvex remarkand onStarlike separateMappings holomorphyin Several Complex Variables |year=20061998 |last1=JarnickiGong |first1=MarekSheng |last2isbn=Pflug |first2=Peter |journal=Studia Mathematica |volume=174 |issue=3 |pages=309–317 |s2cid=15660985978-94-010-6191-9 }}

*{{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 |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|page=146}}

{{PlanetMath attribution|title=univalent analytic function|urlname=UnivalentAnalyticFunction}}