univalent function

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

One can prove that if $G$ and $\backslash Omega$ are two open connected sets in the complex plane, and

:$f:\; G\; \backslash to\; \backslash Omega$

is a univalent function such that $f(G)\; =\; \backslash Omega$ (that is, $f$ is surjective), then the derivative of $f$ is never zero, $f$ is invertible, and its inverse $f^\{-1\}$ is also holomorphic. More, one has by the chain rule

:$(f^\{-1\})\text{'}(f(z))\; =\; \backslash frac\{1\}\{f\text{'}(z)\}$

for all $z$ in $G.$

For real analytic functions, unlike for complex analytic (that is, holomorphic) functions, these statements fail to hold. For example, consider the function

:$f:\; (-1,\; 1)\; \backslash to\; (-1,\; 1)\; \backslash ,$

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

• {{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}}

{{Reflist}}

• {{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 |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}}

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Category:Analytic functions

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