Univalent function

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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.[1][2]

Examples

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The function   is univalent in the open unit disc, as   implies that  . As the second factor is non-zero in the open unit disc,   so   is injective.

Basic properties

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One can prove that if   and   are two open connected sets in the complex plane, and

 

is a univalent function such that   (that is,   is surjective), then the derivative of   is never zero,   is invertible, and its inverse   is also holomorphic. More, one has by the chain rule

 

for all   in  

Comparison with real functions

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

 

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

See also

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Note

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  1. ^ (Conway 1995, p. 32, 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.")
  2. ^ (Nehari 1975)

References

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  • Conway, John B. (1995). "Conformal Equivalence for Simply Connected Regions". Functions of One Complex Variable II. Graduate Texts in Mathematics. Vol. 159. doi:10.1007/978-1-4612-0817-4. ISBN 978-1-4612-6911-3.
  • "Univalent Functions". Sources in the Development of Mathematics. 2011. pp. 907–928. doi:10.1017/CBO9780511844195.041. ISBN 9780521114707.
  • Duren, P. L. (1983). Univalent Functions. Springer New York, NY. p. XIV, 384. ISBN 978-1-4419-2816-0.
  • Gong, Sheng (1998). Convex and Starlike Mappings in Several Complex Variables. doi:10.1007/978-94-011-5206-8. ISBN 978-94-010-6191-9.
  • Jarnicki, Marek; Pflug, Peter (2006). "A remark on separate holomorphy". Studia Mathematica. 174 (3): 309–317. arXiv:math/0507305. doi:10.4064/SM174-3-5. S2CID 15660985.
  • Nehari, Zeev (1975). Conformal mapping. New York: Dover Publications. p. 146. ISBN 0-486-61137-X. OCLC 1504503.

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