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
[edit]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
[edit]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
[edit]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
[edit]- Biholomorphic mapping – Bijective holomorphic function with a holomorphic inverse
- De Branges's theorem – Statement in complex analysis; formerly the Bieberbach conjecture
- Koebe quarter theorem – Statement in complex analysis
- Riemann mapping theorem – Mathematical theorem
- Nevanlinna's criterion – Characterization of starlike univalent holomorphic functions
Note
[edit]- ^ (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.")
- ^ (Nehari 1975)
References
[edit]- 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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