Univalent function

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

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

Basic properties

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

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

given by ƒ(x) = x3. 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(εω) = f(ε) (where ω is a primitive cube root of unity and ε is a positive real number smaller than the radius of G as a neighbourhood of 0).

See also

Note

  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

  • 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. 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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