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.

Examples

The function [math]\displaystyle{ f \colon z \mapsto 2z + z^2 }[/math] is univalent in the open unit disc, as [math]\displaystyle{ f(z) = f(w) }[/math] implies that [math]\displaystyle{ f(z) - f(w) = (z-w)(z+w+2) = 0 }[/math]. As the second factor is non-zero in the open unit disc, [math]\displaystyle{ z = w }[/math] so [math]\displaystyle{ f }[/math] is injective.

Basic properties

One can prove that if [math]\displaystyle{ G }[/math] and [math]\displaystyle{ \Omega }[/math] are two open connected sets in the complex plane, and

[math]\displaystyle{ f: G \to \Omega }[/math]

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

[math]\displaystyle{ (f^{-1})'(f(z)) = \frac{1}{f'(z)} }[/math]

for all [math]\displaystyle{ z }[/math] in [math]\displaystyle{ G. }[/math]

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

[math]\displaystyle{ f: (-1, 1) \to (-1, 1) \, }[/math]

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

• Biholomorphic mapping
• De Branges's theorem
• Koebe quarter theorem
• Riemann mapping theorem
• Schlicht function

References

• Conway, John B. (1995). "Conformal Equivalence for Simply Connected Regions". Functions of One Complex Variable II. Graduate Texts in Mathematics. 159. doi:10.1007/978-1-4612-0817-4. ISBN 978-1-4612-6911-3. https://books.google.com/books?id=yV74BwAAQBAJ&pg=PA32. 
• "Univalent Functions". Sources in the Development of Mathematics. 2011. pp. 907–928. doi:10.1017/CBO9780511844195.041. ISBN 9780521114707. https://doi.org/10.1017/CBO9780511844195.041. 
• 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. 
• Nehari, Zeev (1975). Conformal mapping. New York: Dover Publications. p. 146. ISBN 0-486-61137-X. OCLC 1504503. https://www.worldcat.org/oclc/1504503. 

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