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

Basic properties

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

f : G → Ω {\displaystyle f:G\to \Omega }

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

( f − 1 ) ′ ( f ( z ) ) = 1 f ′ ( z ) {\displaystyle (f^{-1})'(f(z))={\frac {1}{f'(z)}}}

for all z {\displaystyle z} in G . {\displaystyle G.}

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

f : ( − 1 , 1 ) → ( − 1 , 1 ) {\displaystyle f:(-1,1)\to (-1,1)\,}

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

See also

• Biholomorphic mapping– Bijective holomorphic function with a holomorphic inversePages displaying short descriptions of redirect targets
• 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

• Conway, John B. (1995). . Functions of One Complex Variable II. Graduate Texts in Mathematics. Vol.159. doi:. ISBN978-1-4612-6911-3.
• . Sources in the Development of Mathematics. 2011. pp.907–928. doi:. ISBN9780521114707.
• Duren, P. L. (1983). Univalent Functions. Springer New York, NY. p.XIV, 384. ISBN978-1-4419-2816-0.
• Gong, Sheng (1998). Convex and Starlike Mappings in Several Complex Variables. doi:. ISBN978-94-010-6191-9.
• Jarnicki, Marek; Pflug, Peter (2006). . Studia Mathematica. 174 (3): 309–317. arXiv:. doi:. S2CID.
• Nehari, Zeev (1975). Conformal mapping. New York: Dover Publications. p.146. ISBN0-486-61137-X. OCLC.

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