- Source: 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
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. 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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Kata Kunci Pencarian:
- Fungsi univalen
- Teorema limit seragam
- Univalent function
- Univalent
- De Branges's theorem
- Injective function
- Koebe quarter theorem
- Geometric function theory
- Grunsky matrix
- Riemann mapping theorem
- Loewner differential equation
- Nevanlinna's criterion
