PlanetMath: univalent analytic function

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univalent analytic function

(Definition)

Definition 1 An analytic function on an open set is called univalent if it is one-to-one.

For example mappings of the unit disc to itself $\phi_a : {\mathbb{D}} \rightarrow {\mathbb{D}}$ , where $\phi_a(z) = \frac{z-a}{1 - \bar{a}z}$ , for any $a \in {\mathbb{D}}$ are univalent. The following proposition summarizes some basic properties of univalent functions.

Proposition 1 Suppose that $G,\Omega \subset {\mathbb{C}}$ are regions and $f \colon G \to \Omega$ is a univalent mapping such that $f(G) = \Omega$ (it is onto), then

$f^{-1} \colon \Omega \to G$ (where $f^{-1}(f(z)) = z$ ) is an analytic function and $(f^{-1})'(f(z)) = \frac{1}{f'(z)}$ ,
$f'(z) \not= 0$ for all $z \in G$

Bibliography

1: John B. Conway. Functions of One Complex Variable I. Springer-Verlag, New York, New York, 1978.
2: John B. Conway. Functions of One Complex Variable II. Springer-Verlag, New York, New York, 1995.

"univalent analytic function" is owned by jirka.

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Other names:

univalent function, univalent

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Cross-references: onto, regions, unit disc, mappings, one-to-one, open set, analytic function
There are 6 references to this entry.

This is version 3 of univalent analytic function, born on 2004-02-26, modified 2005-03-07.
Object id is 5633, canonical name is UnivalentAnalyticFunction.
Accessed 6185 times total.

Classification:

AMS MSC:

30C55 (Functions of a complex variable :: Geometric function theory :: General theory of univalent and multivalent functions)

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