|
|
|
|
univalent analytic function
|
(Definition)
|
|
|
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 $z \in G$
- 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.
|
|
(view preamble | get metadata)
| Other names: |
univalent function, univalent |
|
|
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 7373 times total.
Classification:
| AMS MSC: | 30C55 (Functions of a complex variable :: Geometric function theory :: General theory of univalent and multivalent functions) |
|
|
|
|
|
|
Pending Errata and Addenda
|
|
|
|
|
|
|
|
|
|
|
