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univalent analytic function
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.
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