univalent analytic function
Definition.
An analytic function^{} on an open set is called univalent^{} if it is onetoone.
For example mappings of the unit disc to itself ${\varphi}_{a}:\mathbb{D}\to \mathbb{D}$, where ${\varphi}_{a}(z)=\frac{za}{1\overline{a}z}$, for any $a\in \mathbb{D}$ are univalent. The following summarizes some basic of univalent functions.
Proposition.
Suppose that $G\mathrm{,}\mathrm{\Omega}\mathrm{\subset}\mathrm{C}$ are regions and $f\mathrm{:}G\mathrm{\to}\mathrm{\Omega}$ is a univalent mapping such that $f\mathit{}\mathrm{(}G\mathrm{)}\mathrm{=}\mathrm{\Omega}$ (it is onto), then

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${f}^{1}:\mathrm{\Omega}\to G$ (where ${f}^{1}(f(z))=z$) is an analytic function and ${({f}^{1})}^{\prime}(f(z))=\frac{1}{{f}^{\prime}(z)}$,

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${f}^{\prime}(z)\ne 0$ for all $z\in G$
References
 1 John B. Conway. . SpringerVerlag, New York, New York, 1978.
 2 John B. Conway. . SpringerVerlag, New York, New York, 1995.
Title  univalent analytic function 

Canonical name  UnivalentAnalyticFunction 
Date of creation  20130322 14:12:06 
Last modified on  20130322 14:12:06 
Owner  jirka (4157) 
Last modified by  jirka (4157) 
Numerical id  6 
Author  jirka (4157) 
Entry type  Definition 
Classification  msc 30C55 
Synonym  univalent function 
Synonym  univalent 