univalent analytic function

Definition.

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 summarizes some basic of univalent functions.

Proposition.

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 $(f^{-1})^{\prime}(f(z))=\frac{1}{f^{\prime}(z)}$ ,
•

$f^{\prime}(z)\not=0$ for all $z\in G$

References

1 John B. Conway. . Springer-Verlag, New York, New York, 1978.
2 John B. Conway. . Springer-Verlag, New York, New York, 1995.

Title	univalent analytic function
Canonical name	UnivalentAnalyticFunction
Date of creation	2013-03-22 14:12:06
Last modified on	2013-03-22 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