# univalent analytic function

###### Definition.

An analytic function on an open set is called 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 UnivalentAnalyticFunction 2013-03-22 14:12:06 2013-03-22 14:12:06 jirka (4157) jirka (4157) 6 jirka (4157) Definition msc 30C55 univalent function univalent