univalent analytic function
UnivalentAnalyticFunction
2004-02-26 13:41:34
2005-03-07 20:24:40
Definition
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\begin{defn}
An analytic function on an open set is called {\em univalent} if it is one-to-one.
\end{defn}
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 \PMlinkescapetext{proposition} summarizes some
basic \PMlinkescapetext{properties} of univalent functions.
\begin{prop}
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
\begin{itemize}
\item $f^{-1} \colon \Omega \to G$ (where $f^{-1}(f(z)) = z$) is an analytic
function and $(f^{-1})'(f(z)) = \frac{1}{f'(z)}$,
\item $f'(z) \not= 0$ for all $z \in G$
\end{itemize}
\end{prop}
\begin{thebibliography}{9}
\bibitem{Conway:complexI}
John~B. Conway.
{\em \PMlinkescapetext{Functions of One Complex Variable I}}.
Springer-Verlag, New York, New York, 1978.
\bibitem{Conway:complexII}
John~B. Conway.
{\em \PMlinkescapetext{Functions of One Complex Variable II}}.
Springer-Verlag, New York, New York, 1995.
\end{thebibliography}