Koebe 1/4 theoremKoebe14Theorem2004-06-07 14:28:502005-03-07 05:05:54Theorem% this is the default PlanetMath preamble. as your knowledge
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\newtheorem*{defn}{Definition}\begin{thm}[Koebe]
Suppose $f$ is a schlicht function (univalent function on the unit disc
such that $f(0) = 0$ and $f'(0) = 1$) and ${\mathbb{D}} \subset {\mathbb{C}}$ is the unit disc in the complex plane, then
\begin{equation*}
f({\mathbb{D}}) \supseteq \{ w \mid \lvert w \rvert < 1/4 \} .
\end{equation*}
\end{thm}
That is, if a univalent function on the unit disc maps 0 to 0 and has derivative
1 at 0, then the image of the unit disc contains the ball of radius $1/4$. So for any $w \notin f({\mathbb{D}})$ we have that $\lvert w \rvert \geq 1/4$. Furthermore, if we look at the Koebe function, we can see that the constant $1/4$ is sharp and cannot be improved.
\begin{thebibliography}{9}
\bibitem{Conway:complexII}
John~B. Conway.
{\em \PMlinkescapetext{Functions of One Complex Variable II}}.
Springer-Verlag, New York, New York, 1995.
\end{thebibliography}