# Koebe 1/4 theorem

###### Theorem (Koebe).

Suppose $f$ is a schlicht function (univalent function on the unit disc such that $f(0)=0$ and $f^{\prime}(0)=1$) and ${\mathbb{D}}\subset{\mathbb{C}}$ is the unit disc in the complex plane, then

 $f({\mathbb{D}})\supseteq\{w\mid\lvert w\rvert<1/4\}.$

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.

## References

• 1 John B. Conway. . Springer-Verlag, New York, New York, 1995.
Title Koebe 1/4 theorem Koebe14Theorem 2013-03-22 14:23:57 2013-03-22 14:23:57 jirka (4157) jirka (4157) 8 jirka (4157) Theorem msc 30C45 Köbe 1/4 theorem Koebe one-fourth theorem Köbe one-fourth theorem SchlichtFunctions