# Koebe distortion theorem

###### Theorem (Koebe).

Suppose $f$ is a schlicht function (univalent function on the unit disc such that $f(0)=0$ and $f^{\prime}(0)=1$) then

 $\frac{1-\lvert z\rvert}{(1+\lvert z\rvert)^{3}}\leq\lvert f^{\prime}(z)\rvert% \leq\frac{1+\lvert z\rvert}{(1-\lvert z\rvert)^{3}},$

and

 $\frac{\lvert z\rvert}{(1+\lvert z\rvert)^{2}}\leq\lvert f(z)\rvert\leq\frac{% \lvert z\rvert}{(1-\lvert z\rvert)^{2}}.$

Equality holds for one of the four inequalities at some point $z\not=0$ if and only if $f$ is a rotation of the Koebe function.

###### Theorem.

If $K$ is a compact subset of a region $G\subset{\mathbb{C}}$, then there is a constant $M$ (depending on $K$) such that for every univalent function $f$ on $G$ and ever pair of points $z,w\in K$ we have

 $\frac{1}{M}\leq\frac{\lvert f^{\prime}(z)\rvert}{\lvert f^{\prime}(w)\rvert}% \leq M.$

## References

• 1 John B. Conway. . Springer-Verlag, New York, New York, 1995.
Title Koebe distortion theorem KoebeDistortionTheorem 2013-03-22 14:23:25 2013-03-22 14:23:25 jirka (4157) jirka (4157) 7 jirka (4157) Theorem msc 30C45 distortion theorem generalized distortion theorem Köbe distortion theorem SchlichtFunctions