Koebe distortion theorem

Theorem (Koebe).

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

$$\frac{1-|z|}{{(1+|z|)}^3}\le |f^{\prime }(z)|\le \frac{1+|z|}{{(1-|z|)}^3},$$

and

$$\frac{|z|}{{(1+|z|)}^2}\le |f(z)|\le \frac{|z|}{{(1-|z|)}^2}.$$

Equality holds for one of the four inequalities at some point $z\mathrm{\ne }\mathrm{0}$ if and only if $f$ is a rotation of the Koebe function.

Following is a generalized distortion theorem.

Theorem.

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

$$\frac{1}{M}\le \frac{|f^{\prime }(z)|}{|f^{\prime }(w)|}\le M.$$