Koebe 1/4 theorem

Theorem (Koebe).

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

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 $w\notin f(𝔻)$ we have that $|w|\ge 1/4$. Furthermore, if we look at the Koebe function, we can see that the constant $1/4$ is sharp and cannot be improved.

Title	Koebe 1/4 theorem
Canonical name	Koebe14Theorem
Date of creation	2013-03-22 14:23:57
Last modified on	2013-03-22 14:23:57
Owner	jirka (4157)
Last modified by	jirka (4157)
Numerical id	8
Author	jirka (4157)
Entry type	Theorem
Classification	msc 30C45
Synonym	Köbe 1/4 theorem
Synonym	Koebe one-fourth theorem
Synonym	Köbe one-fourth theorem
Related topic	SchlichtFunctions