Koebe 1/4 theorem

Theorem (Koebe).

Suppose f is a schlicht functionMathworldPlanetmath (univalent functionMathworldPlanetmath on the unit disc such that f(0)=0 and f(0)=1) and DC is the unit disc in the complex planeMathworldPlanetmath, 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 wf(𝔻) we have that |w|1/4. Furthermore, if we look at the Koebe function, we can see that the constant 1/4 is sharp and cannot be improved.


  • 1 John B. Conway. . Springer-Verlag, New York, New York, 1995.
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