Koebe 1/4 theorem
Theorem (Koebe).
Suppose is a schlicht function (univalent function
on the unit discsuch that and ) and 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 derivative1 at 0, then the image of the unit disc contains the ball of radius . So for any we have that . Furthermore, if we look at the Koebe function, we can see that the constant is sharp and cannot be improved.
References
- 1 John B. Conway..Springer-Verlag, New York, New York, 1995.