Koebe quarter theorem

Every injective holomorphic function on the unit disk that fixes zero and has derivative one there has an image containing the disk of radius one quarter centered at zero.

Univalent function: A holomorphic function that is injective on its domain. The theorem applies precisely to normalized univalent functions on the unit disk.
Holomorphic function: A complex-valued function that is complex differentiable at every point of an open domain. The theorem concerns holomorphic maps of the unit disk.
Ludwig Bieberbach: A German mathematician who initiated the coefficient conjecture for normalized univalent functions in 1916. His coefficient conjecture prompted techniques closely connected to sharp mapping estimates.
Riemann mapping theorem: A theorem stating that every simply connected proper plane domain is conformally equivalent to the unit disk. The quarter theorem gives a size guarantee after a conformal map is normalized.
Normalized univalent function: A univalent holomorphic function on the unit disk satisfying f(0)=0 and f′(0)=1. These normalization conditions define the class for which the quarter-radius guarantee holds.
Injective function: A function that maps distinct inputs to distinct outputs. Injectivity is the condition that makes the theorem’s image guarantee possible.
Bieberbach conjecture: The conjecture that the nth Taylor coefficient of a normalized univalent function has modulus at most n. The quarter theorem belongs to the same classical study of extremal univalent functions.
Conformal radius: A scale associated with a simply connected plane domain and a conformal map from the unit disk. The theorem bounds the inradius of a normalized conformal image in terms of its conformal scale.
Koebe function: The function f(z)=z/(1−z)², an extremal normalized univalent function on the unit disk. Its image omits the ray extending left from −1/4, showing the bound is sharp.
Unit disk: The set of complex numbers whose modulus is less than one. It is the domain on which the normalized functions in the theorem are defined.

Linked from 4 pages