In complex analysis, a branch of mathematics, the Koebe 1/4 theorem states the following: Koebe Quarter Theorem.
The image of an injective analytic function
onto a subset of the complex plane contains the disk whose center is
.The theorem is named after Paul Koebe, who conjectured the result in 1907.
The theorem was proven by Ludwig Bieberbach in 1916.
The example of the Koebe function shows that the constant
in the theorem cannot be improved (increased).
A related result is the Schwarz lemma, and a notion related to both is conformal radius.
Suppose that is univalent in
, the complement of the image of the disk
Its area is given by Since the area is positive, the result follows by letting
The above proof shows equality holds if and only if the complement of the image of
has zero area, i.e. Lebesgue measure zero.
This result was proved in 1914 by the Swedish mathematician Thomas Hakon Grönwall.
The Koebe function is defined by Application of the theorem to this function shows that the constant
in the theorem cannot be improved, as the image domain
with radius larger than
The rotated Koebe function is with
a complex number of absolute value
The Koebe function and its rotations are schlicht: that is, univalent (analytic and one-to-one) and satisfying
Then This follows by applying Gronwall's area theorem to the odd univalent function Equality holds if and only if
is a rotated Koebe function.
This result was proved by Ludwig Bieberbach in 1916 and provided the basis for his celebrated conjecture that
, proved in 1985 by Louis de Branges.
Applying an affine map, it can be assumed that so that In particular, the coefficient inequality gives that
Applying the coefficient inequality to
gives so that The Koebe distortion theorem gives a series of bounds for a univalent function and its derivative.
It is a direct consequence of Bieberbach's inequality for the second coefficient and the Koebe quarter theorem.
be a univalent function on
is a Koebe function