Rotated Koebe function

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 f : D → C from the unit disk D onto a subset of the complex plane contains the disk whose center is f(0) and whose radius is |f′(0)|/4.

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 1/4 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 |z| > 1. Then

In fact, if r > 1, the complement of the image of the disk |z| > r is a bounded domain X(r). Its area is given by

Since the area is positive, the result follows by letting r decrease to 1. The above proof shows equality holds if and only if the complement of the image of g 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 1/4 in the theorem cannot be improved, as the image domain f(D) does not contain the point z = −1/4 and so cannot contain any disk centred at 0 with radius larger than 1/4.

The rotated Koebe function is

with α a complex number of absolute value 1. The Koebe function and its rotations are schlicht: that is, univalent (analytic and one-to-one) and satisfying f(0) = 0 and f′(0) = 1.

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