Bloch's theorem (complex variables)

The set of Bloch is a statement of function theory, which was proved in 1925 by the French mathematician André Bloch. The sentence specifies a limit on the complexity of the image area of holomorphic functions.

Motivation

It was a field. Then a non-constant holomorphic function is an open mapping, which means that for each pixel, there is a circular disk that is in the picture. The set of Bloch sharpens this statement to the effect that there is a circular disk of a certain size in the image area ( up to normalization ) is independent of the function.

Statement

If the unit disc and a holomorphic function is then the image area contains a circular disk of radius

Consequences

• It is an area and holomorphic with a. Then contains a circular disk of radius with
• A non - constant entire contains (on all holomorphic ) function circular disks of arbitrarily large radii. The centers of the circles are different but depending on the radius, so it will not always covered, for example, is
• The Little Theorem Picard can be proved with the help of the set of Bloch, if you do not want to rely on the results of the uniformization.

Landau constant

The set of Bloch gives a lower bound for the radius. This raises the question of the optimal constant, so then, what is the largest circular disk that is present in every case. For this purpose, whether for the supremum of all possible radii of circular disks that are used in space, defined as:

Landau, the constant is then defined as

The exact size of the constant is not known, but there are the following assessments:

Where is the Euler gamma function called.

The upper limit found Raphael Robinson in 1937 ( unpublished) and Hans Rademacher 1942, which also suggested that the upper bound corresponds to the actual value of the landau between constant. This conjecture is an open problem until now.

Bloch constant

The condition in the set of Bloch implies according to the implicit function theorem that an unspecified specific area will be mapped even biholomorphic to its image. Therefore, it is obvious that the same question with the additional condition that the place in the image area circular disk space must be biholomorphes image of an area, also be investigated.

Bloch himself scored the assessment

It is for the supremum of all possible radii of circular disks in which biholomorphes image of a part of the area are defined:

Bloch's constant is then defined as

The precise value of the Bloch constant is also not known to have been found so far, the estimations

The upper limit found LV Ahlfors and Grunsky H. 1937. They suggested also that this limit corresponds to the actual value of the bloch between constant. Also, this assumption has not yet been proven.