Bloch's theorem (complex variables)
- Bloch's theorem (complex variables)
Bloch's theorem, named after André Bloch is as follows.
Let "f" be a holomorphic function on a region (i.e. an open, connected subset of C) that includes as a subset the closed unit disk |"z"| ≤ 1. Suppose "f"(0) = 0 and "f" ′(0) = 1. Then for some subset "S" of the open unit disk |"z"| < 1, "f" is injective on "S" and "f"("S") contains a disk of radius 1/72.
Bloch's constant
The lower bound 1/72 is not the best possible, whose exact value, called "Bloch's constant", is not known.
References
* Albert II Baernstein, Jade P. Vinson: "Local minimality results to the Bloch and Landau constants" in: "Quasiconformal mappings and analysis", Springer, New York 1998
* André Bloch: "Les théorèmes de M.Valiron sur les fonctions entières et la théorie de l'uniformisation." in: "Annales de la faculté des sciences de l'Université de Toulouse. Série 3." 17/1925, S. 1-22, ISSN|0240-2963
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