Bloch's theorem (complex variables)

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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.

[edit] Bloch's constant

The lower bound 1/72 is not the best possible, whose exact value, called "Bloch's constant", is not known.

[edit] 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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