Bloch's theorem (complex analysis)

Bloch's theorem, named after André Bloch is as follows.

Let f be a holomorphic function on a region (i.e. and open subset of C) that includes as a subset the closure of the unit disk |z} < 1. Suppose f(0) = 0 and f &prime(0) = 1. Then for some subset S of the open unit disk, f is injective on S and f(S) contains a disk of radius 1/72.

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