Bloch's theorem (complex variables)
Let f be a holomorphic function in the unit disk |z| ≤ 1. Suppose that |f′(0)| = 1. Then there exists a disc of radius b and an analytic function φ in this disc, such that f(φ(z)) = z for all z in this disc. Here b > 1/72 is an absolute constant.
If f is a holomorphic function in the unit disc with the property |f′(0)| = 1, then the image of f contains a disc of radius l, where l ≥ b is an absolute constant.
This theorem is named after Edmund Landau.
Bloch's theorem was inspired by the following theorem of Georges Valiron:
Theorem. If f is a non-constant entire function then there exist discs D of arbitrarily large radius and analytic functions φ in D such that f(φ(z)) = z for z in D.
Bloch's theorem corresponds to Valiron's theorem via the so-called Bloch's Principle.
Bloch's and Landau's constants
The lower bound 1/72 in Bloch's theorem is not the best possible. The number B defined as the supremum of all b for which this theorem holds, is called the Bloch's constant. Bloch's theorem tells us B ≥ 1/72, but the exact value of B is still unknown.
The similarly defined optimal constant L in Landau's theorem is called the Landau's constant. Its exact value is also unknown.
The best known bounds for B at present are
In their paper, Ahlfors and Grunsky conjectured that their upper bounds are actually the true values of B and L.
- Ahlfors, Lars Valerian; Grunsky, Helmut (1937). "Über die Blochsche Konstante". Mathematische Zeitschrift. 42 (1): 671–673. doi:10.1007/BF01160101. S2CID 122925005.
- Baernstein, Albert II; Vinson, Jade P. (1998). "Local minimality results related to the Bloch and Landau constants". Quasiconformal mappings and analysis. Ann Arbor: Springer, New York. pp. 55–89.
- Bloch, André (1925). "Les théorèmes de M.Valiron sur les fonctions entières et la théorie de l'uniformisation". Annales de la Faculté des Sciences de Toulouse. 17 (3): 1–22. doi:10.5802/afst.335. ISSN 0240-2963.
- Chen, Huaihui; Gauthier, Paul M. (1996). "On Bloch's constant". Journal d'Analyse Mathématique. 69 (1): 275–291. doi:10.1007/BF02787110. S2CID 123739239.