Bloch’s constant

Bloch’s theorem can be stated in the following way:

Bloch’s Theorem.

Let F be the set of all functionsMathworldPlanetmath f holomorphic on a region containing the closure of the disk D={zC:|z|<1} and satisfying f(0)=0 and f(0)=1. For each fF let β(f) be the supremum of all numbers r such that there is a disk SD on which f is injective and f(S) contains a disk of radius r. Let B be the infimum of all β(f), for fF. Then B1/72.

The constant B is usually referred to as Bloch’s constant. Nowadays, better bounds are known and, in fact, it has been conjectured that B has the following tantalizing form


where Γ(x) is the gamma functionDlmfDlmfMathworldPlanetmath.


  • 1 John B. Conway, Functions of One Complex Variable I, Second Edition, 1978, Springer-Verlag, New York.
Title Bloch’s constant
Canonical name BlochsConstant
Date of creation 2013-03-22 15:58:04
Last modified on 2013-03-22 15:58:04
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 5
Author alozano (2414)
Entry type Definition
Classification msc 32H02
Related topic LandausConstant