*N.B. A detailed on-line essay by S. Finch
was the starting point for this entry.*

Let be the set of Complex analytic functions defined on an open region containing the closure of the unit disk
satisfying and . For each in , let be the Supremum of all
numbers such that there is a disk in on which is One-to-One and such that contains a disk
of radius . In 1925, Bloch (Conway 1978) showed that . Define Bloch's constant by

Ahlfors and Grunsky (1937) derived

They also conjectured that the upper limit is actually the value of ,

(Le Lionnais 1983).

**References**

Conway, J. B. *Functions of One Complex Variable, 2nd ed.* New York: Springer-Verlag, 1989.

Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/bloch/bloch.html

Le Lionnais, F. *Les nombres remarquables.* Paris: Hermann, p. 25, 1983.

Minda, C. D. ``Bloch Constants.'' *J. d'Analyse Math.* **41**, 54-84, 1982.

© 1996-9

1999-05-26