Bloch Constant

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.