Bloch Constant
N.B. A detailed on-line essay by S. Finchwas 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 allnumbers 
such that there is a disk 
in 
on which is One-to-One and such that 
contains a diskof 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).See also Landau Constant
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.
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