Bloch Constant

Let  be the set of complex analytic functions  defined on an open region containing the set closure of the unit disk  satisfying  and . For each  in , let  be the supremum of all numbers  such that there is a subregion  in  on which  is one-to-one and such that  contains a disk of radius . In 1925, Bloch (Conway 1989) showed that .

Define Bloch's constant by

(1)

Ahlfors and Grunsky (1937) derived

(2)

Bonk (1990) proved that , which was subsequently improved to  (Chen and Gauthier 1996; Xiong 1998; Finch 2003, p. 456).

Ahlfors and Grunsky (1937) also conjectured that the upper limit is actually the value of ,

			(3)
			(4)
			(5)

(OEIS A085508; Le Lionnais 1983).

REFERENCES:

Ahlfors, L. V. and Grunsky, H. "Über die Blochsche Konstante." Math. Zeit. 42, 671-673, 1937.

Bonk, M. "On Bloch's Constant." Proc. Amer. Math. Soc. 110, 889-894, 1990.

Chen, H. and Gauthier, P. M. "On Bloch's Constant." J. d'Analyse Math. 69, 275-291, 1996.

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

Finch, S. R. "Bloch-Landau Constants." §7.1 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 456-459, 2003.

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

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

Sloane, N. J. A. Sequence A085508 in "The On-Line Encyclopedia of Integer Sequences."

Xiong, C. "Lower Bound of Bloch's Constant." Nanjing Daxue Xuebao Shuxue Bannian Kan 15, 174-179, 1998.