Landau Constant

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  contains a disk of radius . Then

This constant is called the Landau constant, or the Bloch-Landau constant. Robinson (1938, unpublished) and Rademacher (1943) derived the bounds

(OEIS A081760), where  is the gamma function, and conjectured that the second inequality is actually an equality.

REFERENCES:

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

Rademacher, H. "On the Bloch-Landau Constant." Amer. J. Math. 65, 387-390, 1943.

Sloane, N. J. A. Sequence A081760 in "The On-Line Encyclopedia of Integer Sequences."