## Landau 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 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 Robinson (1938, unpublished) and Rademacher (1943) derived the bounds

where is the Gamma Function, and conjectured that the second inequality is actually an equality,

Finch, S. Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/bloch/bloch.html
Rademacher, H. On the Bloch-Landau Constant.'' Amer. J. Math. 65, 387-390, 1943.