Landau Constant

Let F be the set of complex analytic functions f defined on an open region containing the closure of the unit disk D={z:|z|<1} satisfying f(0)=0 and df/dz(0)=1. For each f in F, let l(f) be the supremum of all numbers r such that f(D) contains a disk of radius r. Then

 L=inf{l(f):f in F}.

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 Robinson (1938, unpublished) and Rademacher (1943) derived the bounds


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

See also

Bloch Constant, Landau-Kolmogorov Constants, Landau-Ramanujan Constant

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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."

Referenced on Wolfram|Alpha

Landau Constant

Cite this as:

Weisstein, Eric W. "Landau Constant." From MathWorld--A Wolfram Web Resource.

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