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Landau Constant

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 Robinson (1938, unpublished) and Rademacher (1943) derived the bounds

1/2<L<=(Gamma(1/3)Gamma(5/6))/(Gamma(1/6))=0.5432589...

(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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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."

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Landau Constant

Cite this as:

Weisstein, Eric W. "Landau Constant." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LandauConstant.html

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