Landau's constants

In complex analysis, a branch of mathematics, Landau's constants (Edmund Landau 1929) are certain mathematical constants that describe the behaviour of holomorphic functions defined on the unit disk. Consider the set F of all those holomorphic functions f on the unit disk for which

$f'(0) = 1.\,$

We define L_f to be the radius of the largest disk contained in the image of f, and B_f to be the radius of the largest disk that is the biholomorphic image of a subset of a unit disk.

Landau's constants are then defined as the infimum of L_f or B_f, where f is any holomorphic function or any injective holomorphic function on the unit disk with

$f'(0) = 1.\,$

The three resulting constants are abbreviated L, B, and A (for injective functions), respectively.

The exact values of L, B, and A are not known, but it is known that

$0.4330 + 10^{-14} < B < 0.472 \,\!$

$0.5 < L \le 0.5432589... \,\!$ (sequence A081760 in OEIS)

$0.5 < A \le 0.7853$

See also

• Table of selected mathematical constants

References

• Landau, Edmund (1929), "Über die Blochsche Konstante und zwei verwandte Weltkonstanten", Mathematische Zeitschrift 30 (1), doi:10.1007/BF01187791

External links

• Weisstein, Eric W., "Landau Constant", MathWorld.