Landau's constants

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In complex analysis, Landau's constants 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

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