Landau's constants

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

<math>f'(0) = 1.\,</math>

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

<math>f'(0) = 1.\,</math>

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

<math>0.4330 + 10^{-14} < B < 0.472 \,\! </math>

<math>0.5 < L < 0.544 \,\!</math>

<math>0.5 < A \le 0.7853.</math>

External links

• Eric W. Weisstein, Landau Constant at MathWorld.