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

${\displaystyle f'(0)=1.\,}$

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

${\displaystyle f'(0)=1.\,}$

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

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

${\displaystyle {{\sqrt {3}} \over {4}}=0.4330127019....<B\leq {\left({1 \over {\sqrt {1+{\sqrt {3}}}}}\right)\left({{\Gamma \left({1 \over 3}\right)\Gamma \left({11 \over 12}\right)} \over {\Gamma \left({1 \over 4}\right)}}\right)}<0.472<{1 \over 2}<L}$

B is the Bloch's constant.

${\displaystyle 0.5<L\leq {{\Gamma {{1} \over {3}}\Gamma {{5} \over {6}}} \over {\Gamma {{1} \over {6}}}}=0.543258965342...\,\!}$ (sequence A081760 in the OEIS)

${\displaystyle 0.5<A\leq 0.7853}$