Bloch's and Landau's Constants
The lower bound 1/72 in Bloch's theorem is not the best possible. The number B defined as the supremum of all b for which this theorem holds, is called the Bloch's constant. Bloch's theorem tells us B ≥ 1/72, but the exact value of B is still unknown.
The similarly defined optimal constant L in Landau's theorem is called the Landau's constant. Its exact value is also unknown.
The best known bounds for B at present are
where Γ is the Gamma function. The lower bound was proved by Chen and Gauthier, and the upper bound dates back to Ahlfors and Grunsky. They also gave an upper bound for the Landau constant.
In their paper, Ahlfors and Grunsky conjectured that their upper bounds are actually the true values of B and L.
Read more about this topic: Bloch's Theorem (complex Variables)