Ahlfors–Grunsky conjecture for Bloch’s constant

Determine whether Bloch’s constant equals the upper bound proposed by Ahlfors and Grunsky, namely \(\Gamma(1/3)\Gamma(11/12)/(\Gamma(1/4)\sqrt{1+\sqrt{3})\).

Background

Bloch’s constant $B$ is bounded below by $\sqrt{3}/4$ and above by the Ahlfors–Grunsky expression $\Gamma(1/3)\Gamma(11/12)/(\Gamma(1/4)\sqrt{1+\sqrt{3})$. Ahlfors and Grunsky conjectured that this upper bound is the exact value of $B$.

The paper improves the known explicit lower bound for $B$ but leaves a positive gap between the improved lower bound and the conjectured Ahlfors–Grunsky value; thus the equality remains unresolved.