Let $\phi$ be a univalent (i .e., holomorphic and injective) function on the unit disc. Consider the growth gate of the lengths of the images of circles $|z|=r$ as $r$ goes to 1: $$ \limsup_{r\to 1-}\frac{\log \int_0^{2\pi}|\phi'(re^{i\theta})|d\theta}{|\log(1-r)|}, $$ and denote by $\gamma$ the supremum of this quantity over all bounded univalent $\phi$.
Beliaev and Smirnov describe the work on upper bounds for $\gamma$, as of 2004:
Conjectural value of $\gamma=B(1)$ is $1/4$, but existing estimates are quite far. The first result in this direction is due to Bieberbach [7] who in 1914 used his area theorem to prove that $\gamma\leq 1/2$. <...> Clunie and Pommerenke in [16] proved that $\gamma \leq 1 / 2 − 1 / 300$ <...> Carleson and Jones [13] <...> used Marcinkiewicz integrals to prove $\gamma< 0.49755$. This estimate was improved by Makarov and Pommerenke [43] to $\gamma< 0.4886$ and then by Grinshpan and Pommerenke [21] to $γ<0.4884$. The best current estimate is due to Hedenmalm and Shimorin [24] who quite recently proved that $B(1)<0.46.$
I guess the latter estimate is still the best as of now.