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][1] 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.


  [1]: https://www.unige.ch/~smirnov/papers/ecm-publ.pdf