Let $\Sigma$ be the class of univalent (injective) holomorphic functions on $\mathbb{C}\backslash \mathbb{D}$ where $\mathbb{D}$ is the closed unit disk. Analogous to the famous Bieberbach conjecture is the problem of finding sharp bounds for the coefficient $b\_n$ of the functions $$g(z) = z + b\_0 + b\_1 z^{-1} + b\_2 z^{-2} + \cdots$$ in $\Sigma.$ However this problem is significantly more complicated than the Bieberbach conjecture since there are no good candidate for the extremal functions here. The initial conjecture $$|b\_n| \leq \frac{2}{n+1}$$ is false and, as far as I know, we only get the sharp bounds for $b\_2$ et $b\_3$, namely $$|b\_2| \leq \frac{2}{3} \quad \text{and} \quad |b\_3| \leq \frac{1}{2} + e^{-6}.$$ There are also some theorems that give more informations if we add extra conditions on the function $g$ but those ones do not interest me here. My question is quite general :



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> What are the recent results concerning the sharp bounds for the coefficients $b\_n$ in the general case ? For example do we know it for $|b\_4|$ or are we still stuck ? Are there some new candidate for the extremal functions ?
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Surprinsingly these informations are difficult to get in the literature since $\Sigma$ is much less studied than $\mathcal{S}$. Thanks for any help.



Coefficient problem in the class $\Sigma$
The latest results are not very recent. Most of them are mentioned in the paper



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> MR1162188 Carleson, Lennart; Jones, Peter W. On coefficient problems for univalent functions and conformal dimension. Duke Math. J. 66 (1992), no. 2, 169–206.
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Let $B\_n=\sup\_\Sigma |b\_n|$. The question is about the order of decrease of $B\_n$.
An easy estimate is $B\_n\leq n^{-1/2}$, and the old result of Clunie and Pommerenke
says $B\_n\leq Cn^{-0.503}$. I am not sure whether this has been ever improved, if it was, then by very little. Carleson and Jones proved that $\gamma:=-\lim \log B\_n/\log n$ exists. (So by Clunie-Pommerenke, $\gamma>0.5$). The extremal functions are apparently conformal maps onto complements of some Julia sets, or other self-similar fractals. Using Julia sets, Carleson and Jones were able to show
that $\gamma\leq 0.79$,( computer-assisted), and conjectured that in fact $\gamma=3/4$. Probably this conjecture is unpublished but it has been discussed among the specialists in the 1990s. I don't know of any substantial progress since then.


