Coefficient problem in the class $\Sigma$

Question

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 :

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 ?

Surprinsingly these informations are difficult to get in the literature since $\Sigma$ is much less studied than $\mathcal{S}$. Thanks for any help.

Answer 1

The latest results are not very recent. Most of them are mentioned in the paper

MR1162188 Carleson, Lennart; Jones, Peter W. On coefficient problems for univalent functions and conformal dimension. Duke Math. J. 66 (1992), no. 2, 169–206.

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 Probably this conjecture is unpublished but it has been discussed among the specialists in the 1990s.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.