# Coefficient problem in the class $\Sigma$

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.

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 Probably this conjecture is unpublished but it has been discussed among the specialists in the 1990s.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