Coefficient estimates for the class \(\Sigma\) (Q1073950)

Let \(\Sigma\) be the class of functions \(f(z)=z+\sum^{\infty}_{n=0}b_ nz^{-n}\) that are analytic and univalent for \(| z| >1\). For each \(n\geq 2\), \textit{Y. J. Leung} and the reviewer [Proc. Am. Math. Soc. 94, 659-664 (1985; Zbl 0584.30013)] proved that there is a finite value t, depending on n, such that the inequality \(Re\{tb_ 1+b_ n\}\leq t\) is valid for all functions in \(\Sigma\). As soon as this inequality is valid for one value of t, it is easy to see that it remains valid for all larger ones. Denote by \(A_ n\) the best (smallest) value for t. In this article the author shows that \(A_ 3\leq 2\), \(A_ 5\leq (27+8\sqrt{3})/12\), \(A_ 7\leq 5.5\), \(A_ 9<8\), and \(A_{11}<10\). The area theorem reduces the estimates to when \(b_ 1\) is near 1. Then the author makes profound use of Grunsky's inequalities. About the same time, Leung proved that \((e^ 4+3)/(e^ 4-1)\leq A_ 3\leq 2\) and conjectured that \(A_ 3=(e^ 4+3)/(e^ 4-1)\). Very recently, Ozawa has verified this conjecture.