On coefficient estimates and conjectures for the class \(\Sigma\) (Q800522)

Let \(\Sigma\) denote the family of univalent functions \(f(z)=z+\sum^{\infty}_{n=0}b_ nz^{-n}\) in \(D=\{z: | z| >1\}.\) Using the basic lemma with constraint from the paper of \textit{W. E. Kirwan} and \textit{G. Schober}, Math. Z. 180, 19-40 (1982; Zbl 0498.30024) and some results of Jenkins, Garabedian and Schiffer and Kubota several very nice inequalities for combinations of the coefficients of the functions from \(\Sigma\) are determined. Especially the conjecture of Kirwan that \(Re\{nb_ 1-b_ n\}\leq n,\quad n\geq 2\) for \(f\in\Sigma \) is examined. The paper is ended by four conjectures connected with Kirwan's conjecture. - This is a very interesting paper.