The second coefficient body of \(\Sigma\) (Q1085300)

Let \(\Sigma\) denote the set of univalent functions \[ f(z)=z+\sum^{\infty}_{n=0}b_ nz^{-n}\quad in\quad | z| >1. \] Its second coefficient body is the compact set \(\{(b_ 1,b_ 2):\) \(f\in \Sigma \}\). This coefficient body is shown to be strictly convex, and its boundary is parametrized. After rotations, the problem reduces to maximizing the functional \(Re\{b_ 2+\lambda b_ 1\}\) over \(\Sigma\) for each fixed complex \(\lambda\). The solution basically follows the lines given by \textit{Y. J. Leung} and the reviewer [Ann. Acad. Sci. Fenn., Ser. A I 11, 39-62 (1986; Zbl 0603.30022)] except that uniqueness up to translation of extremal functions is deduced directly and skillfully, without recourse to Jenkins' general coefficient theorem.