Univalent functions maximizing \(\text{Re}[f(\zeta_ 1)+f(\zeta_ 2)]\) (Q1922861)

The author studies the extremal problem. \({\mathcal R} h(\zeta_1) + {\mathcal R} h(\zeta_2) =\) maximum for \(h \in S\) and \(\zeta_1\), \(\zeta_2\) in the unit disc \(\Delta\). It is a straightforward matter to write down a corresponding variational equation and in particular to see that an extremal function suitably extended maps \(\partial \Delta\) onto a trajectory arc of the quadratic differential \[ \left( {f (\zeta_1)^2 \over f(\zeta_1) - w} + {f (\zeta_2)^2 \over f (\zeta_2) - w} \right) \left( {dw \over w} \right)^2. \] The author gives some qualitative results mostly of the form that under certain subsidiary conditions the Koebe-function and its rotations are or are not solutions of the problem.