Abstract

We study the problem maxh∈Sℜ[h(z1)+h(z2)] with z1,z2 in Δ. We show that no rotation of the Koebe function is a solution for this problem except possibly its real rotation, and only when z1=z¯2 or z1,z2 are both real, and are in a neighborhood of the x-axis. We prove that if the omitted set of the extremal function f is part of a straight line that passes through f(z1) or f(z2) then f is the Koebe function or its real rotation. We also show the existence of solutions that are not unique and are different from the Koebe function or its real rotation. The situation where the extremal value is equal to zero can occur and it is proved, in this case, that the Koebe function is a solution if and only if z1 and z2 are both real numbers and z1z2<0.