On a functional on the class of pairs of functions

The problem about the range $\Delta$ of the functional $\Phi$ is solved by the method of internal variations on the class of pairs of functions univalent in the “disk–exterior of the disk” system. We have obtained a system of functional-differential equations for pairs of functions $(f(z),F(\zeta))\in\mathfrak M'$ that are in correspondence with nonsingular boundary points $\Phi_0$ of the set $\Delta$. Each equation from the system contains a parameter which is a root of an algebraic equation of the sixth degree.