On an extremal problem for nonoverlapping domains

The paper considers the problem of finding the range of the functional $I=J(f(z_0), \overline{f(z_0)}, F(\zeta_0), \overline{F(\zeta_0)})$ defined on the class $\mathfrak{M}$ of functions pairs $(f(z), F(\zeta))$ that are univalent in the system of the disk and the interior of the disk, using the method of internal variations. We establish that the range of this functional is bounded by the curve whose equation is written in terms of elliptic integrals, depending on the parameters of the functional $I$.