The maximum of the product of conformal, radii of nonoverlapping domains in the disk

Some Geometric Properties of Extremal Decompositions

“…In [3], the author proved that the quadratic differential determining the extremal system of domains in Problem 4 has at least two distinct zeros on the circle T = {|z| = 1}. Theorem 1 supplements this assertion.…”

“…. , m. Then (see the proof of Theorem 2 in [3]), for all sufficiently small ε > 0 we have b j ∈ G j ⊂ U ζ , j = 1, . .…”

“…, n. Find the maximum of the functional Obviously, the functional I 1 in Problem 2 is bounded and invariant with respect to rotation transformations z → e iϕ z, ϕ ∈ R. It is well known that the functional I 2 is invariant with respect to linear-fractional transformations of C, whereas the functional I 3 is invariant with respect to hyperbolic motions of the disk U . The functionals I 2 and I 3 are bounded as well (see, e.g., [3]). From the theory of normal families it follows that systems of points and domains on which the functionals in Problems 2-4 attain their maxima do exist.…”