On the reduced modulus of the complex sphere

Transfinite Diameter with Respect to the Neumann Function

“…Conversely, if the points of Z * are extremal n-points, then for any other nonextremal collection Z inequalities (11) and (12) hold for a sufficiently small r.…”

“…In [10, p. 364] for a finitely connected domain G and in [11] for G = C z the following asymptotic formula was proved: …”

“…If, for a collection Z * = {z * k } n k=1 of points in E, there is a positive δ such that for an arbitrary r, 0 < r < δ, and any other collection Z = {z k } n k=1 of points in E either cap C(G, r, Z * ) ≥ cap C(G, r, Z) (11) or I(u n (z, Z * ), G r ) ≤ I(u n (z, Z), G r ),…”

“…In the present paper, by applying the technique of generalized condensers, developed by Dubinin and his school [9][10][11][12], we obtain representations for the transfinite diameter with respect to the Neumann function in terms of the condenser capacity and the Dirichlet integral of a certain function. As corollaries of these representations, some estimates on the diameter in question for a set located in the exterior of the unit disk are established.…”

“…Let one of inequalities (11) and (12) be valid. By using (7) and (10), we obtain Therefore, Z * is an extremal collection of points.…”