On the existence of compacta of minimal capacity in the theory of rational approximation of multi-valued analytic functions

摘要

For an interval E = [a, b] on the real line, let mu be either the equilibrium measure, or the normalized Lebesgue measure of E, and let V-mu denote the associated logarithmic potential. In the present paper, we construct a function f which is analytic on E and possesses four branch points of second order outside of E such that the family of the admissible compacta of f has no minimizing elements with regard to the extremal theoretic-potential problem, in the external field equals V-mu. (C) 2015 Elsevier Inc. All rights reserved.