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

For an interval E=[a,b]E=[a,b] on the real line, let μμ be either the equilibrium measure, or the normalized Lebesgue measure of EE, and let VμVμ denote the associated logarithmic potential. In the present paper, we construct a function  ff which is analytic on EE and possesses four branch points of second order outside of EE such that the family of the admissible compacta of ff has no minimizing elements with regard to the extremal theoretic-potential problem, in the external field equals V−μV−μ.