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

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For an interval E=[a,b] on the real line, let μ be either the equilibrium measure, or the normalized Lebesgue measure of E, and let Vμ 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−μ.