On the existence of compacta of minimal capacity in the theory of rational approximation of multi-valued analytic functions
For an interval $E=[a,b]$ on the real line, let $μ$ be either the equilibrium measure, or the normalized Lebesgue measure of $E$, and let $V^μ$ 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 $V^{-μ}$.
math.CV↗