Extremal problems involving logarithmic and Green capacity (Q1309325)

The principal result of this paper is an extensive generalization of a result conjectured orally at a seminar by V. V. Kozhevnikov in 1980, and independently by \textit{D. Gaier} in a joint work with \textit{W. K. Hayman} [Constr. Approx. 7, No. 4, 453-467 (1991; Zbl 0729.30008)]. Let \(D\) be a Green domain in \(\mathbb{C}\), and for any compact subset \(K\) of \(D\) let \(V(K,D)\) denote the Green energy of \(K\) with respect to \(D\). Let \(\text{cap} (K,D) =e^{-V(K,D)}\), and for \(r>1\) let \(\text{cap} (K,r)=\text{cap} (K,rD)\). Similarly, let \(V(K, \infty)\) denote the logarithmic energy of a compact set \(K\), and let \(\text{cap} (K, \infty)=e^{-V(K,\infty)}\). Suppose that \(D\) is a simply connected plane domain, symmetric about the real axis, with \(0 \in D \neq \mathbb{C}\). Suppose that \(K\) is a compact, connected set with positive capacity, that \(0 \leq a<b<\beta\), that \(a \in K\), and that \(K \subseteq rD\) for all \(r>b/ \beta\). It is proved that if \(\text{cap} (K,D)=\text{cap} ([a,b],D)\), then \(\text{cap} (K,r) \geq \text{cap} ([a,b],r)\) whenever \(1<r \leq \infty\); and that if equality holds for some \(r\) then either \(K=[a,b]\) or \(a=0\) and \(K=e^{i \theta}[a,b]\). There are several interesting related results.