Bounds in capacity inequalities for two sheeted spheres

Abstract

Take a pair of two disjoint nonpolar compact subsets A and B of the complex plane C = Ĉ\{∞}, the complex sphere less the point at infinity, with connected complement Ĉ\(A ∪ B) and a simple arc γ in Ĉ\(A ∪ B). We form the two sheeted covering surface Ĉ_γ of Ĉ by pasting Ĉ\γ with another copy Ĉ\γ crosswise along γ. Embed A and B in Ĉ_γ either in the same sheet or in the different sheets and consider the variational 2-capacity cap(A, Ĉ_γ\B) of A contained in the open subset Ĉ_γ\B of Ĉ_γ. Concerning the relation between the above capacity and the variational 2-capacity cap(A, Ĉ\B) of A contained in the open subset Ĉ\B of Ĉ, we will establish the following capacity inequality for the two sheeted cover and its base:

0 < cap (A, Ĉ_γ\B) < 2 · cap(A, Ĉ\B),

where the bound 2 in the above inequality is the best possible in the sense that, for any 0 < τ < 2, there is a triple of A, B, and γ such that cap(A, Ĉ_γ\B) > τ · cap(A, Ĉ\B), where A and B may in the same sheet or in the different sheets.