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 Ĉγ of Ĉ by pasting Ĉ\γ with another copy Ĉ\γ crosswise along γ. Embed A and B in Ĉγ either in the same sheet or in the different sheets and consider the variational 2-capacity cap(A, Ĉγ\B) of A contained in the open subset Ĉγ\B of Ĉγ. 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, Ĉγ\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, Ĉγ\B) > τ · cap(A, Ĉ\B), where A and B may in the same sheet or in the different sheets.
Citation
Mitsuru Nakai. "Bounds in capacity inequalities for two sheeted spheres." Kodai Math. J. 30 (2) 223 - 236, June 2007. https://doi.org/10.2996/kmj/1183475513
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