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