For E,F\(\subset \overline{R}^ n\), \(n\geq 2\), let \(\Delta\) (E,F) stand for the family of all curves joining E to F in \(R^ n\) and let M(\(\Delta\) (E,F)) denote its n-modulus. Let \(e_ 1=(1,0,...,0)\in R^ n\) and for \(x\in R^ n\setminus \{0,e_ 1\}\) set \[ p(x)=\inf_{E,F}M(\Delta (E,F)) \] where the infimum is taken over all pairs E, F of connected sets with \(0,e_ 1\in E\) and 0,\(\infty \in F\). For \(n=2\) the problem of evaluating p(x) explicitly in terms of well- known functions was suggested by O. Teichmüller and its solution was found by M. Schiffer (1946) and H. Wittich (1949) (see Chapter 5 in G. V. Kuz’mina’s book “Moduli of families of curves and quadratic differentials” (1982; Zbl 0491.30013), esp. p. 192). By performing a spherical symmetrization we see that \(p(x)\geq \tau_ n(| x-e_ 1|)\) holds for \(| x-e_ 1| \leq | x|\) where \(\tau_ n\) stands for the capacity of the Teichmüller ring in \(R^ n.\)
In this paper the following upper bounds of p(x) are proved when \(| x-e_ 1| \leq | x|:\)
(1) \(p(x)\leq 2\tau_ n(| x-e_ 1|)\) for \(x\in R^ n\setminus B^ n(-e_ 1,2),\)
(2) \(p(x)\leq 4\tau_ n(| x-e_ 1|)\) for \(| x| \geq 1,\)
(3) \(p(x)\leq 2^{n+1}\tau_ n(| x-e_ 1|).\)
Applying these results one can prove estimates for certain conformal invariants. Exploiting these estimates the author derives distortion theorems e.g. for quasiconformal mappings in QED-domains. Finally, the author proves a sharp distortion theorem for Möbius transformations improving an earlier result of A. F. Beardon. (Some applications of the inequalities (1)-(3) have been reported by J. Ferrand in a series of papers see e.g. Complex Analysis, Proc. 13th Rolf Nevanlinna-Colloq., Joensuu/Finl. 1987, Lect. Notes Math. 1351, 110-123 (1988; Zbl 0661.30015).)