## Möbius modulus of ring domains in $$\overline{\mathbb{R}}^n$$.(English)Zbl 1134.30304

A ring $$R=R(C_1,C_2)\subset\overline{\mathbb R}^n$$ is a domain whose complement is the union of two disjoint non-degenerate compact connected sets $$C_1$$ and $$C_2$$. A ring $$R(C_1,C_2)$$ is said to separate the sets $$E$$ and $$F$$ if $$E\subset C_1$$ and $$F\subset C_2$$. The author introduces a new Möbius invariant modulus for ring domains $$R$$ in $$\mathbb R^n$$ as follows: $\text{mod}_MR=\max_{u,v\in\overline{\mathbb R}^n}\min_{x\in C_1,y\in C_2 }\left| \log\frac{| u-y|| x-v|}{| u-x|| y-v|}\right|.$
The author obtains the following sharp estimate.
Theorem. Let $$a,b,c,d\in\overline{\mathbb{R}}^n$$ be distinct points. Then $\max_R\text{mod}_MR=\text{arccosh}(\sigma(a,b,c,d))$ , where the maximum ist taken over all rings $$R$$ which separate the two sets $$\{a,b\}$$ and $$\{c,d\}$$.
This proves a conjecture of M. Vuorinen [Q. J. Math., Oxf. II. Ser. 43, No. 172, 501–514 (1992; Zbl 0766.30014)].

### MSC:

 30C35 General theory of conformal mappings 30C75 Extremal problems for conformal and quasiconformal mappings, other methods

Zbl 0766.30014
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