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On the problem of maximizing the product of powers of conformal radii nonoverlapping domains. (English. Russian original) Zbl 1069.30042

J. Math. Sci., New York 122, No. 6, 3641-3647 (2004); translation from Zap. Nauchn. Semin. POMI 286, 103-114 (2002).
Summary: A sharp estimate of the product \[ \prod_{k=1}^4 R^{\alpha_k^2} (D_k,b_k) \] (as usual, \(R(D,b)\) denotes the conformal radius of a domain \(D\) with respect to a point \(b\in D\)) in the family of all quadruples of nonoverlapping simply connected domains \(\{D_k\}\), \(b_k\in D_k\), \(k=1,\dots,4\), is obtained. Here, \(\{b_1,\dots,b_4\}\) are four arbitrary distinct points on \(\overline{\mathbb C}\), \(\alpha_1= \alpha_2= 1\), \(\alpha_3= \alpha_4= \alpha\), and \(\alpha\) is an arbitrary positive number. The proof involves the solution of the problem on maximizing a certain conformal invariant, which is related to the problem under consideration.

MSC:

30C75 Extremal problems for conformal and quasiconformal mappings, other methods
31A15 Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions
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