Weakened problem on extremal decomposition of the complex plane
Journal Title: Математичні Студії - Year 2019, Vol 51, Issue 1
Abstract
The paper deals with the problem of the maximum of the functional rγ(B0,0)∏k=1nr(Bk,ak), where B0,..., Bn, n⩾2, are pairwise disjoint domains in C¯¯¯¯, a0=0, |ak|=1, k∈{1,…,n} and γ∈(0,n] (r(B,a) is the inner radius of the domain B⊂C¯¯¯¯ with respect to a). We show that the functional attains its maximum at a configuration of the domains Bk and the points ak possessing rotational n-symmetry. The proof is due to Dubinin [1] for γ=1 and to Kuz'mina [3] for 0≤γ≤1. Subsequently, Kovalev [4] solved this problem for n⩾5 under the additional assumption that the angles between neighbouring line segments [0,ak] do not exceed 2π/γ−−√. In the paper, we obtain some estimate of the functional for γ∈(1,n].
Authors and Affiliations
A. Bakhtin, I. Denega
Keywords
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- EP ID EP584887
- DOI 10.15330/ms.51.1.35-40
- Views 55
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