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Generalized (n, d)-ray systems of points and inequalities for nonoverlapping domains and open sets

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Ukrainian Mathematical Journal Aims and scope

We solve the extremal problem of finding the maximum of the functional

$$ \prod\limits_{k = 1}^n {\prod\limits_{p = 1}^{{m_k}} {r\left( {{B_{k,p}},{a_{k,p}}} \right)}, } $$

where

$$ {m_k} \in \mathbb{N},\quad \sum\limits_{k = 1}^n {{m_k} = m,\quad n,\,m \in \mathbb{N},\quad 0 < \left| {{a_{k,\,1}}} \right| < \ldots < \left| {{a_{k,\,{m_k}}}} \right| < \infty }, $$
$$ \arg {a_{k,\,1}} = \arg {a_{k,\,2}} = \ldots = \arg {a_{k,\,{m_k}}} = :{\theta_k},\quad k = \overline {1,n}, $$
$$ 0 = {\theta_1} < {\theta_2} < \ldots < {\theta_n} < {\theta_{n + 1}}: = 2\pi, $$

and r(B, a) is the inner radius of a domain B with respect to a point aB. The points a k,p , \( k = \overline {1,n} \), \( p = \overline {1,{m_k}} \), are not fixed. Some generalizations of these results are also considered.

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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 63, No. 7, pp. 867–879, July, 2011.

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Bakhtin, A.K., Targonskii, A.L. Generalized (n, d)-ray systems of points and inequalities for nonoverlapping domains and open sets. Ukr Math J 63, 999–1012 (2011). https://doi.org/10.1007/s11253-011-0560-6

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  • DOI: https://doi.org/10.1007/s11253-011-0560-6

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