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Extremal problems of nonoverlapping domains with free poles on a circle

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Abstract

Let α1, α2 > and let r(B, a) be the interior radius of the domain B lying in the extended complex plane

relative to the point aB. In terms of quadratic differentials, we give a complete description of extremal configurations in the problem of maximization of the functional \(\left( {\frac{{r(B_1 ,a_1 ) r(B_3 ,a_3 )}}{{\left| {a_1 - a_3 } \right|^2 }}} \right)^{\alpha _1 } \left( {\frac{{r(B_2 ,a_2 ) r(B_4 ,a_4 )}}{{\left| {a_2 - a_4 } \right|^2 }}} \right)^{\alpha _2 } \) defined on all collections consisting of points a 1, a 2, a 3, a 4 ∈ {z ∈ ℂ: |z| = 1} and pairwise-disjoint domains B 1, B 2, B 3, B 4

such that a 1B 1, a 1B 2, a 3B 3, and a 4B 4.

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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 7, pp. 867–886, July, 2006.

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Bakhtin, A.K. Extremal problems of nonoverlapping domains with free poles on a circle. Ukr Math J 58, 981–1000 (2006). https://doi.org/10.1007/s11253-006-0118-1

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Keywords

  • Harmonic Function
  • Conformal Mapping
  • Extremal Problem
  • Quadratic Differential
  • Circular Domain