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Extremal decomposition of the complex plane with restrictions for free poles

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Abstract

The problems of extremal decomposition with free poles on a circle are well known in the geometric theory of functions of a complex variable. One of such problems is the problem of maximum of the functional

$$ {I}_n\left(\upgamma \right)={r}^{\upgamma}\left({B}_0,0\right)\prod \limits_{k=1}^nr\left({B}_k{,}_{ak}\right), $$

where γ ∈ (0, n], B0, B1, B2,...,B n , n ≥ 2, are pairwise disjoint domains in \( \overline{\mathrm{C}},{a}_0=0,\left|{a}_k\right|=1,k=\overline{1,n} \) are different points of the circle, r(B, a) is the inner radius of the domain B ⊂ \( \overline{\mathrm{C}} \) relative to the point a ∈ B. We consider a more general problem, in which the restriction \( \left|{a}_k\right|=1,k=\overline{1,n}, \) is replaced by a more general condition.

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Authors and Affiliations

  1. Institute of Mathematics of the NAS of Ukraine, Kiev, Ukraine

    Aleksandr K. Bakhtin

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  1. Aleksandr K. Bakhtin
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Correspondence to Aleksandr K. Bakhtin.

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Translated from Ukrains’kiǐ Matematychnyǐ Visnyk, Vol. 14, No. 3, pp. 309–329 July–September, 2017.

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Bakhtin, A.K. Extremal decomposition of the complex plane with restrictions for free poles. J Math Sci 231, 1–15 (2018). https://doi.org/10.1007/s10958-018-3801-5

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