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Separating transformation and extremal problems on nonoverlapping simply connected domains

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Abstract

We consider the well-known problem of maximum of the functional

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

where B0, …, Bn are pairwise disjoint domains in \( \overline{\mathrm{\mathbb{C}}} \), a0 = 0, |ak| = 1, \( k=\overline{1,n} \), are different points of the circle, γ ∈ (0, n], and r(B, a) is the inner radius of the domain \( B\subset \overline{\mathrm{\mathbb{C}}} \) relative to the point a. In the case of simply connected domains for n=2, 3, and 4, we have obtained the solution of this problem for the maximum interval of values of the parameter γ.

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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’kiĭ Matematychnyĭ Visnyk, Vol. 14, No. 4, pp. 456–471 October–December, 2017.

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Bakhtin, A.K. Separating transformation and extremal problems on nonoverlapping simply connected domains. J Math Sci 234, 1–13 (2018). https://doi.org/10.1007/s10958-018-3976-9

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  • DOI: https://doi.org/10.1007/s10958-018-3976-9

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