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 Zap. Nauchn. Sem. LOMI, 1976, Volume 59, Pages 60–80 (Mi znsl2085)

Maximum of the fourth diameter in the family of continua with prescribed capacity

G. V. Kuz'mina

Abstract: We obtain a complete solution of the problem of the maximum of the fourth diameter
$$d_4(E)=\{\max_{z_k,z_r\in E}\prod_{1\leqslant k\leqslant l\leqslant4}|z_k-z_l|\}^{1/6}$$
in the family of continua with capacity 1. Let $E(0,e^{i\alpha},e^{-i\alpha})$, $0<\alpha<\pi/2$, be a continuum of minimum capacity containing the points $0$, $e^{i\alpha}$, $e^{-i\alpha}$; $H(\alpha)=\operatorname{cap}E(0,e^{i\alpha},e^{-i\alpha})$. Let $c(\alpha)$ be the common point of three analytic arcs which form $E(0,e^{i\alpha},e^{-i\alpha})$. One shows that the indicated maximum is realized by the continuum $\mathscr E=ż:H(\alpha_0)z^2\in E(0,e^{i\alpha},e^{-i\alpha})\}$ where $\alpha_0$, $0<\alpha_0<\pi/2$, is a solution of the equation $c(\alpha)=\frac13\cos\alpha$. Any other extremal continuum of the gives problem is an image of $\mathscr E$ under the mapping $z\to e^{i\gamma}z+C$ ($\gamma$ is a real and $C$ is a complex constant). One finds the value of the required maximum. The paper contains a brief exposition of the proof of this result.

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English version:
Journal of Soviet Mathematics, 1978, 10:2, 241–256

Bibliographic databases:

UDC: 517.54

Citation: G. V. Kuz'mina, “Maximum of the fourth diameter in the family of continua with prescribed capacity”, Boundary-value problems of mathematical physics and related problems of function theory. Part 9, Zap. Nauchn. Sem. LOMI, 59, "Nauka", Leningrad. Otdel., Leningrad, 1976, 60–80; J. Soviet Math., 10:2 (1978), 241–256

Citation in format AMSBIB
\Bibitem{Kuz76} \by G.~V.~Kuz'mina \paper Maximum of the fourth diameter in the family of continua with prescribed capacity \inbook Boundary-value problems of mathematical physics and related problems of function theory. Part~9 \serial Zap. Nauchn. Sem. LOMI \yr 1976 \vol 59 \pages 60--80 \publ "Nauka", Leningrad. Otdel. \publaddr Leningrad \mathnet{http://mi.mathnet.ru/znsl2085} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=444939} \zmath{https://zbmath.org/?q=an:0389.30019|0347.30015} \transl \jour J. Soviet Math. \yr 1978 \vol 10 \issue 2 \pages 241--256 \crossref{https://doi.org/10.1007/BF01566605}