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 Zap. Nauchn. Sem. POMI, 2001, Volume 276, Pages 253–275 (Mi znsl1420)

Problems of extremal decomposition of the Riemann sphere

G. V. Kuz'mina

St. Petersburg Department of V. A. Steklov Institute of Mathematics, Russian Academy of Sciences

Abstract: We apply a variant of the method of the extremal metric to some problems concerning extremal decompositions and related problems. Let $\mathbf a=\{a_1,…,a_n\}$ be a system of distinct points on $\overline{\mathbb C}$ and let $\mathscr D(\mathbf a)$ be the family of all systems $\mathbb D=\{D_1,…,D_n\}$ of nonoverlapping simply connected domains on $\overline{\mathbb C}$ such that $a_k\in D_k, k=1,…,n$. Let
$$J(a)=\max\limits_{\mathbb D\subset\mathscr D(\mathbf a)}\{2\pi\sum_{k=1}^nM(D_k,a_k)-\frac2{n-1}\sum_{1\le k<l\le n}\log|a_k-a_l|\},$$
where $M(D_k,a_k)$ is the reduced module of the domain $D_k$ with respect to the point $a_k$. At present, the problem concerning the value $\max\limits_{\mathbf a}J(a)$ was solved completely for $n=2,3,4$. In this work, we continue the previous author's investigations and consider the case $n=5$. In addition, we consider the problem concerning the maximum of the sum
$$\alpha^2\{M(D_0,0)+M(D_{n+1},\infty)\}+\sum_{k=1}^nM(D_k,a_k)$$
in the family $\mathscr D(\mathbf a)$ introduced above, where $\mathbf a=\{0,a_1,…,a_n,\infty\}$, $a_k$, $k=1,…,n$, are arbitrary points of the circle $|z|=1$, and $\alpha$ is a positive number. We prove that if $\alpha/n\le1/\sqrt8$, then the maximum is attained $\alpha$ only for systems of equidistant points of the circle $|z|=1$. For $\alpha/n=1/\sqrt8$, this result was obtained earlier by Dubinin who applied the method of symmetrization. It is shown that if $n\ge2$, where $\alpha/n\ge1/2$ is an even number, then equidistant points of the circle $|z|=1$ do not realize the indicated maximum.

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English version:
Journal of Mathematical Sciences (New York), 2003, 118:1, 4880–4894

Bibliographic databases:

UDC: 517.54

Citation: G. V. Kuz'mina, “Problems of extremal decomposition of the Riemann sphere”, Analytical theory of numbers and theory of functions. Part 17, Zap. Nauchn. Sem. POMI, 276, POMI, St. Petersburg, 2001, 253–275; J. Math. Sci. (N. Y.), 118:1 (2003), 4880–4894

Citation in format AMSBIB
\Bibitem{Kuz01} \by G.~V.~Kuz'mina \paper Problems of extremal decomposition of the Riemann sphere \inbook Analytical theory of numbers and theory of functions. Part~17 \serial Zap. Nauchn. Sem. POMI \yr 2001 \vol 276 \pages 253--275 \publ POMI \publaddr St.~Petersburg \mathnet{http://mi.mathnet.ru/znsl1420} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1850371} \zmath{https://zbmath.org/?q=an:1071.30022} \transl \jour J. Math. Sci. (N. Y.) \yr 2003 \vol 118 \issue 1 \pages 4880--4894 \crossref{https://doi.org/10.1023/A:1025580802209} 

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This publication is cited in the following articles:
1. G. V. Kuz'mina, “The method of extremal metric in extremal decomposition problems with free parameters”, J. Math. Sci. (N. Y.), 129:3 (2005), 3843–3851
2. G. V. Kuz'mina, “Problems of extremal decomposition of the Riemann sphere. III”, J. Math. Sci. (N. Y.), 133:6 (2006), 1676–1685
3. Bakhtin A.K., “Piecewise separating transformation and extremal problems with free poles”, Doklady Mathematics, 72:3 (2005), 854–856
4. V. N. Dubinin, “Capacities of condensers, generalizations of Grötzsch Lemmas, and symmetrization”, J. Math. Sci. (N. Y.), 143:3 (2007), 3053–3068
5. Bakhtin A.K., Targonskii A.L., “Generalized (N, D)-Ray Systems of Points and Inequalities for Nonoverlapping Domains and Open Sets”, Ukrainian Math J, 63:7 (2011), 999–1012
6. Targonskii A., “Extremal Problems on the Generalized (N, D)-Equiangular System of Points”, Analele Stiint. Univ. Ovidius C., 22:2 (2014), 239–251
7. J. Math. Sci. (N. Y.), 222:5 (2017), 645–689
8. G. V. Kuz'mina, “On an extremal metric approach to extremal decomposition problems”, J. Math. Sci. (N. Y.), 225:6 (2017), 980–990
9. Targonskii A., “Extremal Problem on (2N, 2M-1)-System Points on the Rays.”, Analele Stiint. Univ. Ovidius C., 24:2 (2016), 283–299
10. A. K. Bakhtin, I. V. Denega, “Sharp estimates of products of inner radii of non-overlapping domains in the complex plane”, Probl. anal. Issues Anal., 8(26):1 (2019), 17–31