Weakened problem on extremal decomposition of the complex
plane

A. K. Bakhtin; I. V. Denega

doi:10.15330/ms.51.1.35-40

The paper deals with the problem of the maximum of the functional $$r^\gamma\left(B_0,0\right)\prod\limits_{k=1}^n r\left(B_k,a_k\right),$$ where $B_{0}$,..., $B_{n}$, $n\geqslant 2$, are pairwise disjoint domains in $\overline{\mathbb{C}},$ $a_0=0,$ $|a_{k}|=1$, $k\in\{1,\ldots,n\}$ and $\gamma\in (0, n]$ ($r(B,a)$ is the inner radius of the domain $B\subset\overline{\mathbb{C}}$ with respect to $a$). We show that the functional attains its maximum at a configuration of the domains $B_{k}$ and the points $a_{k}$ possessing rotational $n$-symmetry. The proof is due to Dubinin [1] for $\gamma = 1$ and to Kuz'mina [3] for $0 \le\gamma\le 1$. Subsequently, Kovalev [4] solved this problem for $n\geqslant5$ under the additional assumption that the angles between neighbouring line segments $[0, a_{k}]$ do not exceed $2\pi / \sqrt{\gamma}$. In the paper, we obtain some estimate of the functional for $\gamma\in(1,\,n]$.

1. V.N. Dubinin, Symmetrization method in geometric function theory of complex variables, Uspekhy Matem. Nauk, 49 (1994), №1, 3.76. (in Russian); Engl.transl.: Russian Math. Surveys, 1 (1994), 1. 79.

2. V.N. Dubinin, The separating transformation of domains and problems on the extremal partition, Zap. Nauchn. Sem. LOMI, 168 (1988), 48.66. (in Russian)

3. G.V. Kuzmina, The method of extremal metric in extremal decomposition problems with free parameters, J. Math. Sci., 129 (2005), №3, 3843.3851.

4. L.V. Kovalev, On the problem of extremal decomposition with free poles on a circle, Dalnevostochnyi Mat. Sb., (1996), №2, 96.98. (in Russian)

5. M.A. Lavrentev, On the theory of conformal mappings, Tr. Sci. Inst AN USSR, 5 (1934), 159.245. (in Russian)

6. G.M. Goluzin, Geometric theory of functions of a complex variable, Amer. Math. Soc. Providence, R.I., 1969.

7. J. Jenkins, Univalent functions and conformal mapping, Moscow: Publishing House of Foreign Literature, 256, 1962. (in Russian)

8. V.N. Dubinin, Condenser capacities and symmetrization in geometric function theory, Birkhauser/Springer, Basel, 2014.

9. A.K. Bakhtin, G.P. Bakhtina, Yu.B. Zelinskii, Topological-algebraic structures and geometric methods in complex analysis, Zb. prats of the Inst. of Math. of NASU, 2008. (in Russian)

10. A.K. Bakhtin, I.V. Denega, Addendum to a theorem on extremal decomposition of the complex plane, Bulletin de la societe des sciences et des lettres de Lodz, Recherches sur les deformations, 62 (2012), №2, 83-92.

11. I.V. Denega, Generalization of some extremal problems on non-overlapping domains with free poles, Annales universitatis Mariae Curie-Skladovska, LXVII (2013), №1, 11–22.

12. Ja.V. Zabolotnij, Determination of the maximum of a product of inner radii of pairwise nonoverlapping domains, Dopov. Nats. Akad. Nauk Ukr., (2016), №3, 7–13. (in Ukrainian)

13. A. Bakhtin, I. Dvorak, I. Denega, Separating transformation and extremal decomposition of the complex plane, Bulletin de la societe des sciences et des lettres de Lodz, Recherches sur les deformations, LXVI (2016), №2, 13–20.

14. A. Bakhtin, L. Vygivska, I. Denega, N-radial systems of points and problems for non-overlapping domains, Lobachevskii Journal of Mathematics, 38 (2017), №2, 229–235.

15. A.K. Bakhtin, Ya.V. Zabolotnii, Estimates of a product of the inner radii of nonoverlapping domains, J. Math. Sci., 221 (2017), №5, 623–629.

16. A.K. Bakhtin, L.V. Vygivska, I.V. Denega, Inequalities for the internal radii of non-overlapping domains, J. Math. Sci., 220 (2017), №5, 584–590.

17. I.V. Denega, Ya.V. Zabolotnii, Estimates of products of inner radii of non-overlapping domains in the complex plane, Complex Variables and Elliptic Equations, 62 (2017), №11, 1611–1618.

18. G. Polya, G. Szego, Isoperimetric inequalities in mathematical physics, M.: Fizmatgiz, 1962.

	Weakened problem on extremal decomposition of the complex plane
Author	A. K. Bakhtin¹, I. V. Denega² abahtin@imath.kiev.ua¹, iradenega@gmail.com² Institute of Mathematics of National Academy of Sciences of Ukraine, Department of Complex analysis and Potential Theory, Kyiv, Ukraine
Abstract	The paper deals with the problem of the maximum of the functional $$r^\gamma\left(B_0,0\right)\prod\limits_{k=1}^n r\left(B_k,a_k\right),$$ where $B_{0}$,..., $B_{n}$, $n\geqslant 2$, are pairwise disjoint domains in $\overline{\mathbb{C}},$ $a_0=0,$ $\|a_{k}\|=1$, $k\in\{1,\ldots,n\}$ and $\gamma\in (0, n]$ ($r(B,a)$ is the inner radius of the domain $B\subset\overline{\mathbb{C}}$ with respect to $a$). We show that the functional attains its maximum at a configuration of the domains $B_{k}$ and the points $a_{k}$ possessing rotational $n$-symmetry. The proof is due to Dubinin [1] for $\gamma = 1$ and to Kuz'mina [3] for $0 \le\gamma\le 1$. Subsequently, Kovalev [4] solved this problem for $n\geqslant5$ under the additional assumption that the angles between neighbouring line segments $[0, a_{k}]$ do not exceed $2\pi / \sqrt{\gamma}$. In the paper, we obtain some estimate of the functional for $\gamma\in(1,\,n]$.
Keywords	extremal problems with free poles; inner radius of domain; non-overlapping domains; complex plane; transfinite diameter; theorem on minimizing of the area; Cauchy inequality
DOI	doi:10.15330/ms.51.1.35-40
Reference	1. V.N. Dubinin, Symmetrization method in geometric function theory of complex variables, Uspekhy Matem. Nauk, 49 (1994), №1, 3.76. (in Russian); Engl.transl.: Russian Math. Surveys, 1 (1994), 1. 79. 2. V.N. Dubinin, The separating transformation of domains and problems on the extremal partition, Zap. Nauchn. Sem. LOMI, 168 (1988), 48.66. (in Russian) 3. G.V. Kuzmina, The method of extremal metric in extremal decomposition problems with free parameters, J. Math. Sci., 129 (2005), №3, 3843.3851. 4. L.V. Kovalev, On the problem of extremal decomposition with free poles on a circle, Dalnevostochnyi Mat. Sb., (1996), №2, 96.98. (in Russian) 5. M.A. Lavrentev, On the theory of conformal mappings, Tr. Sci. Inst AN USSR, 5 (1934), 159.245. (in Russian) 6. G.M. Goluzin, Geometric theory of functions of a complex variable, Amer. Math. Soc. Providence, R.I., 1969. 7. J. Jenkins, Univalent functions and conformal mapping, Moscow: Publishing House of Foreign Literature, 256, 1962. (in Russian) 8. V.N. Dubinin, Condenser capacities and symmetrization in geometric function theory, Birkhauser/Springer, Basel, 2014. 9. A.K. Bakhtin, G.P. Bakhtina, Yu.B. Zelinskii, Topological-algebraic structures and geometric methods in complex analysis, Zb. prats of the Inst. of Math. of NASU, 2008. (in Russian) 10. A.K. Bakhtin, I.V. Denega, Addendum to a theorem on extremal decomposition of the complex plane, Bulletin de la societe des sciences et des lettres de Lodz, Recherches sur les deformations, 62 (2012), №2, 83-92. 11. I.V. Denega, Generalization of some extremal problems on non-overlapping domains with free poles, Annales universitatis Mariae Curie-Skladovska, LXVII (2013), №1, 11–22. 12. Ja.V. Zabolotnij, Determination of the maximum of a product of inner radii of pairwise nonoverlapping domains, Dopov. Nats. Akad. Nauk Ukr., (2016), №3, 7–13. (in Ukrainian) 13. A. Bakhtin, I. Dvorak, I. Denega, Separating transformation and extremal decomposition of the complex plane, Bulletin de la societe des sciences et des lettres de Lodz, Recherches sur les deformations, LXVI (2016), №2, 13–20. 14. A. Bakhtin, L. Vygivska, I. Denega, N-radial systems of points and problems for non-overlapping domains, Lobachevskii Journal of Mathematics, 38 (2017), №2, 229–235. 15. A.K. Bakhtin, Ya.V. Zabolotnii, Estimates of a product of the inner radii of nonoverlapping domains, J. Math. Sci., 221 (2017), №5, 623–629. 16. A.K. Bakhtin, L.V. Vygivska, I.V. Denega, Inequalities for the internal radii of non-overlapping domains, J. Math. Sci., 220 (2017), №5, 584–590. 17. I.V. Denega, Ya.V. Zabolotnii, Estimates of products of inner radii of non-overlapping domains in the complex plane, Complex Variables and Elliptic Equations, 62 (2017), №11, 1611–1618. 18. G. Polya, G. Szego, Isoperimetric inequalities in mathematical physics, M.: Fizmatgiz, 1962.
Pages	35-40
Volume	51
Issue	1
Year	2019
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

Weakened problem on extremal decomposition of the complex plane