Some inequalities for inner radii of pair-wise disjoint domains and open sets (in Ukrainian)

A. L. Targonskii
Zhytomyr Ivan Franko State University
Sharp estimates of product of inner radii for pairwise disjoint domains and open sets are obtained. In particular, we solve an extremal problem in the case of arbitrary finite number of rays containing arbitrary even number of free poles.
inner radius of domain; quadratic differential; piecewise-separating transformation; Green’s function; radial systems of points; logarithmic capacity; variational formula; non-overlapping domains
1. Lavrent’ev M.A. On the theory of conformal mappings// Tr. Fiz.-Mat. Inst. Akad. Nauk SSSR. – 1934. – V.5. – P. 159–245. (in Russian)

2. Goluzin G.M. Geometric theory of functions of a complex variable. – Nauka, 1966, Moscow. (in Russian)

3. Bakhtina G.P. Variational methods and quadratic differentials in problems for disjoint domains. – Author’s Abstract of the Candidate-Degree Thesis (Physics and Mathematics), 1975, Kiev. (in Russian)

4. Dubinin V.N. Separating transformation of domains and problems of extremal division// Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Ros. Akad. Nauk. – 1988. – V.168. – P. 48–66. (in Russian)

5. Dubinin V.N. Method of symmetrization in the geometric theory of functions of a complex variable// Usp. Mat. Nauk. – 1994. – V.49, ¹1(295). – P. 3–76. (in Russian)

6. Dubinin V.N. Asymptotic representation of the modulus of a degenerating condenser and some its applications// Zap. Nauchn. Sem. Peterburg. Otdel. Mat. Inst. – 1997. – V.237. – P. 56–73. (in Russian)

7. Dubinin V.N. Capacities of condensers and symmetrization in geometric function theory of complex variables. – Dal’nayka, Vladivostok, 2009. (in Russian)

8. Bakhtin A.K., Bakhtina G.P., Zelinskii Yu.B. Topological-algebraic structures and geometric methods in complex analysis// Proceedings of the Institute of Mathematics of NAS of Ukraine. – 2008. – V.73. – 308 p. (in Russian)

9. Bakhtin A.K. Inequalities for the inner radii of nonoverlapping domains and open sets// Ukr. Mat. Zh. – 2009. – V.61, ¹5. – P. 596-–610. (in Ukrainian); English translation: Ukr. Math. J. – 2009. – V.61, .5. – P. 716–733.

10. Bakhtin A.K., V’un V.E., Trokhimchuk Yu.Yu. Inequalities for the inner radii of nonoverlapping domains// Dopov. Nats. Akad. Nauk Ukr. – 2007. – ¹8. – P. 7–10. (in Russian)

11. Targonskii A.L. Extremal problems of partially nonoverlapping domains on a Riemann sphere// Dopov. Nats. Akad. Nauk Ukr. – 2008. – ¹9. – P. 31–36. (in Russian)

12. Hayman W.K. Multivalent functions, Cambridge University, Cambridge, 1960.

13. Kuz’mina G.V. Problem of extremal division of a Riemann sphere// Zap. Nauchn. Sem. Peterburg. Otdel. Mat. Inst. – 2001. – V.276. – P. 253–275. (in Russian)

14. Emel’yanov E.G. On the problem of the maximum of the product of powers of conformal radii of disjoint domains// Zap. Nauchn. Sem. Peterburg. Otdel. Mat. Inst. – 2002. – V.286. – P. 103–114. (in Russian)

15. Jenkins J.A. Univalent functions and conformal mapping. – Springer, Berlin, 1962.

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