## Abstract

Let α_{1}, α_{2} > and let *r*(*B, a*) be the interior radius of the domain *B* lying in the extended complex plane

relative to the point *a* ∈ *B*. In terms of quadratic differentials, we give a complete description of extremal configurations in the problem of maximization of the functional \(\left( {\frac{{r(B_1 ,a_1 ) r(B_3 ,a_3 )}}{{\left| {a_1 - a_3 } \right|^2 }}} \right)^{\alpha _1 } \left( {\frac{{r(B_2 ,a_2 ) r(B_4 ,a_4 )}}{{\left| {a_2 - a_4 } \right|^2 }}} \right)^{\alpha _2 } \) defined on all collections consisting of points *a*
_{1}, *a*
_{2}, *a*
_{3}, *a*
_{4} ∈ {*z* ∈ ℂ: |*z*| = 1} and pairwise-disjoint domains *B*
_{1}, *B*
_{2}, *B*
_{3}, *B*
_{4} ⊂

such that *a*
_{1} ∈ *B*
_{1}, *a*
_{1} ∈ *B*
_{2}, *a*
_{3} ∈ *B*
_{3}, and *a*
_{4} ∈ *B*
_{4}.

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## References

- 1.
M. A. Lavrent’ev, “On the theory of conformal mappings,”

*Tr. Fiz.-Mat. Inst. Akad. Nauk SSSR*,**5**, 159–245 (1934). - 2.
G. M. Goluzin,

*Geometric Theory of Functions of Complex Variables*[in Russian], Nauka, Moscow (1966). - 3.
N. A. Lebedev,

*Principle of Areas in the Theory of Univalent Functions*[in Russian], Nauka, Moscow (1975). - 4.
J. A. Jenkins,

*Univalent Functions and Conformal Mappings*, Springer, Berlin (1958). - 5.
G. P. Bakhtina,

*Variational Methods and Quadratic Differentials in Problems of Nonoverlapping Domains*[in Russian], Author’s Abstract of the Candidate-Degree Thesis (Physics and Mathematics), Kiev (1975). - 6.
G. V. Kuz’mina,

*Moduli of Families of Curves and Quadratic Differentials*[in Russian], Nauka, Leningrad (1980). - 7.
V. N. Dubinin,

*A Method for Symmetrization in the Geometric Theory of Functions*[in Russian], Doctoral-Degree Thesis (Physics and Mathematics), Vladivostok (1988). - 8.
V. N. Dubinin, “Separating transformation of domains and problems of extremal separation,”

*Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Akad. Nauk SSSR*,**168**, 48–66 (1988). - 9.
V. N. Dubinin, “A symmetrization method in the geometric theory of functions of complex variables,”

*Usp. Mat. Nauk*,**49**, No. 1 (295), 3–76 (1994). - 10.
V. N. Dubinin,

*Capacities of Condensers in the Geometric Theory of Functions*[in Russian], Far-Eastern University, Vladivostok (2003). - 11.
G. V. Kuz’mina, “Method of extremal metric in the problem of maximum of the product of powers of conformal radii of nonoverlapping domains in the presence of free parameters,”

*Zap. Nauchn. Sem. Peterburg. Otdel. Mat. Inst. Ros. Akad. Nauk*,**302**, 52–67 (2003). - 12.
E. G. Emel’yanov, “On the problem of maximum of the product of powers of conformal radii of nonoverlapping domains,”

*Zap. Nauchn. Sem. Peterburg. Otdel. Mat. Inst. Ros. Akad. Nauk*,**286**, 103–114 (2002). - 13.
L. V. Kovalev, “On the interior radii of axially symmetric nonoverlapping domains,”

*Izv. Vyssh. Uchebn. Zaved., Ser. Mat.*, No. 6, 82–87 (2000). - 14.
A. K. Bakhtin, “Some problems in the theory of nonoverlapping domains,”

*Ukr. Mat. Zh.*,**51**, No. 6, 723–731 (1999). - 15.
A. K. Bakhtin, “On some problems in the theory of nonoverlapping domains,” in:

*Abstracts of the International Conference on Complex Analysis and Potential Theory*, Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (2001), p. 64. - 16.
A. K. Bakhtin, “On the product of interior radii of symmetric nonoverlapping domains,”

*Ukr. Mat. Zh.*,**49**, No. 11, 1454–1464 (1997). - 17.
W. K. Hayman,

*Multivalent Functions*, Cambridge University, Cambridge (1958). - 18.
J. A. Jenkins, “Some uniqueness results in the theory of symmetrization,”

*Ann. Math.*,**61**, No. 1, 106–115 (1955). - 19.
J. A. Jenkins, “Some uniqueness results in the theory of symmetrization. II,”

*Ann. Math.*,**75**, No. 2, 223–230 (1962). - 20.
P. L. Duren and M. Schiffer, “A variation method for function schlicht in annulus,”

*Arch. Ration. Mech. Anal.*,**9**, 260–272 (1962). - 21.
M. A. Schiffer, “A method of variation within the family of simple functions,”

*Proc. London Math. Soc.*,**44**, 432–449 (1938). - 22.
V. N. Dubinin, “Symmetrization method in problems of nonoverlapping domains,”

*Mat. Sb.*, 128, No. 1, 110–123 (1985).

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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 7, pp. 867–886, July, 2006.

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Bakhtin, A.K. Extremal problems of nonoverlapping domains with free poles on a circle.
*Ukr Math J* **58, **981–1000 (2006). https://doi.org/10.1007/s11253-006-0118-1

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### Keywords

- Harmonic Function
- Conformal Mapping
- Extremal Problem
- Quadratic Differential
- Circular Domain