Koebe domain in the Montel class

Authors

  • Gavrilyuk M.N. Kuban State University, Krasnodar, Российская Федерация

UDC

517.54

Abstract

Study subclass of the well-known Montel class of regular simple functions, determinated by additional metric condition. Using the module method in form of general theorem of extreme partition of the plane on non-overlapping admissible domains, solve the problem of finding Koebe domain in the pointed class. In terms of moduli domains of extreme partition, which includes extremal configuration, inner and upper boundary of Koebe domain are described. These boundaries are determinated by special equations. For points of inner boundary obtained extreme mapping of the unit disc to the simply connected domain, associated with some quadratic differential. For upper boundary extremal mappings do not exist, but the boundary can not be improved.

Keywords:

conformal mapping, quadratic differential, extremal decomposition, extremal mapping

Author Info

Mikhail N. Gavrilyuk

канд. физ.-мат. наук, доцент кафедры теории функций Кубанского государственного университета

e-mail: mngavril@gmail.com

References

  1. Jenkins J. Odnolistnye funktsii i konformnye otobrazheniya [Univalent functions and conformal mappings]. Moscow, Inostrannaya literatura Pub., 1962, 262 p. (In Russian)
  2. Kuzmina G.V. Moduli semeystv krivykh i kvadratichnye differentsialy [The modules of families of curves and quadratic differentials]. Trudy Matematicheskogo instituta AN SSSR [Proc. of the Institute of Mathematics of USSR Academy of Sciences], 1980, vol. 139, p. 240. (In Russian)
  3. Jenkins J.A. On the existence of certain general extremal metrics 1,11. Ann. Math, 1957, vol. 66, iss. 2, pp. 440-453; \emph{Tokoru Math. J.}, 1993, vol. 45, iss. 2, pp. 249-257.
  4. Solynin A.Yu. Moduli i ekstremal'no-metricheskie problemy [Modules and Extreme-Metric Problems]. Algebra i analiz [Algebra and Analysis], 1999, vol. 11, no. 1, pp. 3-86.
  5. Vasil'ev A. Moduli of families of curves for conformal and quasiconformal mappings. Lecture Notes in Math., 2002, vol. 1788. Springer-Verlag, Berlin. doi: 10.1007/b83857
  6. Gavrilyuk M.N., Solynin A.Yu. Primenenie problem modulya k nekotorym ekstremal'nym zadacham [Applications module problems for certain extremal problems]. Dep. VINITI, no. 3072-83, 139 p. (In Russian).
  7. Krzyz J., Zlotkiewiez E. Koebe sets for univalent functions with two preassigned points. Ann. Acad. Sci. Fenn. Ser. A Math. 1971, vol. 487, pp. 45-62.
  8. Kuzmina G.V. Metod moduley i ekstremal'nye zadachi v klasse $\sum \left( r \right)$ [Modules method and extremal problems in the class $\sum \left( r \right) $]. Zapiski nauchnykh seminarov POMI RAN [Notes of scientific seminars of the St. Petersburg Department of the V.A. Steklov Institute of Mathematics of Russian Academy of Sciences], 2013, vol. 418, pp. 136-152. (In Russian)
  9. Kuzmina G.V. Ob odnoy probleme modulya dlya semeystv krivykh [On a problem of a module for families of curves]. Preprint LOMI R-6-83, Leningrad, 1983, 43 p. (In Russian)
  10. Solynin A.Yu. Ob ekstremal'nykh razbieniyakh ploskosti ili kruga na dve nenalegayushchie oblasti [On extremal partitions of a plane or a circle into two nonoverlapping domains]. Dep. in VINITI, no. 7800-84, 17 p. (In Russian)

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Issue

2017. No. 1

Pages

45-50

Submitted

2016-12-16

Published

2017-03-30

How to Cite

Gavrilyuk M.N. Koebe domain in the Montel class. Ecological Bulletin of Research Centers of the Black Sea Economic Cooperation, 2017, no. 1, pp. 45-50. (In Russian)

Copyright (c) 2017 Gavrilyuk M.N.

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