Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)

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Zherdev A. V. An Asymptotic Relation for Conformal Radii of Two Nonoverlapping Domains. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2018, vol. 18, iss. 3, pp. 274-283. DOI: 10.18500/1816-9791-2018-18-3-274-283

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An Asymptotic Relation for Conformal Radii of Two Nonoverlapping Domains

Zherdev Andrey V., Saratov State University

We consider a family of continuously varying closed Jordan curves given by a polar equation, such that the interiors of the curves form an increasing or decreasing chain of domains. Such chains can be described by the Löwner–Kufarev differential equation. We deduce an integral representation of a driving function in the equation.Using this representation we obtainan a symptotic formula, which establishes a connection between conformal radii of bounded and unbounded components of the complement of the Jordan curve when the bounded component is close to the unit disk.

  1. Lebedev N. A. Printsip ploshchadei v teorii odnolistnykh funktsii [The area principle in the theory of univalent functions]. Moscow, Nauka, 1975. 336 p. (in Russian).
  2. Hamilton D. H. Conformal welding. Handbook of complex analysis: geometric function theory. Ed. R. Kühnau. Amsterdam, North Holland, 2002. 548 p.
  3. Grong E., Gumenyuk P., Vasil’ev A. Matching univalent functions and conformal welding. Ann. Acad. Sci. Fenn. Math., 2009, vol. 34, pp. 303—314.
  4. Bishop C. J. Conformal welding and Koebe’s theorem. Ann. Math., 2007, vol. 166, pp. 613– 656. DOI: https://doi.org/10.4007/annals.2007.166.613
  5. Prokhorov D. V. Conformal welding for domains close to a disk. Anal. Math. Phys., 2011, vol. 1, pp. 101—114. DOI: https://doi.org/10.1007/s13324-011-0007-0
  6. Prokhorov D. V. Asymptotic conformal welding via Löwner—Kufarev evolution. Comput. Methods Funct. Theory, 2013, vol. 13, no. 1, pp. 37—46. DOI: https://doi.org/10.1007/s40315-012-0002-y
  7. Marshall D. E. Conformal Welding for Finitely Connected Regions. Comput. Methods Funct. Theory, 2012, vol. 11, pp. 655–669. DOI: https://doi.org/10.1007/BF03321879
  8. Löwner K. Untersuchungen über schlichte konforme Abbildungen des Einheitskreises. I. Math. Ann., 1923, vol. 89, pp. 103–121.
  9. Kufarev P. P. Ob odnoparametricheskikh semeistvakh analiticheskikh funktsii [On one-parametric families of analytic functions]. Mat. Sbornik, 1943, vol. 13(55), no. 1, pp. 87–118 (in Russian).
  10. Pommerenke Ch. Uber die Subordination analytischer Funktionen. J. Reine Angew. Math., 1965, vol. 218, pp. 159–173. DOI: https://doi.org/10.1515/crll.1965.218.159
  11. Pommerenke Ch. Univalent functions. Gottingen, Vandenhoeck and Ruprecht, 1975. 376 p.
  12. Siryk G. V. O konformnom otobrazhenii blizkikh oblastei [On a conformal mapping of near domains]. Uspekhi Mat. Nauk, 1956, vol. 11, no. 5(71), pp. 57–60 (in Russian).
  13. Lavrentyev M. A., Shabat B. V. Metody teorii funktsii kompleksnogo peremennogo [Methods of Function Theory of a Complex Variable]. Moscow, Nauka, 1965. 716 p. (in Russian).
  14. Pommerenke Ch. Boundary Behaviour of Conformal Maps. Berlin, Springer-Verlag, 1992. 300 p. DOI: https://doi.org/10.1007/978-3-662-02770-7
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