Ukrainian Mathematical Journal

, Volume 70, Issue 9, pp 1456–1466 | Cite as

On the Equicontinuity of One Family of Inverse Mappings in Terms of Prime Ends

  • R. R. Salimov
  • E. A. Sevost’yanov

For a class of mappings satisfying upper modular estimates with respect to the families of curves, we study the local behavior of the corresponding inverse mappings. In terms of prime ends, we prove that the families of these homeomorphisms are equicontinuous (normal) in the closure of a given domain.


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  1. 1.
    V. Gutlyanskii, V. Ryazanov, and E. Yakubov, “The Beltrami equations and prime ends,” Ukr. Mat. Visn., 12, No. 1, 27–66 (2015).MathSciNetzbMATHGoogle Scholar
  2. 2.
    D. A. Kovtonyuk and V. I. Ryazanov, “On the theory of prime ends for space mappings,” Ukr. Mat. Zh., 67, No. 4, 467–479 (2015); English translation : Ukr. Math. J., 67, No. 4, 528–541 (2015).Google Scholar
  3. 3.
    O. Martio, V. Ryazanov, U. Srebro, and E. Yakubov, Moduli in Modern Mapping Theory, Springer Science + Business Media, New York (2009).zbMATHGoogle Scholar
  4. 4.
    V. Ya. Gutlyanskii, V. I. Ryazanov, U. Srebro, and E. Yakubov, The Beltrami Equation: A Geometric Approach, Springer, New York (2012).zbMATHGoogle Scholar
  5. 5.
    E. A. Sevost’yanov, “On the equicontinuity of homeomorphisms with unbounded characteristic,” Mat. Trudy, 15, No. 1, 178–204 (2012).Google Scholar
  6. 6.
    R. Näkki, “Prime ends and quasiconformal mappings,” J. Anal. Math., 35, 13–40 (1979).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    F. W. Gehring and O. Martio, “Quasiextremal distance domains and extension of quasiconformal mappings,” J. Anal. Math., 24, 181–206 (1985).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    W. Hurewicz and H. Wallman, Dimension Theory, Princeton Univ. Press, Princeton (1948).zbMATHGoogle Scholar
  9. 9.
    K. Kuratowski, Topology, Vol. 2, Academic Press, New York (1968).Google Scholar
  10. 10.
    S. Saks, Theory of the Integral, Państwowe Wydawnictwo Naukowe, Warsaw (1937).zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • R. R. Salimov
    • 1
  • E. A. Sevost’yanov
    • 2
  1. 1.Institute of MathematicsUkrainian National Academy of SciencesKyivUkraine
  2. 2.I. Franko Zhitomir State UniversityZhitomirUkraine

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