We study space mappings with branching that satisfy modulus inequalities. For classes of these mappings, we obtain several sufficient conditions for the normality of families.
Similar content being viewed by others
References
V. N. Ryazanov and E. A. Sevost’yanov, “Equicontinuous classes of ring Q-homeomorphisms,” Sib. Mat. Zh., 48, No. 6, 1361–1376 (2007).
C. J. Bishop, V. Ya. Gutlyanskii, O. Martio, and M. Vuorinen, “On conformal dilatation in space,” Int. J. Math. Math. Sci., 22, 1397–1420 (2003).
P. M. Tamrazov, “Moduli and extremal metrics in nonoriented and twisted Riemannian manifolds,” Ukr. Mat. Zh., 56, No. 10, 1388–1398 (1998).
J. Väisälä, Lectures on n-Dimensional Quasiconformal Mappings, Springer, Berlin (1971).
O. Martio, V. Ryazanov, U. Srebro, and E. Yakubov, “Mappings with finite length distortion,” J. D. Annal. Math., 93, 215–236 (2004).
A. L. Gol’berg, “Quasiconformal mappings and radii of normal systems of neighborhoods,” Ukr. Mat. Zh., 51, No. 11, 1566–1568 (1999).
N. V. Zorii, “Modulus, functional, and potential characteristics of condensers in a domain; relations between them,” Ukr. Mat. Zh., 44, No. 5, 604–613 (1992).
A. Ignat’ev and V. Ryazanov, “Finite mean oscillation in the theory of mappings,” Ukr. Mat. Vestn., 2, No. 3, 395–417 (2005).
O. Martio, V. Ryazanov, U. Srebro, and E. Yakubov, “Q-homeomorphisms,” Cont. Math., 364, 193–203 (2004).
V. I. Ryazanov, “Closure of classes of quasiconformal mappings with integral constraints,” Ukr. Mat. Zh., 43, No. 4, 399–404 (1991).
O. Martio, S. Rickman, and J. Väisälä, “Definitions for quasiregular mappings,” Ann. Acad. Sci. Fenn. Ser. Al., 448, 1–40 (1969).
S. Rickman, “Quasiregular mappings,” Results Math. Relat. Areas, 26, No. 3 (1993).
O. Martio, S. Rickman, and J. Väisälä, “Distortion and singularities of quasiregular mappings,” Ann. Acad. Sci. Fenn. Ser. Al., 465, 1–13 (1970).
F. W. Gehring, “Symmetrization of rings in space,” Trans. Amer. Math. Soc., 101, 499–519 (1961).
F. W. Gehring, “Quasiconformal mappings,” Complex Anal. Appl., 2 (1976).
M. Vuorinen, Conformal Geometry and Quasiregular Mappings, Springer, Berlin (1988).
K. Kuratowski, Topology [Russian translation], Vol. 2, Mir, Moscow (1969).
G. T. Whyburn, Analytic Topology, American Mathematical Society, Providence, RI (1942).
S. Saks, Theory of the Integral, Dover, New York (1937).
F. John and L. Nirenberg, “On functions of bounded mean oscillation,” Commun. Pure Appl. Math., 14, 415–426 (1961).
Author information
Authors and Affiliations
Additional information
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 10, pp. 1389–1400, October, 2008.
Rights and permissions
About this article
Cite this article
Sevost’yanov, E.A. On the normality of families of space mappings with branching. Ukr Math J 60, 1618–1632 (2008). https://doi.org/10.1007/s11253-009-0157-5
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-009-0157-5