Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On a class of quasiconformal functions in Banach spaces

Author: Su Shing Chen
Journal: Proc. Amer. Math. Soc. 37 (1973), 545-548
MSC: Primary 32H15; Secondary 32K05
MathSciNet review: 0318518
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A quasiconformal function f on a domain D in a complex Banach space E is defined as a function on D such that for every holomorphic mapping $ \Phi $ from the unit disk $ \Delta $ into D the composite mapping $ f \circ \Phi $ is quasiconformal in the usual sense. With respect to the Kobayashi-Kiernan pseudo distance on D, Schwarz's lemma, Liouville's theorem and the little Picard theorem are obtained for quasiconformal functions. A maximum modulus principle is also obtained for quasiconformal functions.

References [Enhancements On Off] (What's this?)

  • [1] L. V. Ahlfors, Lectures on quasiconformal mappings, Van Nostrand Math. Studies, no. 10, Van Nostrand, Princeton, N.J., 1966. MR 34 #336. MR 0200442 (34:336)
  • [2] E. Hille and R. S. Phillips, Functional analysis and semi-groups, Amer. Math. Soc. Colloq. Publ., vol. 31, Amer. Math. Soc., Providence, R.I., 1957. MR 19, 664. MR 0089373 (19:664d)
  • [3] S. Hitotumatu, On quasi-conformal functions of several complex variables, J. Math. Mech. 8 (1959), 77-94. MR 21 #1398. MR 0102608 (21:1398)
  • [4] P. J. Kiernan, Quasiconformal mappings and Schwarz's lemma, Trans. Amer. Math. Soc. 148 (1970), 185-197. MR 41 #467. MR 0255807 (41:467)
  • [5] S. Kobayashi, Hyperbolic manifolds and holomorphic mappings, Pure and Appl. Math., no. 2, Dekker, New York, 1970. MR 43 #3503. MR 0277770 (43:3503)
  • [6] L. Nachbin, Topology on spaces of holomorphic mappings, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 47, Springer-Verlag, Berlin and New York, 1969. MR 40 #7787. MR 0254579 (40:7787)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 32H15, 32K05

Retrieve articles in all journals with MSC: 32H15, 32K05

Additional Information

Keywords: Quasiconformal function, Kobayashi pseudo distance, Schwarz lemma, Liouville theorem, the little Picard theorem, maximum modulus principle
Article copyright: © Copyright 1973 American Mathematical Society

American Mathematical Society