Proceedings of the Japan Academy, Series A, Mathematical Sciences

Explicit quasiconformal extensions and Löwner chains

Ikkei Hotta

Full-text: Open access


In this paper we construct Löwner chains which enable us to derive quasiconformal extension criteria for typical classes of univalnet functions. This method also provides us explicit quasiconformal extensions.

Article information

Proc. Japan Acad. Ser. A Math. Sci. Volume 85, Number 8 (2009), 108-111.

First available in Project Euclid: 2 October 2009

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 30C62: Quasiconformal mappings in the plane 30C45: Special classes of univalent and multivalent functions (starlike, convex, bounded rotation, etc.)

Univalent functions quasiconformal mapping Löwner(Loewner) chains


Hotta, Ikkei. Explicit quasiconformal extensions and Löwner chains. Proc. Japan Acad. Ser. A Math. Sci. 85 (2009), no. 8, 108--111. doi:10.3792/pjaa.85.108.

Export citation


  • J. Becker, Löwnersche Differentialgleichung und quasikonform fortsetzbare schlichte Funktionen, J. Reine Angew. Math. 255 (1972), 23–43.
  • J. Becker, Conformal mappings with quasiconformal extensions, in Aspects of contemporary complex analysis (Proc. NATO Adv. Study Inst., Univ. Durham, Durham, 1979), Academic Press, London, 1980, pp. 37–77.
  • J. E. Brown, Quasiconformal extensions for some geometric subclasses of univalent functions, Internat. J. Math. Math. Sci. 7 (1984), no. 1, 187–195.
  • M. Fait, J. G. Krzy$\.z$ and J. Zygmunt, Explicit quasiconformal extensions for some classes of univalent functions, Comment. Math. Helv. 51 (1976), no. 2, 279–285.
  • Y. C. Kim and T. Sugawa, A note on Bazilevi$\v c$ function, Taiwanese J. Math. (to appear).
  • J. G. Krzy$\.z$ and A. K. Soni, Close-to-convex functions with quasiconformal extension, in Analytic functions, $B\l$aż$ejewko 1982 ($B\laż$ejewko, 1982), Lecture Notes in Math., 1039, Springer, Berlin, 1983, pp. 320–327.
  • R. Kühnau, Bemerkung zur quasikonformen Fortsetzung, Ann. Univ. Mariae Curie-$Sk\l<$>odowska Sect. A 56 (2002), 53–55.
  • S. S. Miller and P. T. Mocanu, Differential subordinations, Dekker, New York, 2000.
  • K. S. Padmanabhan and S. Kumar, On a class of subordination chains of univalent function, J. Math. Phys. Sci. 25 (1991), no. 4, 361–368.
  • C. Pommerenke, Univalent functions, Vandenhoeck & Ruprecht, Göttingen, 1975.
  • S. Ruscheweyh, An extension of Becker's univalence condition, Math. Ann. 220 (1976), no. 3, 285–290.
  • T. Sugawa, Holomorphic motions and quasiconformal extensions, Ann. Univ. Mariae Curie-$Sk\l$odowska Sect. A 53 (1999), 239–252.