Hamada, Hidetaka;Honda, Tatsuhiro;Shon, Kwang Ho

  • Received : 2012.11.08
  • Published : 2013.07.31


Let $f$ be a harmonic mapping on the unit disc ${\Delta}$ in $\mathbb{C}$. We give some condition for $f$ to be a quasiconformal homeomorphism on ${\Delta}$ and to have a quasiconformal extension to the whole plane $\bar{\mathbb{C}}$. We also obtain quasiconformal extension results for starlike harmonic mappings of order ${\alpha}{\in}(0,1)$.


harmonic mapping;quasiconformal extension;starlike domain


  1. L. V. Ahlfors, Quasiconformal reflections, Acta Math. 109 (1963), 291-301.
  2. Y. Avciand E. Z lotkiewicz, On harmonic univalent mappings, Ann. Univ. Mariae Curie-Sklodowska Sect. A 44 (1990), 1-7.
  3. A. A. Brodskii, Quasiconformal extension of biholomorphic mappings, Theory of mappings and approximation of functions, 30-34, "Naukova Dumka", Kiev, 1983.
  4. P. Curt, G. Kohr, and M. Kohr, Homeomorphic extension of strongly spirallike mappings in $\mathbb{C}^n$, Sci. China. Math. 53 (2010), no. 1, 87-100.
  5. J. Clunie and T. Sheil-Small, Harmonic univalent functions, Ann. Acad. Sci. Fenn. Ser. A. I Math. 9 (1984), 3-25.
  6. M. Fait, J. G. Krzyz, and J. Zygmunt, Explicit quasiconformal extensions for some classes of univalent functions, Comment. Math. Helv. 51 (1976), no. 2, 279-285.
  7. A. Ganczar, On harmonic univalent mappings with small coefficients, Demonstratio Math. 34 (2001), no. 3, 549-558.
  8. A. Ganczar, Explicit quasiconformal extensions of planar harmonic mappings, J. Comput. Anal. Appl. 10 (2008), no. 2, 179-186.
  9. I. Graham, H. Hamada, and G. Kohr, Radius problems for holomorphic mappings on the unit ball in $\mathbb{C}^n$, Math. Nachr. 279 (2006), no. 13-14, 1474-1490.
  10. H. Hamada and G. Kohr, Loewner chains and quasiconformal extension of holomorphic mappings, Ann. Polon. Math. 81 (2003), no. 1, 85-100.
  11. H. Hamada and G. Kohr, Quasiconformal extension of biholomorphic mappings in several complex variables, J. Anal. Math. 96 (2005), 269-282.
  12. H. Hamada and G. Kohr, Univalence criterion and quasiconformal extension of holomorphic mappings, Manuscripta Math. 141 (2013), no. 1-2, 195-209.
  13. J. M. Jahangiri, Harmonic functions starlike in the unit disk, J. Math. Anal. Appl. 235 (1999), no. 2, 470-477.
  14. O. Lehto and K. I. Virtanen, Quasiconformal Mappings in the Plane, Springer-Verlag, New York-Heidelberg, Second Edition, 1973.
  15. H. Lewy, On the non-vanishing of the Jacobian in certain one-to-one mappings, Bull. Amer. Math. Soc. 42 (1936), no. 10, 689-692.
  16. T. Sheil-Small, Complex Polynomials, Cambridge Studies in Advanced Mathematics, 75, Cambridge University Press, Cambridge, 2002.
  17. H. Silverman, Harmonic univalent functions with negative coefficients, J. Math. Anal. Appl. 220 (1998), no. 1, 283-289.

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  1. Pluriharmonic mappings in Cn and complex Banach spaces vol.426, pp.2, 2015,


Supported by : National Research Foundation of Korea(NRF)