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http://dx.doi.org/10.4134/BKMS.2013.50.4.1377

QUASICONFORMAL EXTENSIONS OF STARLIKE HARMONIC MAPPINGS IN THE UNIT DISC

Hamada, Hidetaka (Faculty of Engineering Kyushu Sangyo University)
Honda, Tatsuhiro (Faculty of Engineering Hiroshima Institute of Technology)
Shon, Kwang Ho (Department of Mathematics Pusan National University)

Publication Information

Bulletin of the Korean Mathematical Society / v.50, no.4, 2013 , pp. 1377-1387 More about this Journal

Abstract

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 ${\alpha}{\in}(0,1)$ .

Keywords

harmonic mapping; quasiconformal extension; starlike domain;

Citations & Related Records

Reference

1	L. V. Ahlfors, Quasiconformal reflections, Acta Math. 109 (1963), 291-301. DOI
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. DOI ScienceOn
5	J. Clunie and T. Sheil-Small, Harmonic univalent functions, Ann. Acad. Sci. Fenn. Ser. A. I Math. 9 (1984), 3-25. DOI
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. DOI
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. DOI ScienceOn
10	H. Hamada and G. Kohr, Loewner chains and quasiconformal extension of holomorphic mappings, Ann. Polon. Math. 81 (2003), no. 1, 85-100. DOI
11	H. Hamada and G. Kohr, Quasiconformal extension of biholomorphic mappings in several complex variables, J. Anal. Math. 96 (2005), 269-282. DOI
12	H. Hamada and G. Kohr, Univalence criterion and quasiconformal extension of holomorphic mappings, Manuscripta Math. 141 (2013), no. 1-2, 195-209. DOI
13	J. M. Jahangiri, Harmonic functions starlike in the unit disk, J. Math. Anal. Appl. 235 (1999), no. 2, 470-477. DOI ScienceOn
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. DOI
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. DOI ScienceOn