Explicit quasiconformal extensions and Löwner chains. (English) Zbl 1183.30016

Let \(\mathcal A\) be the class of all analytic functions \(f\) on the unit disk \(\mathbb{D}\) normalized so that \(f(0)=f'(0)-1=0\), and, for \(k\in (0,1)\), let \(U(k) = \{ w \in \mathbb {C}: |w-c_k|< r_k\}\) with \(r_k=2k/(1-k^2)\) and \(c_k=(1+k^2)/(1-k^2)\,\). The author starts with the following well-known result.
Theorem A. For \(f \in \mathcal{A}\) let \(h(z)\) represent one of the quantities \(z f'(z)/f(z)\), \(1+ f''(z)/f'(z)\), and \(f'(z)\,\). If \(h(z) \in U(k)\) for all \(z \in \mathbb{D}\,\), then the function \(f\) can be extended to a \(k\)-quasiconformal automorphism of the complex plane.
In this connection, the papers of J. E. Brown [Int. J. Math. Math. Sci. 7, 187–195 (1984; Zbl 0566.30011)], K. S. Padmanabhan and S. Kumar [J. Math. Phys. Sci. 25, No.4, 361–368 (1991; Zbl 0770.30019)], and T. Sugawa [Ann. Univ. Mariae Curie-Skłodowska, Sect. A 53, 239–252 (1999; Zbl 1031.30011)] should be mentioned. The author improves this theorem by giving an explicit construction of the quasiconformal extension. The proof makes use of Löwner chains.


30C62 Quasiconformal mappings in the complex plane
30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
Full Text: DOI


[1] J. Becker, Löwnersche Differentialgleichung und quasikonform fortsetzbare schlichte Funktionen, J. Reine Angew. Math. 255 (1972), 23-43. · Zbl 0239.30015 · doi:10.1515/crll.1972.255.23
[2] 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. · Zbl 0491.30012
[3] J. E. Brown, Quasiconformal extensions for some geometric subclasses of univalent functions, Internat. J. Math. Math. Sci. 7 (1984), no. 1, 187-195. · Zbl 0566.30011 · doi:10.1155/S0161171284000193
[4] 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. · Zbl 0332.30010 · doi:10.1007/BF02568157
[5] Y. C. Kim and T. Sugawa, A note on Bazilevi\(\v c\) function, Taiwanese J. Math. (to appear). · Zbl 1180.30015
[6] J. G. Krzy\(\.z\) and A. K. Soni, Close-to-convex functions with quasiconformal extension, in Analytic functions, \(B\l\)aż\(ejewko 1982 (\)B\(ejewko, 1982) , Lecture Notes in Math., 1039, Springer, Berlin, 1983, pp. 320-327.\)
[7] R. Kühnau, Bemerkung zur quasikonformen Fortsetzung, Ann. Univ. Mariae Curie-\(Sk\l&lt;\)&gt;odowska Sect. A 56 (2002), 53-55.
[8] S. S. Miller and P. T. Mocanu, Differential subordinations , Dekker, New York, 2000. · Zbl 0954.34003
[9] 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. · Zbl 0770.30019
[10] C. Pommerenke, Univalent functions , Vandenhoeck & Ruprecht, Göttingen, 1975.
[11] S. Ruscheweyh, An extension of Becker’s univalence condition, Math. Ann. 220 (1976), no. 3, 285-290. · Zbl 0318.30015 · doi:10.1007/BF01431098
[12] T. Sugawa, Holomorphic motions and quasiconformal extensions, Ann. Univ. Mariae Curie-\(Sk\l\)odowska Sect. A 53 (1999), 239-252. · Zbl 1031.30011
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.