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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.

MSC:

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