Quasiconformal extensions for some geometric subclasses of univalent functions (Q1059174)

Let S denote the set of functions f which are analytic and univalent in \(U=\{z: | z| <1\}\) and satisfy \(f(0)=0\) and \(f'(0)=1\). Let \(S^*\), C and R denote the subsets of S consisting of starlike mappings, convex mappings and functions satisfying Re\(f'(z)>0\) for \(| z| <1\), respectively. The author proves that certain additional conditions on functions in \(S^*\), C or R imply that the function have a \((1+k)/(1-k)\)- quasiconformal extension to the extended complex plane. Examples of the conditions treated by the author are given by the subordinations \[ zf'(z)/f(z)\prec (1+kz)/(1-kz)\quad and\quad 1+(zf''(z))/f'(z)\prec (1+kz)/(1-kz) \] and by \(| z^ 2(\lambda f'(z)-1)| \leq k\) for \(| z| <1\) where \(\lambda\) is a complex number. A few similar results are obtained for certain meromorphic univalent functions. As pointed out by the author, some of his results follow from a general theorem for quasiconformal extensions due to \textit{J. Becker} [J. Reine Angew. Math. 255, 23-43 (1972; Zbl 0239.30015)], but here all extensions are given by simple explicit formulas for f(z) when \(| z| \geq 1\) in terms of f(z) for \(| z| <1\). The results in this paper are comparable to those obtained by \textit{J. G. Krzyż} [Comment. Math. Helv. 51, 99-104 (1976; Zbl 0323.30022)] and by \textit{M. Fait}, \textit{J. G. Krzyż} and \textit{J. Zygmunt} [ibid. 51, 279-285 (1976; Zbl 0332.30010)].