International Journal of Mathematics and Mathematical Sciences
Volume 7 (1984), Issue 1, Pages 187-195
Quasiconformal extensions for some geometric subclasses of univalent functions
Department of Mathematics, Purdue University, West Lafayette 47907, Indiana, USA
Received 7 November 1983
Copyright © 1984 Johnny E. Brown. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Let denote the set of all functions which are analytic and univalent in the unit disk normalized so that . Let and be those functions in for which is starlike and convex, respectively. For , let denote the subclass of functions in which admit -quasiconformal extensions to the extended complex plane. Sufficient conditions are given so that a function belongs to or . Functions whose derivatives lie in a half-plane are also considered and a Noshiro-Warschawski-Wolff type sufficiency condition is given to determine which of these functions belong to . From the main results several other sufficient conditions are deduced which include a generalization of a recent result of Fait, Krzyz and Zygmunt.