International Journal of Mathematics and Mathematical Sciences
Volume 7 (1984), Issue 1, Pages 187-195

Quasiconformal extensions for some geometric subclasses of univalent functions

Johnny E. Brown

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 S denote the set of all functions f which are analytic and univalent in the unit disk D normalized so that f(z)=z+a2z2+. Let S and C be those functions f in S for which f(D) is starlike and convex, respectively. For 0k<1, let Sk denote the subclass of functions in S which admit (1+k)/(1k)-quasiconformal extensions to the extended complex plane. Sufficient conditions are given so that a function f belongs to SkS or SkC. 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 Sk. From the main results several other sufficient conditions are deduced which include a generalization of a recent result of Fait, Krzyz and Zygmunt.