Abstract

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.