Quasiconformal extensions for some geometric subclasses of univalent functions

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 + a 2 z 2 + … . Let S ∗ and C be those functions f in S for which f ( D ) is starlike and convex, respectively. For 0 ≤ k < 1 , let S k denote the subclass of functions in S which admit ( 1 + k ) / ( 1 − k ) -quasiconformal extensions to the extended complex plane. Sufficient conditions are given so that a function f belongs to S k ⋂ S ∗ or S k ⋂ C . 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 S k . From the main results several other sufficient conditions are deduced which include a generalization of a recent result of Fait, Krzyz and Zygmunt.