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.