International Journal of Mathematics and Mathematical Sciences
Volume 7 (1984), Issue 1, Pages 187-195
doi:10.1155/S0161171284000193

Quasiconformal extensions for some geometric subclasses of univalent functions

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 0≤k<1, let Sk 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 Sk⋂S∗ or Sk⋂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 Sk. From the main results several other sufficient conditions are deduced which include a generalization of a recent result of Fait, Krzyz and Zygmunt.