Quasiconformal extensions for some geometric subclasses of univalent functions

Abstract

LetSdenote the set of all functionsfwhich are analytic and univalent in the unit diskDnormalized so thatf(z)=z+a2z2+…. LetS∗andCbe those functionsfinSfor whichf(D)is starlike and convex, respectively. For0≤k<1, letSkdenote the subclass of functions inSwhich admit(1+k)/(1−k)-quasiconformal extensions to the extended complex plane. Sufficient conditions are given so that a functionfbelongs toSk⋂S∗orSk⋂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 toSk. From the main results several other sufficient conditions are deduced which include a generalization of a recent result of Fait, Krzyz and Zygmunt.