Abstract

In [2], MacGregor found the radius of convexity of the functions f(z)=z+a2z2+a3z3+, analytic and univalent such that |f(z)1|<1. This paper generalized MacGregor's theorem, by considering another univalent function g(z)=z+b2z2+b3z3+ such that |f(z)g(z)1|<1 for |z|<1. Several theorems are proved with sharp results for the radius of convexity of the subfamilies of functions associated with the cases: g(z) is starlike for |z|<1, g(z) is convex for |z|<1, Re{g(z)}>α(α=0,1/2).