In [2], MacGregor found the radius of convexity of the functions f ( z ) = z + a 2 z 2 + a 3 z 3 + … , analytic and univalent such that | f ′ ( z ) − 1 | < 1 . This paper generalized MacGregor's theorem, by considering another univalent function g ( z ) = z + b 2 z 2 + b 3 z 3 + … 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 , \alpha $"> Re { g ′ ( z ) } > α ( α = 0 , 1 / 2 ) .