International Journal of Mathematics and Mathematical Sciences
Volume 3 (1980), Issue 3, Pages 483-489
doi:10.1155/S0161171280000361

The radius of convexity of certain analytic functions. II

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).