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
Department of Mathematics, University of South Florida, Tampa 33620, Florida, USA
Received 7 August 1979
Copyright © 1980 J. S. Ratti. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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).