International Journal of Mathematics and Mathematical Sciences
Volume 3 (1980), Issue 3, Pages 483-489

The radius of convexity of certain analytic functions II

J. S. Ratti

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.


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).