International Journal of Mathematics and Mathematical Sciences
Volume 6 (1983), Issue 1, Pages 59-68
doi:10.1155/S0161171283000034
Convolutions of prestarlike functions
1Department of Mathematics, University of Khartoum, Khartoum, Sudan
2Department of Mathematics, College of Charleston, Charleston, South Carolina 29424, USA
Received 2 October 1982
Copyright © 1983 O. P. Ahuja and H. Silverman. 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
The convolution of two functions f(z)=∑n=0∞anzn and g(z)=∑n=0∞bnzn defined as (f∗g)(z)=∑n=0∞anbnzn. For f(z)=z−∑n=2∞anzn and g(z)=z/(1−z)2(1−γ), the extremal function for the class of functions starlike of order γ, we investigate functions h, where h(z)=(f∗g)(z), which satisfy the inequality |(zh′/h)−1|/|(zh′/h)+(1-2α)|<β, 0≤α<1, 0<β≤1 for all z in the unit disk. Such functions f are said to be γ-prestarlike of order α and type β. We characterize this family in terms of its coefficients, and then determine extreme
points, distortion theorems, and radii of univalence, starlikeness, and convexity. All results are sharp.