Convolutions of prestarlike functions - O. P. Ahuja; H. Silverman (International Journal of Mathematics and Mathematical Sciences)
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.