Convolutions of prestarlike functions
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...
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Wiley
1983-01-01
|
| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/S0161171283000034 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Summary: | 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. |
|---|---|
| ISSN: | 0161-1712 1687-0425 |