Convolutions of prestarlike functions

The convolution of two functions and defined as . For 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 |( z h ′ / h ) − 1|/|( z h ′ / h ) + (1 − 2 α )| < 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.