The convolution of two functions f ( z ) = ∑ n = 0 ∞ a n z n and g ( z ) = ∑ n = 0 ∞ b n z n defined as ( f ∗ g ) ( z ) = ∑ n = 0 ∞ a n b n z n . For f ( z ) = z − ∑ n = 2 ∞ a n z n 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.