SOME FAMILIES OF p-VALENT FUNCTIONS WITH NEGATIVE COEFFICIENTS

We introduce two subclasses T ∗(p, α, j) and C(p, α, j) (0 ≤ α < p − j + 1, 1 ≤ j ≤ p, p ∈ N = {1, 2, . . .}) of p-valent starlike and p-valent convex functions with negative coefficents. In this paper we obtain coefficient inequalities, distortion theorems, extreme points and integral operators for functions belonging to the classes T ∗(p, α, j) and C(p, α, j). We also determine the radii of close-to-convexity and convexity for the functions belonging to the class T ∗(p, α, j). Also we obtain several results for the modified Hadamard products of functions belonging to the classes T ∗(p, α, j) and C(p, α, j).