Generalizations of modified-Hadamard products of p-valent functions with negative coefficients

Let T (n, p) denote the class of functions of the form f (z) = z p − ∑ ∞ k=n ak+pz k+p (ak+p ≥ 0; p, n ∈ N ) which are analytic and p-valent in the open unit disc U = {z : |z| < 1}. For functions f j (z) ( j = 1, 2) belonging to T (n, p), generalizations of the modified-Hadamard product of f1(z) and f2(z) represented by ( f1∆ f2) (r, s; z) (r, s ∈ R) are introduced. In this paper, we obtain several results of these generalizations of the modified-Hadamard products of functions in the classes Tn(p, α) and Cn(p, α). Also we obtain the order of p-valent starlikeness of a p-valent convex function of order α (0 ≤ α < p), with negative coefficients. c © 2007 Elsevier Ltd. All rights reserved.