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

Let T ( n , p ) denote the class of functions of the form f ( z ) = z p − ∑ k = n ∞ a k + p z k + p ( a k + 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 f 1 ( z ) and f 2 ( z ) represented by ( f 1 Δ f 2 ) ( 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 T n ( p , α ) and C n ( 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.