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

Let T(n,p)T(n,p) denote the class of functions of the form f(z)=zp−∑k=n∞ak+pzk+p(ak+p≥0;p,n∈N) which are analytic and pp-valent in the open unit disc U={z:|z|<1}U={z:|z|<1}. For functions fj(z)(j=1,2) belonging to T(n,p)T(n,p), generalizations of the modified-Hadamard product of f1(z)f1(z) and f2(z)f2(z) represented by (f1Δf2)(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)Tn(p,α) and Cn(p,α)Cn(p,α). Also we obtain the order of pp-valent starlikeness of a pp-valent convex function of order α(0≤α