Inclusion and Neighborhood Properties of a Certain Subclasses of P-valent Functions with Negative Coefficients

By means of Ruscheweyh derivative operator , we introduced and investigated two new subclasses of p-valent analytic functions.The various results obtained here for each of these function class include coefficient bounds and distortion inequalities, associated inclusion relations for the (n, θ)-neighborhoods of subclasses of analytic and multivalent functions with negative coefficients, which are defined by means of non-homogenous differential equation. 1 Introductin Let Tp(n) denote the class of functions of the form: f(z) = z − ∞ ∑ k=n+p akz k (ak ≥ 0; p, n ∈ N = {1, 2, ....}), (1.1) which are analytic and p-valent in the open unit disc U = {z : |z| < 1}. The modified Hadamard product (or convolution) of the function f(z) given by (1.1) and the function g(z) ∈ Tp(n) given by g(z) = z − ∞ ∑ k=n+p bkz k (bk ≥ 0; p, n ∈ N) (1.2) is defined by (f ∗ g)(z) = z − ∞ ∑ k=n+p akbkz k = (g ∗ f)(z). (1.3) 2000 Mathematics Subject Classifications. 30C45.