Inclusion Properties for Classes of p -Valent Functions

Making use of a di ﬀ erential operator, which is de ﬁ ned here by means of the Hadamard product


Introduction
Let A p be the class of functions which are analytic and p-valent in U = ϰ ϰ < 1 .
If E and G are analytic in U, E is subordinate to G, E ≺ G if there exists an analytic function ω 0 = 0 and ω ϰ < 1 such that E ϰ = G ω ϰ .Furthermore, if G is univalent in U, then (see [1,2]) For functions E ϰ ∈ A p , given by (1) and G ϰ ∈ A p defined by the Hadamard product of E and G is given by For E ϰ ∈ A p , denote by S * p ζ and K p ζ the classes of p-valently starlike and convex functions of order ζ and 0 ≤ ζ < p, respectively (see [3,4]), satisfying It follows from ( 5) and ( 6) that See Goodman [5].
Also, denote by C p η, ζ and C * p η, ζ the classes of p-valently close-to-convex and quasi-convex functions of order η and type ζ satisfying, respectively (see [6][7][8] (with p = 1), It follows from ( 8) and ( 9) that Dziok and Srivastava [9] used the hypergeometric function (see Srivastava and Karlsson [10]) and defined the linear operator where Setting the function we define a function D * n p,λ E ϰ in terms of the Hadamard product (or convolution) by 16), it can be easy to verify that Using the operator H n p,λ α 1 , we introduce the subclasses.
We note that
The following lemma due to Miller and Mocanu is required to prove the results.

Conclusion
Using the hypergeometric function (see Srivastava and Karlsson [10]) and Hadamard product, we defined an operator for p-valent functions.This operator generalizes many other operators for special values of its parameters and has two recurrence relations and then defined four classes related to starlike, convex, close-to-convex, and quasi-toconvex p-valent functions.We used Miller and Mocanu lemma [11] for second differential inequalities to obtain inclusion relations for these classes and also for the generalized Libera integral operator.

Future Studies
The authors suggest to obtain the inclusion results for the classes using the following lemma according to Jack [13] instead of Lemma 1. Jack's lemma [13] state that, if ω ϰ is analytic function in U, with ω 0 = 0, ω ϰ attains its maximum value on the circle ϰ = r < 1 at a point ϰ 0 ∈ U and ξ ≥ 1; then, ϰ 0 ω′ ϰ 0 = ξωϰ 0 64