ON CERTAIN CLASSES OF P -VALENT FUNCTIONS DEFINED BY MULTIPLIER TRANSFORMATION AND DIFFERENTIAL OPERATOR

. In this paper, we discuss the p -valent functions that satisfy the diﬀerential subordinations z ( Ip ( r,λ ) f ( z ))( j +1) ( p − j )( Ip ( r,λ ) f ( z ))( j ) ≺ a +( aB +( A − B ) β ) z a (1+ Bz ) . We also obtain coeﬃcient inequalities, extreme points, integral representation and arithmetic mean. Further we investigate some interesting properties of operators deﬁned on A p ( r,j,β,a,A,B ).

where These functions are analytic in the unit disk D (For details see [1], [5]).
Definition 1.1.A function f ∈ A p is said to be in the class S * p (α) , p-valently starlike functions of order α, if it satisfies Re zf (z) f (z) > α, (0 ≤ α < p, z ∈ D).We note that S * p (0) = S * p , the class of p-valently starlike functions in D. A function f ∈ A p is said to be in the class C p (α) of p-valently convex of order α, if it satisfies Let h(z) be analytic and h(0 The class S * p (h) and a corresponding convex class C p (h) are defined by Ma and Minda [6].But results about the convex class can be obtained easily from the corresponding result of functions in S * p (h).If then the classes reduce to the usual classes of starlike and convex functions.If , then the classes reduce to the classes of strongly starlike and convex function of order α that consists of univalent functions f ∈ A satisfying or equivalently we have Obradovič and Owa [8], Silverman [11], Obradovič and Tuneski [9] and Tuneski [12] have studied the properties of classes of functions defined in terms of the ratio of 1 + zf (z) and zf (z) f (z) .
Definition 1.2.A function f ∈ A p is said to be p-valent Bazilevic of type η and order α if there exists a function g ∈ S * p such that Re zf (z) for some η(η ≥ 0) and α(0 ≤ α < p).We denote by B p (η, α), the subclass of A p consisting of all such functions.In particular, a function in B p (1, α) = B p (α) is said to be p-valently close-to-convex of order α in ∆.
Definition 1.3.[7] For two functions f and g, analytic in ∆ we say f is subordinate to g denoted by f ≺ g if there exists a Schwarz function w(z), analytic in ∆ with w(0) = 1 and |w(z)| < 1, such that f (z) = g(w(z)), z ∈ ∆.In particular, if the function g is univalent in ∆, the above subordination is equivalent to . Also, we say that g is superordinate to f .
Using the techniques of Cho and Srivastava [4],Cho and Kim [3] and Uralegadi and Somanatha [12] we define the following transformation.Definition 1.4.We define the multiplier transformation operator We note that Sǎlǎgean derivative operators [9] is closely related to the operators I p (r, λ) when the coefficient of f (z) is positive.Also note that the class I 1 (r, 1) = I r [12] ,I 1 (r, λ) = I λ r the classes studied in [4] and [3].
a n z n we have where n, p ∈ N, p > j, and where and We say that f (z) is superordinate to h(z) if f (z) satisfies the following where h(z) is analytic in ∆ and h(0) = 1.
We note that if By Definition 1.2., if g(z) ∈ S * , univalent starlike and j = r = 0 and ) class Bazilevic function of type η = 2 and order α = 1.

MAIN RESULTS
In this section we obtain sharp coefficient estimates for functions in A p (r, j, β, a, A, B).
Theorem 2.1.Let f (z) be of the form (1). Then f ∈ A p (r, j, β, a, A, B) if and only if Proof.The function f (z) of the form (1) can be expressed as where m = n − p + 1 and k m = (a,m)(b,m) (c,m)m!, and also we have for all r, j ∈ N 0 Let f (z) ∈ A p (r, j, a, β, A, B) then where We choose the values of z on the real axis and letting z → 1 − then we have and Conversely, we assume that the condition (10) holds true.Hence it is sufficient to show that f ∈ A p (r, j, β, a, A, B), that is to prove that

But we have
and so proof is complete.The inequality (10) is sharp for the function with q ≥ 1 + p.
Corollary 2.2.Let f ∈ A p (r, j, β, a, A, B) then we have In the next theorem we prove that the class A p (r, j, β, α, A, B) is closed under linear combination.r, j, β, a, A, B). r, j, β, a, A, B).

Then the function
Proof.We have Hence we obtain Now we prove that the class A p (r, j, β, a, A, B) is closed under arithmetic mean.
Then the function Proof.Since f j (z) ∈ A p (r, j, β, a, A, B), then by (10) we have Then the function f (z) ∈ A p (r, j, β, a, A, B) if and only if it can be expressed in the form where µ m ≥ 0 and Proof.Suppose that f can be expressed in the form (15) then we have Therefore we conclude the result.Conversely, let f ∈ A p (r, j, β, a, A, B) since by (10) we may set µ n then we can write Remark 2.6.The extreme points of the class A p (r, j, β, a, A, B) are the function f p (z), f m+p (z), m ≥ 1 + p as in Theorem 2.1.
In the following theorem, we obtain the integral representation for A p (r, j, β, a, A, B).

Proof. Set z(I
Then we have For obtaining the second representation let X = {x : |x| = 1} then we have R−aBQ(z) = xz, z ∈ ∆ and then we conclude that Thus L c (f (z)) ∈ A p (r, j, β, a, A, B).
Denote by A the class of functions f (z) = z − ∞ n=2 a n z n analytic in D = {z ∈ C : |z| < 1}.Also denote A p the class of all analytic functions of the form then the classes reduce to the usual classes of starlike and convex functions of order α.If h(z) = 1+Az 1+Bz , − 1 ≤ B < A ≤ 1, then the classes reduce to the class of Janowski starlike function S * p [A, B] defined by