On Sandwich Theorems of Analytic Functions Involving Noor Integral Operator

2 2 f (z) z a z ... . = + + Let F and G be analytic functions in the unit disk U. The function F is subordinate to G, written F G , if G is univalent F(0) = G(0) and F(U)⊂G(U). In general, given two functions F(z) and G(z), which are analytic in U, the function F(z) is said to be subordination to G(z) in U if there exists a function h(z), analytic in U with h(0) and | h(z) | 1 < for all z∈U such that F(z) = G(h(z)) for all z∈U. Let C C → 2 : φ and let h be univalent in U. If p is analytic in U and satisfies the differential subordination (p(z)),zp '(z)) h(z) φ .


INTRODUCTION
Let H be the class of functions analytic in U and H[a, n] be the subclass of H consisting of functions of the form: n n 1 n n 1 f (z) a a z a z ...
Let A be the subclass of H consisting of functions of the form: 2 2 f (z) z a z ... .

= + +
Let F and G be analytic functions in the unit disk U. The function F is subordinate to G, written F G , if G is univalent F(0) = G(0) and F(U)⊂G(U). In general, given two functions F(z) and G(z), which are analytic in U, the function F(z) is said to be subordination to G(z) in U if there exists a function h(z), analytic in U with h(0) and | h(z) | 1 < for all z∈U such that F(z) = G(h(z)) for all z∈U.
Let C C → 2 : ϕ and let h be univalent in U. If p is analytic in U and satisfies the differential subordination Then p is called a solution of the differential subordination. The univalent function q is called a dominant of the solutions of the differential subordination, p q . If p and (p(z)),zp '(z)) ϕ are univalent in U and satisfy the differential superordination h(z) (p(z)),zp '(z)) ϕ then p is called a solution of the differential superordination.
An analytic function q is called subordinant of the solution of the differential superordination if q p . Denote by D : A A α → the operator defined by: where, (*) refers to the Hadamard product or convolution. Then implies that: We note that D 0 f(z) = f(z) and D'f (z) zf '(z) The operator D n f is called Ruscheweyh derivative of nth order of f. Noor [1] defined and studied an integral operator Note that 0 The operator I n f(z) is called the Noor Integral of n-th order of f. Using (1), (2) and a well-known identity for D n f we have: Using hypergeometric functions 2 1 F , (2) becomes: The following definitions can be found in [2] .  > .
In the present study, we apply a method based on the differential subordination in order to obtain subordination results involving Noor Integral operator for a normalized analytic function f: .
In order to prove our subordination and superordination results, we need the following definition and lemmas in the sequel.  [4] . Let q(z) be univalent in the unit disk U and let θ and φ be analytic in a domain D containing q(U) with φ(w) ≠ 0 when w∈q(U). Set: Q(z) : zq '(z) (q(z)) , h(z) : (q(z)) Q(z) = φ = θ + Suppose that: • Q(z) is starlike univalent in U and then p(z) q(z) and q(z) is the best dominant.
Lemma 2: Shanmugam, et al. [5] . Let q(z) be convex univalent in the unit disk U and Ψ and γ in C with: then p(z) q(z) and q is the best dominant. Lemma 3: Bulboaca [6] . Let q(z) be convex univalent in the unit disk U and ϑ and υ be analytic in a domain D containing q(U). Suppose that: • zq '(z) (q(z)) ϕ is starlike univalent in U and  is univalent in U and (q(z)) zq '(z) (q(z)) (p(z)) zp '(z) (p(z)) ϑ + ϕ ϑ + ϕ then q(z) p(z) and q(z) is the best subordinant. [3] . Let q(z) be convex univalent in the unit disk U and C ∈ γ . Further, assume that

SANDWICH RESULTS
By making use of Lemmas 1 and 2, we prove the following subordination results.
We find that Q(z) is starlike univalent in U and that: Then the relation (6) follows by an application of Lemma 1.  is analytic in U and the subordination z[I f (z)]' z(I f (z))'' z(I g(z))' n n n {1 + [1 + -]} I g(z) (I (z))' I g(z) n n n q(z) + zq'(z), C γ γ ∈ holds then: