Generalization of Differential Operator

Abstract: The main objective of this study was to generalize a differential operator. The generalized differential operator reduced to many known operators studied by various authors. New classes containing this generalized operator were studied and characterization of these classes was obtained. Further, subordination and superordination results involving this operator were studied and obtained the sandwich theorem.

Some of relations for the differential operator (2) are discussed in the next lemma.
Lemma 1: Let f ∈A. Then: In the following definitions, new classes of analytic functions containing the differential operator (2) are introduced: Then f(z) ∈ k , S λ δ (µ) if and only if: 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 and G, which are analytic in U, the function F is said to be subordinate to G in U if there exists a function h, analytic in U with: h(0) = 0 and |h(z)|<1for all z ∈ U Such that: F(z) = G (h(z)) for all z ∈ U Let φ: 2 → and let h be univalent in U. If p is analytic in U and satisfies the differential subordination φ(p(z)), zp′(z)) ≺ h(z) 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 φ(p(z)), zp′(z)) are univalent in U and satisfy the differential superordination h(z) ≺ φ (p(z)), zp′(z)) 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. Let Φ be an analytic function in a domain containing f(U), Φ(0) = 0 and Φ′(0) > 0. The This concept was introduced by [3] and established that a function f ∈ A is univalent if and only if f is Φlike for some Φ.
Lemma 3 [6] : Let q(z) be convex univalent in the unit disk U and ϑ and ϕ be analytic in a domain D containing q(U). Suppose that:  We also note that the assertion (3) is sharp and the extremal function is given by: Then the function f belongs to the class k , S ( ) λ δ µ .
In the same way we can verify the following results: Then the function f belongs to the class k , C ( ) λ δ µ .
Also we have the following inclusion results: Proof: By theorem 2.
Moreover, we introduce the following distortion theorems. In the same way we can get the following results.
Theorem 8: Let the functions f∈A and (6) holds. Then for z∈U and 0 ≤ µ < 1: Also, we have the following distortion results. In the same way we can get the following results.

RESULTS
By making use of lemmas 2 and 3, we prove the following subordination and superordination results involving the differential operator (2).
Theorem 11: Let q(z) ≠ 0 be univalent in U such that zq '(z) q(z) is starlike univalent in U and: zq ''(z) zq '(z) 1 0 , , , 0 q(z) q '(z) q(z) and q(z) is the best dominant. covered the well known operators and many interesting results such as the coefficient estimates, distortion results and the sandwich theorem are found. Further many other results are yet to be studied.