Differential sandwich results for Wanas operator of analytic functions

In the present article, we determine some subordination and superordination results involving Wanas operator for certain normalized analytic functions defined in the unit disk U. These results are applied to establish sandwich results. Our results extend corresponding previously known results.


Introduction
Denote by H = H (U) the collection of analytic functions in the unit disk U = {z ∈ C : |z| < 1} and assume that H [a, n] be the subclass of H consisting of functions of the form: f (z) = a + a n z n + a n+1 z n+1 + ... (a ∈ C, n ∈ N = {1, 2, ...}).
Also, let A be the subclass of H consisting of functions of the form: (1) f (z) = z + ∞ n=2 a n z n .
Now we recall the principal of subordination between analytic functions, let the functions f and g be analytic in U, we say that the function f is subordinate to g, if there exists a Schwarz function w analytic in U with w(0) = 0 and |w(z)| < 1 (z ∈ U) such that f (z) = g (w(z)). This subordination is indicated by f ≺ g or f (z) ≺ g(z) (z ∈ U). Furthermore, if the function g is univalent in U, then we have the following equivalent (see [8]), f (z) ≺ g(z) ⇐⇒ f (0) = g(0) and f (U) ⊂ g(U).

It is readily verified from (3) that
Very recently, Rahrovi [12], Attiya and Yassen [3], Seoudy [14], Wanas and Majeed [25] and Srivastava and Wanas [21] have obtained sandwich results for certain classes of analytic functions. Motivated by aforementioned works to investigate sufficient condition for f based on Wanas differential operator we define a new subclasses of normalized analytic functions satisfying the following: and where q 1 and q 2 are given univalent functions in U with q 1 (0) = q 2 (0) = 1.
To establish our main results, we need the following definition and lemmas.

Lemma 1 ([8]
). Let q be univalent in the unit disk U and let θ and φ be analytic in a domain D containing then ξ ≺ q and q is the best dominant of (5).

If ξ is analytic in U and
then ξ ≺ q and q is the best dominant of (6).

Lemma 4 ([5]
). Let q be convex univalent in the unit disk U and let θ and φ be analytic in a domain D containing q (U). Suppose that then q ≺ ξ and q is the best subordinant of (8).

Main Results
Theorem 1. Let q be convex univalent in U with q (0) = 1, σ ∈ C\{0}, γ > 0 and suppose that q satisfies If f ∈ A satisfies the subordination and q is the best dominant of (10).
It is clear that Q (z) is starlike univalent in U, Thus, by Lemma 1, we get ξ (z) ≺ q (z). By using (16), we obtain the desired result.
Theorem 3. Let q be convex univalent in U with q (0) = 1, γ > 0 and and q is the best subordinant of (18).
Concluding the results of differential subordination and superordination, we state the following "sandwich results".