APPLICATIONS OF FRACTIONAL DERIVATIVE ON A DIFFERENTIAL SUBORDINATIONS AND SUPERORDINATORS FOR ANALYTIC FUNCTIONS ASSOCIATED WITH DIFFERENTIAL OPERATOR

The purpose of this paper is to derive subordination and superordination results involving fractional derivative of differential operator for analytic functions in the open unit disk. These results are applied to obtain sandwich results. Our results extend corresponding previously known results.


Introduction and Preliminaries
Let H = H(U ) denote the class of analytic functions in the open unit disk U = {z ∈ C : |z| < 1}. For a ∈ C and n ∈ N, let 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.
Also, let A be the subclass of H consisting of functions of the form: (1. 1) f (z) = z + ψ(r, s, t; z) : C 3 × U → C. If p and ψ(p(z), zp (z), z 2 p (z); z) are univalent functions in U and if p satisfies the second-order differential superordination (1.2) h(z) ≺ ψ(p(z), zp (z), z 2 p (z); z), then p is called a solution of the differential superordination (1.2). An analytic function q is called a subordinate of (1.2), if q ≺ p for all p satisfying (1.2). An univalent subordinatq that satisfies q ≺q for all the subordinants q of (1.2) is called the best subordinant. Miller and Mocanu [6] obtained conditions on the functions h, q and ψ for which the following implication holds: Ali et al. [1] have used the results of Bulboacǎ [3] to obtain sufficient conditions for certain normalized analytic functions to satisfy where q 1 and q 2 are given univalent functions in U with q 1 (0) = q 2 (0) = 1. Also, Tuneski [16] obtain sufficient condition for starlikeness of f ∈ A in terms of the quantity f (z)f (z) (f (z)) 2 . Shanmugam et al. [14], Goyal et al. [4], Wanas [17,18] and Attiya and Yassen [2] have obtained sandwich results for certain classes of analytic functions. Definition 1.1 ([9]). For f ∈ A the operator I n,m λ 1 ,λ 2 , ,d is defined by I n,m λ 1 ,λ 2 , ,d : and R n f (z) denotes the Ruscheweyh derivative operator [10] given by If f given by (1.1), then we easily find that 15]). The fractional derivative of order δ, 0 ≤ δ < 1, of a function f is defined by where the function f is analytic in a simply-connected region of the z-plane containing the origin and the multiplicity of (z − t) −δ is removed by requiring log(z − t) to be real, when Re(z − t) > 0.
From Definition 1.1 and Definition 1.2, we have It follows from (1.3) that . In order to prove our results, we make use of the following known results.   ([5]). Let q 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)φ(q(z)) and h(z) = θ(q(z)) + Q(z). Suppose that If p is analytic in U , with p(0) = q(0), p(U ) ⊂ D and then p ≺ q and q is the best dominant of (1.5).

Lemma 1.2 ([6]
). Let q be a convex univalent function in U and let α ∈ C, β ∈ C \{0}, with If p is analytic in U and then p ≺ q and q is the best dominant of (1.6).

Lemma 1.3 ([6]). Let q be a convex univalent function in U and let
which implies that q ≺ p and q is the best subordinant of (1.7).

Lemma 1.4 ([3]).
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 ≺ p and q is the best subordinant of (1.8).

Subordination Results
If f ∈ A satisfies the subordination and q is the best dominant of (2.2).

It is clear that Q(z) is starlike univalent in
Thus, by Lemma 1.1, we get p(z) ≺ q(z). By using (2.8), we obtain the desired result.
and q is the best subordinant of (3.1).
Proof. Define the function p by Then the function p is analytic in U and p(0) = 1. Differentiating (3.3) logarithmically with respect to z, we get After some computations and using (1.4), we find that From (3.1) and (3.4), we have Thus, an application of Lemma 1.3, with α = 1 and β = σ (1−δ)γ , we obtain the results.