Subordination and Superordination Properties for Certain Family of Analytic Functions Associated with Mittag–Leffler Function

We obtain new outcomes of analytic functions linked with operator Hα,βη,k(f) defined by Mittag–Leffler function. Moreover, new theorems of differential sandwich-type are obtained.


Basic Definitions and Preliminaries
Let A define the class of analytic functions in the open unit disk U = {z ∈ C : |z| < 1} and let H[a, n] be the subclass of A, which is f (z) = a + a n z n + a n+1 z n+1 + a n+2 z n+2 + a n+3 z n+3 + . . .
(a ∈ C), Furthermore, let H be the subclass of A of all the functions f (z) ∈ H normalized by f (z) = z + ∞ ∑ n=2 a n z n .

Definition 3.
If ψ : C 4 × U −→ C and h(z) be univalent in U. If p(z) is analytic in U, and satisfies the third-order differential subordination then p(z) is called a solution of the differential subordination and q(z) is called a dominant of the solutions of the differential subordination as well as a dominant if p(z) ≺ q(z) for all p(z) satisfying (5). q(z) that satisfies q(z) ≺ q(z) for all dominants of (5) is called the best dominant of (5).
Analogous to the second order differential super-ordinations introduced by Miller and Mocanu [20], Tang et al. [21] defined the differential super-ordinations as follows: Let ψ : C 4 × U −→ C and the function h(z) be analytic in U. If functions p(z) and ψ(p(z), zp (z), z 2 p (z), z 3 p (z)) are univalent in U, and satisfy the following third-order differential super-ordination then p(z) is called a solution of the differential superordination and q(z) is called a subordinant of the solutions of the differential super-ordinations as well as a subordinant if p(z) ≺ q(z) for all p(z) satisfying Equation (6). A univalent subordinant q(z) that satisfies q(z) ≺ q(z) for all super-ordinations of (6) is the best superordinant.

Theorem 2.
Let q(z) ∈ H[a, n] and ψ ∈ Ψ n [Ω, q(z)]. If p(z) ∈ Q(a) and ψ(p(z), zp (z), z 2 p (z), z 3 p (z); z) is univalent in U and Here, we study a certain family of admissible functions by using the third-order differential subordination and superordination given by Antonino and Miller [22] and Tang et al. [21]-see also Attiya et al. [23]-we obtain new results of subordination and superordination properties of analytic functions linked with the operator H η,k α,β ( f ).

Definition 7.
Let Ω ⊆ C and q(z) ∈ Q . The class of admissible functions Ψ Γ [Ω, q(z)] consists of those functions φ : C 4 × U −→ C that satisfy the admissibility condition φ(a 1 , a 2 , a 3 , ]. If f (z) ∈ H and q(z) ∈ Q 1 satisfy: which implies Now, we define the parameters a 1 , a 2 , a 3 , and a 4 as Then, transformation ψ : by using the relations from (9) to (12), we have therefore, we recompute (8) as then, the proof is completed by showing that the admissibility condition for φ ∈ Ψ Γ [Ω, q(z)] is equivalent to the admissibility condition for ψ as given in Definition 3, since we also note that and by Theorem 1, p(z) ≺ q(z).
In a similar way, we define the parameters a 1 , a 2 , a 3 , and a 4 as follows: Let Ω ⊆ C and q(z) ∈ Q. The class of admissible functions Ψ Γ [Ω, q(z)] consists of those functions φ : C 4 × U −→ C that satisfy the admissibility condition ]. If f (z) ∈ H and q(z) ∈ Q 1 satisfy the following conditions: which implies Parameters a 1 , a 2 , a 3 and, a 4 as The transformation ψ : by using the relations from (17) to (20), we have we recompute (16) as This completes the proof by showing that the admissibility condition for φ ∈ Ψ Γ [Ω, q(z)] is equivalent to the admissibility condition for ψ as given in Definition 3, since we also note that and hence by Theorem 1, p(z) ≺ q(z).
If Ω = C is simply connected to the domain, then Ω = h(U) for some conformal mapping h(z) of U onto Ω. In this case, the class Ψ Γ [h(U), q(z)] is written as Ψ Γ [h, q]; the following theorem is a direct consequence of Theorems 3 and 4.
]. If f (z) ∈ H and q(z) ∈ Q 1 satisfy the following conditions: ]. If f (z) ∈ H and q(z) ∈ Q 1 satisfy the following conditions: Re The next corollaries extend Theorems 3 and 4, when the behavior of q(z) on ∂U is not known.
Proof. By using Theorem 3 that q(z) is a dominant of (24). Since q(z) satisfies (26), it is also a solution of (24) and therefore q(z) will be dominated by all dominants. Hence, q(z) is the best dominant.
Moreover, in a similar way, using Theorem 4, we have Theorem 8. Let h(z) be univalent in U. Let φ : C 4 × U −→ C. Suppose that the differential equation has a solution q(z) with q(0) = 1, which satisfies (15). If f (z) ∈ H satisfies (25) and and q(z) is the best dominant of (29).