Differential Subordination and Superordination Results Associated with Mittag–Lefﬂer Function

: In this paper, we derive a number of interesting results concerning subordination and superordination relations for certain analytic functions associated with an extension of the Mittag– Lefﬂer function.

Let λ and h be two analytic functions in U, suppose Φ(r, s, t; z) : C 3 × U → C.
Due to the importance of Mittag-Leffler function, it is involved in many problems in natural and applied science.
Attiya [10] noted that From (6) follows (see [10]) and In order to derive our results, we will use the following known definitions and lemmas.
Definition 1. Ref [4]. Denote by µ the set of all functions f that are analytic and injective on with f (ζ) = 0 for ζ ∈ ∂U \ E( f ).
In this paper we drive a number of interesting results concerning subordination and superordination relations for the operator H γ,k α,β ( f )(z). Also, some of interesting sandwich results of the operator H γ,k α,β ( f )(z) have been obtained.

Subordination and Superordination Results with
If f ∈ A satisfies the following subordination relation and µ(z) is the best dominant of (14).
Proof. Define the function λ by The function λ is analytic in U and λ(0) = 1. Differentiating the function λ with respect to z logarithmically, we have In the resulting equation by using the identity (7), we have Therefore, It follows from (14) that Thus, an application of Lemma 3 with ψ = 1 and κ = ρ δ , we obtain (15). In view of (8), and by using the similar method of proof the Theorem 1, we get the proof of Theorem 2.
and µ(z) is the best dominant of (18).
Suppose that Also, if f ∈ A satisfies the following subordination relation: where and µ(z) is the best dominant of (20).
and µ(z) is the best dominant of (20).
and µ 1 and µ 2 are respectively the best subordinate and best dominant.

Remark 2.
By specifying the function Ω and selecting the particular values of α, β, γ and k we can derive a number of known results. Some of them are given below.

Conclusions
We obtained a number of interesting results concerning subordination and superordination relations for the operator H