Certain Admissible Classes of Multivalent Functions

In such a case we write f(z) ≺ F(z). If F is univalent in U, then f(z) ≺ F(z) if and only if f(0) = F(0) and f(U) ⊂ F(U) (see [1–3]; see also several recent works [4–8] dealing with various properties and applications of the principle of differential subordination and the principle of differential superordination). We denote byF the set of all functions q that are analytic and injective on U \ E(q), where


Introduction
Let H(U) be the class of functions analytic in the open unit disk Denote by H[, ] the subclass of H(U) consisting of functions of the form with Also let A() be the class of all analytic and -valent functions of the form ( ∈ N = {1, 2, 3, . ..} ;  ∈ U) .
Let  and  be members of the function class H(U).The function () is said to be subordinate to (), or the function () is said to be superordinate to (), if there exists a function (), analytic in U with such that  () =  ( ()) .
We further let the subclass of F for which (0) =  be denoted by F() and write In order to prove our results, we will make use of the following classes of admissible functions.
In this paper, we determine the sufficient conditions for certain admissible classes of multivalent functions so that where  > 0 and  1 and  2 are given univalent functions in U with In addition, we derive several differential sandwich-type results.A similar problem for analytic functions involving certain operators was studied by Aghalary et al. [9], Ali et al. [10], Aouf et al. [11], Kim and Srivastava [12], and other authors (see [13][14][15]).In particular, unlike the earlier investigation by Aouf and Seoudy [16], we have not used any operators in our present investigation.Nevertheless, for the benefit of the targeted readers of our paper, in addition to oft-cited paper [11], we have included several further citations of recent works (see, e.g., [17][18][19][20][21]) in which various families of linear operators were applied in conjunction with the principle of differential subordination and the principle of differential superordination for the study of analytic or meromorphic multivalent functions.

A Set of Subordination Results
Unless otherwise mentioned, we assume throughout this paper that  ∈ N,  > 0,  ∈ U, and all power functions are tacitly assumed to denote their principal values. whenever where  ∈ U,  ∈ U \ (), and  ≧ 1.For simplicity, we write Hence (25) becomes The proof is completed if it can be shown that the admissibility condition for  ∈ Φ[Ω, , , ] is equivalent to the admissibility condition for  as given in Definition 1.We note that The asserted result is now deduced from the fact that   () ≺ ().
If  ∈ A() satisfies condition (36), then Proof.The proof of Theorem 10 is similar to the proof of a known result [2, p. 30, Theorem 2.3d] and is, therefore, omitted.
Proof.Following the same arguments in [2, p. 31, Theorem 2.3e], we deduce that  is a dominant from Theorems 7 and 10.
Since  satisfies (45), it is also a solution of (36) and, therefore,  will be dominated by all dominants.Hence  is the best dominant.
In the special case when

Superordination and Sandwich-Type Results
In this section we investigate the dual problem of differential subordination, that is, differential superordination of multivalent functions.For this purpose, the class of admissible functions is given in the following definition.
Definition 16.Let Ω be a set in C and  ∈ H with   () ̸ = 0.The class Φ  [Ω, , , ] of admissible functions consists of those functions  : C 3 × U → C that satisfy the following admissibility condition: whenever where  ∈ U,  ∈ U, and  ≧ 1.For convenience, we write which evidently completes the proof of Theorem 17.
Proceeding similarly as in Section 2, the following result can be derived as an immediate consequence of Theorem

Definition 5 .
Let Ω be a set in C and  ∈ F 1 ∩ H.The class Φ[Ω, , , ] of admissible functions consists of those functions  : C 3 × U → C that satisfy the following admissibility condition: