Certain Admissible Classes of Multivalent Functions

Abstract

We investigate some applications of the differential subordination and the differential superordination of certain admissible classes of multivalent functions in the open unit disk . Several differential sandwich-type results are also obtained.

1. Introduction

Let be the class of functions analytic in the open unit disk Denote by the subclass of consisting of functions of the form with Also let be the class of all analytic and -valent functions of the form Let and be members of the function class . The function is said to be subordinate to , or the function is said to be superordinate to , if there exists a function , analytic in with such that In such a case we write . If is univalent in , then if and only if and (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 by the set of all functions that are analytic and injective on , where and are such that We further let the subclass of for which be denoted by and write

In order to prove our results, we will make use of the following classes of admissible functions.

Definition 1 (see [2, p. 27, Definition 2.3a]). Let be a set in ,  , and . The class of admissible functions consists of those functions that satisfy the following admissibility condition: whenever where , and . We write simply as .
In particular, if then , and . In this case, we set . Moreover, in the special case, when we set , the class is simply denoted by .

Definition 2 (see [3, p. 817, Definition 3]). Let be a set in with . The class of admissible functions consists of those functions that satisfy the following admissibility condition: whenever where , , and . In particular, we write simply as .
In our investigation we need the following lemmas which are proved by Miller and Mocanu (see [2] and [3]).

Lemma 3 (see [2, p. 28, Theorem 2.3b]). Let with . If the analytic function given by satisfies the inclusion relationship then .

Lemma 4 (see [3, p. 818, Theorem 1]). Let with . If and the function is univalent in , then implies that .

In this paper, we determine the sufficient conditions for certain admissible classes of multivalent functions so that where and and are given univalent functions in 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 [13–15]). 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., [17–21]) 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.

2. A Set of Subordination Results

Unless otherwise mentioned, we assume throughout this paper that , , , and all power functions are tacitly assumed to denote their principal values.

Definition 5. Let be a set in and . The class of admissible functions consists of those functions that satisfy the following admissibility condition: whenever where , , and . For simplicity, we write

Theorem 6. Let . If satisfies the condition then

Proof. We begin by defining the analytic function in by Then, in view of (27), we get Further computations show that
We now define the transformations from to by and suppose that The proof will make use of Lemma 3. Indeed, by using (27) to (31), we obtain Hence (25) becomes The proof is completed if it can be shown that the admissibility condition for is equivalent to the admissibility condition for as given in Definition 1. We note that and hence . By Lemma 3, we thus obtain which evidently proves Theorem 6.

If is a simply connected domain, then for some conformal mapping of onto . In this case, the class is written, for convenience, as . The following result is an immediate consequence of Theorem 6.

Theorem 7. Let . If satisfies the condition, then

Putting in Theorem 7, we obtain the following corollary.

Corollary 8. Let . If satisfies the condition then

Our next result is an extension of Theorem 6 to the case where the behavior of on is not known.

Corollary 9. Let and suppose that the function is univalent in with . Also let for some , where . If and then

Proof. Theorem 6 readily yields The asserted result is now deduced from the fact that .

Theorem 10. Let the functions and be univalent in , with , and set
Also let satisfy one of the following conditions:(1) for some or(2)there exists such that for all .
If 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.

The next theorem yields the best dominant of differential subordination (36).

Theorem 11. Let the function be univalent in . Also let . Suppose that the differential equation has a solution with and satisfies one of the following conditions:(1) and ;(2)the function is univalent in and for some ; or(3)the function is univalent in and there exists such that for all . If satisfies (36), then and is the best dominant.

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 particular case when , in view of Definition 5, the class of admissible functions, denoted by , is described below.

Definition 12. Let be a set in and . The class of admissible functions consists of those functions such that whenever , , and for all real and .

Corollary 13. Let . If satisfies the condition then

In the special case when the class is simply denoted by .

Corollary 14. Let . If satisfies the condition then

Corollary 15. If and satisfies the condition then

Proof. Corollary 15 follows from Corollary 14 upon setting

3. 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 be a set in and with . The class of admissible functions consists of those functions that satisfy the following admissibility condition: whenever where , , and . For convenience, we write

Theorem 17. Let . If , is univalent in , then implies that

Proof. From (32) and (61), we find that We also see from (30) that the admissibility condition for the function class is equivalent to the admissibility condition for as given in Definition 2. Hence . Thus, by Lemma 4, we have which evidently completes the proof of Theorem 17.

If is a simply connected domain, then for some conformal mapping of onto . In this case, the class is written simply as .

Proceeding similarly as in Section 2, the following result can be derived as an immediate consequence of Theorem 17.

Theorem 18. Let the function be analytic in and . If , is univalent in , then implies that

Putting in Theorem 18, we obtain the following corollary.

Corollary 19. Let the function be analytic in and . If , is univalent in , then implies that

Theorems 17 and 18 can only be used to obtain subordinants of the differential superordination of the form (61) or (66). The following theorem proves the existence of the best subordinant of (66) for a specified .

Theorem 20. Let the function be analytic in and . Suppose that the differential equation has a solution . If ,, is univalent in , then implies that and is the best subordinant.

Proof. The proof is similar to the proof of Theorem 11. We, therefore, omit the details involved.

Combining Theorems 7 and 18, we obtain the following sandwich-type theorem.

Theorem 21. Let the functions and be analytic in , the function univalent in with If , is univalent in , then implies that

Upon setting in Theorem 21, we get the following result.

Corollary 22. Let the functions and be analytic in , the function univalent in with and . If , is univalent in , then implies that

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.