Certain Admissible Classes of Multivalent Functions

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