Subordination and Superordination for Multivalent Functions Associated with the Dziok-Srivastava Operator
© Nak Eun Cho et al. 2011
Received: 21 September 2010
Accepted: 26 January 2011
Published: 15 February 2011
Subordination and superordination preserving properties for multivalent functions in the open unit disk associated with the Dziok-Srivastava operator are derived. Sandwich-type theorems for these multivalent functions are also obtained.
Let be the open unit disk in the complex plane , and let denote the class of analytic functions defined in For and , let consist of functions of the form . Let and be members of . The function is said to be subordinate to , or is said to be superordinate to , if there exists a function analytic in , with and such that . In such a case, we write or . If the function is univalent in , then if and only if and (cf. [1, 2]). Let , and let be univalent in . The subordination is called a first-order differential subordination. It is of interest to determine conditions under which arises for a prescribed univalent function . The theory of differential subordination in is a generalization of a differential inequality in , and this theory of differential subordination was initiated by the works of Miller, Mocanu, and Reade in 1981. Recently, Miller and Mocanu  investigated the dual problem of differential superordination. The monograph by Miller and Mocanu  gives a good introduction to the theory of differential subordination, while the book by Bulboacă  investigates both subordination and superordination. Related results on superordination can be found in [5–23].
By using the theory of differential subordination, various subordination preserving properties for certain integral operators were obtained by Bulboacă , Miller et al. , and Owa and Srivastava . The corresponding superordination properties and sandwich-type results were also investigated, for example, in . In the present paper, we investigate subordination and superordination preserving properties of functions defined through the use of the Dziok-Srivastava linear operator (see (1.9) and (1.10)), and also obtain corresponding sandwich-type theorems.
that can be verified by direct calculations (see, e.g., ). The linear operator includes various other linear operators as special cases. These include the operators introduced and studied by Carlson and Shaffer , Hohlov (, also see [34, 35]), and Ruscheweyh , as well as works in [27, 37].
2. Definitions and Lemmas
Recall that a domain is convex if the line segment joining any two points in lies entirely in , while the domain is starlike with respect to a point if the line segment joining any point in to lies inside . An analytic function is convex or starlike if is, respectively, convex or starlike with respect to 0. For , analytically, these functions are described by the conditions or , respectively. More generally, for , the classes of convex functions of order and starlike functions of order are, respectively, defined by or . A function is close-to-convex if there is a convex function (not necessarily normalized) such that . Close-to-convex functions are known to be univalent.
The following definitions and lemmas will also be required in our present investigation.
Definition 2.1 (see [1, page 16]).
then is called a solution of differential subordination (2.1). A univalent function is called a dominant of the solutions of differential subordination (2.1), or more simply a dominant, if for all satisfying (2.1). A dominant that satisfies for all dominants of (2.1) is said to be the best dominant of (2.1).
Definition 2.2 (see [3, Definition 1, pages 816-817]).
then is called a solution of differential superordination (2.2). An analytic function is called a subordinant of the solutions of differential superordination (2.2), or more simply a subordinant, if for all satisfying (2.2). A univalent subordinant that satisfies for all subordinants of (2.2) is said to be the best subordinant of (2.2).
Definition 2.3 (see [1, Definition 2.2b, page 21]).
Lemma 2.4 (cf. [1, Theorem 2.3i, page 35]).
One of the points of importance of Lemma 2.4 was its use in showing that every convex function is starlike of order 1/2 (see e.g., [38, Theorem 2.6a, page 57]). In this paper, we take an opportunity to use the technique in the proof of Theorem 3.1.
Lemma 2.5 (see [39, Theorem 1, page 300]).
Lemma 2.6 (see [1, Lemma 2.2d, page 24]).
Lemma 2.7 (see [3, Theorem 7, page 822]).
Lemma 2.8 (see [3, Lemma B, page 822]).
3. Main Results
by virtue of subordination condition (3.4). This contradicts the above observation that . Therefore, subordination condition (3.4) must imply the subordination given by (3.16). Considering , we see that the function is the best dominant. This evidently completes the proof of Theorem 3.1.
We next prove a dual result to Theorem 3.1, in the sense that subordinations are replaced by superordinations.
The first part of the proof is similar to that of Theorem 3.1 and so we will use the same notation as in the proof of Theorem 3.1.
we can prove easily that is a subordination chain as in the proof of Theorem 3.1. Therefore according to Lemma 2.7, we conclude that superordination condition (3.27) must imply the superordination given by (3.31). Furthermore, since the differential equation (3.29) has the univalent solution , it is the best subordinant of the given differential superordination. This completes the proof of Theorem 3.2.
Combining Theorems 3.1 and 3.2, we obtain the following sandwich-type theorem.
Since given by (3.3) in Theorem 3.1 satisfies the inequality , condition (3.40) means that is a close-to-convex function in (see ) and hence is univalent in . Furthermore, by using the same techniques as in the proof of Theorem 3.1, we can prove the convexity (univalence) of and so the details may be omitted. Therefore, from Theorem 3.3, we obtain Corollary 3.4.
For the choice , with , (3.48) reduces to the well-known Bernardi integral operator . The following is a sandwich-type result involving the generalized Libera integral operator .
The remaining part of the proof is similar to that of Theorem 3.3 (a combined proof of Theorems 3.1 and 3.2) and is therefore omitted.
By using the same methods as in the proof of Corollary 3.4, the following result is obtained.
This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (no. 2010-0017111) and grants from Universiti Sains Malaysia and University of Delhi. The authors are thankful to the referees for their useful comments.
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