# Subordination and Superordination for Multivalent Functions Associated with the Dziok-Srivastava Operator

- Nak Eun Cho
^{1}, - Oh Sang Kwon
^{2}, - Rosihan M Ali
^{3}Email author and - V Ravichandran
^{3, 4}

**2011**:486595

https://doi.org/10.1155/2011/486595

© Nak Eun Cho et al. 2011

**Received: **21 September 2010

**Accepted: **26 January 2011

**Published: **15 February 2011

## Abstract

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.

## 1. Introduction

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 [3] investigated the dual problem of differential superordination. The monograph by Miller and Mocanu [1] gives a good introduction to the theory of differential subordination, while the book by Bulboacă [4] 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 [24], Miller et al. [25], and Owa and Srivastava [26]. The corresponding superordination properties and sandwich-type results were also investigated, for example, in [4]. 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., [27]). The linear operator includes various other linear operators as special cases. These include the operators introduced and studied by Carlson and Shaffer [32], Hohlov ([33], also see [34, 35]), and Ruscheweyh [36], 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]).

with is analytic in and satisfies .

Lemma 2.6 (see [1, Lemma 2.2d, page 24]).

A function defined on is a subordination chain (or Löwner chain) if is analytic and univalent in for all , is continuously differentiable on for all , and for .

Lemma 2.7 (see [3, Theorem 7, page 822]).

Furthermore, if has a univalent solution , then is the best subordinant.

Lemma 2.8 (see [3, Lemma B, page 822]).

## 3. Main Results

We first prove the following subordination theorem involving the operator defined by (1.10).

Theorem 3.1.

Moreover, the function is the best dominant.

Proof.

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.

Theorem 3.2.

Moreover, the function is the best subordinant.

Proof.

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.

Theorem 3.3.

Moreover, the functions and are the best subordinant and the best dominant, respectively.

need to be univalent in may be replaced by another condition in the following result.

Corollary 3.4.

Moreover, the functions and are the best subordinant and the best dominant, respectively.

Proof.

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 [40]) 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.

By taking , , , , and in Theorem 3.3, we have the following result.

Corollary 3.5.

Moreover, the functions and are the best subordinant and the best dominant, respectively.

For the choice , with , (3.48) reduces to the well-known Bernardi integral operator [41]. The following is a sandwich-type result involving the generalized Libera integral operator .

Theorem 3.6.

Moreover, the functions and are the best subordinant and the best dominant, respectively.

Proof.

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.

Corollary 3.7.

Moreover, the functions and are the best subordinant and the best dominant, respectively.

Taking , , , and in Corollary 3.7, we have the following result.

Corollary 3.8.

Moreover, the functions and are the best subordinant and the best dominant, respectively.

## Declarations

### Acknowledgments

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.

## Authors’ Affiliations

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