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.

The Dziok-Srivastava linear operator is a particular instance of a linear operator defined by convolution. For , let denote the class of functions

(1.1)

that are analytic and -valent in the open unit disk with . The Hadamard product (or convolution) of two analytic functions

(1.2)

is defined by the series

(1.3)

For complex parameters and , the generalized hypergeometric function is given by

(1.4)

where is the Pochhammer symbol (or the shifted factorial) defined (in terms of the Gamma function) by

(1.5)

To define the Dziok-Srivastava operator

(1.6)

via the Hadamard product given by (1.3), we consider a corresponding function

(1.7)

defined by

(1.8)

The Dziok-Srivastava linear operator is now defined by the Hadamard product

(1.9)

This operator was introduced and studied in a series of recent papers by Dziok and Srivastava ([27–29]; see also [30, 31]). For convenience, we write

(1.10)

The importance of the Dziok-Srivastava operator from the general convolution operator rests on the relation

(1.11)

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]).

Let , and let be univalent in . If is analytic in and satisfies the differential subordination

(2.1)

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]).

Let , and let be analytic in . If and are univalent in and satisfy the differential superordination

(2.2)

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]).

Denote by the class of functions that are analytic and injective on , where

(2.3)

and are such that for .

Lemma 2.4 (cf. [1, Theorem 2.3i, page 35]).

Suppose that the function satisfies the condition

(2.4)

for all real and , where is a positive integer. If the function is analytic in and

(2.5)

then in .

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]).

Let with , and let with . If for , then the solution of the differential equation

(2.6)

with is analytic in and satisfies .

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

Let with , and let be analytic in with and . If is not subordinate to , then there exists points and , for which ,

(2.7)

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]).

Let , , and set . If is a subordination chain and , then

(2.8)

implies that

(2.9)

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

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

The function , with and , is a subordination chain if and only if

(2.10)

3. Main Results

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

Theorem 3.1.

Let . For , , let

(3.1)

Suppose that

(3.2)

where

(3.3)

Then the subordination condition

(3.4)

implies that

(3.5)

Moreover, the function is the best dominant.

Proof.

Let us define the functions and , respectively, by

(3.6)

We first show that if the function is defined by

(3.7)

then

(3.8)

Logarithmic differentiation of both sides of the second equation in (3.6) and using (1.11) for yield

(3.9)

Now, differentiating both sides of (3.9) results in the following relationship:

(3.10)

We also note from (3.2) that

(3.11)

and, by using Lemma 2.5, we conclude that differential equation (3.10) has a solution with . Let us put

(3.12)

where is given by (3.3). From (3.2), (3.10), and (3.12), it follows that

(3.13)

In order to use Lemma 2.4, we now proceed to show that for all real and . Indeed, from (3.12),

(3.14)

where

(3.15)

For given by (3.3), we can prove easily that the expression given by (3.15) is positive or equal to zero. Hence, from (3.14), we see that for all real and . Thus, by using Lemma 2.4, we conclude that for all . That is, defined by (3.6) is convex in . Next, we prove that subordination condition (3.4) implies that

(3.16)

for the functions and defined by (3.6). Without loss of generality, we also can assume that is analytic and univalent on and for . For this purpose, we consider the function given by

(3.17)

Note that

(3.18)

This shows that the function

(3.19)

satisfies the condition for all . Furthermore,

(3.20)

Therefore, by virtue of Lemma 2.8, is a subordination chain. We observe from the definition of a subordination chain that

(3.21)

Now suppose that is not subordinate to ; then, by Lemma 2.6, there exist points and such that

(3.22)

Hence,

(3.23)

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.

Let . For , , let

(3.24)

Suppose that

(3.25)

where is given by (3.3). Further, suppose that

(3.26)

is univalent in and . Then the superordination

(3.27)

implies that

(3.28)

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.

Now let us define the functions and , respectively, by (3.6). We first note that if the function is defined by (3.7), then (3.9) becomes

(3.29)

After a simple calculation, (3.29) yields the relationship

(3.30)

Then by using the same method as in the proof of Theorem 3.1, we can prove that for all . That is, defined by (3.6) is convex (univalent) in . Next, we prove that the subordination condition (3.27) implies that

(3.31)

for the functions and defined by (3.6). Now considering the function defined by

(3.32)

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.

Let . For , , , let

(3.33)

Suppose that

(3.34)

where is given by (3.2). Further, suppose that

(3.35)

is univalent in and . Then

(3.36)

implies that

(3.37)

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

The assumption of Theorem 3.3 that the functions

(3.38)

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

Corollary 3.4.

Let . For , , let

(3.39)

and , be as in (3.33). Suppose that condition (3.34) is satisfied and

(3.40)

where is given by (3.3). Then

(3.41)

implies that

(3.42)

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

Proof.

In order to prove Corollary 3.4, we have to show that condition (3.40) implies the univalence of and

(3.43)

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.

Let . Let

(3.44)

Suppose that

(3.45)

and is univalent in and . Then

(3.46)

implies that

(3.47)

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

Next consider the generalized Libera integral operator defined by (cf. [37, 41–43])

(3.48)

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.

Let . Let

(3.49)

Suppose that

(3.50)

where

(3.51)

If is univalent in and , then

(3.52)

implies that

(3.53)

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

Proof.

Let us define the functions and by

(3.54)

respectively. From the definition of the integral operator given by (3.48), it follows that

(3.55)

Then, from (3.49) and (3.55),

(3.56)

Setting

(3.57)

and differentiating both sides of (3.51) result in

(3.58)

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.

Let and

(3.59)

Suppose that condition (3.50) is satisfied and

(3.60)

where is given by (3.51). Then

(3.61)

implies that

(3.62)

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.

Let . Let

(3.63)

Suppose that

(3.64)

where is given by (3.51), and is univalent in and . Then,

(3.65)

implies that

(3.66)

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