Research Article | Open Access

# Third-Order Differential Subordination and Superordination Results for Meromorphically Multivalent Functions Associated with the Liu-Srivastava Operator

**Academic Editor:**Om P. Ahuja

#### Abstract

There are many articles in the literature dealing with the first-order and the second-order differential subordination and superordination problems for analytic functions in the unit disk, but only a few articles are dealing with the above problems in the third-order case (see, e.g., Antonino and Miller (2011) and Ponnusamy et al. (1992)). The concept of the third-order differential subordination in the unit disk was introduced by Antonino and Miller in (2011). Let Ω be a set in the complex plane . Also let be analytic in the unit disk and suppose that . In this paper, we investigate the problem of determining properties of functions that satisfy the following third-order differential superordination: . As applications, we derive some third-order differential subordination and superordination results for meromorphically multivalent functions, which are defined by a family of convolution operators involving the Liu-Srivastava operator. The results are obtained by considering suitable classes of admissible functions.

#### 1. Introduction, Definitions, and Preliminaries

Let be the class of functions which are analytic in the open unit disk: For and , let and suppose that .

Let and be members of the analytic function class . The function is said to be subordinate to , or is superordinate to , if there exists a Schwarz function , analytic in with such that In such a case, we write Furthermore, if the function is univalent in , then we have the following equivalence (see, for details, [1]):

Let denote the class of functions of the form which are analytic and multivalent in the punctured unit disk:

For the function given by (7) and the function given by the Hadamard product (or convolution) of the functions and is defined by

For parameters and , the generalized hypergeometric function is defined by (see, for example, [2, 3]) where denotes the Pochhammer symbol defined, in terms of Gamma function, by

Recently, Tang et al. [4] introduced a function defined by

In particular, when , we obtain which was introduced and studied by Liu and Srivastava [5].

Corresponding to the function given by (13), we consider a convolution operator defined by the following Hadamard product (or convolution): For the sake of convenience, we write

It is easily verified from definition (16) that

We note that, for , the operator reduces to the Liu-Srivastava operator (see [5, 6]; see also [7]), while the Liu-Srivastava operator is the meromorphic analogous of the Dziok-Srivastava operator (see [8–10]; see also [11, 12]), which includes (as its special cases) the meromorphic analogous of the Carlson-Shaffer convolution operator (see [13, 14]), the meromorphic analogous of the Ruscheweyh derivative operator (see [15]), and the operator studied by Uralegaddi and Somanatha [16].

Let be any set in . Also let be analytic in and suppose that . Recently, Antonino and Miller [17] have extended the theory of second-order differential subordinations in introduced by Miller and Mocanu [1] to the third-order case. They determined properties of functions that satisfy the following third-order differential subordination:

We will now recall some definitions and a theorem due to Antonino and Miller [17], which are required in our next investigations.

*Definition 1 (see [17], p. 440, Definition 1). *Let and be univalent in . If is analytic in and satisfies the following third-order differential subordination:
then is called a solution of the differential subordination. A univalent function is called a dominant of the solutions of the differential subordination or, more simply, a dominant if for all satisfying (22). A dominant that satisfies for all dominants of (22) is said to be the best dominant.

*Definition 2 (see [17], p. 441, Definition 2). *Let denote the set of functions that are analytic and univalent on the set , where
is such that
for . Further, let the subclass of for which be denoted by and

*Definition 3 (see [17], p. 449, Definition 3). *Let be a set in , , and . The class of admissible functions consists of those functions that satisfy the following admissibility condition:
whenever
where , , and .

Theorem 4 (see [17], p. 449, Theorem 1). *Let with . Also let and satisfy the following conditions**
where , , and . If is a set in , and
**
then
*

In this paper, following the theory of second-order differential superordinations in the unit disk introduced by Miller and Mocanu [18], we consider the dual problem of determining properties of functions that satisfy the following third-order differential superordination: In other words, we determine the conditions on , , and for which the following implication holds true: where is any set in .

If either or is a simply connected domain, then (32) can be rephrased in terms of superordination. If is univalent in , and if is a simply connected domain with , then there is a conformal mapping of onto such that . In this case, (32) can be rewritten as follows: If is also a simply connected domain with , then there is a conformal mapping of onto such that . In addition, if the function is univalent in , then (33) can be rewritten as There are three key ingredients in the implication relationship (33): the differential operator , the set , and the “dominating” function . If two of these entities were given, one would hope to find conditions on the third entity so that (33) would be satisfied. In this paper, we start with a given set and a given function , and we then determine a set of “admissible” operators so that (33) holds true.

We first introduce the following definition.

*Definition 5. *Let and the function be analytic in . If the functions and
are univalent in and satisfy the following third-order differential superordination:
then is called a solution of the differential superordination. An analytic function is called a subordinant of the solutions of the differential superordination or more simply a subordinant if for satisfying (37). A univalent subordinant that satisfies the condition
for all subordinants of (37) is said to be the best subordinant. We note that the best subordinant is unique up to a rotation of .

For a set in , with and as given in Definition 5, we suppose that (37) is replaced by
Although this more general situation is a “differential containment,” yet we also refer to it as a differential superordination, and the definitions of solution, subordinant, and best subordinant as given above can be extended to this more general case.

We will use the following lemma [[17], p. 445, Lemma D] from the theory of third-order differential subordinations in to determine subordinants of the third-order differential superordinations.

Lemma 6 (see [17]). *Let , and let be analytic in with and . If is not subordinate to , then there exists points and , and an for which ,*(i)*,
*(ii) * and ,*(iii) *,*(iv) *,
*(v) *.*

#### 2. Admissible Functions and a Fundamental Result

We next define the class of admissible functions referred to in the preceding section.

*Definition 7. *Let be a set in , and . The class of admissible functions consists of those functions that satisfy the following admissibility condition:
whenever
where , , and .

If and , then the admissibility condition (41) reduces to the following form:

If and with , then the admissibility condition (41) reduces to the following form:
whenever , , and

The next theorem is a foundation result in the theory of the third-order differential superordinations in .

Theorem 8. *Let and . If
**
is univalent in and satisfy the following conditions:
**
then
**
implies that
*

*Proof. *Suppose that
Then, by the above lemma, there exists points and , and an that satisfy conditions (i)–(v) of the above lemma. Using these conditions with , , , , and in Definition 7, we obtain
which contradicts (48), so we have

In the special case when is a simply connected domain and is a conformal mapping of onto , we denote this class by . The following result is an immediate consequence of Theorem 8.

Theorem 9. *Let . Also let the function be analytic in and suppose that . If satisfies condition (47) and
**
is univalent in , then
**
implies that
*

Theorems 8 and 9 can only be used to obtain subordinants of the third-order differential superordination of the forms (48) or (54).

Theorem 10. *Let the function be analytic in and let . Suppose that the differential equation
**
has a solution . If , , and
**
is univalent in , then (54) implies that
**
and is the best subordinant.*

* Proof. *Since , by applying Theorem 9, we deduce that is a subordinant of (54). Since satisfies (56), it is also a solution of the differential superordination (54). Therefore, all subordinants of (54) will be subordinate to . It follows that will be the best subordinant of (54).

In the next two sections, by making use of the third-order differential subordination results of Antonino and Miller [17] in the unit disk and the third-order differential superordination results in obtained in Section 2 (see, for details, Theorems 8, 9, and 10), we determine certain appropriate classes of admissible functions and investigate some third-order differential subordination and differential superordination properties of meromorphically multivalent functions associated with the operator defined by (16). It should be remarked in passing that, in recent years, several authors obtained many interesting results involving various linear and nonlinear convolution operators associated with (second-order) differential subordination and superordination, and the interested reader may refer to several earlier works including (for example) [19] to [20–23].

#### 3. Third-Order Differential Subordination of the Operator

We first define the following class of admissible functions, which are required in proving the differential subordination theorem involving the operator defined by (16).

*Definition 11. *Let be a set in and . The class of admissible functions consists of those functions that satisfy the following admissibility condition:
whenever
where , , , and .

Theorem 12. *Let . If the functions and satisfy the following conditions:
**
then
*

*Proof. *Define the analytic function in by
Then, differentiating (64) with respect to and using (18), we have
Further computations show that
We now define the transformation from to by
Let
The proof will make use of Theorem 4. Using (64) to (67), we find from (70) that
Hence, clearly, (62) becomes
We note that
Thus, the admissibility condition for in Definition 11 is equivalent to the admissibility condition for as given in Definition 3 with . Therefore, by using (61) and Theorem 4, we have
or, equivalently,
which evidently completes the proof of Theorem 12.

Our next result is an extension of Theorem 12 to the case where the behavior of on is not known.

Corollary 13. *Let and let the function be univalent in with . Suppose also that for some , where . If the functions and satisfy the following conditions:
**
then
*

*Proof. *We note from Theorem 12 that
The result asserted by Corollary 13 is now deduced from the following subordination property:

If is a simply connected domain, then for some conformal mapping of onto . In this case, the class is written as . The following two results are immediate consequences of Theorem 12 and Corollary 13.

Theorem 14. *Let . If the functions and satisfy the following conditions:
**
then
*

Corollary 15. *Let and let the function be univalent in with . Suppose also that for some , where . If the functions and satisfy the following conditions:
**
then
*

Our next theorem yields the best dominant of the differential subordination (70).

Theorem 16. *Let the function be univalent in . Also let and be given by (70). Suppose that the differential equation
**
has a solution with , which satisfies condition (61). If the function satisfies condition (81) and the function
**
is analytic in , then
**
and is the best dominant.*

* Proof. *By applying Theorem 12, we deduce that is a dominant of (81). Since satisfies (85), it is also a solution of (81). Therefore, will be dominated by all dominants. Hence is the best dominant.

In view of Definition 11, in the particular case when , the class of admissible functions, denoted simply by , is described below.

*Definition 17. *Let be a set in , , and . The class of admissible functions consists of those functions such that
whenever , and for all and .

Corollary 18. *Let . If the function satisfies the following conditions:
**
then
*

In the special case when the class is denoted, for brevity, by . Corollary 18 can now be rewritten in the following form.

Corollary 19. *Let . If the function satisfies the following conditions:
**
then
*

Corollary 20. *Let with and . If the function satisfies the following conditions:
**
then
*

*Proof. *Corollary 20 follows from Corollary 19 by setting

Corollary 21. *Let , , and . If the function satisfies the following conditions**
then
*

*Proof. *Let
where
In order to use Corollary 18, we need to show that ; that is, the admissibility condition (88) is satisfied. This follows easily, since