Research Article | Open Access

# Differential Subordination and Superordination for Srivastava-Attiya Operator

**Academic Editor:**Khalil Ezzinbi

#### Abstract

Due to the well-known Srivastava-Attiya operator, we investigate here some results relating the *p*-valent of the operator with differential subordination and subordination. Further, we obtain some interesting results on sandwich-type theorem for the same.

#### 1. Introduction and Motivation

Let be the class of analytic functions in the open unit disc and let be the subclass of consisting functions of the form , with and . For two functions and analytic in , the function is subordinate to , or superordinate to , written as if there exists a function , analytic in with and such that . In particular, if the function is univalent in , then is equivalent to and .

Let and . If and are univalent and satisfies the second-order differential subordination then is called a solution of the differential subordination. The univalent function is called a dominant if for all satisfying (1.1). Miller and Mocanu discussed many interesting results containing the above mentioned subordination and also many applications of the field of differential subordination in [1]. In that direction, many differential subordination and differential superordination problems for analytic functions defined by means of linear operators were investigated. See [2β11] for related results.

Let denote the class of functions of the form
which are analytic and *p*-valent in . For satisfying (1.2), let the generalized Srivastava-Attiya operator [12] be denoted by
where
with
and the symbol (*) denotes the usual Hadamard product (or convolution). From the equations, we can see that
Note that for in (1.6), coincides with the Srivastava-Attiya operator [13]. Further, observe that for proper choices of and , the operator coincides with the following: (i),
(ii) [14],(iii) [15, 16],(iv) [17],(v) [18].

Since the above mentioned operator, the generalized Srivastava-Attiya operator, reduces to the well-known operators introduced and studied in the literature by suitably specializing the values of and and also in view of the several interesting properties and characteristics of well-known differential subordination results, we aim to associate these two motivating findings and obtain certain other related results. Further, we consider the differential superordination problems associated with the same operator. In addition, we also obtain interesting sandwich-type theorems.

The following definitions and theorems were discussed and will be needed to prove our results.

*Definition 1.1 (see [1], Definition 2.2b, page 21). *Denote by the set of all functions that are analytic and injective on where
and are such that for . Further let the subclass of for which be denoted by , , and .

*Definition 1.2 (see [1], Definition 2.3a, page 27). *Let be a set in , and let be a positive integer. The class of admissible functions consists of those functions that satisfy the admissibility condition whenever *,* and
, and . Let .

*Definition 1.3 (see [19], Definition 3, page 817). *Let be a set in *, * with *.* The class of admissible functions consists of those functions that satisfy the admissibility condition *βwhenever **,* and
, and . Let .

Theorem 1.4 (see [1], Theorem 2.3b, page 28). * Let with . If the analytic function satisfies
**
then .*

Theorem 1.5 (see [19], Theorem 1, page 818). * Let with . If and is univalent in , then
**
implies .*

#### 2. Subordination Results Associated with Generalized Srivastava-Attiya Operator

*Definition 2.1. *Let be a set in and . The class of admissible functions consists of those functions that satisfy the admissibility condition:
whenever
, and .

Theorem 2.2. *Let . If satisfies
**
then
*

* Proof. *The following relation obtained in [13]
is equivalent to
and hence
Define the analytic function in by
and then we get
Further, let us define the transformations from to by
Let
The proof will make use of Theorem 1.4. Using (2.8) and (2.9), from (2.12) we obtain
Hence (2.3) becomes
Note that
and since the admissibility condition for is equivalent to the admissibility condition for as given in Definition 1.2, hence , and by Theorem 1.4,
or

In the case , we have the following example.

*Example 2.3. *Let the class of admissible functions consist of those functions that satisfy the admissibility condition:
, and and . If satisfies
then

If is a simply connected domain, then for some conformal mapping of onto and the class is written as . The following result follows immediately from Theorem 2.2.

Theorem 2.4. *Let . If satisfies
**
then
*

The next result occurs when the behavior of on is not known.

Corollary 2.5. *Let , be univalent in and . Let for some where . If and
**
then
*

* Proof. *From Theorem 2.2, we see that and the proof is complete.

Theorem 2.6. *Let and be univalent in , with and set and . Let satisfy one of the following conditions: *(1)*, for some , or*(2)*there exists such that , for all . ** If satisfies (2.21), then
*

* Proof. *The proof is similar to the one in [1] and therefore is omitted.

The next results give the best dominant of the differential subordination (2.21).

Theorem 2.7. *Let be univalent in . Let . Suppose that the differential equation
**
has a solution with and satisfy one of the following conditions: *(1)* and ,*(2)* is univalent in and , for some , or*(3)* is univalent in and there exists such that , for all . **If satisfies (2.21), then
**
and is the best dominant.*

* Proof. *Following the same arguments in [1], we deduce that is a dominant from Theorem 2.4 and Theorem 2.6. Since satisfies (2.26), it is also a solution of (2.21) and therefore will be dominated by all dominants. Hence is the best dominant.

*Definition 2.8. *Let be a set in and . The class of admissible functions consists of those functions that satisfy the admissibility condition:
whenever
, and .

Theorem 2.9. *Let . If satisfies
**
then
*

* Proof. *Define the analytic function in by
Using the relations (2.5) and (2.32), we get
Further, let us define the transformations from to by
Let
The proof will make use of Theorem 1.4. Using (2.32) and (2.33), from (2.35) we obtain
Hence (2.30) becomes
Note that
and since the admissibility condition for is equivalent to the admissibility condition for as given in Definition 1.2, hence , and by Theorem 1.4,
or

In the case , we have the following example.

*Example 2.10. *Let the class of admissible functions consist of those functions that satisfy the admissibility condition:
, and and . If satisfies
then

If is a simply connected domain, then for some conformal mapping of onto and the class is written as . The following result follows immediately from Theorem 2.9.

Theorem 2.11. *Let . If satisfies
**
then
*

*Definition 2.12. *Let be a set in and . The class of admissible functions consists of those functions that satisfy the admissibility condition:
whenever
, and .

Theorem 2.13. *Let . If satisfies
**
then
*

* Proof. *Define the analytic function in by
Differentiating (2.50) yields
From the relation (2.5) we get
and hence
Further computations show that
Let us define the transformations from to by
Let
The proof will make use of Theorem 1.4. Using (2.50), (2.53) and (2.54), from (2.56) we obtain
Hence (2.48) becomes
Note that
and since the admissibility condition for is equivalent to the admissibility condition for as given in Definition 1.2, hence and by Theorem 1.4,
or

If is a simply connected domain, then for some conformal mapping of onto and the class is written as . The following result follows immediately from Theorem 2.13.

Theorem 2.14. *Let . If satisfies
**
then
*

#### 3. Superordination Results Associated with Generalized Srivastava-Attiya Operator

*Definition 3.1. *Let be a set in and with . The class of admissible functions consists of those functions that satisfy the admissibility condition:
whenever
, and .

Theorem 3.2. *Let . If , and
**
is univalent in , then
**
implies that
*

* Proof. *From (2.13) and (3.4), we have
From (2.10), we see that the admissibility condition for is equivalent to the admissibility condition for as given in Definition 1.3. Hence , and by Theorem 1.5, or

If is a simply connected domain, then for some conformal mapping of onto and the class is written as . The next result follows immediately from Theorem 3.2.

Theorem 3.3. *Let be analytic in and . If , and
**
is univalent in , then
**
and then
*

Theorems 3.2 and 3.3 can only be used to obtain subordinants for differential superordination of the form (3.4) and (3.9). The following theorems prove the existence of the best subordinant of (3.9) for certain .

Theorem 3.4. *Let be analytic in and . Suppose that the differential equation
**
has a solution . If , , , and
**
is univalent in , then
**
implies that
**
and is the best subordinant.*

* Proof. *The result can be obtained by similar proof of Theorem 2.7.

The next result, the sandwich-type theorem follows from Theorems 2.4 and 3.3.

Corollary 3.5. *Let ββand ββbe analytic in , and let ββbe univalent function in , ββwith and . If , , and
**
is univalent in , then
**
implies that
*

*Definition 3.6. *Let be a set in and with *.* The class of admissible functions consists of those functions that satisfy the admissibility condition:
whenever
, and .

The following result is associated with Theorem 2.9.

Theorem 3.7. *Let . If , , and
**
is univalent in , then
**
implies that
*

* Proof. *From (2.36) and (3.21), we have
From (2.34), we see that the admissibility condition for is equivalent to the admissibility condition for as in Definition 1.3. Hence , and by Theorem 1.5, or

If is a simply connected domain, then for some conformal mapping of onto and the class is written as . The next result follows immediately from Theorem 3.7.

Theorem 3.8. *Let , and let be analytic on , and let . If , , and
**
is univalent in , then
**
implies that
*

Combining Theorems 2.11 and 3.8, we obtain the following sandwich-type theorem.

Corollary 3.9. *Let and be analytic in , let be univalent function in , with , and . If , , and
**
is univalent in , then
**
implies that
*

*Definition 3.10. *Let be a set in and , and . The class of admissible functions consists of those functions that satisfy the admissibility condition:
whenever
, and .

The following result is associated with Theorem 2.13.

Theorem 3.11. *Let . If , and
**
is univalent in , then
**
implies that
*

* Proof. *From (2.57) and (3.34), we have
From (2.55), we see that the admissibility condition for is equivalent to the admissibility condition for as in Definition 1.3. Hence , and by Theorem 1.5, or

If is a simply connected domain, then for some conformal mapping of onto and the class is written as . The next result follows immediately from Theorem 3.11 as in the previous section.

Theorem 3.12. *Let , let be analytic in , and let . If and
**
is univalent in , then
**
implies that
*

Combining Theorems 2.14 and 3.12, we obtain the following sandwich-type theorem.

Corollary 3.13. *Let ββand be analytic in , let be univalent function in , with , and . If , and
**
is univalent in , then
**
implies that
*

Other work related to certain operators concerning the subordination and superordination can be found in [20β25].

#### Acknowledgment

The work presented here was partially supported by UKM-ST-FRGS-0244-2010.

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Copyright Β© 2011 Maisarah Haji Mohd and Maslina Darus. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.