## Nonlinear Problems: Analytical and Computational Approach with Applications

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# Sandwich-Type Theorems for a Class of Multiplier Transformations Associated with the Noor Integral Operators

**Academic Editor:**Muhammad Aslam Noor

#### Abstract

We obtain some subordination- and superordination-preserving properties for a class of multiplier transformations associated with Noor integral operators defined on the space of normalized analytic functions in the open unit disk. The sandwich-type theorems for these transformations are also considered.

#### 1. Introduction

Let denote the class of analytic functions in the open unit disk . For and nonnegative integer , let We also denote by the subclass of with the usual normalization .

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 , and such that . In such a case, we write or . If the function is univalent in , then we have if and only if and (cf. [1]).

*Definition 1.1 (see [1]). *Let and let be univalent in . If is analytic in and satisfies the differential subordination:
then is called a solution of the differential subordination. The univalent function is called a dominant of the solutions of the differential subordination, or more simply a dominant if for all satisfying (1.2). A dominant that satisfies for all dominants of (1.2) is said to be the best dominant.

*Definition 1.2 (see [2]). *Let and let be analytic in . If and are univalent in and satisfy the 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 all satisfying (1.3). A univalent subordinant that satisfies for all subordinants of (1.3) is said to be the best subordinant.

*Definition 1.3 (see [2]). *We denote by the class of functions that are analytic and injective on , where
and are such that for .

Following Komatu [3], we introduce the integral operator defined by
where the symbol stands the Gamma function. We also note that the operator defined by (1.5) can be expressed by the series expansion as follows:
Obviously, we have, for ,
In particular, the operator is closely related to the multiplier transformation studied earlier by Flett [4]. Various interesting properties of the operator have been studied by Jung et al. [5] and Liu [6]. We also note from (1.6) that we can define the operator for any real number .

Let
and let be defined such that
where the symbol stands for the Hadamard product (or convolution). Then, motivated essentially by the Noor integral operator [7] (see also [8–11]), we now introduce the operator , which are defined here by
In view of (1.9) and (1.10), we obtain the following relations:

Making use of the principle of subordination between analytic functions, Miller et al. [12] investigated some subordination theorems involving certain integral operators for analytic functions in (see, also [13]). Moreover, Miller and Mocanu [2] considered differential superordinations, as the dual concept of differential subordinations (see also [14]). In the present paper, we obtain the subordination- and superordination-preserving properties of the multiplier transformations defined by (1.10) with the sandwich-type theorems.

The following lemmas will be required in our present investigation.

Lemma 1.4 (see [15]). *Suppose that the function satisfies the condition:
**
for all real and , where is a positive integer. If the function is analytic in and
**
then in .*

Lemma 1.5 (see [16]). *Let with and let with . If , then the solution of the differential equation:
**
is analytic in and satisfies .*

Lemma 1.6 (see [1]). *Let with and let be analytic in with and . If is not subordinate to , then there exist points and , for which ,
*

Lemma 1.7 (see [2]). *Let , let , and set . If is a subordination chain and , then
**
implies that
**
Furthermore, if has a univalent solution , then is the best subordinant.*

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

Lemma 1.8 (see [17]). *The function with and . Suppose that ia analytic in for all , is continuously differentiable on for all . If satisfies
**
for some positive constants and and
**
then is a subordination chain.*

#### 2. Main Results

Firstly, we begin by proving the following subordination theorem involving the multiplier transformation defined by (1.10).

Theorem 2.1. *Let . Suppose that
**
where
**
Then the subordination:
**
implies that
**
Moreover, the function is the best dominant.*

*Proof. *Let us define the functions and , respectively, by

We first show that if the function is defined by
then
Taking the logarithmic differentiation on both sides of the second equation in (2.5) and using (1.11) for , we obtain
which, in conjunction with (2.8), yields the relationship:
From (2.1), we have
and by using Lemma 1.5, we conclude that the differential equation (2.9) has a solution with .

Let us put
where is given by (2.2). From (2.1), (2.9), and (2.11), we obtain
Now we proceed to show that for all real and . From (2.11), we have
where
For given by (2.2), we can prove easily that the expression given by (2.14) is positive or equal to zero. Moreover, the quadratic expression by in (2.14) is a perfect square for the assumed value of . Hence from (2.13), we see that for all real and . Thus, by using Lemma 1.4, we conclude that for all . That is, is convex in .

Next, we prove that the subordination condition (2.3) implies that
for the functions and defined by (2.5). Without loss of generality, we can assume that is analytic and univalent on and that . Now we consider the function given by
We note that
This shows that the function
satisfies the condition for all . By using the well-known growth and distortion theorems for convex functions, it is easy to check that the first part of Lemma 1.8 is satisfied. Furthermore, we have
since is convex and . Therefore, by virtue of Lemma 1.8, is a subordination chain. We observe from the definition of a subordination chain that
This implies that

Now suppose that is not subordinate to , then by Lemma 1.6, there exists points and such that
Hence we have
by virtue of the subordination condition (2.3). This contracts the above observation that . Therefore, the subordination condition (2.3) must imply the subordination given by (2.15). Considering , we see that the function is the best dominant. This evidently completes the proof of Theorem 2.1.

*Remark 2.2. *We note that given by (2.2) in Theorem 2.1 satisfies the inequality .

We next prove a dual problem of Theorem 2.1, in the sense that the subordinations are replaced by superordinations.

Theorem 2.3. *Let . Suppose that
**
where is given by (2.2), and the function is univalent in and . Then the superordination:
**
implies that
**
Moreover, the function is the best subordinant.*

*Proof. *Let us define the functions and , respectively, by (2.5). We first note that, if the function is defined by (2.6), by using (2.8), then we obtain
After a simple calculation, (2.27) yields the relationship:
Then by using the same method as in the proof of Theorem 2.1, we can prove that for all . That is, defined by (2.6) is convex (univalent) in .

Next, we prove that the subordination condition (2.27) implies that
for the functions and defined by (2.5). Now consider the function defined by
Since is convex and , we can prove easily that is a subordination chain as in the proof of Theorem 2.1. Therefore according to Lemma 1.7, we conclude that the superordination condition (2.27) must imply the superordination given by (2.29). Furthermore, since the differential equation (2.27) has the univalent solution , it is the best subordinant of the given differential superordination. Therefore we complete the proof of Theorem 2.3.

If we combine this Theorems 2.1 and 2.3, then we obtain the following sandwich-type theorem.

Theorem 2.4. *Let . Suppose that
**
where is given by (2.2), and the function is univalent in and . Then the subordination relation:
**
implies that
**
Moreover, the functions and are the best subordinant and the best dominant, respectively.*

The assumption of Theorem 2.4, that the functions need to be univalent in , may be replaced by another conditions in the following result.

Corollary 2.5. *Let . Suppose that the condition (2.31) is satisfied and
**
where is given by (2.2). Then the subordination relation:
**
implies that
**
Moreover, the functions and are the best subordinant and the best dominant, respectively.*

*Proof. *In order to prove Corollary 2.5, we have to show that the condition (2.34) implies the univalence of and . Since from Remark 2.2, the condition (2.34) means that is a close-to-convex function in (see [18]) and hence is univalent in . Furthermore, by using the same techniques as in the proof of Theorem 2.4, we can prove the convexity (univalence) of and so the details may be omitted. Therefore, by applying Theorem 2.4, we obtain Corollary 2.5.

By setting and in Theorem 2.4, so that , we deduce the following consequence of Theorem 2.4.

Corollary 2.6. *Let . Suppose that
**
and the function is univalent functions in and . Then the subordination relation:
**
implies that
**
Moreover, the functions and are the best subordinant and the best dominant, respectively.*

If we take and in Theorem 2.4, then we easily to lead to the following result.

Corollary 2.7. *Let . Suppose that
**
and the function is univalent functions in and . Then the subordination relation:
**
implies that
**
Moreover, the functions and are the best subordinant and the best dominant, respectively.*

The proof of Theorem 2.8 below is similar to that of Theorem 2.4 by using (1.12) and so the details may be omitted.

Theorem 2.8. *Let . Suppose that
**
where is given by (2.2) with , and the function is univalent in and . Then the subordination relation:
**
implies that
**
Moreover, the functions and are the best subordinant and the best dominant, respectively.*

By using a similar method given in the proof of Theorems 2.4 and 2.8, we have the corresponding two theorems below.

Theorem 2.9. *Let with the additional condition . Suppose that
**
where is given by
**
and the function is univalent in and . Then the subordination relation:
**
implies that
**
Moreover, the functions and are the best subordinant and the best dominant, respectively.*

Theorem 2.10. *Let with the additional condition . Suppose that
**
where is given by (2.47) with , and the function is univalent in and . Then the subordination relation:
**
implies that
**
Moreover, the functions and are the best subordinant and the best dominant, respectively.*

#### Acknowledgment

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. 2011–0007037).

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#### Copyright

Copyright © 2012 Nak Eun Cho and Khalida Inayat Noor. 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.