Abstract
The purpose of the paper is to investigate several subordination- and superordination-preserving properties of a class of integral operators, which are defined on the space of analytic functions in the open unit disk. The sandwich-type theorem for these integral operators is also presented. Moreover, we consider an application of the subordination and superordination theorem to the Gauss hypergeometric function.
1. Introduction
Let be the class of functions analytic in the unit disk , and denote by the class of analytic functions in and usually normalized, that is, . The function is said to be subordinate to , or is said to be superordinate to , if there exists a function such that In this case, we write If the function is univalent in , then we have (cf. [1])
Definition 1.1 (Miller and Mocanu [1]). Let and let be univalent in . If is analytic in and satisfies the following 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 the differential subordination (1.5). A dominant that satisfies for all dominants of (1.5) is said to be the best dominant.
Definition 1.2 (Miller and Mocanu [2]). Let and let be analytic in . If and are univalent in and satisfy the following differential superordination: then is called a solution of the differential superordination. An analytic function is called a subordinat of the solutions of the differential superordination or, more simply, a subordinant if for all satisfying the differential superordination (1.6). A univalent subordinat that satisfies for all subordinats of (1.6) is said to be the best subordinat.
Definition 1.3 (Miller and Mocanu [2]). We denote by the class of functions that are analytic and injective on , where
and are such that
We define the family of integral operators as follows:
where each of the functions and belong to the class A and the parmeters , , , were so constrained that the integral operators in (1.9) exist.
Throughout this paper, we will denote by the following analytic function class:
This integral operator defined by (1.9) has been extensively studied by authors [3–6] with suitable restriction on the parameters and .
In particular, if we take we get the integral operator defined by Bulboacă [7–12] and if we put in (1.9), we will get the results in [13].
In the present paper, we obtain the subordination- and superordination-preserving properties of the integral operator defined by (1.9) with the sandwich-type theorem. We also consider an interesting application of our main results to the Gauss hypergeometric function.
The following lemmas will be required in our present investigation.
Lemma 1.4 (Miller and Mocanu [14]). Suppose that the function satisfies the following condition: for all real and for all If the function is analytic in and then
Lemma 1.5 (Miller and Mocanu [15]). Let with and let with . If then the solution , , of the following differential equation: is analytic in and satisfies the inequality given by
Lemma 1.6 (Miller and Mocanu [1]). Let with and let be analytic in with If is not subordinate to , then there exist points for which
Our next lemmas deal with the notion of subordination chain. A function defined on is called the subordination chain (or Löwner chain) if is analytic and univalent in for all is continuously differentiable on for all and
Lemma 1.7 (Miller and Mocanu [2]). Let and Also set If is a subordination chain and then implies that Furthermore, if has a univalent solution , then is the best subordinat.
Lemma 1.8 (Pommerenke [16]). The function with is a subordination chain if and only if
2. Main Results
Our first subordination is contained in Theorem 2.1. To short the formulas in this section, let us denote
Theorem 2.1. Let . Suppose that where Then the following subordination relation: implies that where is the integral operator defined by (1.9). Moreover, the function is the best dominant.
Proof. Let us define the functions and by
respectively. Then
We first show that, if the function is defined by
then
In terms of the function , the definition (1.9) readily yields
We also have
By a simple calculation in conjunction with (2.10) and (2.11), we obtain the following relationship:
We also see from (2.2) that
and, by using Lemma 1.5, we conclude that the differential equation (2.12) has a solution with
Let us put
where is given by (2.3). From (2.2), (2.12), and (2.15), we obtain
Now we proceed to show that
Indeed, from (2.15), we have
where
For given by (2.3), we note that the coefficient of is in the quadratic expression for defined by (2.19) is greater than or equal to zero. Moreover, the discriminant of in (2.19) is represented by
which, for the assumed value of given by (2.3), yields
and so the quadratic expression for in given by (2.19) is a perfect square. We thus see from (2.18) that
Hence, by using Lemma 1.4, we conclude that
that is, the function defined by (2.6) is convex in .
Next, we prove that the subordination condition (2.4) implies that
for the functions and defined by (2.6). For this purpose, we consider the function given by
Since is convex in and , we obtain
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 exist points and such that
Hence, we have
by virtue of the subordination condition (2.4). This contradicts the above observation that
Therefore, the subordination condition (2.4) must imply the subordination given by (2.24). 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 (2.3) in Theorem 2.1 satisfies the following inequality .
Theorem 2.3. Let . Suppose that where is given by (2.3), and that the function is univalent in and such that , where is the integral operator defined by (1.9). Then the following superordination relation: implies that Moreover, the function is the best subordinat.
Proof. The first part of the proof is similar to that of Theorem 2.1 and so we will use the same notation as in the proof of Theorem 2.1. Now let us define the functions and , as before, by (2.6). We first note that, by using (2.3) and (2.11), we obtain
After a simple calculation, (2.35) yields the following relationship:
where function is defined by (2.8). Then, by using the same method as in the proof of Theorem 2.1, we can prove that
that is, defined by (2.6) is convex (univalent) in .
Next, we prove that the superordination condition (2.33) implies that
For this purpose, we 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 (2.33) must imply the superordination given by (2.38). Furthermore, since the differential equation (2.35) has the univalent solution , it is the best subordinat of the given differential subordination. We thus complete the proof of Theorem 2.3.
If we suitably combine Theorems 2.1 and 2.3, then we obtain the following sandwich-type theorem.
Theorem 2.4. Let . Suppose that where is given by (2.3), and that the function is univalent in and such that , where is the integral operator defined by (1.9). Then the following subordination relation: implies that Moreover, the functions and are the best subordinat and the best dominant, respectively.
The assumption of Theorem 2.4, that the functions and need to be univalent in , may be replaced by another condition in the following result.
Corollary 2.5. Let . Suppose that the condition (2.49) is satisfied and that and where is given by (2.3). Then the following subordination relation: implies that where is the integral operator defined by (1.9). Moreover, the functions and are the best subordinat and the best dominant, respectively.
Proof. In order to prove Corollary 2.5, we have to show that the condition (2.43) implies the univalence of each of the functions and .
Since , just as in Remark 2.2, the condition (2.43) means that is a close-to-convex function in (see [17]), 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 are being omitted here. Thus, by applying Theorem 2.4, we readily obtain Corollary 2.5.
By setting in Theorem 2.4, so that , we deduce the following consequence of Theorem 2.4.
Corollary 2.6. Let . Suppose that and that the function is univalent in and , where is the integral operator defined by (1.9) with . Then the following subordination relation: implies that Moreover, the functions and are the best subordinat and the best dominant, respectively.
If we take in Theorem 2.4, then we are easily led to the following result.
Corollary 2.7. Let . Suppose that and that the function is univalent in and , where is the integral operator defined by (1.9) with . Then the following subordination relation: implies that Moreover, the functions are the best subordinat, respectively.
3. Application to the Gauss Hypergeometric Function
We begin by recalling that the Gauss hypergeometric function is defined by (see, for details, [18] and [19, Chapter 14]) where denotes the Pochhammer symbol (or the shifted factorial) defined (for and in terms of the Gamma function) by For this useful special function, the following Eurlerian integral representation is fairly well known [19, page. 293]: In view of (3.3), we set and , so that the definition (1.9) yields Moreover, we note that and for the condition (2.2) becomes . Therefore, by applying Theorem 2.1 with , , we obtain the following result.
Theorem 3.1. Let with , , . Suppose that where Then the following subordination relation: implies that where is the integral operator defined by (1.9). Moreover, the function is the best dominant.
For we get and Theorem 3.1 becomes the following Corollary.
Corollary 3.2. Let and . Then the following subordination relation: implies that where the integral operator is defined by (1.9).
We state the following result as the dual result of Theorem 3.1, which can be obtained by similarly applying Theorem 2.3.
Theorem 3.3. Under the assumption of Theorem 3.1, suppose also that the function is univalent in and that , where is the integral operator defined by (1.9). Then the following superordination relation: implies that Moreover, the function is the best subordinat.
If we set in (1.9) and , then we get Therefore by applying Theorem 2.1, we obtain the following result.
Theorem 3.4. Let with , , . Suppose that and where Then the following subordination relation: implies that where is the integral operator defined by (1.9). Moreover, the function is the best dominant.
By taking in Theorem 3.1, we are led to Corollary 3.5.
Corollary 3.5. Let with , , . Then the following subordination relation: implies that where the integral operator is defined by (1.9).
We state the following result as the dual result of Theorem 3.4, which can be obtained by similarly applying Theorem 2.3.
Theorem 3.6. Under the assumption of Theorem 3.4, suppose also that the function is univalent in and that , where is the integral defined by (1.9) if we take . Then the following superordination relation: implies that Moreover, the function is the best subordinat.
Acknowledgment
This paper was supported by the Science Research Program in Science College in Dammam University, Saudi Arabia.