On Certain Analogues of Noor Integral Operators Associated with Fractional Integrals

Fardi, Mojtaba; Amini, Ebrahim; Al-Omari, Shrideh

doi:https://doi.org/10.1155/2024/4565581

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Research Article | Open Access

Volume 2024 | Article ID 4565581 | https://doi.org/10.1155/2024/4565581

On Certain Analogues of Noor Integral Operators Associated with Fractional Integrals

Mojtaba Fardi,¹Ebrahim Amini,²and Shrideh Al-Omari³

Academic Editor: Humberto Rafeiro

Received25 Oct 2023

Revised08 Jan 2024

Accepted11 Jan 2024

Published22 Jan 2024

Abstract

In this paper, we employ a -Noor integral operator to perform a -analogue of certain fractional integral operator defined on an open unit disc. Then, we make use of the Hadamard convolution product to discuss several related results. Also, we derive a class of convex functions by utilizing the -fractional integral operator and apply the inspired presented theory of the differential subordination, to geometrically explore the most popular differential subordination properties of the aforementioned operator. In addition, we discuss an exciting inclusion for the given convex class of functions. Over and above, we investigate the -fractional integral operator and obtain some applications for the differential subordination.

1. Introduction

The theory of quantum calculus and its applications has been applied in several branches of mathematics, engineering sciences, and physics. Hence, many researchers have used -calculus to study discrete dynamical systems, discrete stochastic processes, -deformed super algebras, -transform analysis, and so on. In literature, differentiation and integration of function are formulated by using the quantum theory of calculus (or -calculus) [1–3]. The Jackson -calculus is also involved in various areas of science including fractional -calculus, optimal control, nonlinear integrodifferential equations, -difference, and -integral equations [4–7]. Ismail et al. [8] are the first to employ the theory of -calculus for investigating the geometric function theory. Srivastava in [9] points out some comprehensive reviews and applications in the geometric function theory of -calculus and discusses many important applications of starlike functions. Arif et al. in [10, 11] derive some properties of multivalent functions by using -calculus. Authors in [12] discuss -calculus and the Salagean operator to obtain differential subordination results. Aouf and Mostafa [13] used differential subordination to define a new subclass of analytic functions with -analogue fractional differential operator. Mahmood and Sokół [14] apply properties of the Ruscheweyh -differential operator for a subclass of analytic functions and study some of its applications. Kanas and Raducanu [15] investigate -analogues of the Ruscheweyh operator by using the Hadamard product; see, for some details, [14, 16] and [11, 17].

Let be a real or complex value and be unit disc , . Then, the -difference operator is defined by [1]

The -differentiation rules may be wrote as

Let consist of analytic functions in the unit disc of the form . For and , we use . Therefore, the function has the expansion of the form

Note that every function is normalized by and . The class of univalent functions in is denoted by . In particular, is the class of starlike functions, is the class of convex functions, and is the class of close-to-convex function [18, 19]. Recently, authors in [20] used the convolution to introduce three new subclasses of starlike functions, convex functions, and close-to-convex functions with the novel Borel distribution operator.

Let be given by (4) and . Then, the convolution of and is denoted by , which is a function in given by

We say that the function is subordinate to in and write for if there exists a Schwartz analytic function in such that and and [21]. In particular, if the function is univalent in , then if and only if

Ma and Minda [22] studied the class of starlike and convex functions by using the principle of differential subordination. Those differential subordinations provide interesting results when they are used to study new sets of univalent functions [23–25].

Making use of equations (4) and (1), we can easily obtain that where

It is clear that . For , the -factorial is given by [3]

In addition, with , the -Pochhammer symbol has the form [3]

Note that when .

For , the -analogue of the gamma function is presented as

Now, we define a -analogue Noor integral operator as follows: where is defined by the relation

Hence, is of the form (4). Therefore,

This, by taking , shows that the operator defined in (14) reduces to the familiar Noor integral operator of [26, 27].

Now, by using the idea of Cho and Aouf [28], we introduce the -fractional Riemann-Liouville integral of order () as follows.

Definition 1 (see [28]). The -fractional Riemann-Liouville integral of order () is defined for a function by where is an analytic function in .
Many other useful studies are introduced in the field of analytic functions including the fractional integral operator and its applications (see [29–31]).
In this paper, we introduce the -fractional integral operator by using the -Noor integral operator. This operator is based on the -fractional Riemann-Liouville integral of order (). Also, by using a newly defined -fractional integral operator, we introduce a subclass of analytic functions and prove that is a convex set. Furthermore, several exciting subordination results of the -fractional integral operator are obtained.

2. Preliminary Lemmas

Lemma 2 ([19], Theorem 8.9, p. 254). Let and be a convex function. If , then for all .

Lemma 3 ([32], Theorem 10, p. 259). Let , , be the analytic function in from the following form: If , then the function is an analytic function that satisfies the inequality

Lemma 4 ([33], Lemma 2, p. 2, [34]). Let be the analytic function in from the following form: If , then

3. Definition and Coefficient Bounds

We introduce a -fractional integral of the operator .

Definition 5. Let , , and . The -fractional integral of the operator is defined by the following: We note that and .
Now, by taking , we have Now, we study the subclass of analytic function by using the new operator.

Definition 6. Let the function , , , and . We define the subclass of functions which satisfy the inequality By allowing in Definition 6, the class is denoted by .

Example 1. In this example, we show that the set is nonempty.
Since , we can find a function such that By using assertion (21) for , we have So, we get From (24) and (26) and using the series expansion of , we obtain that This, indeed, implies that Therefore, we obtain that the function which is a member of the set .

Theorem 7. Let , , and . Then, the class is convex.

Proof. Consider two functions and from which are given in the form It is sufficient to show that the function with belongs to . Since , we get By using the -difference operator, we obtain So, we get which implies Thus, the desired results are obtained.

Taking into Theorem 7 leads to the following corollary.

Corollary 8. Let and . Then, the class is convex.

Lemma 9. Let , , , and such that be a convex function. If with and then

Proof. The differential equation has a unique solution (38). Since , then, in view of (4), we have So, the differential subordination (37) can be rewritten in the form Applying Lemma 2 gives This proves (39). The proof of Lemma 9 is completed.

Note that when , then . For example, we have

Remark 10. Plots of the suggested functions , , , and in the unit disc are illustrated in Figures 1 and 2. These plots show that these suggested functions are convex in the unit disc .

An analytic function in (37) is said to be a solution of the differential subordination. The analytic function is a dominant of the solution of the differential subordination (37), if for all satisfying (37).

A dominant is said to be the best dominant of (37) if it satisfies for all dominants of (37). The best dominant is unique up to the rotation of .

Theorem 11. Let , with , and such that defined in (36) is a convex function. If , , the following differential subordination implies

Proof. In view of the definition , we have . Since the -derivative rule in (2) holds, we get Now, by using the , we for can obtain the differential equation as follows: From (45) and (48), we have By using the notation we obtain . This implies that Now, applying Lemma 9, we obtain This means that the differential subordination (46) is established. This completes the proof of Theorem 11.

By using Remark 10, we can obtain the following corollary.

Corollary 12. Let with . If , , the following differential subordination implies where is given by (22) and .

In the next theorem, we derive an exciting inclusion for the class .

Theorem 13. Let and , , and such that , defined in (36), is a convex function. If is a convex function, then we have the following inclusion: where .

Proof. In view of Theorem 11, we obtain the following: where .
Now, by applying Lemma 9, we obtain . Indeed, where From the convexity of and using the fact that is symmetric with respect to the real axis, we have This completes the proof of Theorem 13.

By using Remark 10, we can obtain the following corollaries.

Corollary 14. Let and , . If , then we have the following inclusion: where .

Corollary 15. Let and , . If , then we have the following inclusion: where .

Corollary 16. Let and , . If , then we have the following inclusion: where .

Corollary 17. Let and ,. If , then we have the following inclusion: where .

Theorem 18. Let , with , and such that , defined in (36), is a convex function. If satisfies then we have the following result

Proof. Denoted by , we obtain By applying -derivative and using rule (2), we derive By applying Lemma 9, we get This implies the differential subordination (70). This completes the proof of Theorem 18.

By using Remark 10, we can obtain the following corollary.

Corollary 19. Let , with . If satisfies then we have the following result: where .

4. Applications

In this section, we obtain some interesting applications involving the -analogue differential subordination.

Theorem 20. Let , , and , such that , defined in (36), is a convex function. If satisfies then we have the following result:

Proof. Let . By putting in the assertions (67) and using (82), we obtain By applying Theorem 18 and Lemma 9, we for derive By using the principle of differential subordination, we have Taking into account that , we write Since , for and , we establish inequality (83).
To prove the sharpness of (83), we, for , define For this function, we have When , we get This completes the proof of Theorem 20.

We can obtain the following corollaries by using Remark 10.

Corollary 21. Let . If satisfies then we have the following result:

Corollary 22. Let . If satisfies then we have the following result:

Theorem 23. Let , , and such that , defined in (36), is a convex function. If satisfies then we have the following result:

Proof. Similar to the proof of Theorem 20, for , the differential subordination (92) is equivalent to Therefore, By using Remark 10, we can obtain the following corollaries.

Corollary 24. Let . If satisfies then we have the following result:

Corollary 25. Let . If satisfies then we have the following result:

Corollary 26. Let . If satisfies then we have the following result:

Theorem 27. Let , , , and such that , defined in (36), is a convex function. If satisfies then we have the following result: where

Proof. Let . Then, by using the differential subordinations (67) for and (112), we infer By applying Theorem 18 and Lemma 9, we have Also, we define a function in the following form: By using the differential subordination (112), we write Similar to the proof of Theorem 20, we derive where It is clear that Therefore, by applying Lemma 3, we have By applying Lemma 4 and using the fact that , we calculate By using inequality (104), we derive By using assertions (106) and (107), we obtain This completes the proof of Theorem 27.

We can obtain the following corollaries by using Remark 10.

Corollary 28. Let , . If satisfies then we have the following result: where

Corollary 29. Let , . If satisfies then we have the following result: where

5. Conclusions

In this paper, the topics related to applications in the geometric function theory of -calculus are presented. The proposed -differential operator was applied to introduce a -analogue of a fractional integral operator, and the geometric behavior of the operator is also investigated using the principle of differential subordination. Several interesting results of the -analogue fractional integral operator are obtained here by following the differential subordination method. A new class of convex functions, , are defined, and an inclusion for the class is obtained. The fractional integral operator is defined on open unit disc , and some properties of differential subordination are studied. Therefore, the results obtained in this research could be further used for writing the dual theory of differential subordination which is added to the study of the -fractional integral operator.

Data Availability

There is no data availability statement to be declared.

Conflicts of Interest

The authors declare no competing interests regarding the publication of the article.

Authors’ Contributions

All authors contributed equally and significantly in writing this paper. They read and approved the final manuscript.

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