Abstract

In this article, we develop a novel framework to study for a new class of preinvex functions depending on arbitrary nonnegative function, which is called -polynomial preinvex functions. We use the -polynomial preinvex functions to develop -analogues of the Ostrowski-type integral inequalities on coordinates. Different features and properties of excitement for quantum calculus have been examined through a systematic way. We are discussing about the suggestions and different results of the quantum inequalities of the Ostrowski-type by inferring a new identity for -differentiable function. However, the problem has been proven to utilize the obtained identity, we give -analogues of the Ostrowski-type integrals inequalities which are connected with the -polynomial preinvex functions on coordinates. Our results are the generalizations of the results in earlier papers.

1. Introduction

Calculus is an imperative study of the derivatives and integrals. The classical derivative was convoluted with the strength regulation kind kernel, and eventually, this gave upward thrust to new calculus referred to as the quantum calculus. In mathematics, quantum calculus (named -calculus) is the study of calculus without limits. The interest in this subject has exploded, and the -calculus has in the last twenty years served as a bridge between mathematics and physics. The -calculus has numerous applications in various fields of mathematics, for example, dynamical systems, number theory, combinatorics, special functions, fractals, and also for scientific problems in some applied areas such as computer science, quantum mechanics, and quantum physics. Jackson [1] defined the -analogue of derivative and integral operator as well as provided some of their applications. It is imperative to mention that quantum integral inequalities are more practical and informative than their classical counterparts. It has been mainly due to the fact that quantum integral inequalities can describe the hereditary properties of the processes and phenomena under investigation. Historically, the subject of quantum calculus can be traced back to Euler and Jacobi, but in recent decades, it has experienced a rapid development. As a result, new generalizations of the classical concepts of quantum calculus have been initiated and reviewed in many literature. Tariboon and Ntouyas [2, 3] proposed the quantum calculus concepts on finite intervals and obtained several -analogues of classical mathematical objects, which inspired many other researchers to study the subject in depth, and as a consequence, numerous novel results concerning quantum analogues of classical mathematical results have been launched. Noor et al. [4] obtained new -analogues of inequality utilizing the first-order -differentiable convex function.

Inequality plays an irreplaceable role in the development of mathematics. Very recently, many new inequalities such as the Hermite-Hadamard-type inequality [5–9], Petrović-type inequality [10], Pólya-Szegö and Ćebyšev-type inequalities [11], Ostrowski-type inequality [12], reverse Minkowski inequality [13], Jensen-type inequality [14–16], Bessel function inequality [17], trigonometric and hyperbolic functions inequalities [18], fractional integral inequality [19–22], complete and generalized elliptic integrals inequalities [23–28], generalized convex function inequality [29–31], and mean values inequality [32–34] have been discovered by many researchers. While the concept of classical convexity has been brought into a streamline by mathematical inequalities [35–50]. In fact, convex function and its connection with mathematical inequalities have wide applications in the estimation of some parameters in scientific observations and calculations [51–65]. In recent years, the classical concept of convexity has been extended and generalized in different directions, one of the important generalization of convexity is the invexity, which was studied by Hanson [66]; this work has greatly expanded the role of invexity in optimization. In [67, 68], the authors introduced a class of functions, which is called preinvexity as a generalization of convex functions.

Now, we recall the classical and well-known Hermite-Hadamard inequality [69], which can be stated as for all if is a convex function.

Ostrowski [70] established an integral inequality for continuous and differentiable function as follows.

Theorem 1 (See [70]). Let be continuous and differentiable on such that for all . Then, one has for all with the best possible constant 1/4.
The inequality (2) can be described in an identical kind as

Latif et al. [71] generalized the Ostrowski inequality (2) to the coordinated convex function by establishing an identity as follows.

Theorem 2 (See [71]). Let and be continuous and differentiable on such that .
Then the identity holds for all , where Noor et al. [4] presented the Ostrowski-type inequality for quantum calculus.

Theorem 3 (See [4]). Let and be continuous such that is integrable on . Then The following quantum integral version of the Hermite-Hadamard-type inequality for the coordinated convex function was proved by Alp and Sarıkaya [72].

Theorem 4 (See [72]). Let , , and be a coordinated convex function on . Then one has Kalsoom et al. [73] found the quantum integral inequality for two parameters function on the finite rectangle.

Next, we present the definitions of -derivative and integral, and their two known results.

Definition 5. Let and be a continuous function. Then, the partially -derivative, -derivative, and -derivative at for the function are defined by respectively. The function is said to be partially -, -, and -differentiable on if , , and exist for all .

Definition 6. Let and be a continuous function. Then the -integral of the function on is defined by for .

Theorem 7. Let and be a continuous function. Then, we have the identities for .

Theorem 8. Let and be continuous functions. Then, the identities holds for .

Very recently, Toplu et al. [74] improved the Hermite-Hadamard inequality (1) by investigating the -polynomial convexity. The main purpose of the article is to introduce the notion of -polynomial preinvex function, provide a new generalized quantum integral identity, establish new quantum analogues of Ostrowski-type inequalities for the -polynomial preinvex function on coordinates, and generalize and unify the previous known results.

2. Discussions and Main Results

In the beginning of this section, we introduce the definition of -polynomial prevexity.

Definition 9 (See [75]). Let be a continuous bi-function. Then, is said to be invex if for all and .

Definition 10 (See [67]). The function is said to be preinvex if for all and .

Definition 11. Let . Then, the nonnegative function is said to be -polynomial preinvex if for all and .
Note that if , then the definition of -polynomial preinvex function reduce to the definition of preinvex function.
If we take , then we have 2-polynomial preinvex function inequality

Proposition 12. Let and be an arbitrary family of -polynomial preinvex functions and . If is nonempty, then is an interval and is an -polynomial preinvex function on .

Proof. Let and . Then, we have This completes the proof.

In order to establish new quantum analogues of the Ostrowski-type inequalities on coordinates for the -polynomial preinvex function, we need a key lemma, which we present in this section.

Lemma 13. Let , be the interior of , and be mixed partial -differentiable on such that is continuous and integrable on for . Then, we have the identity where

Proof. Considering it follows from the definitions of partial -derivative and -integral that Note that From (20)–(23) we get Multiplying both sides of equality (25) by leads to Similarly, we have Therefore, Lemma 13 follows from (19), (27), (28), (29), and (30).

Theorem 14. Let , , be mixed partial differentiable on (the interior of ) such that its mixed partial -derivatives is continuous and integrable on for . If is an -polynomial preinvex function on the coordinates on and for , then we have where is defined in Lemma 13.

Proof. It follows from (19) and the properties of the -polynomial preinvexity function of the function on coordinates that Considering the first integral in view of the Definition 6 for , we get From (33) and computing the -integral, we get Computing the -integral and utilizing the fact , inequality (35) leads to the conclusion that Analogously, we also have Making use of the inequalities (36), (37), (38), and (39) and the fact that we get the desired inequality (31).

Theorem 15. Let with , and be mixed partial -differentiable on such that its mixed partial -derivative is continuous and integrable on for . If is an -polynomial preinvex function on the coordinates on and , then we have where is defined in Lemma 13.

Proof. It follows from (19), the Hölder inequality and the property of -polynomial preinvexity of the function on coordinates that for all .
From the -polynomial preinvexity of the function , we get Computing the -integral on the right-hand side of (43), we have In view of the Definition 6 for , we obtain Therefore, we get Similarly, computing the -integral and utilizing the fact on the right-hand side of (46), one has Analogously, we also can get Therefore, the desired inequality (41) follows from (47), (48), (49), and (50) and the fact that where and are the - and -analogues of and , respectively.

Theorem 16. Let be mixed partial -differentiable over such that its partial -derivative is continuous and integrable on . If is an -polynomial preinvex function on the coordinates on , then we have the inequality where is defined in Lemma 13.

Proof. It follows from (19), the power mean inequality and the property of -polynomial preinvexity of the function on coordinates that for all .
By similar argument as in Theorem 14, we can prove that Now by making use of the inequalities (54), (55), (56), and (57) and the fact that we get the desired inequality (52).

3. Conclusion

In the article, we have introduced a new class of preinvex functions which is named -polynomial preinvex functions and discovered a new quantum integral identity involving second-order mixed partial differentiable function. By using the obtained quantum integral identity as an auxiliary result, we have established several -Ostrowski-type inequalities for the class of -polynomial preinvex functions on coordinates, which have improved and unified many previously known results. Our given ideas and approaches may lead to a lot of follow-up research for the interested readers.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Authors’ Contributions

H. Kalsoom introduced the Definition 11, carried out the proof of Proposition 12 and Lemma 13, and drafted the manuscript. M. Idrees carried out the proof of Theorem 14. D. Baleanu carried out the proof of Theorem 15. Y.-M. Chu provided the main idea of the manuscript, carried out the proof of Theorem 16, completed the final revision and submitted the article. All authors read and approved the final manuscript.

Acknowledgments

The authors would like to express their sincere thanks to the editor and the anonymous reviewers for their helpful comments and suggestions. This work was supported by the Natural Science Foundation of China (Grant Nos. 61673169, 11971142, 11871202, 11301127, 11701176, 11626101, and 11601485).