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Innovative Applications of Fractional Calculus

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Volume 2021 |Article ID 5577340 | https://doi.org/10.1155/2021/5577340

Hua Wang, Humaira Kalsoom, Hüseyin Budak, Muhammad Idrees, "-Hermite–Hadamard Inequalities for Generalized Exponentially -Preinvex Functions", Journal of Mathematics, vol. 2021, Article ID 5577340, 10 pages, 2021. https://doi.org/10.1155/2021/5577340

-Hermite–Hadamard Inequalities for Generalized Exponentially -Preinvex Functions

Academic Editor: Ahmet Ocak Akdemir
Received13 Feb 2021
Revised07 Mar 2021
Accepted26 Mar 2021
Published19 Apr 2021

Abstract

In this article, we introduce a new extension of classical convexity which is called generalized exponentially -preinvex functions. Also, it is seen that the new definition of generalized exponentially -preinvex functions describes different new classes as special cases. To prove our main results, we derive a new -integral identity for the twice -differentiable function. By using this identity, we show essential new results for Hermite–Hadamard-type inequalities for the -integral by utilizing differentiable exponentially -preinvex functions. The results presented in this article are unification and generalization of the comparable results in the literature.

1. Introduction and Preliminaries

In mathematics, quantum calculus is equivalent to usual infinitesimal calculus without the concept of limits or the investigation of calculus without limits (quantum is from the Latin word “quantus,” and literally, it means how much, in Swedish “Kvant”). It has two major branches: -calculus and -calculus. And both of them were worked out by Cheung and Kac [1] in the early twentieth century. In the same era, Jackson started to work on quantum calculus or -calculus, but Euler and Jacobi had already figured out this type of calculus. A number of studies have recently been widely used in the field of -analysis, beginning with Euler, due to the vast necessity for mathematics that models of quantum computing -calculus exist in the framework between physics and mathematics. In 2013, Tariboon and Ntouyas introduced the -difference operator [2, 3]. This inspired other researchers, and as a consequence, numerous novel results concerning quantum analogues of classical mathematical results have already been launched in the literature. In various mathematical fields, it has many applications, such as theory of numbers, combinations, orthogonal polynomials, basic hypergeometric functions and other subjects, quantum mechanics, physics, and the principle of relativity. Many important aspects of quantum calculus are covered in the articles by Humaira et al. [47]. The quantum calculus is currently a subfield of the more general scientific field of time-scale calculus. New developments have recently been made in the research and methodology of dynamic derivatives on time scales. The research offers a consolidation and application of traditional differential and difference equations. Moreover, it is a unification of the discrete theory with the continuous theory, from the theoretical perspective. Recently, in 2020, Bermudo et al. introduced the notion of the -derivative and integral [8]. For more details, see [915] and references cited therein.

The discussion and application of convex functions has become a very rich source of motivational material in pure and applied science. This vision not only promoted new and profound results in many branches of mathematical and engineering sciences but also provided a comprehensive framework for the study of many problems. Many scholars have studied various classes of convex sets and convex functions; see [16, 17]. The concept of convexity has been extended in several directions, since these generalized versions have significant applications in different fields of pure and applied sciences. One of the convincing examples on extensions of convexity is the introduction of invex function, which was introduced by Hanson [18] Weir and Mond [19] explored the idea of preinvex functions and actualized it to the foundation of adequate optimality conditions and duality in nonlinear programming.

The Hermite–Hadamard inequality was introduced by Hermite and Hadamard; see [20]. It is one of the most recognized inequalities in the theory of convex functional analysis, which is stated as follows.

Let be a convex mapping and with . Then,

If is concave, both inequalities hold in the reverse direction.

The important objective of this paper is to introduce an exponentially generalized definition of -preinvex functions. Furthermore, the new -integral identity is determined. By using this new identity, we proved many new estimates of bounds for it, essentially based on the concept of quantum calculus.

2. Preliminaries

In this section, we derive a new definition of the generalized exponentially -preinvex function. Also, we present all necessary concepts related to quantum calculus.

First of all, let be a nonempty set, be a continuous function, and and be two continuous functions.

Definition 1. A set is supposed to be -invex concerning and with some fixed iffor all and .
If , the above equation is called the convex set, and is an invex set; however, the reverse is not possible.

Example 1. Consider andAs one can see, is also an invex set for , but not a convex set.

Definition 2. A function is said to be a generalized exponentially -preinvex function if there exist and , , and nonpositive such thatfor all and and for some fixed .

Remark 1. In Definition 2,(1)If we choose or , then the definition of the generalized exponentially -preinvex function is converted into the definition of the generalized -preinvex function(2)If we choose and , then we get the definition of -preinvexity(3)If we choose , , and , then we get the definition of -convexity(4)If we choose and , we get the definition in [21](4)If we choose , then we have the definition of exponentially -preinvex functions, stated as follows

Definition 3. A function is called exponentially -preinvex if there exist , , and nonpositive such thatfor all and and for some fixed .
Many researchers proved several results about the importance and development in the theory of exponentially convex functions and their applications. For more details, see [2225] and references cited therein.
Jackson derived the -Jackson integral in [12] from 0 to for as follows:provided the sum converges absolutely.
The -Jackson integral in a generic interval was given by Jackson in [12] and defined as follows:

Definition 4. (see [3]). We suppose that is an arbitrary function. Then, the -derivative of at is defined as follows:Since is an arbitrary function from to , . The function is said to be -differentiable on if exists for all . If in (3), then , where is a familiar -derivative of at defined by the following expression (see [1]):

Definition 5. (see [8]). We suppose that is an arbitrary function; then, the -derivative of at is defined as follows:

Definition 6. (see [3]). We suppose that is an arbitrary function; then, the -definite integral on is defined as follows:In [10], Alp et al. established the -Hermite–Hadamard inequalities for convexity, which are defined as follows.

Theorem 1. Let be a convex differentiable function on and . Then, -Hermite–Hadamard inequalities are as follows:

On the contrary, the following new description and related Hermite–Hadamard-form inequalities were given by Bermudo et al.

Definition 7. (see [8]). Let be an arbitrary function. Then, the -definite integral on is defined as

Theorem 2. (see [8]). Let be a convex function on and . Then, -Hermite–Hadamard inequalities are as follows:

From Theorems 1 and 2, one can achieve the following inequalities.

Corollary 1. (see [8]). For any convex function and , we haveand

Alp and Sarikaya, by using the area of trapezoids, introduced the following generalized quantum integral which we will call as -integral.

Definition 8. (see [11]). Let be an arbitrary function. For ,where .

Theorem 3. (-Hermite–Hadamard; see [11]). Let be a convex continuous function on and . Then, we have

Definition 9. (see [11]). For any real number , the analogue of is defined as

Definition 10. (see [11]). Let . Then, is defined by

3. A New -Integral Identity

In this section, we present a new -integral identity.

Lemma 1. For with , let there be an arbitrary function such that is -integrable on . Then, one has

Proof. We suppose thatMultiplying both sides of the above equality by , we get the required result.

4. Hermite–Hadamard Inequalities for Generalized Exponentially -Preinvex Functions

Theorem 4. We assume that the conditions of Lemma 1 with and hold. If is a generalized exponentially -preinvex function and , then for some fixed , we havewhere

Proof. By utilizing conditions of Lemma 1 and the famous power mean inequality, we obtainwhereanddue to for all and .
We proved our result.

Theorem 5. We assume that the conditions of Lemma 1 with and hold. If is a generalized exponentially -preinvex function and with , then for some fixed , we obtain

Proof. By utilizing conditions of Lemma 1 and the famous Hölder inequality, we obtainThis completes the proof.

Theorem 6. We assume that the conditions of Lemma 1 with and hold. If is a generalized exponentially -preinvex function and with , then for some fixed , we obtain

Proof. By utilizing conditions of Lemma 1 and the famous Hölder inequality, we obtainwhereWe proved our result.

Theorem 7. We assume that the conditions of Lemma 1 with and hold. If is a generalized exponentially -preinvex function and with , then for some fixed , we obtainwhere

Proof. By utilizing conditions of Lemma 1 and the famous Hölder inequality, we obtainApplying the definition of quantum integral, we getThis completes the proof.

Theorem 8. We assume that the conditions of Lemma 1 with and hold. If is a generalized exponentially -preinvex function and with , then for some fixed , we obtainwhere