q-Hermite–Hadamard Inequalities for Generalized Exponentially (s, m; η)-Preinvex Functions

School of Mathematics and Statistics, Changsha University of Science and Technology, Changsha 410114, China Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China Department of Mathematics, Faculty of Science and Arts, Düzce University, Yörük, Turkey Zhejiang Province Key Laboratory of Quantum Technology and Device, Department of Physics, Zhejiang University, Hangzhou 310027, China


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: q-calculus and h-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 q-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 q-analysis, beginning with Euler, due to the vast necessity for mathematics that models of quantum computing q-calculus exist in the framework between physics and mathematics. In 2013, Tariboon and Ntouyas introduced the κ 1 Dq-difference operator [2,3]. is 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. [4][5][6][7]. e 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. e 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 κ 2 Dq-derivative and integral [8]. For more details, see [9][10][11][12][13][14][15] and references cited therein. e discussion and application of convex functions has become a very rich source of motivational material in pure and applied science. is 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]. e 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.
e 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.
e important objective of this paper is to introduce an exponentially generalized definition of (s, m; η)-preinvex functions. Furthermore, the new mκ2 q-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.

Preliminaries
In this section, we derive a new definition of the generalized exponentially (s, m; η)-preinvex function. Also, we present all necessary concepts related to quantum calculus.
en, q-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 9 (see [11]). For any real number n, the q analogue of n is defined as Definition 10 (see [11]). Let k, p > 0. en, B q (k, p) is defined by

A New mκ 2 q-Integral Identity
In this section, we present a new mκ 2 q-integral identity.

Hermite-Hadamard Inequalities for Generalized Exponentially (s, m; η)-Preinvex Functions
Theorem 4. We assume that the conditions of Lemma 1 with χ ≥ 1 and α ∈ R hold. If | mκ 2 D 2 q 5| u is a generalized exponentially (s, m; η)-preinvex function and u ≥ 1, then for some fixed s, m ∈ (0, 1], we have where Proof. By utilizing conditions of Lemma 1 and the famous power mean inequality, we obtain where and due to 2 1− s − k s ≥ 0 for all k ∈ [0, 1] and s ∈ (0, 1]. We proved our result.

Journal of Mathematics
Theorem 5. We assume that the conditions of Lemma 1 with χ ≥ 1 and α ∈ R hold. If | mκ 2 D 2 q 5| u is a generalized exponentially (s, m; η)-preinvex function and u > 1 with p − 1 + u − 1 � 1, then for some fixed s, m ∈ (0, 1], we obtain Proof. By utilizing conditions of Lemma 1 and the famous Hölder inequality, we obtain is completes the proof. □ Theorem 6. We assume that the conditions of Lemma 1 with χ ≥ 1 and α ∈ R hold. If | mκ 2 D 2 q 5| u is a generalized exponentially (s, m; η)-preinvex function and u > 1 with p − 1 + u − 1 � 1, then for some fixed s, m ∈ (0, 1], we obtain Proof. By utilizing conditions of Lemma 1 and the famous Hölder inequality, we obtain where We proved our result. □ Theorem 7. We assume that the conditions of Lemma 1 with χ ≥ 1 and α ∈ R hold. If | mκ 2 D 2 q 5| u is a generalized exponentially (s, m; η)-preinvex function and u > 1 with p − 1 + u − 1 � 1, then for some fixed s, m ∈ (0, 1], we obtain where Proof. By utilizing conditions of Lemma 1 and the famous Hölder inequality, we obtain Journal of Mathematics Applying the definition of quantum integral, we get is completes the proof. □ Theorem 8. We assume that the conditions of Lemma 1 with χ ≥ 1 and α ∈ R hold. If | mκ 2 D 2 q 5| u is a generalized exponentially (s, m; η)-preinvex function and u > 1 with p − 1 + u − 1 � 1, then for some fixed s, m ∈ (0, 1], we obtain where Proof. By utilizing conditions of Lemma 1 and Hölder's inequality, we have Applying the definition of quantum integral, we get is completes the proof.
Journal of Mathematics Theorem 9. We assume that the conditions of Lemma 1 with χ ≥ 1 and α ∈ R hold. If | mκ 2 D 2 q 5| u is a generalized exponentially (s, m; η)-preinvex function and u > 1 with p − 1 + u − 1 � 1, then for some fixed s, m ∈ (0, 1], we obtain where Proof. By utilizing conditions of Lemma 1 and the famous Hölder inequality, we obtain Applying the definition of quantum integral, we get is completes the proof.

Conclusion
In this article, we established the new definition of generalized exponentially (s, m; η)-preinvex functions and proved a new modified mκ 2 q-integral identity. Using this new identity, we have been able to obtain new estimates of the quantum bounds applying the concept of generalized exponentially (s, m; η)-preinvex functions. It is worth to mention here that if we take χ � e, then all of the main results reduce to the results for exponentially (s, m; η)-preinvex functions. For further research, we could expand the inequality-based analysis to other fields, including the inequality-based theory, quantum calculus, machine learning, robotics, weather forecasting, and optimizations.

Data Availability
Data sharing is not applicable to this paper as no datasets were generated or analyzed during the current study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.