Quantum Hermite-Hadamard type inequalities for generalized strongly preinvex functions

: In accordance with the quantum calculus, the quantum Hermite-Hadamard type inequalities shown in recent findings provide improvements to quantum Hermite-Hadamard type inequalities. We acquire a new q κ 1 -integral and q κ 2 -integral identities, then employing these identities, we establish new quantum Hermite-Hadamard q κ 1 -integral and q κ 2 -integral type inequalities through generalized higher-order strongly preinvex and quasi-preinvex functions. The claim of our study has been graphically supported, and some special cases are provided as well. Finally, we present a comprehensive application of the newly obtained key results. Our outcomes from these new generalizations can be applied to evaluate several mathematical problems relating to applications in the real world. These new results are significant for improving integrated symmetrical function approximations or functions of some symmetry degree.


Introduction
In mathematics, the 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").The renowned mathematician Euler was the genius who introduced the analysis q-calculus in the eighteenth century by integrating the parameter q into Newton's work of infinite series.At the beginning of the twentieth century, Jackson [1] has started a study of q-calculus and described quantum-definite integrals.The topic of quantum calculus has very long origins in the past.But to keep up with times, it has undergone rapid growth over the past few decades.I believe this strongly because it is a bridge between mathematics and physics, which is useful when dealing with physics.To get more information, please check the application and results of Ernst [2], Gauchman [3], and Kac and Cheung [4] in the theory of quantum calculus and theory of inequalities in quantum calculus.In previous papers, the authors Ntouyas and Tariboon [5,6] investigated how quantum-derivatives and quantum-integrals are solved over the intervals of the form [κ 1 , κ 2 ] ⊂ R and set several quantum analogs.Our investigation here is motivated essentially by the fact that basic (or q-)Hölder inequality, Hermite-Hadamard inequality and Ostrowski inequality, Cauchy-Bunyakovsky-Schwarz, Gruss, Gruss-Cebysev and other integral inequalities that use classical convexity.Also, Noor et al. [7], Sudsutad et al. [8], and Zhuang et al. [9], played an active role in the study, and some integral inequalities have been established which give quantum analog for the right part of Hermite-Hadamard inequality by using q-differentiable convex and quasi-convex functions.Many mathematicians have done studies in q-calculus analysis, the interested reader can check [10][11][12][13][14][15][16][17][18].
Srivastava [19] presented (or q-)calculus and fractional q-calculus and their applications in geometric function theory of complex analysis.There is also a clear connection between the classical q-analysis, which we used here, and the so-called (p, q)-analysis.We emphasize that the results for the q-analogues, which we discussed in this article for 0 < q < 1, can be easily (and probably trivially) converted into the corresponding results for the (p, q)-analogues (with 0 < q < p ≤ 1) by making a few obvious parametric and argument changes, with the additional parameter p being superfluous.Inspired and motivated by some of the above applications in the field of q-calculus.However, new q-Hermite-Hadamard-type inequalities for quantum integrals on finite intervals has not been studied yet.This gap is the motivation and inspiration for this research.
The discussion and application of convex functions have become a prosperous source of motivational material in pure and applied science.This vision promoted new and profound results in many branches of mathematical and engineering sciences and provided a comprehensive framework for the study of many problems.This discovery produced new and profound results in many mathematical and engineering sciences branches and provided a systematic structure for analyzing many issues in many fields.Many scholars have studied the various classes of convex sets and convex functions.A mapping K : I ⊆ R → R is considered convex if the mapping K satisfies the following inequality: One of the most famous inequalities in the theory of Convex Functional Analysis, Hermite-Hadamard established by Hermite and Hadamard in [20].It has a very fascinating geometric
representation with many significant applications.The extraordinary inequality states that, if K : I ⊆ R → R is a convex mapping on the interval I of real numbers and κ 1 , κ 2 ∈ I with κ 1 < κ 2 .Then, For K to be concave, both inequalities hold in the inverted direction.Many mathematicians have paid considerable attention to the Hermite-Hadamard inequality due to its quality and integrity in mathematical inequality.For significant developments, modifications, and consequences regarding the Hermite-Hadamard uniqueness property and general convex function definitions, the interested reader would like to refer to [21][22][23][24][25][26][27][28] and references therein.
It is noted that quasi-convex functions are a generalization of the convex function class since there are quasi-convex functions that are not convex.Weir et al. [29] introduced the concept of preinvex functions, which were then used in non-linear programming to describe appropriate optimal conditions and duality.Polyak [30] considered and studied the idea of strongly convex functions, which makes an essential contribution to the adaptation of most machine learning models that require the resolution of some form of optimization problem and areas concerned.Zu et al. [31] researched convergence by using iterative techniques based on the strong convex functional theory to resolve variational inequalities and equilibrium issues.Nikodem et al. discovered the new and innovative implementation of the inner product area's characterization with strongly convex functions in [32].
Throughout this paper, we are using continuous bifunctions µ(., .): Weir and Mond [29], has been introduced the definition of invex sets and preinvex mapping.
Note that, the invex set I µ is also called µ-connected set.If µ(κ 2 , κ 1 ) = κ 2 − κ 1 , then the invex set I µ is a convex set, but the reverse is not true.Definition 1.2.Let a mapping K : Here, we would like to point out that Humaira et al. [11] has introduced and studied generalized higher-order strong preinvex functions, which play a crucial role in studying the theory of optimization and related fields.Definition 1.3.A function K : I µ ⊂ R n → R is considered generalized higher-order strong preinvex function of order θ > 0 with modulus χ ≥ 0, if Properties that belong to generalized higher-order strongly preinvex functions are more robust versions of well-known properties of preinvex functions.Let us note the definition of the following generalized higher-order strongly quasi-preinvex functions.Definition 1.4.[11] A function K : I µ ⊂ R n → R is considered generalized higher-order strong preinvex function of order θ > 0 with modulus χ ≥ 0, if Remark 1.The notion of generalized higher-order strongly quasi-preinvexity strengthens the concept of quasi-preinvexity.
Several fundamental inequalities that are well known in classical analysis, like Hölder inequality, Ostrowski inequality, Cauchy-Schwarz inequality, Grüess-Chebyshev inequality, Grüess inequality.Using classical convexity, other fundamental inequalities have been proven and applied to q-calculus.Our objective is to develop new Hermite-Hadamard type inequalities by using quantum calculus and to support this claim graphically.

Preliminaries of q-calculus and some inequalities
In this section, we discuss some required definitions of quantum calculus and important left and right sides bonds of quantum Hermite-Hadamard integral type inequalities.
Jackson derived the q-Jackson integral in [1] from 0 to κ 2 for q ∈ (0, 1) as follows: provided the sum converge absolutely.The q-Jackson integral in a generic interval [κ 1 , κ 2 ] was given by in [1] and defined as follows: Definition 2.1.[5] We suppose that K : [κ 1 , κ 2 ] → R is an arbitrary function.Then q κ 1 -derivative of K at κ ∈ [κ 1 , κ 2 ] is defined as follows: defined by the expression (see [4]) The lemma below is play key part to calculate q κ 1 -derivatives.

Hermite-Hadamard-type inequalities for quantum integrals
We are now providing some new Hermite-Hadamard-type inequalities for functions whose absolute value of first q κ 1 -, q κ 2 -derivatives are generalized higher-order strongly preinvex functions.To prove our main results, we will initially suggest the following useful lemmas.
Proof.Taking modulus on Eq (3.1) and using the power-mean inequality, we have

.10)
Since κ 1 D q K σ is generalized higher-order strongly preinvex function for σ ≥ 1, we have by using Definition 2.1 and Definition 2.3 and Similarly, we also observe that We also have where ρ 1 (q) , ρ 2 (q) and ρ 4 (q) are defined in Theorem 3.3.
Corollary 3. As q → 1 − in Theorem 3.3, we get the inequality Corollary 4. Suppose that the assumptions of Theorem 3.3 with σ = 1 and letting q → 1 − , we obtain the inequality Theorem 3.4.If we assume all the conditions of Lemma 3.1, then the following inequality, shows that κ 1 D q K σ is generalized higher-order strongly preinvex function of order θ > 0 with modulus χ ≥ 0 on where m! q p−m+1 + 1 q m+1 − 1 .
Proof.Taking modulus on Eq (3.1) and using Hölder's inequality, we have We now evaluate the integrals involved in (3.22).We observe that Using the generalized higher-order strongly preinvexity of and similarly, we get Making use of (3.23) and (3.28) in (3.22), we get the required result.□ Theorem 3.5.If we assume all the conditions of lemma 3.1, then the following inequality, shows that κ 1 D q K σ is generalized higher-order strongly quasi-preinvex function of order θ > 0 with modulus where ρ 5 (q) = max κ 1 D q K (κ 1 ) σ , κ 1 D q K (κ 2 ) σ , ρ 3 (q) and ρ 4 (q) are defined in Theorem 3.3.
Proof.The desired inequality (3.36) can be obtained by applying the strategy used in the proof of Theorem 3.3 and taking into account the Lemma 3.8.□ Corollary 8.If σ = 1 together with the assumptions of Theorem 3.6, we obtain where ρ 1 (q), ρ 2 (q) and ρ 4 (q) are defined in Theorem 3.3.
Theorem 3.7.If we assume all the conditions of Lemma 3.2, then the following inequality, shows that κ 2 D q K σ is generalized higher-order strongly preinvex function of order θ > 0 with modulus χ ≥ 0 on where ρ 3 (p, q) is defined in Theorem 3.4.
Proof.The desired inequality (3.38) can be obtained by applying the strategy used in the proof of Theorem 3.4 and taking into account the Lemma 3.8.□ Theorem 3.8.If we assume all the conditions of Lemma 3.2, then the following inequality, shows that κ 2 D q K σ is generalized higher-order strongly quasi-preinvex function of order θ > 0 with modulus where ρ 9 (q) = max κ 2 D q K (κ 1 ) σ , κ 2 D q K (κ 2 ) σ .
Proof.The desired inequality (3.39) can be obtained by applying the strategy used in the proof of Theorem 3.5 and taking into account the Lemma 3.8.□

Conclusions
In this research, the generalized class of preinvex functions has been considered.We also obtained attractive quantum analogs of new Hermite-Hadamard type inequalities for generalized higher-order strongly preinvex and quasi-preinvex functions.New integral identities for q κ 1 -and q κ 2 -differentiable functions were proven, which played an important part in obtaining quantum estimates of Hermite-Hadamard type inequalities for q κ 1 -and q κ 2 -differentiable generalized higher-order strongly preinvex and quasi-preinvex functions.Our study's claim has been graphically supported.Finally, the innovative definition of generalized higher-order strongly preinvex functions has potential applications in parallelogram law of L p -spaces in functional analysis and opening new avenues for future study.Moreover, Srivastava [19] we presented (or q-) calculus and fractional q-calculus and their applications in geometric function theory of complex analysis.There is also a clear connection between the classical q-analysis, which we used here, and the so-called (p, q)-analysis.We emphasize that the results for the q-analogues, which we discussed in this article for 0 < q < 1, can be easily (and probably trivially) converted into the corresponding results for the (p, q)-analogues (with 0 < q < p ≤ 1) by making a few obvious parametric and argument changes, with the additional parameter p being superfluous.