This article proposes a new approach based on quantum calculus framework employing novel classes of higher order strongly generalized -convex and quasi-convex functions. Certain pivotal inequalities of Simpson-type to estimate innovative variants under the -integral and derivative scheme that provides a series of variants correlate with the special Raina’s functions. Meanwhile, a -integral identity is presented, and new theorems with novel strategies are provided. As an application viewpoint, we tend to illustrate two-variable -integral identities and variants of Simpson-type in the sense of hypergeometric and Mittag–Leffler functions and prove the feasibility and relevance of the proposed approach. This approach is supposed to be reliable and versatile, opening up new avenues for the application of classical and quantum physics to real-world anomalies.
In order to calculate the derivatives of real functions, classical calculus employs limits. On the other hand, the calculus without limits is known as quantum calculus or -calculus. According to history, Euler derived the fundamental formulations in -calculus in the eighteenth century. However, Jackson  was the first to develop the conceptions of the definite -derivative and -integral. In addition, Andrews  examined and inspected a number of studies on quantum calculus. The aforementioned consequences stimulated a more intense discussion of quantum theory in the twentieth century.
The evolution of the research of -calculus has been presented by its potential utilities in cosmology, multiple hypergeometric functions, Bernoulli and Euler polynomials, Mock theta functions, and more specifically in the study of analytic and harmonic univalent functions. The consensus of scholars who employ -calculus are physicists, see [3,4]. Baxter  presented the exact solutions of numerous frameworks in statistical mechanics. Bettaibi and Mezlini formulated certain -heat and -wave equations as in ref. . Moreover, several researchers, including Andrews , Gauchman , and Kac and Cheung  have been compensated for their efforts in proving and proposing new definitions and formulations. The topic of -theory has become a remarkable trend for many scientists in recent years, and novel results have been explored in previous research [9,10]. Numerous special function theories [11,12] are being assembled within the context of -calculus, mechanothermodynamics, translimiting states, and generalization of experimental data to analyze the quantum calculus in respect of general energy states [13,14].
With the assistance of special functions and convexity theory, we aim to create the applications of the -calculus to modify Simpson-type inequality in a two-variable formulation. However, in light of the -calculus, the current article could be the first to examine special functions with the correlation of two-variable formulation.
Convex functions have significant applications in a variety of interesting and engrossing fields of study, and they have also played a pivotal role, including coding theory, optimization, physics, information theory, engineering, and inequality theory. Mathematicians have proposed various novel versions of convex functions in the relevant literature [15,16,17].
Let and a mapping is said to be convex on , if the inequality
In the field of applied analysis, mathematical inequalities are considered as a prevalent mechanism for collecting descriptive and analytical description. A consistent increase in significance has evolved to address the prerequisites for widespread use of these variants [20,21,22]. Furthermore, various generalizations have been established by several authors that involve convex functions, such as Hermite–Hadamard, Trapezoid type, Opial, Ostrowski, Grüss, and the supremely illustrious Simpson’s inequality.
A mapping is four times continuously differentiable and . Then, one has the following inequality:
Our intention is to establish the novel -integral identity of Simpson-type within a class of generalized -convex functions in two variable forms. Kalsoom et al.  established the quantum integral Simpson type inequality for convex function on co-ordinates as follows:
 Assume that a mapping having a mixed partial -differentiable function defined on (the interior of ) with to be continuous and integrable on with and , then
This article investigates and presents a novel idea of higher-order strongly generalized -convex and quasi-convex functions in the sense of Raina’s function. Considering the novel auxiliary identity that correlates with the Raina function and the -calculus theory, numerous new Simpson-type inequalities are apprehended via the aforesaid classes of functions derived in two variable forms. Additionally, this suggested scheme in -calculus theory connected with Definitions 2.5 and 2.7 introduced new results for Simpson-type inequalities in hypergeometric and Mittag–Leffler sense. Finally, our findings may stimulate further investigation into special relativity theory and quantum theory.
Let be a non-empty closed set in and a continuous function.
Noor  introduced a class of non-convex mappings known as -convex functions.
 A mapping on the -convex set is said to be -convex, if
Observe that every convex mapping is -convex, but converse does not hold in general.
In ref. , Raina contemplated the subsequent class of function
A non-empty set is said to be generalized -convex set, if
We now define the generalized -convex function presented by Vivas-Cortez et al. .
 Let a set and we say that a function is generalized -convex, if
We say that a function is higher order strongly generalized -convex having , if
Some remarkable special cases are discussed as follows:
II. Taking then the generalized higher-order strongly -convex mappings reduces to generalized strongly -convex mappings, that is,
We say that a function is higher order strongly generalized -convex having , if
III. Letting then Definition 2.5 changes to higher-order strongly convex mappings.
We say that a function is higher order strongly convex having , if
We say that a function is higher order strongly generalized -quasi-convex having , if
For an exceptional appropriate selections of the Raina’s function and , one can attain several earlier and new classes of higher-order generalized strongly convex and quasi-convex mappings. This demonstrates that the new idea involving Raina’s function is wide and modifying one.
In addition, we highlight some key concepts and definitions in the -analog for one and two-variables.
Let , and let with constants , .
Suppose that , is a continuous mapping, then one has -derivative of on at which is written as
It is noted that if in equation (2.11), then , where is well-defined -derivative of , i.e., is mentioned as
Suppose that is a continuous function, indicated as , provided that be -differentiable from identified by
Thus, denotes the higher order -differentiable function.
Suppose that is a continuous function and the -integral on is expressed as
Next, if in equation (2.12), so there is an integral formulation of , which is signified as
Suppose that is a continuous function, then the subsequent assertions fulfill:
Suppose that is a continuous function and , then the subsequent assertions fulfill:
In ref. , Kalsoom et al. presented the quantum integral identities in a two-variable context as follows:
 Suppose a mapping in two-variables sense is continuous, then the partial and -derivative at are, respectively, described as:
Suppose a function in two-variables sense is continuous, then the definite -integral on is stated as
Suppose a function in two-variables sense is continuous, then the subsequent assertions fulfill:
Suppose that are continuous mappings of two-variables. Then the subsequent assertions fulfill for
3 A -integral identity for generalized -convex functions associated with Raina’s function
To illustrate the important consequences of this article, we proceed with some integral identities and inequalities for generalized -convex functions with the well-known Raina function.
Throughout this investigation, we utilized the following hypothesis:
Let and is a bounded sequence of positive real numbers.
Suppose that a twice partial -differentiable mapping defined on (the interior of ) having to be continuous and integrable on such that for .
Suppose that Assumptions (I) and (II) are satisfied, then the following equality holds:
In view of definition of partial -derivatives and definite -integrals, one has
By the definition of partial -derivatives and definite -integrals, we have
We observe that