Trapezoidal-Type Inequalities for Strongly Convex and Quasi-Convex Functions via Post-Quantum Calculus

In this paper, we establish new (p,q)κ1-integral and (p,q)κ2-integral identities. By employing these new identities, we establish new (p,q)κ1 and (p,q)κ2- trapezoidal integral-type inequalities through strongly convex and quasi-convex functions. Finally, some examples are given to illustrate the investigated results.


Introduction and Preliminaries
Quantum calculus, often known as q-calculus, is a branch of mathematics that studies calculus without limits. Euler's work on infinite series, which he integrated with Newton's work on parameters, served as the idea for the q-calculus analysis, which was founded in the eighteenth century by famous mathematician Leonhard Euler (1707-1783). In 1910, F. H. Jackson [1] used L. Euler's expertise to define the q-derivative and q-integral of any function on the interval (0, ∞) using the q-calculus of infinite series. Quantum calculus has a very long history. However, to keep up with the times, it has undergone rapid growth over the past few decades. However, in order to stay current, it has experienced tremendous development over the last several decades. I am a strong believer in this as it serves as a link between mathematics and physics, which is useful when working 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] investigated how quantum-derivatives and quantum-integrals are solved over the intervals of the form [κ 1 , κ 2 ] ⊂ R. In addition, they studied the characteristics and specific results of initial value problems in impulsive q-differential equations of the first and second order. Furthermore, set a number of quantum analogs for some well-known effects, for example, Hölder inequality, Hermite-Hadamard inequality and Ostrowski inequality, Cauchy-Bunyakovsky-Schwarz, Gruss, Gruss-Cebysev, and other integral inequalities that use classical convexity. Furthermore, Noor et al. [6], Sudsutad et al. [7], and Zhuang et al. [8] 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. Numerous mathematicians have carried out research in the area of q-calculus analysis; interested readers may check the works in [9][10][11][12][13][14][15][16][17][18][19].
q-calculus generalization is post-quantum or, often, is referred to as (p, q) calculus. Post-quantum calculus is a recent advancement in the study of quantum calculus that contains p and q-numbers with two independent variables p and q. Quantum calculus Inequality (1) was introduced by C. Hermite [31] and investigated by J. Hadamard [32] in 1893. For K to be concave, both inequalities hold in the inverted direction. Many mathematicians have paid great attention to the inequality of Hermite-Hadamard due to its quality and validity in mathematical inequalities. For significant developments, modifications, and consequences regarding the Hermite-Hadamard uniqueness property and general convex function definitions, for essential details, the interested reader would like to refer to the works in [33][34][35] and references therein.
Different inequalities are used to represent convex functions. Convex functions are responsible for several well-known inequalities. Strongly convexity is a reinforcement of the concept of convexity; some aspects of strongly convex functions are just "stronger versions" of known convex properties. Polyak [36] introduced the strongly convex function as just stronger versions of properties of convex functions. Moreover, Nikodem et al. [37] introduced new characterizations of inner product spaces among normed spaces involving the notion of strong convexity.
Note that quasi-convex functions are a generalization of the convex function class, as there are quasi-convex functions that are not convex.

Remark 1.
The notion of strongly quasi-convexity strengthens the concept of quasi-convexity.
Latif et al. [22] proved quantum estimates of (p, q)-trapezoidal integral inequalities through convex and quasi-convex functions Several fundamental inequalities that are well known in classical analysis, like Hölder inequality, Ostrowski inequality, Cauchy-Schwarz inequality, Grüess-Chebyshev inequality, and Grüess inequality. Using classical convexity, other fundamental inequalities have been proven and applied to q-calculus.
Our objective is to develop improved trapezoidal type inequalities by using postquantum calculus and to support this claim graphically.

q-Derivatives and Integrals
In this section, we discuss some required definitions of q, (p, q)-Calculus and important quantum integral inequalities for Hermite-Hadamard on left and right sides bonds. Throughout this paper, we are using constants 0 < q < 1 and 0 < q < p ≤ 1.
The [m] q integers are known as q-integers and are written as [m] q = m, f or m = 1.
The [m] q ! and m i q ! are denoted as q-factorial and q-binomial, respectively, and are written as follows: In the early twentieth century, the Reverend Frank Hilton Jackson made major contributions to the classical concept of a derivative of a function at a point, which allowed for a more straightforward study of ordinary calculus and number theory in these investigations. Jackson is responsible for numerous seminal studies in the subject, including that in [1], in addition to creating the q-analogs of certain major results discovered in these disciplines.
The classic Jackson approach is given below.
provided the sum converge absolutely. The q-Jackson integral in a generic interval [κ 1 , κ 2 ] is defined as follows: Whenever q approaches 1, the number theory, deduction, and ordinary integration findings become polynomial expressions in a real variable q.
The following lemma is play key part to calculate q κ 1 -derivatives.

Definition 4 ([5]
). We suppose that K : [κ 1 , κ 2 ] → R be an arbitrary function, then the q κ 1definite integral on [κ 1 , κ 2 ] is described as below The following properties are very important in quantum calculus: The following is useful results for evaluating such qκ 1 -integrals.

(p, q)-Derivatives and Integrals
In this section, we review some fundamental notions and symbols of (p, q)-calculus. The [m] p,q integers are known as (p, q) integers and are written as The [m] p,q ! and m i p,q ! are denoted as (p, q)-factorial and (p, q)-binomial, respectively, and are written as follows: Definition 8 ([20]). The (p, q) κ 1 -derivative of mapping K : [κ 1 , κ 2 ] → R is given as As K is an arbitrary function from [κ 1 , κ 2 ] to R, so for κ = κ 1 , we define κ 1 D p,q K(κ 1 ) = lim As K is an arbitrary function from [κ 1 , Remark 3. If we take κ 1 = 0 and κ = κ 2 = 1 in (15), then we have Similarly, by taking κ = κ 1 = 0 and κ 2 = 1 in (16), then we obtain that In [21], Kunt et al. proved the following Hermite-Hadamard-type inequalities for convex functions via (p, q) κ 1 integral: Lemma 3. We have the following equalities: Proof. From Definition 11, we have Similarly, we can compute the second integral by using the Definition 10, for more details see in [18].
The main objective of this paper is to present some new (p, q) estimates of trapezoidal type inequalities for strongly convex and quasi-convex functions and show the relationship between the results given herein. Some examples are given to illustrate the investigated results. Finally, conclusion part is given at the end.

Trapezoidal Type Inequalities for (p, q)-Quantum Integrals
We are now providing new trapezoidal type inequalities for functions whose absolute value of first (p, q) κ 1 -and (p, q) κ 2 -derivatives are strongly convex functions with modulus χ ≥ 1. To prove our main results, we will initially suggest the following useful lemmas. κ 2 ). Then, the following identity holds: Proof. By using Definitions 8 and 10, we have We observe that Similarly, The equalities (20)-(23) give Multiplying both sides of (24) by q(κ 2 −κ 1 ) 2 , we get (18).

Theorem 7.
If we suppose that all of the criteria of Lemma 4 are satisfied, then the resulting inequality, shows that κ 1 D p,q K σ is a strongly convex functions on [κ 1 , κ 2 ] with modulus χ ≥ 1 for σ ≥ 1, then Proof. Taking modulus on Equation (18) and using the power-mean inequality, we have Using the strongly convexity of κ 1 D p,q K σ on [κ 1 , κ 2 ], we obtain By using Definition 10, we get 2 1 Applying (29)- (32) in (28), we get Similarly, we also observe that We also have Applying (36)- (38) in (27), we obtain the desired inequality.

Corollary 2.
If σ = 1 together with the assumptions of Theorem 7, we obtain where W 1 (p, q), W 2 (p, q) and W 4 (p, q) are defined in Theorem 7.

Corollary 3.
As p = 1 and q → 1 − in Theorem 7, we get the inequality Corollary 4. Suppose that the assumptions of Theorem 7 with σ = 1, p = 1 and letting q → 1 − , we obtain the inequality Theorem 8. If we suppose that all of the criteria of Lemma 4 are satisfied, then the resulting inequality, shows that κ 1 D p,q K σ 2 is a strongly convex functions on [κ 1 , κ 2 ] with modulus χ ≥ 1 where Proof. Taking modulus on Equation (18) and using Hölder inequality, we have We now evaluate the integrals involved in (43). We observe that . (47) Using the strongly convexity of κ 1 D p,q K σ 2 on [κ 1 , κ 2 ], we obtain and similarly, we get Making use of (44) and (49) in (43), we get the required result.
Theorem 9. If we suppose that all of the criteria of Lemma 4 are satisfied, then the resulting inequality shows that κ 1 D p,q K σ is a strongly quasi-convex functions on [κ 1 , κ 2 ] with modulus and W 4 (p, q), W 5 (p, q) are defined in Theorem 7.

. (60)
From the graph below, it is obvious that the LHS is less than or equal to the RHS. Therefore, the inequality (50) is valid for every strongly quasi-convex functions.

Conclusions
Convex functions are represented in terms of different inequalities. Many of the well-known inequalities are consequences of convex functions. Strong convexity is a strengthening of the notion of convexity; some properties of strongly convex functions are just stronger versions of known properties of convex functions. In this research, we identified new results that are used to calculate (p, q) κ 1 and (p, q) κ 2 -trapezoidal integral-type inequalities through strongly convex and quasi-convex functions. Furthermore, some examples were presented to illustrate the outcome of the research.