On generalizations of quantum Simpson’s and quantum Newton’s

1 Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Bangkok, 10800, Thailand 2 Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210023, China 3 Department of Mathematics, Faculty of Science and Arts, Düzce University, Düzce-TURKEY 4 Department of Mathematics, Government College University Lahore, Pakistan 5 Department of Medical research, China Medical University Hospital, China, Medical University, Taichung, Taiwan 6 Mathematics Department, Faculty of Science and Technology, Suan Dusit University, Bangkok, 10300, Thailand

In the literature, there are several estimations linked to these quadrature laws, one of which is known as Simpson's inequality: Theorem 1.1. Suppose that Π : [π 1 , π 2 ] → R is a four times continuously differentiable mapping on (π 1 , π 2 ) , and let Π (4) ∞ = sup x∈(π 1 ,π 2 ) Π (4) (x) < ∞. Then, one has the inequality 1 3 Many authors have concentrated on Simpson's type inequalities for different classes of functions in recent years. Since convexity theory is an effective and efficient method for solving a large number of problems that exist within various branches of pure and applied mathematics, some mathematicians have worked on Simpson's and Newton's type results for convex mappings. Dragomir et al. [1], presented new Simpson's type inequalities and their applications to numerical integration quadrature formulas. Furthermore, Alomari et al. in [2] derive some Simpson's type inequalities for s-convex functions. Following that, in [3], Sarikaya et al. discovered variants of Simpson's type inequalities dependent on convexity. The authors given some Newton's type inequalities for harmonic and pharmonic convex functions in [4,5]. Iftikhar et al. also have new Newton's type inequalities for functions whose local fractional derivatives are generalized convex in [6].
On the other hand, in the domain of q analysis, many works are being carried out as initiated by Euler in order to attain adeptness in mathematics that constructs quantum computing q calculus considered as a relationship between physics and mathematics. In different areas of mathematics, it has numerous applications such as combinatorics, number theory, basic hypergeometric functions, orthogonal polynomials, and other sciences, as well as mechanics, the theory of relativity, and quantum theory [7,8]. Quantum calculus also has many applications in quantum information theory, which is an interdisciplinary area that encompasses computer science, information theory, philosophy, and cryptography, among other areas [9,10]. Apparently, Euler invented this important branch of mathematics. He used the q parameter in Newton's work on infinite series. Later, in a methodical manner, the q-calculus, calculus without limits, was firstly given by Jackson [11,12]. In 1966, Al-Salam [13] introduced a q-analogue of the q-fractional integral and q-Riemann-Liouville fractional. Since then, related research has gradually increased. In particular, in 2013, Tariboon [14] introduced the π 1 D q -difference operator and q π 1 -integral. In 2020, Bermudo et al. [15] introduced the notion of π 2 D q derivative and q π 2 -integral. Sadjang [16] generalized to quantum calculus and introduced the notions of post-quantum calculus, or briefly (p, q)-calculus. Soontharanon et al. [17] introduced the fractional (p, q)-calculus later on. In [18], Tunç and Göv gave the post-quantum variant of π 1 D qdifference operator and q π 1 -integral. Recently, in 2021, Chu et al. [19] introduced the notions of π 2 D p,q derivative and (p, q) π 2 -integral.
Inspired by this ongoing studies, we offer some new quantum parameterized Simpson's and Newton's type inequalities for convex functions using the notions of quantum derivatives and integrals.
The structure of this paper is as follows: Section 2 provides a quick review of the ideas of q-calculus, as well as some related works. In Section 3, we present two integral identities that aid in the proof of the key conclusions. We prove quantum Simpson's and quantum Newton's inequalities in sections 4 and 5, respectively. Section 6 finishes with a few suggestions for future research.

Preliminaries of q-calculus and some inequalities
In this section, we first present some known definitions and related inequalities in q-calculus. Set the following notation(see, [8]): Jackson [11] defined the q-integral of a given function Π from 0 to π 2 as follows: provided that the sum converges absolutely. Moreover, he defined the q-integral of a given function over the interval [π 1 , π 2 ] as follows: Definition 2.1. [14] We consider the mapping Π : [π 1 , π 2 ] → R. Then, the q π 1 -derivative of Π at x ∈ [π 1 , π 2 ] is defined by the the following expression
In [22,27], the authors proved quantum Hermite-Hadamard type inequalities and their estimations by using the notions of q π 1 -derivative and q π 1 -integral.

Crucial identities
To obtain the key results of this paper, we prove three separate identities in this section. Let's begin with the following crucial Lemma.
[6] q , then we have the following identity: which is proved by Iftikhar et al. in [41].
such that π 1 D q Π is continuous and integrable on [π 1 , π 2 ], then we have the following identity: Proof. By the fundamental properties of quantum integrals, we have By applying the same steps in the proof of Lemma 3.1 for rest of this proof, one can obtain the desired identity (3.7).
[8] q in Lemma 3.2, then we obtain the following identity which is proved by Erden et al. in [44].
Remark 3.6. If we take λ = µ = ν = 1 [2] q , in Lemma 3.2, then we obtain [42, Lemma 3.1]. Corollary 3.1. If we take the limit q → 1 − in Lemma 3.2, then we obtain the following new identity For brevity, let us prove another lemma that will be used frequently in the main results.
Lemma 3.3. The following quantum integrals holds for λ, µ, ν ≥ 0: ) ) ) ) ) (3.21) Proof. By the definition of q-integral, we have and so This gives the proof of the equality (3.13). The others can be calculated in similar way.

Generalizations of Simpson's type inequalities for quantum integrals with two parameters
In this section, we prove a new generalization of quantum Simpson's, Midpoint and Trapezoid type inequalities for quantum differentiable convex functions.
Proof. By taking the modulus in Lemma 3.1 and using the convexity of π 1 D q Π , we obtain which is the desired inequality.  [6] q in Theorem 4.1, then we obtain the following inequality which is proved by Ifitikhar et al. [41].
Proof. By taking the modulus in Lemma 3.1 and using the power mean inequality, we have By using the convexity of π 1 D q Π p 1 , we have where A 1 (q) , A 2 (q) , B 1 (q) and B 2 (q) are defined in Remark 4.4. The above inequality is proved by Ifitikhar et al. [41]. Theorem 4.3. We assume that the given conditions of Lemma 3.1 hold. If the mapping π 1 D q Π p 1 , , then the following inequality holds: and Proof. By taking the modulus in Lemma 3.1 and using the Hölder inequality, we have Since π 1 D q Π p 1 is convex on [π 1 , π 2 ], we have This completes the proof.
which is established by Iftikhar et al. in [41].

Generalizations of Newton's type inequalities for quantum integrals with three parameters
Some new generalized versions of quantum Newton's and Trapezoid type inequalities for quantum differentiable convex functions are offered in this section.
Proof. By considering Lemma 3.2 and applying the same method that used in the proof of Theorem 4.1, then we can obtain the desired inequality (5.1).

Conclusions
To sum up, we provided some generalisations of quantum Simpson's and quantum Newton's inequalities for quantum differentiable convex functions with two and three parameters, respectively. It is important to note that by considering the limit q → 1 − and different special choices of the involved parameters in our key results, our results transformed into some new and well-known results. We believe that it is an interesting and innovative problem for future researchers who can obtain similar inequalities for different types of convexity and quantum integrals.