Some New Simpson’s and Newton’s Formulas Type Inequalities for Convex Functions in Quantum Calculus

: In this paper, using the notions of q κ 2 -quantum integral and q κ 2 -quantum derivative, we present some new identities that enable us to obtain new quantum Simpson’s and quantum Newton’s type inequalities for quantum differentiable convex functions. This paper, in particular, generalizes and expands previous ﬁndings in the ﬁeld of quantum and classical integral inequalities obtained by various authors.


Introduction
Thomas Simpson developed crucial techniques for numerical integration and estimation of definite integrals, which became known as Simpson's law during his lifetime (1710-1761). However, J. Kepler used a similar approximation nearly a century before, so it is also known as Kepler's law. Since Simpson's rule includes the three-point Newton-Cotes quadrature rule, estimations based entirely on a three-step quadratic kernel are often referred to as Newton-type results.
(1) Simpson's quadrature formula (Simpson's 1/3 rule): (2) Simpson's second formula or Newton-Cotes quadrature formula (Simpson's 3/8 rule): Within the literature, there are a plethora of estimations correlated with certain quadrature laws, one of which is the following estimation known as Simpson's inequality: Theorem 1. Let a mapping F : [κ 1 , κ 2 ] → R be four times continuously differentiable on (κ 1 , κ 2 ), and let F (4) ∞ = sup x∈(κ 1 ,κ 2 ) F (4) (x) < ∞. Then, one has the inequality 1 3 F (4) generalized the results of [14] and proved HH type inequalities and their left estimates using the κ 2 D p,q -difference operator and (p, q) κ 2 -integral. Inspired by the ongoing studies, we use the q-integral to develop some new quantum Simpson's and quantum Newton's formulas type inequalities for q-differentiable convex functions, and these inequalities can be helpful for finding the bounds of Simpson's and Newton's formulas for numerical integration. We also show that the newly developed inequalities are extensions of some previously known inequalities.
The following is the structure of this paper: Section 2 provides a brief overview of the fundamentals of q-calculus as well as other related studies in this field. In Section 3, we establish two crucial identities that play an essential role in developing the main results of this paper. The Simpson's and Newton's type inequalities for q-differentiable functions via q-integrals are described in Section 4. The relationship between the findings reported here and similar findings in the literature are also taken into account. Section 5 concludes with some recommendations for future research.

Preliminaries of q-Calculus and Some Inequalities
In this section, we first present the definitions and some properties of quantum derivatives and quantum integrals. We also mention some well-known inequalities for quantum integrals. The following notation will be used frequently in this work (see [8]): Jackson defined the q-Jackson integral from 0 to κ 2 for 0 < q < 1 as follows: provided the sum converges absolutely [11].
On the other hand, Bermudo et al. defined a new quantum derivative and a quantum integral, which are called the q κ 2 -derivative and q κ 2 -integral: if it exists and it is finite.
Moreover, we will need to use the subsequent Lemma in our key results: . We have the equality for a ∈ R\{−1}.

Identities
We deal with identities, which is necessary to attain our main estimations in this section. We first establish an identity based on a two steps kernel in the following Lemma.
Lemma 2. Let F : [κ 1 , κ 2 ] → R be a q κ 2 -differentiable function on (κ 1 , κ 2 ) and 0 < q < 1. If κ 2 D q F is continuous and integrable on [κ 1 , κ 2 ], then one has the identity where: Proof. Using Formula (2), from the definition of the function Λ(s), we find that By Definition 3, one also has Now, if we substitute the above Equation into (7), we obtain: Calculating the first integral in the right-hand side of (8) by taking into account the case when κ 1 = 0 of Definition 2, it is found that If we similarly observe the other integrals in the right-hand side of (8), from Definition 4, then we obtain Substituting Expressions (9)-(11) into (8), and later multiplying both sides of the resulting identity by q(κ 2 − κ 1 ), Equality (6) can be captured.
We now observe how an equality comes out when we use the kernel mapping with three sections.
, then one has the identity where: Proof. Using Formula (2), from the definition of the function ∆(s), we find that If the same steps in the proof of Lemma 2 are applied for the rest of this proof, the desired result can be obtained.

Main Results
For brevity, we start this section with the following notations, which will be used in the new results: and

Simpson's Inequalities for q κ 2 -Quantum Integral
In this subsection, we prove some Simpson's type inequalities by using Lemma 2.
Let us start to find some new quantum estimates by using Lemma 2. We first examine a new result for functions whose q κ 2 -derivatives in modulus are convex in the following theorem.
Proof. On taking the modulus in Lemma 2, because of the properties of modulus, we find that To calculate integrals in the right-hand side of (23), using the convexity of κ 2 D q F , it follows that Thus, we obtain Similarly, using Equation (2) in addition to the convexity of κ 2 D q F and Lemma 1, we have By putting (24) and (25) into (23), we attain Inequality (22), which finishes the proof.

Corollary 1.
Under the assumptions of Theorem 4 with q → 1 − , we have the following inequality of Simpson's type for the function whose modulus values of first derivative are convex (see [2]): Now, we observe how the inequalities come out when we use the mappings whose powers of q κ 2 -derivatives in absolute value are convex. Theorem 5. Let F : [κ 1 , κ 2 ] → R be a q κ 2 -differentiable function on (κ 1 , κ 2 ) such that κ 2 D q F is continuous and integrable on [κ 1 , κ 2 ]. If κ 2 D q F p 1 is convex on [κ 1 , κ 2 ] for some p 1 > 1, then we have following inequality for q κ 2 -integrals: where 0 < q < 1 and 1 p 1 + 1 r 1 = 1.
Proof. Applying the well-known Hölder's inequality for quantum integrals to the integrals in the right-hand side of (23), it is found that By using the convexity of κ 2 D q F p 1 , we obtain To calculate the integrals in the right-hand side of (27), if we first use Rule (1), then we obtain Similarly, we have For the other integrals in the right-hand side of (27), using the case when κ 1 = 0 of Lemma 1, we find that Similarly, we get By substituting (28)- (33) into (27), we obtain the desired Inequality (26), which completes the proof. Theorem 6. Let F : [κ 1 , κ 2 ] → R be a q κ 2 -differentiable function on (κ 1 , κ 2 ) such that κ 2 D q F is continuous and integrable on [κ 1 , κ 2 ]. If κ 2 D q F p 1 is convex on [κ 1 , κ 2 ] for some p 1 ≥ 1, then we have following inequality for q κ 2 -integrals: where 0 < q < 1, and A 1 (q), A 2 (q), B 1 (q), B 2 (q) are given as in (12)-(15), respectively.
Proof. Utilizing from the results in the proof of Theorem 4 after applying the well-known Power mean inequality to the integrals in the right-hand side of (23), owing to the convexity of κ 2 D q F p 1 , we find that We also observe that and by using similar operations, we have By substituting (36) and (37) into (35), we attain the required Inequality (34). Hence, the proof is completed.

Corollary 2.
Under the given conditions in Theorem 6 with q → 1 − , we have following inequality given by Alomari et al. (see [2]):

Newton's Inequalities for q κ 2 -Quantum Integral
In this subsection, we establish some new Newton's type inequalities by using Lemma 3.
In the next theorems, we will present new quantum estimates by using the convexity of q κ 2 -derivatives in modulus and Lemma 3.
Proof. On taking the modulus in Lemma 3, we gain From now on, it is sufficient to use the same methods as in the proof of Theorem 4 in order to reach desired result (38).

Remark 1.
Under the given conditions of Theorem 7 with the limit q → 1 − , we have following inequality: which is given by Noor et al. in [5].
Proof. If we apply Hölder's inequality to the expressions in the right-hand side of (39), then we obtain For the rest of this proof, if the same procedure that was used in the proof of Theorem 5 is applied to the above inequality, then Result (40) can be captured. Theorem 9. Let F : [κ 1 , κ 2 ] → R be a q κ 2 -differentiable function on (κ 1 , κ 2 ) such that κ 2 D q F is continuous and integrable on [κ 1 , κ 2 ]. If κ 2 D q F p 1 is convex on [κ 1 , κ 2 ] for some p 1 ≥ 1, then we have the following inequality for q κ 2 -integrals: where 0 < q < 1, and A 3 (q), A 4 (q), A 5 (q), B 3 (q), B 4 (q),B 5 (q) are given as in (16)- (21), respectively.
Proof. If the strategy that was used in the proof of Theorem 6 is applied by taking into account Lemma 3, the desired Inequality (41) can be attained.

Remark 2.
Under the conditions of Theorem 9 with the limit q → 1 − , we obtain the following inequality: , which can be found in [5].

Conclusions
We conclude our work by stating that, using the concepts of quantum derivative and quantum integral, we proved some new Simpson's and Newton's type quantum integral inequalities for quantum differentiable convex functions, and these inequalities can be helpful for finding the bounds of Simpson's and Newton's formulas for numerical integration. It is important to note that by considering the limit q → 1 − in our key results, our results were transformed into some new and well-known results. We believe it is an interesting and creative problem for upcoming researchers who will be able to obtain similar inequalities for various types of convexity and co-ordinated convexity in their future work.  Acknowledgments: The first author would like to thank the Department of Mathematics, Faculty of Science, Kasetsart University.

Conflicts of Interest:
The authors declare no conflict of interest.