Abstract

In this work, two generalized quantum integral identities are proved by using some parameters. By utilizing these equalities, we present several parameterized quantum inequalities for convex mappings. These quantum inequalities generalize many of the important inequalities that exist in the literature, such as quantum trapezoid inequalities, quantum Simpson’s inequalities, and quantum Newton’s inequalities. We also give some new midpoint-type inequalities as special cases. The results in this work naturally generalize the results for the Riemann integral.

1. Introduction

Thomas Simpson has developed crucial methods for the numerical integration and estimation of definite integrals considered as Simpson’s rule during 1710–1761. Nevertheless, a similar approximation was used by J. Kepler almost one hundred years earlier, so it is also known as Kepler’s rule. Simpson’s rule includes the three-point Newton–Cotes quadrature rule, so estimation based on the three-step quadratic kernel is sometimes called 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):

There are a large number of estimations related to these quadrature rules in the literature; one of them is the following estimation known as Simpson’s inequality.

Theorem 1. Suppose that is a four-time continuously differentiable mapping on , and let . Then, one has the inequality

In recent years, many authors have focused on Simpson’s type inequality in various categories of mappings. Specifically, some mathematicians have worked on the results of Simpson’s and Newton’s type in obtaining a convex map because convexity theory is an effective and powerful way to solve a large number of problems from different branches of pure and applied mathematics. For example, Dragomir et al. [1] presented the new Simpson’s inequalities and their applications in quadrature formulas for numerical integration. In addition, some inequalities of Simpson’s type of s-convex functions were determined by Alomari et al. in [2]. Subsequently, Sarikaya et al. noted the variance of Simpson’s type inequality based on convexity in [3]. For the further studies of this area, one can consult [46].

On the contrary, many well-known integral inequalities have been studied in the setup of -calculus using the concept of classical convexity [728].

2. Preliminaries of -Calculus and Some Inequalities

In this section, we first present some known definitions and related inequalities in -calculus. Set the following notation (see [29]):

Jackson [30] defined the -integral of a given function from 0 to as follows:provided that the sum converges absolutely. Moreover, he defined the -integral of a given function over the interval as follows:

Definition 1 (see [31]). We consider the mapping . Then, the -derivative of at is defined by the following expression:If , we define if it exists and it is finite.

Definition 2 (see [32]). We consider the mapping . Then, the -derivative of at is defined byIf , we define if it exists and it is finite.

Definition 3 (see [31]). We consider the mapping . Then, the -definite integral on is defined byIn [11, 21], the authors proved quantum Hermite–Hadamard-type inequalities and their estimations by using the notions of the -derivative and -integral.
On the contrary, in [32], Bermudo et al. gave the following definition and obtained the related Hermite–Hadamard-type inequalities:

Definition 4 (see [32]). We consider the mapping . Then, the -definite integral on is defined by

Theorem 2 (see [32]). Let be a convex function on and . Then, -Hermite–Hadamard inequalities are given as follows:

In [33], Budak proved the left and right bounds of inequality (11).

The present paper aims to generalize the results proved in [13, 33, 34]. The key benefit of our paper is that it includes multiple inequalities at the same time, such as Simpson’s inequalities, Newton’s inequalities, midpoint inequalities, and trapezoidal inequities.

3. Crucial Identities

We deal with the three identities which are necessary to obtain our main results in this section.

Let us start with the following useful lemma.

Lemma 1. If is a -differentiable function on such that is continuous and integrable on , then we have the following identity:where .

Proof. By Definition 2, we see thatAfter applying the fundamental properties of quantum integrals, we deduce thatFrom Definition 4, we conclude thatandWe can obtain the required identity (12) by putting the computed integrals (15)–(17) in (14).

Remark 1. If we assume and in Lemma 1, then we obtain Lemma 2 of [13].

Remark 2. In Lemma 1, by taking the limit , we have Lemma 2.1 of [34] for .

Remark 3. In Lemma 1, if we choose , then we obtain the following identity:which is proved by Budak in Lemma 1 of [33].

Corollary 1. In Lemma 1, if we choose and , then we obtain the following new identity:

Lemma 2. If is a -differentiable function on such that is continuous and integrable on , then we have the following identity:where .

Proof. After applying the fundamental properties of quantum integrals, we deduce thatIf the same steps in the proof of Lemma 1 are applied for the rest of this proof, we can obtain the desired identity (20).

Remark 4. By assuming , , and in Lemma 2, we obtain Lemma 3 of [13].

Remark 5. If we take in Lemma 2, then we recapture identity (18).

Corollary 2. If we take the limit in Lemma 2, then we obtain the following new identity:

Now, we calculate the integrals in the following lemma which will be used in our next section.

Lemma 3. The subsequent quantum integrals are true:

Proof. Case I: let .
By the definition of the -integral, we haveCase II: let .
From the definition of the -integral, we getThis gives the proof of equality (28). In a similar way, we can prove the others.

4. Simpson’s Type Inequalities for Quantum Integrals

An extension of quantum Simpson’s inequalities for quantum differentiable convex functions using the quantum integrals is given in this section.

Theorem 3. We assume that the given conditions of Lemma 1 hold. If the mapping is convex on , then the following Simpson’s type inequality holds:where are given in (28)–(30), respectively.

Proof. By Lemma 1 and the convexity of , we conclude thatwhich is the desired conclusion.

Remark 6. By taking the limit in Theorem 3, we have Theorem 2.1 of [34] for .

Remark 7. If we assume in Theorem 3, then we obtain the following trapezoidal type inequality:which is established by Budak in Theorem 3 of [33].

Corollary 3. In Theorem 3, if we choose and , then we obtain the following midpoint-type inequality:

Remark 8. In Theorem 3, if we assume and , then Theorem 3 reduces to Theorem 4 of [13].

Theorem 4. We assume that the given conditions of Lemma 1 hold. If the mapping , , is convex on , then the following Simpson’s type inequality holds:where , , and are given in (23), (24), and (28)–(30), respectively.

Proof. From Lemma 1 and the power mean inequality, we obtainBy applying the convexity of , we obtainand the proof is completed.

Remark 9. If we take the limit in Theorem 4, then we have Theorem 2.3 of [34] for .

Remark 10. If we assume in Theorem 4, then we obtain the following trapezoidal type inequality:which is established by Budak in Theorem 4 of [33].

Remark 11. If we assume and in Theorem 4, then Theorem 4 reduces to Theorem 6 of [13].

Corollary 4. In Theorem 4, if we choose and , then we obtain the following midpoint-type inequality:

Theorem 5. Assume that the given conditions of Lemma 1 hold. If the mapping , , is convex on , then the following Simpson’s type inequality holds:where and

Proof. From Lemma 1 and the Hölder inequality, we haveBy using the convexity of , we obtainand the proof is completed.

Remark 12. If we take the limit in Theorem 5, then Theorem 5 becomes Theorem 2.2 of [34] for .

Remark 13. If we assume and in Theorem 5, then Theorem 5 becomes Theorem 5 of [13].

5. Newton’s Type Inequalities for Quantum Integrals

In this section, a new extension of quantum Newton’s inequalities for quantum differentiable convex functions is given.

Theorem 6. We assume that the given conditions of Lemma 2 hold. If the mapping is convex on , then the following Newton’s type inequality holds:where are given in (31)–(36), respectively.

Proof. If we consider Lemma 2 and apply the same method that is used in the proof of Theorem 3, then we can obtain the desired inequality (53).

Corollary 5. If we take the limit in Theorem 6, then we obtain the following Newton’s type inequality:where<