# Some Parameterized Quantum Simpson’s and Quantum Newton’s Integral Inequalities via Quantum Differentiable Convex Mappings

Authors: Xue Xiao You; Muhammad Aamir Ali; Hüseyin Budak; Miguel Vivas-Cortez; Shahid Qaisar
Journal: Mathematical Problems in Engineering
 (2021)

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## 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.


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## Body

## 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):(1)∫κ1κ2ℱxdx≈κ2−κ16ℱκ1+4ℱκ1+κ22+ℱκ2.(2)
Simpson’s second formula or Newton–Cotes quadrature formula (Simpson’s 3/8 rule):(2)∫κ1κ2ℱxdx≈κ2−κ18ℱκ1+3ℱ2κ1+κ23+3ℱκ1+2κ23+ℱκ2.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 thatℱ:κ1,κ2⟶ℝ is a four-time continuously differentiable mapping on κ1,κ2, and let ℱ4∞=supx∈κ1,κ2ℱ4x<∞. Then, one has the inequality(3)13ℱκ1+ℱκ22+2ℱκ1+κ22−1κ2−κ1∫κ1κ2ℱxdx≤12880ℱ4∞κ2−κ14.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 [4–6].On the contrary, many well-known integral inequalities have been studied in the setup ofq-calculus using the concept of classical convexity [7–28].


## 2. Preliminaries ofq-Calculus and Some Inequalities

In this section, we first present some known definitions and related inequalities inq-calculus. Set the following notation (see [29]):(4)nq=1−qn1−q=∑k=0n−1qk,q∈0,1.Jackson [30] defined the q-integral of a given function ℱ from 0 to κ2 as follows:(5)∫0κ2ℱxdqx=1−qκ2∑n=0∞qnℱκ2qn,where0<q<1,provided that the sum converges absolutely. Moreover, he defined the q-integral of a given function over the interval κ1,κ2 as follows:(6)∫κ1κ2ℱxdqx=∫0κ2ℱxdqx−∫0κ1ℱxdqx.Definition 1 (see [31]).
We consider the mappingℱ:κ1,κ2⟶ℝ. Then, the qκ1-derivative of ℱ at x∈κ1,κ2 is defined by the following expression:(7)κ1Dqℱx=ℱx−ℱqx+1−qκ11−qx−κ1,x≠κ1.
Ifx=κ1, we define κ1Dqℱκ1=limx⟶κ1κ1Dqℱx if it exists and it is finite.Definition 2 (see [32]).
We consider the mappingℱ:κ1,κ2⟶ℝ. Then, the qκ2-derivative of ℱ at x∈κ1,κ2 is defined by(8)κ2Dqℱx=ℱqx+1−qκ2−ℱx1−qκ2−x,x≠κ2.
Ifx=κ2, we define κ2Dqℱκ2=limx⟶κ2κ2Dqℱx if it exists and it is finite.Definition 3 (see [31]).
We consider the mappingℱ:κ1,κ2⟶ℝ. Then, the qκ1-definite integral on κ1,κ2 is defined by(9)∫κ1κ2ℱxκ1dqx=1−qκ2−κ1∑n=0∞qnℱqnκ2+1−qnκ1=κ2−κ1∫01ℱ1−τκ1+τκ2dqτ.
In [11, 21], the authors proved quantum Hermite–Hadamard-type inequalities and their estimations by using the notions of the qκ1-derivative and qκ1-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 mappingℱ:κ1,κ2⟶ℝ. Then, the qκ2-definite integral on κ1,κ2 is defined by(10)∫κ1κ2ℱxκ2dqx=1−qκ2−κ1∑n=0∞qnℱqnκ1+1−qnκ2=κ2−κ1∫01ℱτκ1+1−τκ2dqτ.Theorem 2 (see [32]).
Letℱ:κ1,κ2⟶ℝ be a convex function on κ1,κ2 and 0<q<1. Then, qκ2-Hermite–Hadamard inequalities are given as follows:(11)ℱκ1+qκ22q≤1κ2−κ1∫κ1κ2ℱxκ2dqx≤ℱκ1+qℱκ22q.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.
Ifℱ:κ1,κ2⊂ℝ⟶ℝ is a qκ2-differentiable function on κ1,κ2 such that κ2Dqℱ is continuous and integrable on κ1,κ2, then we have the following identity:(12)1κ2−κ1∫κ1κ2ℱxκ2dqx−βℱκ2+1−αℱκ1+α−βℱκ1+κ22=κ2−κ1∫01/2qτ−βκ2Dqℱτκ1+1−τκ2dqτ+∫1/21qτ−ακ2Dqℱτκ1+1−τκ2dqτ,where q∈0,1.Proof.
By Definition2, we see that(13)κ2Dqℱτκ1+1−τκ2=ℱqτκ1+1−qτκ2−ℱτκ1+1−τκ21−qκ2−κ1τ.
After applying the fundamental properties of quantum integrals, we deduce that(14)∫01/2qτ−βκ2Dqℱτκ1+1−τκ2dqτ+∫1/21qτ−ακ2Dqℱτκ1+1−τκ2dqτ=∫01/2α−βκ2Dqℱτκ1+1−τκ2dqτ+∫01qτ−ακ2Dqℱτκ1+1−τκ2dqτ=α−β∫01/2ℱqτκ1+1−qτκ2−ℱτκ1+1−τκ21−qκ2−κ1τdqτ+q∫01ℱqτκ1+1−qτκ2−ℱτκ1+1−τκ21−qκ2−κ1dqτ−α∫01ℱqτκ1+1−qτκ2−ℱτκ1+1−τκ21−qκ2−κ1τdqτ.
From Definition4, we conclude that(15)∫01/2ℱqτκ1+1−qτκ2−ℱτκ1+1−τκ21−qκ2−κ1τdqτ=1κ2−κ1∑n=0∞ℱqn+12κ1+1−qn+12κ2−∑n=0∞ℱqn2κ1+1−qn2κ2=1κ2−κ1ℱκ2−ℱκ1+κ22,(16)∫01ℱqτκ1+1−qτκ2−ℱτκ1+1−τκ21−qκ2−κ1τdqτ=1κ2−κ1ℱκ2−ℱκ1,and(17)∫01ℱqτκ1+1−qτκ2−ℱτκ1+1−τκ21−qκ2−κ1τdqτ=1κ2−κ1∑n=0∞qnℱqn+1κ1+1−qn+1κ2−∑n=0∞qnℱqnκ1+1−qnκ2=1κ2−κ11q∑n=1∞qnℱqnκ1+1−qnκ2−∑n=0∞qnℱqnκ1+1−qnκ2=1κ2−κ11q∑n=0∞qnℱqnκ1+1−qnκ2−1qℱκ1−∑n=0∞qnℱqnκ1+1−qnκ2=1κ2−κ11qκ2−κ1∫κ1κ2ℱxκ2dqx−1qℱκ1.
We can obtain the required identity (12) by putting the computed integrals (15)–(17) in (14).Remark 1.
If we assumeβ=1/6 and α=5/6 in Lemma 1, then we obtain Lemma 2 of [13].Remark 2.
In Lemma1, by taking the limit q⟶1−, we have Lemma 2.1 of [34] for m=1.Remark 3.
In Lemma1, if we choose β=α=q/2q, then we obtain the following identity:(18)ℱκ1+qℱκ22q−1κ2−κ1∫κ1κ2ℱxκ2dqx=qκ2−κ12q∫011−2qτκ2Dqℱτκ1+1−τκ2dqτ,which is proved by Budak in Lemma 1 of [33].Corollary 1.
In Lemma1, if we choose β=0 and α=1, then we obtain the following new identity:(19)1κ2−κ1∫κ1κ2ℱxκ2dqx−ℱκ1+κ22=κ2−κ1∫01/2qτκ2Dqℱτκ1+1−τκ2dqτ+∫1/21qτ−1κ2Dqℱτκ1+1−τκ2dqτ.Lemma 2.
Ifℱ:κ1,κ2⊂ℝ⟶ℝ is a qκ2-differentiable function on κ1,κ2 such that κ2Dqℱ is continuous and integrable on κ1,κ2, then we have the following identity:(20)1κ2−κ1∫κ1κ2ℱxκ2dqx−βℱκ2+α−βℱκ1+2κ23+γ−αℱ2κ1+κ23+1−γℱκ1=κ2−κ1∫01/3qτ−βκ2Dqℱτκ1+1−τκ2dqτ+∫1/32/3qτ−ακ2Dqℱτκ1+1−τκ2dqτ+∫231qτ−γκ2Dqℱτκ1+1−τκ2dqτ,where q∈0,1.Proof.
After applying the fundamental properties of quantum integrals, we deduce that(21)∫01/3qτ−βκ2Dqℱτκ1+1−τκ2dqτ+∫1/32/3qτ−ακ2Dqℱτκ1+1−τκ2dqτ+∫2/31qτ−γκ2Dqℱτκ1+1−τκ2dqτ=∫01/2α−βκ2Dqℱτκ1+1−τκ2dqτ+∫02/3γ−ακ2Dqℱτκ1+1−τκ2dqτ+∫01qτ−γκ2Dqℱτκ1+1−τκ2dqτ.
If the same steps in the proof of Lemma1 are applied for the rest of this proof, we can obtain the desired identity (20).Remark 4.
By assumingβ=1/8, α=1/2, and γ=7/8 in Lemma 2, we obtain Lemma 3 of [13].Remark 5.
If we takeβ=α=γ=q/2q in Lemma 2, then we recapture identity (18).Corollary 2.
If we take the limitq⟶1− in Lemma 2, then we obtain the following new identity:(22)1κ2−κ1∫κ1κ2ℱxdx−βℱκ2+α−βℱκ1+2κ23+γ−αℱ2κ1+κ23+1−γℱκ1=κ2−κ1∫01/3τ−βℱ′τκ1+1−τκ2dτ+∫1/32/32τ−αℱ′τκ1+1−τκ2dτ+∫2/31τ−γℱ′τκ1+1−τκ2dτ.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:(23)Ξ11=∫01/2qτ−βdqτ=8β2+q42q−β2,q>2β,β2−q42q,q≤2β,(24)Ξ12=∫1/21qτ−αdqτ=α2−3q42q,q<α,8α2+5q42q−3α2,α≤q≤2α,3q42q−α2,q>2α,(25)Ξ13=∫01/3qτ−βdqτ=2β22q+q92q−β3,q>3β,β3−q92q,q≤3β,(26)Ξ14=∫132/3qτ−αdqτ=α3−q32q,q<3α2,18α2+5q92q−α,3α2≤q≤3α,q32q−α3,q>3α,(27)Ξ15=∫2/31qτ−γdqτ=γ3−5q92q,q<γ,18γ2+13q92q−5γ3,γ≤q≤3γ2,5q92q−γ3,q>3γ2,(28)Ξ1=∫01/2τqτ−βdqτ=2β32q3q+q83q−β42q,q>2β,β42q−q83q,q≤2β,(29)Ξ2=∫01/21−τqτ−βdqτ=Ξ11−Ξ1=8β2+β+q42q−β2−q83q−2β32q3q,q>2β,β2−β+q42q+q83q,q≤2β,(30)Ξ3=∫1/21τqτ−αdqτ=3α42q−7q83q,q<α,2α32q3q−5α42q+9q83q,α≤q≤2α,7q83q−3α42q,q>2α,(31)Ξ4=∫1/211−τqτ−αdqτ=Ξ12−Ξ3=α2−3α+q42q+7q83q,q<α,8α2+5q−5α42q−3α2−9q83q−2α32q3q,α≤q≤2α,3α+q42q−α2−7q83q,q>2α,(32)Ξ5=∫01/3τqτ−βdqτ=2β32q3q+q273q−β92q,q>3β,β92q−q273q,q≤3β,(33)Ξ6=∫01/31−τqτ−βdqτ=Ξ13−Ξ5=18β2+β+q92q−β3−q273q−2β32q3q,q>2β,β3−β+q92q+q273q,q≤2β,(34)Ξ7=∫1/32/3τqτ−αdqτ=α32q−7q273q,q<3α2,2α32q3q−5α92q+q33q,3α2≤q≤3α,7q273q−α32q,q>3α,(35)Ξ8=∫1/32/31−τqτ−αdqτ=Ξ14−Ξ7=α3−q+α32q+7q273q,q<3α2,18α2+5q+5α92q−α−q33q−2α32q3q,3α2≤q≤3α,q+α32q−α3−7q273q,q>3α,(36)Ξ9=∫2/31τqτ−γdqτ=5γ92q−19q273q,q<γ,2γ32q3q−13γ92q+35q273q,γ≤q≤3γ2,19q273q−5γ92q,q>3γ2,(37)Ξ10=∫2/311−τqτ−γdqτ=Ξ15−Ξ9=γ3−5q+γ92q+19q273q,q<γ,18γ2+13q+13γ92q−5γ3−35q273q−2γ32q3q,γ≤q≤3γ2,5q+γ92q−γ3−19q273q,q>3γ2.Proof.
Case I: letq>2β.
By the definition of theq-integral, we have(38)Ξ1=∫01/2τqτ−βdqτ=∫0β/qτβ−qτdqτ+∫β/q1/2τqτ−βdqτ=2∫0β/qτβ−qτdqτ+∫01/2τqτ−βdqτ=2β32q3q+q83q−β42q.
Case II: letq≤2β.
From the definition of theq-integral, we get(39)Ξ1=∫01/2τqτ−βdqτ=∫01/2τβ−qτdqτ=β42q−q83q.
This 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 Lemma1 hold. If the mapping κ2Dqℱ is convex on κ1,κ2, then the following Simpson’s type inequality holds:(40)1κ2−κ1∫κ1κ2ℱxκ2dqx−βℱκ2+1−αℱκ1+α−βℱκ1+κ22≤κ2−κ1Ξ1+Ξ3κ2Dqℱκ1+Ξ2+Ξ4κ2Dqℱκ2,where Ξ1–Ξ4 are given in (28)–(30), respectively.Proof.
By Lemma1 and the convexity of κ2Dqℱ, we conclude that(41)1κ2−κ1∫κ1κ2ℱxκ2dqx−βℱκ2+1−αℱκ1+α−βℱκ1+κ22≤κ2−κ1∫01/2qτ−β|κ2Dqℱτκ1+1−τκ2|dqτ+∫1/21qτ−α|κ2Dqℱτκ1+1−τκ2|dqτ≤κ2−κ1κ2Dqℱκ1∫01/2τqτ−βdqτ+∫1/21τqτ−αdqτ+κ2Dqℱκ2∫01/21−τqτ−βdqτ+∫1/211−τqτ−αdqτ=κ2−κ1Ξ1+Ξ3κ2Dqℱκ1+Ξ2+Ξ4κ2Dqℱκ2,which is the desired conclusion.Remark 6.
By taking the limitq⟶1− in Theorem 3, we have Theorem 2.1 of [34] for s=m=1.Remark 7.
If we assumeβ=α=q/2q in Theorem 3, then we obtain the following trapezoidal type inequality:(42)ℱκ1+qℱκ22q−1κ2−κ1∫κ1κ2ℱxκ2dqx≤κ2−κ1κ2Dqℱκ1q23q+3q3q2q4+κ2Dqℱκ2q21+3q2+2q33q2q4,which is established by Budak in Theorem 3 of [33].Corollary 3.
In Theorem3, if we choose β=0 and α=1, then we obtain the following midpoint-type inequality:(43)ℱκ1+κ22−1κ2−κ1∫κ1κ2ℱxκ2dqx≤κ2−κ1342q3qκ2Dqℱκ1+2q2+2q−142q3qκ2Dqℱκ2.Remark 8.
In Theorem3, if we assume β=1/6 and α=5/6, then Theorem 3 reduces to Theorem 4 of [13].Theorem 4.
We assume that the given conditions of Lemma1 hold. If the mapping κ2Dqℱp1, p1≥1, is convex on κ1,κ2, then the following Simpson’s type inequality holds:(44)1κ2−κ1∫κ1κ2ℱxκ2dqx−βℱκ2+1−αℱκ1+α−βℱκ1+κ22≤κ2−κ1Ξ111−1/p1Ξ1κ2Dqℱκ1p1+Ξ2κ2Dqℱκ2p11/p1+Ξ121−1/p1Ξ3κ2Dqℱκ1p1+Ξ4κ2Dqℱκ2p11/p1,where Ξ11, Ξ12, and Ξ1–Ξ4 are given in (23), (24), and (28)–(30), respectively.Proof.
From Lemma1 and the power mean inequality, we obtain(45)1κ2−κ1∫κ1κ2ℱxκ2dqx−βℱκ2+1−αℱκ1+α−βℱκ1+κ22≤κ2−κ1∫01/2qτ−βdqτ1−1/p1∫01/2qτ−βκ2Dqℱτκ1+1−τκ2p1dqτ1/p1+∫1/21qτ−αdqτ1−1/p1∫1/21qτ−ακ2Dqℱτκ1+1−τκ2p1dqτ1/p1.
By applying the convexity ofκ2Dqℱp1, we obtain(46)1κ2−κ1∫κ1κ2ℱxκ2dqx−βℱκ2+1−αℱκ1+α−βℱκ1+κ22≤κ2−κ1∫01/2qτ−βdqτ1−1/p1×κ2Dqℱκ1p1∫01/2τqτ−βdqτ+κ2Dqℱκ2p1∫01/21−τqτ−βdqτ1/p1+∫1/21qτ−αdqτ1−1/p1×κ2Dqℱκ1p1∫1/21τqτ−αdqτ+κ2Dqℱκ2p1∫1/211−τqτ−αdqτ1/p1=κ2−κ1Ξ111−1/p1Ξ1κ2Dqℱκ1p1+Ξ2κ2Dqℱκ2p11/p1+Ξ121−1/p1Ξ3κ2Dqℱκ1p1+Ξ4κ2Dqℱκ2p11/p1,and the proof is completed.Remark 9.
If we take the limitq⟶1− in Theorem 4, then we have Theorem 2.3 of [34] for s=m=1.Remark 10.
If we assumeβ=α=q/2q in Theorem 4, then we obtain the following trapezoidal type inequality:(47)ℱκ1+qℱκ22q−1κ2−κ1∫κ1κ2ℱxκ2dqx≤qκ2−κ12qq2+q+q32q31−1/p1×κ2Dqℱκ1p1q3q+3q3q2q3+κ2Dqℱκ2p1q1+3q2+2q33q2q3,which is established by Budak in Theorem 4 of [33].Remark 11.
If we assumeβ=1/6 and α=5/6 in Theorem 4, then Theorem 4 reduces to Theorem 6 of [13].Corollary 4.
In Theorem4, if we choose β=0 and α=1, then we obtain the following midpoint-type inequality:(48)1κ2−κ1∫κ1κ2ℱxκ2dqx−ℱκ1+κ22≤κ2−κ1q42q1−1/p1κ2Dqℱκ1p1q83q+κ2Dqℱκ2p1q3q+q282q3q1/p1+2−q42q1−1/p1κ2Dqℱκ1p163q−7q2q82q3q+κ2Dqℱκ2p112−3q42q−63q−7q2q82q3q1/p1.Theorem 5.
Assume that the given conditions of Lemma1 hold. If the mapping κ2Dqℱp1, p1>1, is convex on κ1,κ2, then the following Simpson’s type inequality holds:(49)1κ2−κ1∫κ1κ2ℱxκ2dqx−βℱκ2+1−αℱκ1+α−βℱκ1+κ22≤κ2−κ1Ξ161/r1κ2Dqℱκ1p142q+2q+1κ2Dqℱκ2p142q1/p1+Ξ171/r13κ2Dqℱκ1p142q+2q−1κ2Dqℱκ2p142q1/p1,where p1−1+r1−1=1 and(50)Ξ16=∫01/2qτ−βr1dqτ,Ξ17=∫1/21qτ−αr1dqτ.Proof.
From Lemma1 and the Hölder inequality, we have(51)1κ2−κ1∫κ1κ2ℱxκ2dqx−βℱκ2+1−αℱκ1+α−βℱκ1+κ22≤κ2−κ1∫01/2qτ−βr1dqτ1/r1∫01/2κ2Dqℱτκ1+1−τκ2p1dqτ1/p1+∫1/21qτ−αr1dqτ1/r1∫1/21κ2Dqℱτκ1+1−τκ2p1dqτ1/p1.
By using the convexity ofκ2Dqℱp1, we obtain(52)1κ2−κ1∫κ1κ2ℱxκ2dqx−βℱκ2+1−αℱκ1+α−βℱκ1+κ22≤κ2−κ1∫01/2qτ−βr1dqτ1/r1κ2Dqℱκ1p1∫01/2τdqτ+κ2Dqℱκ2p1∫01/21−τdqτ1/p1+∫1/21qτ−αr1dqτ1/r1κ2Dqℱκ1p1∫1/21τdqτ+κ2Dqℱκ2p1∫1/211−τdqτ1/p1=κ2−κ1Ξ161/r1κ2Dqℱκ1p142q+2q+1κ2Dqℱκ2p142q1/p1+Ξ171/r13κ2Dqℱκ1p142q+2q−1κ2Dqℱκ2p142q1/p1,and the proof is completed.Remark 12.
If we take the limitq⟶1− in Theorem 5, then Theorem 5 becomes Theorem 2.2 of [34] for s=m=1.Remark 13.
If we assumeβ=1/6 and α=5/6 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 Lemma2 hold. If the mapping κ2Dqℱ is convex on κ1,κ2, then the following Newton’s type inequality holds:(53)1κ2−κ1∫κ1κ2ℱxκ2dqx−βℱκ2+α−βℱκ1+2κ23+γ−αℱ2κ1+κ23+1−γℱκ1≤κ2−κ1Ξ5+Ξ7+Ξ9κ2Dqℱκ1+Ξ6+Ξ8+Ξ10κ2Dqℱκ2,where Ξ5–Ξ10 are given in (31)–(36), respectively.Proof.
If we consider Lemma2 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 limitq⟶1− in Theorem 6, then we obtain the following Newton’s type inequality:(54)1κ2−κ1∫κ1κ2ℱxdx−βℱκ2+α−βℱκ1+2κ23+γ−αℱ2κ1+κ23+1−γℱκ1≤κ2−κ1Ξ5∗+Ξ7∗+Ξ9∗κ2Dqℱκ1+Ξ6∗+Ξ8∗+Ξ10∗κ2Dqℱκ2,where(55)Ξ5∗=∫01/3ττ−βdτ=β33+181−β18,Ξ6∗=∫01/31−ττ−βdτ=18β2+β+118−2881−β33,Ξ7∗=∫1/32/3τqτ−αdτ=α33−5α18+19Ξ8∗=∫1/32/31−ττ−αdτ=18α2+5+5α18−α−19−α33,Ξ9∗=∫2/31ττ−γdτ=γ33−13γ18+3581,Ξ10∗=∫2/311−ττ−γdτ=18γ2+13+13γ18−5γ3−3581−γ33.Remark 14.
If we takeβ=1/8, α=1/2, and γ=7/8 in Theorem 6, then Theorem 6 reduces to Theorem 7 of [13].Remark 15.
If we assumeβ=α=γ=q/2q in Theorem 6, then we recapture inequality (42).Theorem 7.
We assume that the given conditions of Lemma2 hold. If the mapping κ2Dqℱp1, p1≥1, is convex on κ1,κ2, then the following Newton’s type inequality holds:(56)1κ2−κ1∫κ1κ2ℱxκ2dqx−βℱκ2+α−βℱκ1+2κ23+γ−αℱ2κ1+κ23+1−γℱκ1≤κ2−κ1Ξ131−1/p1Ξ5κ2Dqℱκ1p1+Ξ6κ2Dqℱκ2p11/p1+Ξ141−1/p1Ξ7κ2Dqℱκ1p1+Ξ8κ2Dqℱκ2p11/p1+Ξ151−1/p1Ξ9κ2Dqℱκ1p1+Ξ10κ2Dqℱκ2p11/p1,where Ξ5–Ξ10 and Ξ13–Ξ15 are given in (31)–(36) and (25)–(27), respectively.Proof.
If we apply the steps used in the proof of Theorem4 and taking into account Lemma 2, we can obtain the required inequality (56).Corollary 6.
If we take the limitq⟶1− in Theorem 7, then we obtain the following Newton’s type inequality:(57)1κ2−κ1∫κ1κ2ℱxdx−βℱκ2+α−βℱκ1+2κ23+γ−αℱ2κ1+κ23+1−γℱκ1≤κ2−κ1Π111−1/p1Ξ5∗κ2Dqℱκ1p1+Ξ6∗κ2Dqℱκ2p11/p1+Π121−1/p1Ξ7∗κ2Dqℱκ1p1+Ξ8∗κ2Dqℱκ2p11/p1+Π131−1/p1Ξ9∗κ2Dqℱκ1p1+Ξ10∗κ2Dqℱκ2p11/p1,where Ξ5∗–Ξ10∗ are defined in Corollary 5 and(58)Π11=∫01/3τ−βdτ=β2+192q−β3,Π12=∫1/32/3τ−αdτ=18α2+518−α,Π13=∫2/31τ−γdτ=18γ2+1318−5γ3.Remark 16.
If we takeβ=1/8, α=1/2, and γ=7/8 in Theorem 7, then Theorem 7 reduces to Theorem 9 of [13].Remark 17.
If we assumeβ=α=γ=q/2q in Theorem 7, then we recapture inequality (47).Theorem 8.
We assume that the given conditions of Lemma2 hold. If the mapping κ2Dqℱp1, p1>1, is convex on κ1,κ2, then the following Newton’s type inequality holds:(59)1κ2−κ1∫κ1κ2ℱxκ2dqx≤−βℱκ2+α−βℱκ1+2κ23+γ−αℱ2κ1+κ23+1−γℱκ1κ2−κ1Ξ181/r1κ2Dqℱκ1p192q+3q+2κ2Dqℱκ2p192q1/p1+Ξ191/r1κ2Dqℱκ1p132q+qκ2Dqℱκ2p132q1/p1+Ξ201/r15κ2Dqℱκ1p192q+3q−2κ2Dqℱκ2p192q1/p1,where p1−1+r1−1=1 and(60)Ξ18=∫01/3qτ−βr1dqτ,Ξ19=∫1/32/3qτ−αr1dqτ,Ξ20=∫2/31qτ−γr1dqτ.Proof.
If we apply the steps used in the proof of Theorem5 and taking into account Lemma 2, we can obtain the required inequality (59).Remark 18.
If we takeβ=1/8, α=1/2, and γ=7/8 in Theorem 8, then Theorem 8 becomes Theorem 8 of [13].


## 6. Conclusions

In this work, using quantum integrals, we developed a new extension of quantum trapezoid, quantum Simpson’s, and quantum Newton’s type estimations for quantum differentiable convex functions. It is also shown that the results presented here are generalizations of the findings presented in [13, 33, 34]. In the special cases of newly developed findings, we also obtained several new Simpson’s type, Newton’s type, midpoint-type, and trapezoidal type inequalities. It is an interesting and innovative problem that future researchers may investigate in order to achieve similar inequalities for convex and coordinated convex functions via different quantum integrals.



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