# Some New Generalized Fractional Newton’s Type Inequalities for Convex Functions

Authors: Jarunee Soontharanon; Muhammad Aamir Ali; Hüseyin Budak; Pinar Kösem; Kamsing Nonlaopon; Thanin Sitthiwirattham
Journal: Journal of Function Spaces
 (2022)

---

## Abstract

In this paper, we establish some new Newton’s type inequalities for differentiable convex functions using the generalized Riemann-Liouville fractional integrals. The main edge of the newly established inequalities is that these can be turned into several new and existing inequalities for different fractional integrals like Riemann-Liouville fractional integrals,k-fractional integrals, Katugampola fractional operators, conformable fractional operators, Hadamard fractional operators, and fractional operators with the exponential kernel without proving one by one. It is also shown that the newly established inequalities are the refinements of the previously established inequalities inside the literature.


---

## Body

## 1. Introduction

The fascinating idea of inequalities has long been a topic of discussion in various mathematical disciplines. Fractional calculus, quantum calculus, operator theory, numerical analysis, operator equations, network theory, and quantum information theory are just a few fascinating applications. This is a very active study topic right now, and the interplay between different areas has enriched it. Numerical integration and definite integral estimation are important aspects of applied sciences. Among the numerical techniques, Simpson’s rules are crucial that can be stated as follows:(1)
Simpson’s1/3 rule:
(1)∫θ1θ2Gϰdϰ≈θ2−θ16Gθ1+Gθ1+θ22+Gθ2(2)
Simpson’s3/8 rule (Newton rule):
(2)∫θ1θ2Gϰdϰ≈θ2−θ18Gθ1+3Gθ1+2θ23+3G2θ1+θ23+Gθ2Researchers have used fractional calculus to develop different fractional integral inequalities that are beneficial in approximation theory due to their importance. Inequalities like Hermite-Hadamard, Simpson’s, midpoint, Ostrowski’s, and trapezoidal inequalities are examples of inequalities that may be used to find the boundaries of numerical integration formulas. The bounds of trapezoidal formula and inequality of Hermite-Hadamard type using the Riemann-Liouville fractional integrals were established in [1]. Set [2] used differentiable convexity and established fractional Ostrowski’s type inequalities. İscan and Wu [3] proved some bounds of numerical integration and inequality of the Hermite-Hadamard type for reciprocal convex functions via Riemann-Liouville fractional integrals. The bounds of midpoint and a new version of fractional inequality of Hermite-Hadamard type were established by Sarikaya and Yildrim in [4]. The bounds for Simpson’s 1/3 formula were obtained by Sarikaya et al. [5] using the general convexity and Riemann-Liouville fractional integral operators. In [6], the authors found some new bounds for Simpson’s 1/3 formula using the Riemann-Liouville fractional integrals. The authors of [7] used s-convexity and found some bounds for Simpson’s 1/3 formula. In 2020, Sarikaya and Ertugral [8] gave a new class of fractional integrals called generalized fractional integrals and established Hermite-Hadamard-type inequalities connected to the newly defined class of integrals. The main advantage of the newly defined class of fractional integral operators is that it can be converted into the classical integral, Riemann-Liouville fractional integrals, k-fractional integrals, Hadamard fractional integrals, etc. In [9], Zhao et al. obtained some bounds for a trapezoidal formula using the reciprocal convex functions and generalized fractional integral operators. Budak et al. [10] established some bounds for Simpson’s 1/3 formula for differentiable convex functions using the generalized fractional integrals. Some bounds for the q-Simpson’s and Newton’s type inequalities were proved by Budak et al. in [11]. Siricharuanun et al. proved some inequalities of Simpson and Newton type by using quantum numbers in [12]. Until recent years, Newton-type inequalities for fractional integrals had not been proven. Recently, Sitthiwirattham et al. [13] used the Riemann-Liouville fractional integrals operators and obtained some bounds for Newton formula.Motivated by the ongoing studies, we obtain some new bounds/inequalities for Newton formula using the convexity and generalized fractional integrals. The main edge of newly established inequalities is that these can be converted into classical Newton inequalities, Riemann-Liouville fractional Newton inequalities and new Newton inequalities fork-fractional integrals without establishing one by one. These results can be helpful in finding the error bounds of Newton formulas in fractional calculus, which is the main motivation of this paper. Moreover, the main difference between the results proved in [11–13] and the results of this paper is that while the papers [11, 12] are derived on Newton type inequalities for quantum integrals and the paper [13] focus on Newton type inequalities for Riemann-Liouville  fractional integrals operators, we prove some inequalities of Newton type by using the generalized fractional integrals. These inequalities generalize the results of the paper [13] and give some new inequalities for k-fractional integrals, Hadamard fractional integrals, conformable fractional integrals, etc.On the other hand, there are many other papers related to our topic. One can consult [14–25] and references therein for more inequalities via fractional integrals. Moreover, several papers focused on the functions of bounded variation to prove some important inequalities such as the Ostrowski type [26], Simpson type [27, 28], trapezoid type [29, 30], and midpoint type [31]. For more applications of fractional calculus in other areas of mathematical sciences, one can consult [32–41].A description of the paper is as follows: In Section2, the fundamentals of fractional calculus, as well as other pertinent research in this field, are briefly discussed. In Section 3, we develop an essential identity that is vital in identifying the key outcomes of the paper. In Section 4, we use generalized fractional integrals to derive some new Newton’s type inequalities for differentiable convex functions. For functions of bounded variation, Section 5 contains certain fractional Newton-type inequalities. Section 6 concludes with some future study ideas.


## 2. Fractional Integrals and Related Inequalities

Several fundamental fractional integral notations and concepts are reviewed in this section. Different fractional integrals are also used to recall various inequalities.Definition 1.
A functionG:I⟶ℝ, where I is an interval in ℝ, is called convex, if it satisfies the inequality
(3)Gtϰ+1−ty≤tGϰ+1−tGy,where ϰ,y∈I and t∈0,1.Definition 2 ([42, 43]).
LetG∈L1θ1,θ2. The Riemann-Liouville fractional integrals (RLFIs) Jθ1+αG and Jθ2−αG of order α>0 with θ1≥0 are defined as follows:
(4)Jθ1+αGϰ=1Γθ1∫θ1ϰϰ−tα−1Gtdt,ϰ>θ1,Jθ2−αGϰ=1Γα∫ϰθ2t−ϰα−1Gtdt,ϰ<θ2,respectively, where the well-known Gamma function is represented by Γ.Definition 3 ([44]).
LetG∈L1θ1,θ2. The k-Riemann-Liouville fractional integrals (KRLFIs) Jθ1+α,kG and Jθ2−,kα,kG of order α,k>0 with θ1≥0 are defined as follows:
(5)Jθ1+α,kGϰ=1kΓkα∫θ1ϰϰ−tα/k−1Gtdt,ϰ>θ1,Jθ2−,kα,kGϰ=1kΓkα∫ϰθ2t−ϰα/k−1Gtdt,ϰ<θ2,respectively, where Γk is the well-known k-Gamma function.Definition 4 ([8]).
LetG∈L1θ1,θ2. The generalized fractional integrals (GRLFIs) Iθ1+φG and Iθ2−φG with θ1≥0 are defined as follows:
(6)Iθ1+φGϰ=∫θ1ϰφϰ−tϰ−tGtdt,ϰ>θ1,Iθ2−φGϰ=∫ϰθ2φt−ϰt−ϰGtdt,ϰ<θ2,respectively, where the mapping is φ:0,∞⟶0,∞. One can consult [8] for further information of function φ.Remark 5.
The GRLFIs are significant because they can be converted into classical Riemann integrals, RLFIs, and KFIs forφλ=λ,φλ=λα/Γα and φλ=λα/k/kΓkα, respectively. For more choices of the function φ, one can recapture the different fractional integrals like Katugampola fractional operators, conformable fractional integrals, Hadamard fractional operators, and fractional operators with the exponential kernel (see [8]).In [45], Ertuğral and Sarikaya used GRLFIs and proved the following Simpson’s type inequalities for differentiable convex functions.Theorem 6.
LetG:I⊂ℝ⟶ℝ be a differentiable function over I∘ and G′∈L1θ1,θ2. If G′ is convex over θ1,θ2, then the following inequality holds:
(7)16Gθ1+4Gθ1+θ22+Gθ2−12Θ1Iθ1+φGθ1+θ22+Iθ2−φGθ1+θ22≤θ2−θ12Θ1ΩtG′θ1+G′θ2,where
(8)Ωt=∫01Θt2−Θ13dt,Θϰ=∫0ϰφθ2−θ1/2ttdt.It is worth mentioning here that the inequality (7) can be turned into classical Simpson’s inequality, RLFIs Simpson’s inequality, and KRLFIs inequality as follows:
(i)
Forφt=t, the following Simpson’s inequality for classical Riemann-integral holds (see [5]):
(9)16Gθ1+4Gθ1+θ22+Gθ2−1θ2−θ1∫θ1θ2Gϰdϰ≤5θ2−θ172G′θ1+G′θ2(ii)
Forφt=tα/Γα, the following Simpson’s inequality for RLFIs holds (see [45]):
(10)16Gθ1+4Gθ1+θ22+Gθ2−Γα+121−αθ2−θ1αJθ1+αGθ1+θ22+Jθ2−αGθ1+θ22≤θ2−θ12ϜαG′θ1+G′θ2,where
(11)Ϝα=231/a+1αα+1+12α+1−13(iii)
Forφt=tα/k/kΓkα, the following Simpson’s inequality for KRLFIs holds (see [45]):
(12)16Gθ1+4Gθ1+θ22+Gθ2−Γkα+k21−α/kθ2−θ1α/kJθ1+α,kGθ1+θ22+Jθ2−α,kGθ1+θ22≤θ2−θ12Ϝα,kG′θ1+G′θ2,where
(13)Ϝα,k=23k/α+1αα+k+k2α+k−13Remark 7.
If we setα=k=1 in (10) and (12), then we obtain the classical Simpson’s inequality (9).


## 3. An Identity

In this section, we prove an integral equality in order to demonstrate the primary findings of the paper. For brevity, we shall use the following notation throughout the paper:(14)Yϰ=∫0ϰφθ2−θ1/3uudu<+∞.Lemma 8.
IfG:I⊂ℝ⟶ℝ is a function such that G is differentiable over I∘ and G′∈L1θ1,θ2, then the following identity holds for GRLFIs:(15)13Y1Iθ1+φG2θ1+θ23+I2θ1+θ2/3+φGθ1+2θ23+Iθ1+2θ2/3+φGθ2−18Gθ1+3G2θ1+θ23+3Gθ1+2θ23+Gθ2=θ2−θ19Y1I1+I2+I3,where
(16)I1=∫01Yt−5Y18G′tθ1+1−t2θ1+θ23dt,I2=∫01Yt−Y12G′t2θ1+θ23+1−tθ1+2θ23dt,I3=∫01Yt−3Y18G′tθ1+2θ23+1−tθ2dt.Proof.
Using the laws of integration by parts and variables change, we have(17)I1=∫01Yt−5Y18G′tθ1+1−t2θ1+θ23dt=3θ2−θ1∫01φθ2−θ1/3ttGtθ1+1−t2θ1+θ23dt−Y1θ2−θ1158G2θ1+θ23+98Gθ1=3θ2−θ1Iθ1+φG2θ1+θ23−Y1θ2−θ1158G2θ1+θ23+98Gθ1.
Also, we have(18)I2=∫01Yt−Y12G′t2θ1+θ23+1−tθ1+2θ23dt=3θ2−θ1I2θ1+θ2/3+φGθ1+2θ23−Y1θ2−θ132Gθ1+2θ23+32G2θ1+θ23,(19)I3=∫01Yt−3Y18G′tθ1+2θ23+1−tθ2dt=3θ2−θ1Iθ1+2θ2/3+φGθ2−Y1θ2−θ198Gθ2+158Gθ1+2θ23.
As a consequence, we may get the resultant equality by adding (17)–(19) and multiplying the resultant one by θ2−θ1/9Y1.


## 4. Newton’s Inequalities for Convex Functions

We will utilize GRLFIs to demonstrate some new Newton’s inequalities for differentiable convex functions in this section. We use the following notations for sake of brevity:(20)A1α=∫01tYt−3Y18dt,A2α=∫01Yt−3Y18dt,A3α=∫01tYt−Y12dt,A4α=∫01Yt−Y12dt,A5α=∫01tYt−5Y18dt,A6α=∫01Yt−5Y18dt.Theorem 9.
IfG′ is a convex function and assumptions of Lemma 8 hold, then we obtain the following Newton’s type inequality:
(21)13Y1Iθ1+φG2θ1+θ23+I2θ1+θ2/3+φGθ1+2θ23+Iθ1+2θ2/3+φGθ2−18Gθ1+3G2θ1+θ23+3Gθ1+2θ23+Gθ2≤θ2−θ127Y1G′θ23A2α−A1α+2A4α−A3α+A6α−A5α+G′θ1A1α+A4α+A3α+2A6α+A5α.Proof.
Using the convexity ofG′ and the modulus in (15), we get
(22)13Y1Iθ1+φG2θ1+θ23+I2θ1+θ2/3+φGθ1+2θ23+Iθ1+2θ2/3+φGθ2−18Gθ1+3G2θ1+θ23+3Gθ1+2θ23+Gθ2≤θ2−θ19∫01Yt−3Y18G′tθ1+2θ23+1−tθ2dt+∫01Yt−Y12G′t2θ1+θ23+1−tθ1+2θ23dt+∫01Yt−5Y18G′tθ1+1−t2θ1+θ23dt=θ2−θ19∫01Yt−3Y18G′3−t3θ2+t3θ1dt+∫01Yt−Y12G′2−t3θ2+1+t3θ1dt+∫01Yt−5Y18G′1−t3θ2+2+t3θ1dt≤θ2−θ19G′θ2∫013−t3Yt−3Y18dt+G′θ1∫01t3Yt−3Y18dt+G′θ2∫012−t3Yt−Y12dt+G′θ1∫011+t3Yt−Y12dt+G′θ2∫011−t3Yt−5Y18dt+G′θ1∫012+t3Yt−5Y18dt=θ2−θ127Y1G′θ23A2α−A1α+2A4α−A3α+A6α−A5α+G′θ1A1α+A4α+A3α+2A6α+A5α.
The proof is now completed.Remark 10.
In Theorem9, we have the following:
(i)
By settingφt=t, we reclaim the inequality established in ([13], Remark 3)(ii)
By settingφt=tα/Γα, we reclaim the inequality established in ([13], Theorem 4)Corollary 11.
By settingφt=tα/k/kΓkα in Theorem 9, we get the following new Newton’s inequality for KRLFIs:
(23)3α/k−1Γkα+1θ2−θ1α/kJθ1+α,kG2θ1+θ23+J2θ1+θ2/3+α,kGθ1+2θ23+Jθ1+2θ2/3+α,kGθ2−18Gθ1+3G2θ1+θ23+3Gθ1+2θ23+Gθ2≤θ2−θ127G′θ23A2α,k−A1α,k+2A4α,k−A3α,k+A6α,k−A5α,k+G′θ1A1α,k+A4α,k+A3α,k+2A6α,k+A5α,k,where
(24)A1α,k=∫01ttα/k−38dt=αα+2k38α+2k/α+kα+2k−316,A2α,k=∫01tα/k−38dt=2αα+k38α+k/α+kα+k−38,A3α,k=∫01ttα/k−12dt=αα+2k12α+2k/α+kα+2k−14,A4α,k=∫01tα/k−12dt=2αα+k12α+k/α+1α+k−12,A5α,k=∫01ttα/k−58dt=αα+2k58α+2k/a+kα+2k−516,A6α,k=∫01tα/k−58dt=2αα+k58α+k/a+kα+k−58.Theorem 12.
IfG′q,q≥1 is a convex function and assumptions of Lemma 8 hold, then we get the following Newton’s type inequality:
(25)13Y1Iθ1+φG2θ1+θ23+I2θ1+θ2/3+φGθ1+2θ23+Iθ1+2θ2/3+φGθ2−18Gθ1+3G2θ1+θ23+3Gθ1+2θ23+Gθ2≤θ2−θ19Y1A21−1/qαG′θ2q3A2α−A1α3+G′θ1qA1α31/q+A41−1/qαG′θ2q2A4α−A3α3+G′θ1qA4α+A3α31/q+A61−1/qαG′θ2A6α−A5α3+G′θ1q2A6α+A5α31/q.Proof.
Applying power mean inequality in (15) after taking the modulus, we have
(26)13Y1Iθ1+φG2θ1+θ23+I2θ1+θ2/3+φGθ1+2θ23+Iθ1+2θ2+φGθ2−18Gθ1+3G2θ1+θ23+3Gθ1+2θ23+Gθ2θ2−θ19Y1∫01Yt−3Y18G′3−t3θ2+t3θ1dt+∫01Yt−Y12G′2−t3θ2+1+t3θ1dt+∫01Yt−5Y18G′1−t3θ2+2+t3θ1dt≤θ2−θ19Y1∫01Yt−3Y18dt1−1/q∫01Yt−3Y18G′3−t3θ2+t3θ1qdt1/q+∫01Yt−Y12dt1−1/q∫01Yt−Y12G′2−t3θ2+1+t3θ1qdt1/q+∫01Yt−5Y18dt1−1/q∫01Yt−5Y18G′1−t3θ2+2+t3θ1qdt1/q.
Using the convexity ofG′q, we have
(27)θ2−θ13Y1Iθ1+φG2θ1+θ23+I2θ1+θ2/3+φGθ1+2θ23+Iθ1+2θ2/3+φGθ2−18Gθ1+3G2θ1+θ23+3Gθ1+2θ23+Gθ2≤θ2−θ19Y1∫01Yt−3Y18dt1−1/q×G′θ2q∫013−t3Yt−3Y18dt+G′θ1q∫01t3Yt−3Y18dt1/q+∫01Yt−Y12dt1−1/q×G′θ2q∫012−t3Yt−Y12dt+G′θ1q∫011+t3Yt−Y12dt1/q+∫01Yt−5Y18dt1−1/q×G′θ2q∫011−t3Yt−5Y18dt+G′θ1q∫012+t3Yt−5Y18dt1/q=θ2−θ19Y1A21−1/qαG′θ2q3A2α−A1α3+G′θ1qA1α31/q+A41−1/qαG′θ2q2A4α−A3α3+G′θ1qA4α+A3α31/q+A61−1/qαG′θ2A6α−A5α3+G′θ1q2A6α+A5α31/q.
Thus, the proof is completed.Remark 13.
In Theorem12, we have the following:
(i)
By settingφt=t, we reclaim the inequality established in ([13], Remark 4)(ii)
By settingφt=tα/Γα, we reclaim the inequality established in ([13], Theorem 5)Corollary 14.
By settingφt=tα/k/kΓkα in Theorem 12, we obtain the following new Newton’s inequality for KRLFIs:
(28)3α/k−1Γkα+1θ2−θ1α/kJθ1+α,kG2θ1+θ23+J2θ1+θ2/3+α,kGθ1+2θ23+Jθ1+2θ2/3+α,kGθ2−18Gθ1+3G2θ1+θ23+3Gθ1+2θ23+Gθ2≤θ2−θ19A21−1/qα,kG′θ2q3A2α,k−A1α,k3+G′θ1qA1α,k31/q+A41−1/qα,kG′θ2q2A4α,k−A3α,k3+G′θ1qA4α,k+A3α,k31/q+A61−1/qα,kG′θ2A6α,k−A5α,k3+G′θ1q2A6α,k+A5α,k31/q.Theorem 15.
IfG′q,q>1 is a convex function and assumptions of Lemma 8 hold, then we have the following Newton’s type inequality:
(29)13Y1Iθ1+φG2θ1+θ23+I2θ1+θ2/3+φGθ1+2θ23+Iθ1+2θ2/3+φGθ2−18Gθ1+3G2θ1+θ23+3Gθ1+2θ23+Gθ2≤θ2−θ19Y1A71/pα,p5G′θ2q+G′θ1q61/q+A81/pα,pG′θ2q+G′θ1q21/q+A91/pα,pG′θ2q+5G′θ1q61/q,where q−1+p−1=1 and
(30)A7α,p=∫01Yt−3Y18pdt,A8α,p=∫01Yt−Y12pdt,A9α,p=∫01Yt−5Y18pdt.Proof.
Applying Hölder’s inequality in (15) after taking the modulus, we have
(31)13Y1Iθ1+φG2θ1+θ23+I2θ1+θ2/3+φGθ1+2θ23+Iθ1+2θ2/3+φGθ2−18Gθ1+3G2θ1+θ23+3Gθ1+2θ23+Gθ2=θ2−θ19Y1∫01Yt−3Y18G′3−t3θ2+t3θ1dt+∫01Yt−Y12G′2−t3θ2+1+t3θ1dt+∫01Yt−5Y18G′1−t3θ2+2+t3θ1dt≤θ2−θ19Y1∫01Yt−3Y18pdt1/p∫01G′3−t3θ2+t3θ1qdt1/q+∫01Yt−Y12pdt1/p∫01G′2−t3θ2+1+t3θ1qdt1/q+∫01Yt−5Y18pdt1/p∫01G′1−t3θ2+2+t3θ1qdt1/q.
From convexity ofG′q,q>1, we obtain
(32)θ2−θ13Y1Iθ1+φG2θ1+θ23+I2θ1+θ2/3+φGθ1+2θ23+Iθ1+2θ2/3+φGθ2−18Gθ1+3G2θ1+θ23+3Gθ1+2θ23+Gθ2≤θ2−θ19Y1∫01Yt−3Y18pdt1/pG′θ2q∫013−t3dt+G′θ1q∫01t3dt1/q+∫01Yt−Y12pdt1/pG′θ2q∫012−t3dt+G′θ1q∫011+t3dt1/q+∫01Yt−5Y18pdt1/pG′θ2q∫011−t3dt+G′θ1q∫012+t3dt1/q=θ2−θ19Y1A71/pα,p5G′θ2q+G′θ1q61/q+A81/pα,pG′θ2q+G′θ1q21/q+A91/pα,pG′θ2q+5G′θ1q61/q.
Thus, the proof is completed.Remark 16.
In Theorem15, we have the following:
(i)
By settingφt=t, we reclaim the inequality established in ([13], Remark 5)(ii)
By settingφt=tα/Γα, we reclaim the inequality established in ([13], Theorem 6)Corollary 17.
By settingφt=tα/kkΓkα in Theorem 15, we obtain the following new Newton’s inequality for KRLFIs:
(33)3a/k−1Γkα+1θ2−θ1α/kJθ1+α,kG2θ1+θ23+J2θ1+θ2/3+α,kGθ1+2θ23+Jθ1+2θ2/3+α,kGθ2−18Gθ1+3G2θ1+θ23+3Gθ1+2θ23+Gθ2≤θ2−θ19A71/pα,p,k5G′θ2q+G′θ1q61/q+A81/pα,k,pG′θ2q+G′θ1q21/q+A91/pα,k,pG′θ2q+5G′θ1q61/q,where q−1+p−1=1 and
(34)A7α,k,p=∫01tα/k−38pdt,A8α,k,p=∫01tα/k−12pdt,A9α,k,p=∫01tα/k−58pdt.


## 5. Fractional Newton-Type Inequality for Functions of Bounded Variation

In this section, we prove a Newton-type inequality for function of bounded variation via generalized fractional integrals.Theorem 18.
LetG:θ1,θ2⟶ℝ be a function of bounded variation on θ1,θ2. Then we have the following Newton-type inequality for generalized fractional integrals:
(35)13Y1Iθ1+φG2θ1+θ23+I2θ1+θ2/3+φGθ1+2θ23+Iθ1+2θ2/3+φGθ2−18Gθ1+3G2θ1+θ23+3Gθ1+2θ23+Gθ2≤524∨θ1θ2G,
where∨cdG denotes the total variation of G on c,d.Proof.
Define the mappingΨφϰ by
(36)Ψφϰ=Y3θ2−θ12θ1+θ23−ϰ−5Y18,forθ1≤ϰ≤2θ1+θ23;Y3θ2−θ1θ1+2θ23−ϰ−Y12,for2θ1+θ23<ϰ≤θ1+2θ23;Y3θ2−θ1θ2−ϰ−3Y18,forθ1+2θ23<ϰ≤θ2.
It follows from that(37)∫θ1θ2ΨφϰdGϰ=∫θ12θ1+θ2/3Y3θ2−θ12θ1+θ23−ϰ−5Y18dGϰ+∫2θ1+θ2/3θ1+2θ2/3Y3θ2−θ1θ1+2θ23−ϰ−Y12dGϰ+∫θ12+θ2/3θ2Y3θ2−θ1θ2−ϰ−3Y18dGϰ.
Integrating by parts, we get(38)∫θ12θ1+θ2/3Y3θ2−θ12θ1+θ23−ϰ−5Y18dGϰ=Y3θ2−θ12θ1+θ23−ϰ−5Y18Gϰθ12θ1+θ2/3+∫θ12θ1+θ2/3φ2θ1+θ2/3−ϰ2θ1+θ2/3−ϰGϰdϰ=−5Y18G2θ1+θ22−3Y18Gθ1+θ1+IφG2θ1+θ23.
Similarly, we have(39)∫2θ1+θ2/3θ1+2θ2/3Y3θ2−θ1θ1+2θ23−ϰ−Y12dGϰ=−Y12Gθ1+2θ22−Y12G2θ1+θ22+2θ1+θ23+IφG2θ1+θ23,(40)∫θ1+2θ2/3θ2Y3θ2−θ1θ2−ϰ−3Y18dGϰ=−3Y18Gθ2−5Y18Gθ1+2θ22+θ1+2θ23+IφGθ2.
By putting the equalities (38)–(40) in (37), we have
(41)13Y1Iθ1+φG2θ1+θ23+2θ1+θ23+IφGθ1+2θ23+θ1+2θ23+IφGθ2−18Gθ1+3G2θ1+θ23+3Gθ1+2θ23+Gθ2=∫θ1θ2ΨφϰdGϰ.
It is well known that ifg,G:θ1,θ2⟶ℝ are such that g is continuous on θ1,θ2 and G is of bounded variation on θ1,θ2, then ∫θ1θ2gtdGt exist and
(42)∫θ1θ2gtdGt≤supt∈θ1,θ2gt∨θ1θ2G.
On the other hand, using (42), we get
(43)13Y1Iθ1+φG2θ1+θ23+2θ1+θ23+IφGθ1+2θ23+θ1+2θ23+IφGθ2−18Gθ1+3G2θ1+θ23+3Gθ1+2θ23+Gθ2=13Y1∫θ1θ2ΨφϰdGϰ≤13Y1∫θ12θ1+θ2/3Y3θ2−θ12θ1+θ23−ϰ−5Y18dGϰ+∫2θ1+θ2/3θ1+2θ2/3Y3θ2−θ1θ1+2θ23−ϰ−Y12dGϰ+∫θ1+2θ2/3θ2Y3θ2−θ1θ2−ϰ−3Y18dGϰ≤13Y1supϰ∈θ1,2θ1+θ2/3Y3θ2−θ12θ1+θ23−ϰ−5Y18∨θ12θ1+θ2/3G+supϰ∈2θ1+θ2/3,θ1+2θ2/3Y3θ2−θ1θ1+2θ23−ϰ−Y12∨2θ1+θ2/3θ1+2θ2/3G+supϰ∈θ1+2θ2/3,θ2Y3θ2−θ1θ2−ϰ−3Y18∨θ1+2θ2/3θ2G=13Y15Y18∨θ12θ1+2θ2/3G+Y12∨2θ1+2θ2/3θ1+2θ2/3G+5Y18∨θ1+2θ2/3θ2G≤524∨θ1θ2G.
This completes the proof.Remark 19.
In Theorem18, we have the following:
(i)
If we takeφt=t, then we recapture the inequality proved in ([46], Corollary 3)(ii)
If we setφt=tα/Γα, then we recapture the inequality established in ([13], Theorem 7)Corollary 20.
If we chooseφt=tα/k/kΓkα, then we obtain the following new Newton’s inequality for KRLFIs:
(44)3α/k−1Γkα+1θ2−θ1α/kJθ1α,k+G2θ1+θ23+J2θ1+θ2/3α,k+Gθ1+2θ23+Jθ1+2θ2/3α,k+Gθ2−18Gθ1+3G2θ1+θ23+3Gθ1+2θ23+Gθ2≤524∨θ1θ2G.


## 6. Conclusion

We demonstrated some new Simpson’s second-type inequalities for differentiable convex functions using Riemann-Liouville fractional integrals. Furthermore, we established fractional Newton-type inequalities for bounded variation functions. It is also shown that the newly established inequalities are an extension of the previously obtained inequalities. It is worth to mentioning here that we can obtain similar inequalities via Katugampola fractional operators, conformable fractional operators, Hadamard fractional operators, and fractional operators with the exponential kernel for different choices of the functionφ. In their future work, future researchers can get similar inequalities for various types of convexity and coordinated convexity on fractals, which is an exciting and new problem.



---

*Source: 6261970-2022-09-02.xml*