Fractional operators with integral inequalities have attracted the interest of several mathematicians. Fractional inequalities are best utilized in mathematical science with their features and wide range of applications in optimization, modeling, engineering and artificial intelligence. In this article, we consider new variants of Simpson-Mercer type inequalities involving the Atangana-Baleanu (A-B) fractional integral operator for s-convex functions. First, an integral identity, which acts as an auxiliary result for the main results is proved in the frame of fractional operator. Employing this new identity, some estimations of Simpson-Mercer type for s-convex functions in the second sense are discussed. In addition, we study various new applications on Modified Bessel functions, special means and q-digamma functions. These applications confirm the effectiveness and validity of the results and also bring a different dimension to the study.
Citation: Muhammad Tariq, Hijaz Ahmad, Soubhagya Kumar Sahoo, Artion Kashuri, Taher A. Nofal, Ching-Hsien Hsu. Inequalities of Simpson-Mercer-type including Atangana-Baleanu fractional operators and their applications[J]. AIMS Mathematics, 2022, 7(8): 15159-15181. doi: 10.3934/math.2022831
[1] | Jia-Bao Liu, Saad Ihsan Butt, Jamshed Nasir, Adnan Aslam, Asfand Fahad, Jarunee Soontharanon . Jensen-Mercer variant of Hermite-Hadamard type inequalities via Atangana-Baleanu fractional operator. AIMS Mathematics, 2022, 7(2): 2123-2141. doi: 10.3934/math.2022121 |
[2] | Maimoona Karim, Aliya Fahmi, Shahid Qaisar, Zafar Ullah, Ather Qayyum . New developments in fractional integral inequalities via convexity with applications. AIMS Mathematics, 2023, 8(7): 15950-15968. doi: 10.3934/math.2023814 |
[3] | Soubhagya Kumar Sahoo, Fahd Jarad, Bibhakar Kodamasingh, Artion Kashuri . Hermite-Hadamard type inclusions via generalized Atangana-Baleanu fractional operator with application. AIMS Mathematics, 2022, 7(7): 12303-12321. doi: 10.3934/math.2022683 |
[4] | Saad Ihsan Butt, Erhan Set, Saba Yousaf, Thabet Abdeljawad, Wasfi Shatanawi . Generalized integral inequalities for ABK-fractional integral operators. AIMS Mathematics, 2021, 6(9): 10164-10191. doi: 10.3934/math.2021589 |
[5] | Jamshed Nasir, Saber Mansour, Shahid Qaisar, Hassen Aydi . Some variants on Mercer's Hermite-Hadamard like inclusions of interval-valued functions for strong Kernel. AIMS Mathematics, 2023, 8(5): 10001-10020. doi: 10.3934/math.2023506 |
[6] | Saad Ihsan Butt, Artion Kashuri, Muhammad Umar, Adnan Aslam, Wei Gao . Hermite-Jensen-Mercer type inequalities via Ψ-Riemann-Liouville k-fractional integrals. AIMS Mathematics, 2020, 5(5): 5193-5220. doi: 10.3934/math.2020334 |
[7] | Shahid Mubeen, Rana Safdar Ali, Iqra Nayab, Gauhar Rahman, Kottakkaran Sooppy Nisar, Dumitru Baleanu . Some generalized fractional integral inequalities with nonsingular function as a kernel. AIMS Mathematics, 2021, 6(4): 3352-3377. doi: 10.3934/math.2021201 |
[8] | Shuang-Shuang Zhou, Saima Rashid, Muhammad Aslam Noor, Khalida Inayat Noor, Farhat Safdar, Yu-Ming Chu . New Hermite-Hadamard type inequalities for exponentially convex functions and applications. AIMS Mathematics, 2020, 5(6): 6874-6901. doi: 10.3934/math.2020441 |
[9] | Thabet Abdeljawad, Muhammad Aamir Ali, Pshtiwan Othman Mohammed, Artion Kashuri . On inequalities of Hermite-Hadamard-Mercer type involving Riemann-Liouville fractional integrals. AIMS Mathematics, 2021, 6(1): 712-725. doi: 10.3934/math.2021043 |
[10] | Areej A. Almoneef, Abd-Allah Hyder, Fatih Hezenci, Hüseyin Budak . Simpson-type inequalities by means of tempered fractional integrals. AIMS Mathematics, 2023, 8(12): 29411-29423. doi: 10.3934/math.20231505 |
Fractional operators with integral inequalities have attracted the interest of several mathematicians. Fractional inequalities are best utilized in mathematical science with their features and wide range of applications in optimization, modeling, engineering and artificial intelligence. In this article, we consider new variants of Simpson-Mercer type inequalities involving the Atangana-Baleanu (A-B) fractional integral operator for s-convex functions. First, an integral identity, which acts as an auxiliary result for the main results is proved in the frame of fractional operator. Employing this new identity, some estimations of Simpson-Mercer type for s-convex functions in the second sense are discussed. In addition, we study various new applications on Modified Bessel functions, special means and q-digamma functions. These applications confirm the effectiveness and validity of the results and also bring a different dimension to the study.
Fractional analysis can be regarded as a generalization of classical analysis. Fractional analysis has been investigated and discussed by several mathematicians and they have explored the fractional derivatives and integrals in variant ways with numerous notations. Although the expressions of these generalized definitions can be transformed into each other, but have variant physical meanings. It is well known that the first fractional integral operator is the Riemann-Liouville fractional integral operator. Recently, fractional calculus has become one of the prestigious fields in research because of its inherent applications in different fields like numerical physical science [1], fluid mechanics [2], biological modeling [3,4], and many more. Fractional calculus owes its starting point to whether or not the importance of the derivative of an integer order could be generalized to a fractional order which is not an integer. Analysis of fractional calculus is the generalization of classical analysis, it has been developed rapidly with the interesting convexity theory attributed to Jensen [5]. Its fertile applications in functional analysis, stochastic theory, finite element method, and optimization theory have made it a fascinating topic for research. Therefore, it acts as an incorporative subject among quantum theory, orthogonal polynomials, combinatorics, etc. These are the principal urges that led to the deep research and improvement of the theory of inequality in literature (see [6,7,8]).
Let 0<γ1≤γ2≤...≤γP and φ=(φ1,φ2,...,φP) be non-negative weights such that ∑Pk=1 φk=1. The Jensen's inequality (see [5]) in the literature, reads as:
If the convexity of Φ holds true on the interval [℘1,℘2], then
Φ(P∑k=1 φk γk)≤P∑k=1 φk Φ(γk), | (1.1) |
for all γk∈[d,D], φk∈[0,1]and(k=1,2,...,P). This inequality helps to obtain new bounds for useful distances in information theory (see [9]).
Mercer [10] in the year 2003, investigated a generalized form of Jensen's inequality, which is famously known as Jensen-Mercer (J-M) inequality:
If Φ is a convex function on [d,D], then
Φ(d+D−P∑k=1 φk γk)≤Φ(d)+Φ(D)−P∑k=1 φk Φ(γk), | (1.2) |
holds true for all γk∈[d,D], φk∈[0,1] and (k=1,2,...,P).
Some generalizations on J-M operator inequalities are proposed by Pečarić et al. in [11]. Later, Niezgoda in [12] proved several generalizations of Mercer's type inequalities to higher dimensions. Kian [13] contemplated the idea of Jensen inequality for superquadratic functions. Moreover, reverse Mercer's type operator inequalities and the aforesaid inequalities for super-quadratic functions have been proposed in [14,15]. Also, the integral and integral mean version have been established in [16]. Cortez et al. (see Remark 3.4 [17]) presented the concept of s-convex variant of J-M inequality in a generalized form. For some recent generalizations on J-M type inequalities, we direct readers to go through [18,19,20] and the references cited therein.
Hudzik et al. [21], introduced the notion of s-convex functions in the first and second sense and also discussed some of their notable features.
Definition 1.1. Let s∈(0,1] and Φ be a real valued function on an interval I=[0,∞). Then ϕ is said to be s-convex in the first sense if
ϕ(φ1γ1+φ2γ2)≤φs1ϕ(γ1)+φs2ϕ(γ2), |
holds true for all γ1,γ2∈I, φ1,φ2≥0 with φs1+φs2=1.
Definition 1.2. Let s∈(0,1] and Φ be a real valued function on an interval I=[0,∞). Then ϕ is said to be s-convex in the second sense if
ϕ(φ1γ1+φ2γ2)≤φs1ϕ(γ1)+φs2ϕ(γ2), |
holds true for all γ1,γ2∈I, φ1,φ2≥0 with φ1+φ2=1.
The class of s-convex function in the first and second sense are denoted by (Φ∈ K1s) and (Φ∈ K2s) respectively. Moreover, they also studied that the class of s-convex function in the second sense is more stronger than the s-convexity in first sense for s∈(0,1). Several features of s-convex function in both senses are discussed taking some examples into consideration. It is interesting to see that if s∈(0,1) and Φ∈ K2s, then Φ is non negative.
For more detail on s-convex function (see [21,22,23,24]) and references cited therein. Chen [23] gave a nice relationship between convex functions and s-convex functions as:
Lemma 1.1. (see [23]) Suppose Φ:[d,D]→ℜ is a convex function. Then
(i) If Φ is non-negative, then it is s-convex for s∈(0,1].
(ii) If Φ is non-positive, then it is s-convex for s∈[1,∞).
Butt et al. [25] investigated the J-M inequality for s-convex functions in Breckner sense (s>0) as follows:
Theorem 1.1. Let φ1,φ2,...,φP be positive probability distributions and P∑ȷ=1φsȷ≤1. If Φ:[d,D]⊂(0,∞)→ℜ is s-Breckner sense convex function, then the following inequality is valid
Φ(d+D−P∑ȷ=1φȷγȷ)≤Φ(d)+Φ(D)−P∑ȷ=1φsȷΦ(γȷ), | (1.3) |
for any increasing sequence {γȷ}Pȷ=1∈[d,D].
The following conditions are necessary:
(i) By considering s=1 in (1.3), we get the J-M inequality (1.1).
(ii) By considering s=1 and φ1=d and φ2=D, in (1.3) it gives the definition of the classical convex function.
The following inequality named as Simpson's inequality, is given in the literature as (see [26,27,28]).
|13{Φ(d)+Φ(D)2+2Φ(d+D2)−1D−d∫DdΦ(x)dx}|≤12880||Φ4||∞(D−d)4, |
where Φ:[d,D]→ℜ is a four times continuously differentiable mapping on (d,D) and ||Φ4||∞=supx∈(d,D)|Φ4(x)|<∞.
Several authors have concentrated on Simpson-type inequalities for different classes of mappings. Precisely, a few mathematicians have dealt with Simpson-type inequalities for convex mappings, since convexity theory is an efficient and stable process for taking care of an extraordinary number of issues that emerge in various branches in pure and applied sciences. For some recent improvements on Simpson type inequality via different convexities and fractional operators see references [29,30,31,32] and cited therein.
Let Φ∈L[d,D]. Then the left and right Riemann-Liouville (R-L) fractional integrals of order δ>0 with d≥0 are defined as follows:
Jδd+Φ(x)= 1Γ(δ)∫xd(x−κ)δ−1 Φ(κ) dκ , x>d, |
and
JδD−Φ(x)= 1Γ(δ)∫Dx(κ−x)δ−1 Φ(κ) dκ , x<D. |
For further details one may (see [33,34]).
In [35], the idea of the J-M inequality has been used by Kian and Moslehian and the following Hermite-Hadamard-Mercer inequality was demonstrated:
Φ(d+D−ω1+ω22)≤1ω2−ω1∫ω2ω1Φ(d+D−κ)dκ≤Φ(d+D−ω1)+Φ(d+D−ω2)2≤Φ(d)+Φ(D)−Φ(ω1)+Φ(ω2)2. | (1.4) |
Recently, the notions of fractional operators have drawn the attention of several researchers. Fractional calculus helps mathematicians solve many real-life problems in a convincing manner than classical calculus. Mainly, there are two types of nonlocal fractional derivatives, the classical Riemann-Liouville and Caputo derivatives with singular kernels and the others with non-singular kernels, which have been introduced recently, such as the Caputo-Fabrizio and Atangana-Baleanu derivatives. However, the fractional derivative operators with non-singular kernels are effective to solve the non-locality of real-world problems in an appropriate manner. The introduction of new notions of fractional operators is a journey to succeed momentum to the fractional calculus and to gain the most efficient operators to the discussion. Now, we recall the notion of the Caputo-Fabrizio fractional operator:
Definition 1.3. Let p∈[1,∞) and [ω1,ω2] be an open subset of ℜ, the Sobolev space Hp(ω1,ω2) is defined by
Hp(ω1,ω2)={Φ∈L2(ω1,ω2):DμΦ∈L2(ω1,ω2),forall|μ|≤p}. |
Definition 1.4. [36] Let Φ∈H1(0,κ), κ>ω, δ∈[0,1] then, the definition of the new Caputo fractional derivative is given as:
CFDδΦ(t)=M(δ)1−δ∫tωΦ′(s)exp[−δ(1−δ)(κ−s)]ds, | (1.5) |
where M(δ) is normalization function with M(0)=M(1)=1.
Moreover, the corresponding Caputo-Fabrizio fractional integral operator is given as:
Definition 1.5. [37] Let Φ∈H1(0,κ), κ>ω, δ∈[0,1]. Then, the definition of Caputo-Fabrizio fractional integrals are given as:
(CFωIδΦ)(t)=1−δM(δ)Φ(t)+δM(δ)∫tωΦ(y)dy, |
and
(CF IδκΦ)(t)=1−δM(δ)Φ(t)+δM(δ)∫κtΦ(y)dy, |
where M(δ) is normalization function with M(0)=M(1)=1.
Recently, Atangana & Baleanu introduced a new fractional operator containing the Mittag-Leffler function in the kernel, that solves the problem of retrieving the original function (a clear advantage over the Caputo-Fabrizio operator). Mittag-Leffler's function is more suitable than a power law in modeling nature and physical phenomena. This attribute has made the A-B operator more effective and helpful. As a result, many researchers have shown keen interest in utilizing this operator. Atangana & Baleanu introduced the derivative operator both in Caputo and Riemann-Liouville sense:
Definition 1.6. [38] Let κ>ω, δ∈[0,1] and Φ∈H1(ω,κ). The new fractional derivative is given as:
ABCωDδt[Φ(t)]=M(δ)1−δ∫tωΦ′(x)Eδ[−δ(t−x)δ(1−δ)]dx, | (1.6) |
where Eδ is the Mittag-Leffler function and is defined as Eδ(−tδ)=∑∞k=0(−t)δkΓ(δk+1).
Definition 1.7. [38] Let Φ∈H1(ω,κ), ω>κ, δ∈[0,1]. The new fractional derivative is given as:
ABRωDδt[Φ(t)]=M(δ)1−δddt∫tωΦ(x)Eδ[−δ(t−x)δ(1−δ)]dx, | (1.7) |
where Eδ is the Mittag-Leffler function.
However, in the same paper the authors also presented corresponding A-B fractional integral operator as:
Definition 1.8. [38] The fractional integral operator with non-local kernel of a function Φ∈H1(ω,κ) is defined as:
ABωItδ{Φ(t)}=1−δM(δ)Φ(t)+δM(δ)Γ(δ)∫tωΦ(y)(t−y)δ−1dy, |
where κ>ω,δ∈[0,1].
In [39], the right hand side of A-B fractional integral operator is presented as follows:
ABκItδ{Φ(t)}=1−δM(δ)Φ(t)+δM(δ)Γ(δ)∫κtΦ(y)(y−t)δ−1dy. |
Here, Γ(δ) is the Gamma function. The positivity of the normalization function M(δ) implies that the fractional A-B integral of a positive function is positive. It is worth noticing that the case when the order δ→1 yields the classical integral and the case when δ→0 provides the initial function.
Sarikaya et al. [33] used Riemann-Liouville fractional integrals to generalise the Hermite-Hadamard inequality. Iscan [40] generalised the conclusions reached by Sarikaya et al. [33] to Hermite-Hadamard-Fejér type inequalities. Chen [41] employed the methods of Sarikaya et al. [33] and used the product of two convex functions to produce fractional Hermite-Hadamard type integral inequalities. Recently, Sun proved different variants of the Hermite-Hadamard inequality via generalized local fractional integral operators with Mittag-Leffler kernel for h-convex functions [42], s-preinvex functions [43] and generalized preinvex fuctions [44]. Fernandez and Mohammed [45] presented a nice relation between Riemann-Liouville fractional integrals and A-B fractional integral operators and also proved new type of Hermite-Hadamard inequalities for convex functions. Ogulmus et al. [46] established Hermite-Hadamard-Mercer type inequalities involving Riemann-Liouville fractional operator. Sahoo et al. [47] presented some mid-point versions of Hermite-Hadamard inequalities via Caputo-Fabrizio fractional operator. Latif et al. [48] worked on Hermite-Hadamard-Fejér type fractional inequalities relating to a convex harmonic function and a positive symmetric increasing function.
The main motivation of this study is to obtain an integral identity employing A-B fractional integral operator, which has a unique place among fractional integral operators, and to generate some new Simpson-Mercer's like inequalities for s-convex functions based on this identity. Considering many special cases of the main findings, the study also includes the applications of the results.
In this section, we study several Simpson-Mercer's like inequalities via the A-B fractional integral operator for differentiable functions on (d,D). For this, we establish a new A-B integral identity that will serve as an auxiliary result to produce subsequent results for improvements. Throughout the paper, let us assume that P1: = (1+κ2)s+(1−κ2)s≤1, for κ∈[0,1].
Lemma 2.1. Suppose the mapping Φ:I=[d,D]→ℜ is differentiable on (d,D) with D>d. If Φ′∈L[d,D], then for all ω1,ω2∈[d,D] and δ>0, the following A-B fractional identity of Simpson-Mercer type holds true:
16{Φ(d+D−ω1)+Φ(d+D−ω2)}+23Φ(d+D−ω1+ω22)+2δ(1−δ)Γ(δ)(ω2−ω1)δΦ(d+D−ω1+ω22)− 2δ−1M(δ)Γ(δ)(ω2−ω1)δ { AB(d+D−ω1)Iδ(d+D−ω1+ω22)Φ(d+D−ω1+ω22) +AB(d+D−ω2)Iδ(d+D−ω1+ω22)Φ(d+D−ω1+ω22)}=ω2−ω12[∫10(κδ2−13)Φ′(d+D−[1+κ2ω1+1−κ2ω2])dκ+∫10(13−κδ2)Φ′(d+D−[1−κ2ω1+1+κ2ω2])dκ], | (2.1) |
for κ∈[0,1].
Proof. Let us assume that
ω2−ω12[∫10(κδ2−13)Φ′(d+D−[1+κ2ω1+1−κ2ω2])dκ+∫10(13−κδ2)Φ′(d+D−[1−κ2ω1+1+κ2ω2])dκ]=ω2−ω12{J1+J2}. | (2.2) |
Implies that
J1=∫10(κδ2−13)Φ′(d+D−[1+κ2ω1+1−κ2ω2])dκ. |
Integrating by parts, we get
J1=(κδ2−13)Φ(d+D−[1+κ2ω1+1−κ2ω2])ω2−ω12|10−∫10Φ(d+D−[1+κ2ω1+1−κ2ω2])ω2−ω12×δκδ−12dκ. |
By substituting the variables, we get
J1=2ω2−ω1{16Φ(d+D−ω1)+13Φ(d+D−ω1+ω22)}−2δM(δ)Γ(δ)(ω2−ω1)δ+1×{ AB(d+D−ω1)Iδ(d+D−ω1+ω22)Φ(d+D−ω1+ω22)−1−δM(δ)Φ(d+D−ω1+ω22)}. |
Similarly,
J2=∫10(13−κδ2)Φ′(d+D−[1−κ2ω1+1+κ2ω2])dκ=2ω2−ω1{16Φ(d+D−ω2)+13Φ(d+D−ω1+ω22)}−2δM(δ)Γ(δ)(ω2−ω1)δ+1{ AB(d+D−ω2)Iδ(d+D−ω1+ω22)Φ(d+D−ω1+ω22)−1−δM(δ)Φ(d+D−ω1+ω22)}. |
By using the values of J1 and J2 in (2.2), we get (2.1).
Corollary 2.1. If we consider δ=1 in (2.1), then we have the following Simpson-Mercer type equality
16{Φ(d+D−ω1)+4Φ(d+D−ω1+ω22)+Φ(d+D−ω2)}−1(ω2−ω1)∫d+D−ω1d+D−ω2Φ(x)dx=ω2−ω12[∫10(κ2−13)Φ′(d+D−[1+κ2ω1+1−κ2ω2])dκ+∫10(13−κ2)Φ′(d+D−[1−κ2ω1+1+κ2ω2])dκ]. |
Remark 2.1. If we set ω1=d and ω2=D in Corollary 2.1, we obtain Lemma 1 in [27].
Theorem 2.1. Taking all the conditions as defined in Lemma 2.1 and assumption P1 into consideration and let |Φ′| be s-convex function on [d,D], for s>0. Then, for all δ>0, the following Simpson-Mercer type inequality for A-B fractional integral holds true for κ∈[0,1]:
|16{Φ(d+D−ω1)+Φ(d+D−ω2)}+23Φ(d+D−ω1+ω22)+2δ(1−δ)Γ(δ)(ω2−ω1)δΦ(d+D−ω1+ω22)− 2δ−1M(δ)Γ(δ)(ω2−ω1)δ { AB(d+D−ω1)Iδ(d+D−ω1+ω22)Φ(d+D−ω1+ω22) +AB(d+D−ω2)Iδ(d+D−ω1+ω22)Φ(d+D−ω1+ω22)}|≤ω2−ω12[2{(23)1+1δ(δδ+1)+1−2δ6(δ+1)}[|Φ′(d)|+|Φ′(D)|]−12s[|Φ′(ω1)|+|Φ′(ω2)|]{J3(δ)+J4(δ)}], | (2.3) |
where
J3(δ)=∫(23)1δ0(13−κδ2)[(1+κ)s+(1−κ)s]dκ, |
and
J4(δ)=∫1(23)1δ(κδ2−13)[(1+κ)s+(1−κ)s]dκ. |
Proof. Employing Lemma 2.1 with the J-M inequality and the s-convexity of |Φ′|, we have
|16{Φ(d+D−ω1)+Φ(d+D−ω2)}+23Φ(d+D−ω1+ω22)+2δ(1−δ)Γ(δ)(ω2−ω1)δΦ(d+D−ω1+ω22)− 2δ−1M(δ)Γ(δ)(ω2−ω1)δ { AB(d+D−ω1)Iδ(d+D−ω1+ω22)Φ(d+D−ω1+ω22) +AB(d+D−ω2)Iδ(d+D−ω1+ω22)Φ(d+D−ω1+ω22)}|≤ ω2−ω12[∫10|κδ2−13||Φ′(d+D−[1+κ2ω1+1−κ2ω2])|dκ+∫10|13−κδ2||Φ′(d+D−[1−κ2ω1+1+κ2ω2])|dκ]≤ ω2−ω12[∫10|κδ2−13|{|Φ′(d)|+|Φ′(D)|−[(1+κ2)s|Φ′(ω1)|+(1−κ2)s|Φ′(ω2)|}dκ+∫10|13−κδ2|{|Φ′(d)|+|Φ′(D)|−[(1−κ2)s|Φ′(ω1)|+(1+κ2)s|Φ′(ω2)|}dκ]≤ ω2−ω12[2∫10|κδ2−13|{|Φ′(d)|+|Φ′(D)|}dκ−12s{|Φ′(ω1)|+|Φ′(ω2)|}∫10|κδ2−13|[(1+κ)s+(1−κ)s]dκ= ω2−ω12[2{∫(23)1δ0(13−κδ2)dκ+∫1(23)1δ(κδ2−13)dκ}{|Φ′(d)|+|Φ′(D)|}−12s{|Φ′(ω1)|+|Φ′(ω2)|}{∫(23)1δ0(13−κδ2)[(1+κ)s+(1−κ)s]dκ+∫1(23)1δ(κδ2−13)[(1+κ)s+(1−κ)s]dκ}]=ω2−ω12[2{(23)1+1δ(δδ+1)+1−2δ6(δ+1)}[|Φ′(d)|+|Φ′(D)|]−12s[|Φ′(ω1)|+|Φ′(ω2)|]{J3(δ)+J4(δ)}], | (2.4) |
which completes the proof.
Remark 2.2. If we consider ω1=d and ω2=D and δ=1 in Theorem 2.1, we obtain Theorem 7 in [27].
Corollary 2.2. Taking the same assumptions as defined in Theorem 2.1 with s=1 into consideration, we get the following Simpson-Mercer type inequality
|16{Φ(d+D−ω1)+Φ(d+D−ω2)}+23Φ(d+D−ω1+ω22)+2δ(1−δ)Γ(δ)(ω2−ω1)δΦ(d+D−ω1+ω22)− 2δ−1M(δ)Γ(δ)(ω2−ω1)δ { AB(d+D−ω1)Iδ(d+D−ω1+ω22)Φ(d+D−ω1+ω22) +AB(d+D−ω2)Iδ(d+D−ω1+ω22)Φ(d+D−ω1+ω22)}|≤ω2−ω12[2{(23)1+1δ(δδ+1)+1−2δ6(δ+1)}[|Φ′(d)|+|Φ′(D)|]−12[|Φ′(ω1)|+|Φ′(ω2)|]{J3(δ)+J4(δ)}]. | (2.5) |
Corollary 2.3. If we consider δ=1 in Theorem 2.1, then we have the following Simpson-Mercer type inequality
|16{Φ(d+D−ω1)+4Φ(d+D−ω1+ω22)+Φ(d+D−ω2)}−1ω2−ω1∫d+D−ω1d+D−ω2Φ(x)dx|≤ω2−ω12[518(|Φ′(d)|+|Φ′(D)|)−12s(|Φ′(ω1)|+|Φ′(ω2)|){16(s+1)(5s+2−13s+1−6)−12(s+2)(5s+2+13s+2−2)+ 5s+2+12(s+1)(s+2)3s+2−2s(2s+7)3(s+1)(s+2)}]. |
In the results to follow hereon, we will give the new upper bounds for the right hand side of the Simpson-Mercer type inequality via s-convexity involving A-B fractional integrals operator.
Theorem 2.2. Taking all the conditions as defined in Lemma 2.1 and assumption P1 into consideration, if |Φ′|q is s-convex function on [d,D], for s>0, then for all δ>0, the following inequality for A-B fractional integral holds true for κ∈[0,1]:
|16{Φ(d+D−ω1)+Φ(d+D−ω2)}+23Φ(d+D−ω1+ω22)+2δ(1−δ)Γ(δ)(ω2−ω1)δΦ(d+D−ω1+ω22)− 2δ−1M(δ)Γ(δ)(ω2−ω1)δ { AB(d+D−ω1)Iδ(d+D−ω1+ω22)Φ(d+D−ω1+ω22) +AB(d+D−ω2)Iδ(d+D−ω1+ω22)Φ(d+D−ω1+ω22)}|≤ ω2−ω12(∫10|κδ2−13|pdκ)1p{(|Φ′(d)|q+|Φ′(D)|q−[|Φ′(ω1)|q+|Φ′(ω1+ω22)|qs+1])1q+(|Φ′(d)|q+|Φ′(D)|q−[|Φ′(ω2)|q+|Φ′(ω1+ω22)|qs+1])1q}, | (2.6) |
where 1p+1q=1 and q>1.
Proof. Employing Lemma 2.1 and the Hölder's inequality, we have
|16{Φ(d+D−ω1)+Φ(d+D−ω2)}+23Φ(d+D−ω1+ω22)+2δ(1−δ)Γ(δ)(ω2−ω1)δΦ(d+D−ω1+ω22)− 2δ−1M(δ)Γ(δ)(ω2−ω1)δ { AB(d+D−ω1)Iδ(d+D−ω1+ω22)Φ(d+D−ω1+ω22) +AB(d+D−ω2)Iδ(d+D−ω1+ω22)Φ(d+D−ω1+ω22)}|≤ ω2−ω12[∫10|κδ2−13||Φ′(d+D−[1+κ2ω1+1−κ2ω2])|dκ+∫10|13−κδ2||Φ′(d+D−[1−κ2ω1+1+κ2ω2])|dκ]≤ ω2−ω12[(∫10|κδ2−13|pdκ)1p(∫10|Φ′(d+D−[1+κ2ω1+1−κ2ω2])|qdκ)1q+ (∫10|κδ2−13|pdκ)1p(∫10|Φ′(d+D−[1−κ2ω1+1+κ2ω2])|qdκ)1q]. | (2.7) |
Since |Φ′|q is s-convex on [d,D], with the J-M inequality, we obtain
∫10|Φ′(d+D−[1+κ2ω1+1−κ2ω2])|qdκ≤ |Φ′(d)|q+|Φ′(D)|q−[|Φ′(ω1)|q+|Φ′(ω1+ω22)|qs+1], | (2.8) |
and
∫10|Φ′(d+D−[1−κ2ω1+1+κ2ω2])|qdκ ≤ |Φ′(d)|q+|Φ′(D)|q−[|Φ′(ω2)|q+|Φ′(ω1+ω22)|qs+1]. | (2.9) |
By using the Eqs (2.8) and (2.9) with (2.7), we get the (2.6). This concludes the proof.
Remark 2.3. If we set ω1=d, ω2=D and δ=1 in Theorem 2.2, we obtain Theorem 8 in [27].
Corollary 2.4. Under the same assumptions as defined in Theorem 2.2 with s=1, we get the following A-B fractional inequality of Simpson-Mercer type:
|16{Φ(d+D−ω1)+Φ(d+D−ω2)}+23Φ(d+D−ω1+ω22)+2δ(1−δ)Γ(δ)(ω2−ω1)δΦ(d+D−ω1+ω22)− 2δ−1M(δ)Γ(δ)(ω2−ω1)δ { AB(d+D−ω1)Iδ(d+D−ω1+ω22)Φ(d+D−ω1+ω22) +AB(d+D−ω2)Iδ(d+D−ω1+ω22)Φ(d+D−ω1+ω22)}|≤ ω2−ω12(∫10|κδ2−13|pdκ)1p{(|Φ′(d)|q+|Φ′(D)|q−[|Φ′(ω1)|q+|Φ′(ω1+ω22)|q2])1q+(|Φ′(d)|q+|Φ′(D)|q−[|Φ′(ω2)|q+|Φ′(ω1+ω22)|q2])1q}. | (2.10) |
Corollary 2.5. Under the same assumptions as defined in Theorem 2.2 with δ=1, we get the following Simpson-Mercer type inequality
|16{Φ(d+D−ω1)+4Φ(d+D−ω1+ω22)+Φ(d+D−ω2)}−1ω2−ω1∫d+D−ω1d+D−ω2Φ(x)dx|≤ ω2−ω112(1+2p+13(p+1))1p{(|Φ′(d)|q+|Φ′(D)|q−[|Φ′(ω1)|q+|Φ′(ω1+ω22)|qs+1])1q+( |Φ′(d)|q+|Φ′(D)|q−[|Φ′(ω2)|q+|Φ′(ω1+ω22)|qs+1])1q}. |
Theorem 2.3. Taking all the conditions as defined in Lemma 2.1 and assumption P1 into consideration, if |Φ′|q is s-convex function on [d,D], for s>0 and q>1, then for all δ>0, the following inequality for A-B fractional integral holds true for κ∈[0,1]:
|16{Φ(d+D−ω1)+Φ(d+D−ω2)}+23Φ(d+D−ω1+ω22)+2δ(1−δ)Γ(δ)(ω2−ω1)δΦ(d+D−ω1+ω22)− 2δ−1M(δ)Γ(δ)(ω2−ω1)δ { AB(d+D−ω1)Iδ(d+D−ω1+ω22)Φ(d+D−ω1+ω22)+AB(d+D−ω2)Iδ(d+D−ω1+ω22)Φ(d+D−ω1+ω22)}|≤ ω2−ω12(∫10|κδ2−13|pdκ)1p×{(|Φ′(d)|q+|Φ′(D)|q−[2s+1−1(s+1)2s|Φ′(ω1)|q+ 1(s+1)2s |Φ′(ω2)|q])1q+ (|Φ′(d)|q+|Φ′(D)|q−;[1(s+1)2s|Φ′(ω1)|q+ 2s+1−1(s+1)2s |Φ′(ω2)|q])1q}, | (2.11) |
where 1p+1q=1 and q>1.
Proof. Employing Lemma 2.1 and the Hölder inequality, we have
|16{Φ(d+D−ω1)+Φ(d+D−ω2)}+23Φ(d+D−ω1+ω22)+2δ(1−δ)Γ(δ)(ω2−ω1)δΦ(d+D−ω1+ω22)− 2δ−1M(δ)Γ(δ)(ω2−ω1)δ { AB(d+D−ω1)Iδ(d+D−ω1+ω22)Φ(d+D−ω1+ω22) +AB(d+D−ω2)Iδ(d+D−ω1+ω22)Φ(d+D−ω1+ω22)}|≤ ω2−ω12[∫10|κδ2−13||Φ′(d+D−[1+κ2ω1+1−κ2ω2])|dκ+∫10|13−κδ2||Φ′(d+D−[1−κ2ω1+1+κ2ω2])|dκ]≤ ω2−ω12[(∫10|κδ2−13|pdκ)1p(∫10|Φ′(d+D−[1+κ2ω1+1−κ2ω2])|qdκ)1q+ (∫10|κδ2−13|pdκ)1p(∫10|Φ′(d+D−[1−κ2ω1+1+κ2ω2])|qdκ)1q]. | (2.12) |
Since |Φ′|q is s-convex on [d,D], with the J-M inequality, we obtain
|Φ′(d+D−[1+κ2ω1+1−κ2ω2])|q≤ |Φ′(d)|q+|Φ′(D)|q−[(1+κ2)s|Φ′(ω1)|q+(1−κ2)s|Φ′(ω2)|q], | (2.13) |
and
|Φ′(d+D−[1−κ2ω1+1+κ2ω2])|q ≤ |Φ′(d)|q+|Φ′(D)|q−[(1−κ2)s|Φ′(ω1)|q+(1+κ2)s|Φ′(ω2)|q]. | (2.14) |
Upon combining the Eqs (2.13) and (2.14) in (2.12) and using the tool of calculus, we get (2.11). This completes the proof.
Remark 2.4. If we set ω1=d, ω2=D and δ=1 in Theorem 2.3, we obtain Theorem 9 in [27].
Corollary 2.6. Under the same assumptions as defined in Theorem 2.3 with s=1, we get the following A-B fractional inequality of Simpson-Mercer type:
|16{Φ(d+D−ω1)+Φ(d+D−ω2)}+23Φ(d+D−ω1+ω22)+2δ(1−δ)Γ(δ)(ω2−ω1)δΦ(d+D−ω1+ω22)− 2δ−1M(δ)Γ(δ)(ω2−ω1)δ { AB(d+D−ω1)Iδ(d+D−ω1+ω22)Φ(d+D−ω1+ω22) +AB(d+D−ω2)Iδ(d+D−ω1+ω22)Φ(d+D−ω1+ω22)}|≤ ω2−ω12(∫10|κδ2−13|pdκ)1p{(|Φ′(d)|q+|Φ′(D)|q−[3|Φ′(ω1)|q+|Φ′(ω2)|q4])1q+(|Φ′(d)|q+|Φ′(D)|q−[|Φ′(ω1)|q+3|Φ′(ω2)|q4])1q}. | (2.15) |
Corollary 2.7. Under the same assumptions as defined in Theorem 2.3 with δ=1, we get the following Simpson-Mercer type inequality:
|16{Φ(d+D−ω1)+4Φ(d+D−ω1+ω22)+Φ(d+D−ω2)}−1ω2−ω1∫d+D−ω1d+D−ω2Φ(x)dx|≤ ω2−ω112(1+2p+13(p+1))1p×{(|Φ′(d)|q+|Φ′(D)|q−[2s+1−1(s+1)2s|Φ′(ω1)|q+ 1(s+1)2s |Φ′(ω2)|q])1q+ (|Φ′(d)|q+|Φ′(D)|q−[1(s+1)2s|Φ′(ω1)|q+ 2s+1−1(s+1)2s |Φ′(ω2)|q])1q}. |
Theorem 2.4. Taking all the conditions as defined in Lemma 2.1 and assumption P1 into consideration, if |Φ′|q is s-convex function on [d,D], for s>0, q≥1, then for all δ>0, the following Simpson- Mercer type inequality for A-B fractional integral {holds true} for κ∈[0,1]:
|16{Φ(d+D−ω1)+Φ(d+D−ω2)}+23Φ(d+D−ω1+ω22)+2δ(1−δ)Γ(δ)(ω2−ω1)δΦ(d+D−ω1+ω22) | (2.16) |
− 2δ−1M(δ)Γ(δ)(ω2−ω1)δ { AB(d+D−ω1)Iδ(d+D−ω1+ω22)Φ(d+D−ω1+ω22) +AB(d+D−ω2)Iδ(d+D−ω1+ω22)Φ(d+D−ω1+ω22)}|≤ω2−ω12J5(δ)×{J6(δ)1q+J7(δ)1q}, | (2.17) |
where
J5(δ)= (∫10|κδ2−13|dκ)1−1q=(∫10|13−κδ2|dκ)1−1q,J6(δ)=∫10|κδ2−13|{|Φ′(d)|q+|Φ′(D)|q−[(1+κ2)s|Φ′(ω1)|q+(1−κ2)s|Φ′(ω2)|q]}dκ, |
and
J7(δ)=∫10|κδ2−13|{|Φ′(d)|q+|Φ′(D)|q−[(1−κ2)s|Φ′(ω1)|q+(1+κ2)s|Φ′(ω2)|q]}dκ. |
Proof. Employing Lemma 2.1 and well-known Power mean inequality, we have
|16{Φ(d+D−ω1)+Φ(d+D−ω2)}+23Φ(d+D−ω1+ω22)+2δ(1−δ)Γ(δ)(ω2−ω1)δΦ(d+D−ω1+ω22) | (2.18) |
− 2δ−1M(δ)Γ(δ)(ω2−ω1)δ { AB(d+D−ω1)Iδ(d+D−ω1+ω22)Φ(d+D−ω1+ω22) +AB(D+D−ω2)Iδ(d+D−ω1+ω22)Φ(d+D−ω1+ω22)}|≤ ω2−ω12[∫10|κδ2−13||Φ′(d+D−[1+κ2ω1+1−κ2ω2])|dκ+∫10|13−κδ2||Φ′(d+D−[1−κ2ω1+1+κ2ω2])|dκ]≤ ω2−ω12[(∫10|κδ2−13|dκ)1−1q(∫10|κδ2−13||Φ′(d+D−[1+κ2ω1+1−κ2ω2])|qdκ)1q+ (∫10|κδ2−13|dκ)1−1q(∫10|κδ2−13||Φ′(d+D−[1−κ2ω1+1+κ2ω2])|qdκ)1q]. | (2.19) |
Since |Φ′|q is s-convex on [d,D], with the J-M inequality, we obtain
|Φ′(d+D−[1+κ2ω1+1−κ2ω2])|q≤ |Φ′(d)|q+|Φ′(D)|q−[(1+κ2)s|Φ′(ω1)|q+(1−κ2)s|Φ′(ω2)|q], | (2.20) |
and
|Φ′(d+D−[1−κ2ω1+1+κ2ω2])|q ≤ |Φ′(d)|q+|Φ′(D)|q−[(1−κ2)s|Φ′(ω1)|q+(1+κ2)s|Φ′(ω2)|q]. | (2.21) |
By using the Eqs (2.20) and (2.21) in (2.19), we get
|16{Φ(d+D−ω1)+Φ(d+D−ω2)}+23Φ(d+D−ω1+ω22)+2δ(1−δ)Γ(δ)(ω2−ω1)δΦ(d+D−ω1+ω22)− 2δ−1M(δ)Γ(δ)(ω2−ω1)δ { AB(d+D−ω1)Iδ(d+D−ω1+ω22)Φ(d+D−ω1+ω22) +AB(d+D−ω2)Iδ(d+D−ω1+ω22)Φ(d+D−ω1+ω22)}|≤ω2−ω12J5(δ)×{J6(δ)1q+J7(δ)1q}, |
which completes the proof.
Remark 2.5. If we consider ω1=d, ω2=D and δ=1 in Theorem 2.4, we obtain Theorem 10 in [27].
Corollary 2.8. Under the same assumptions as defined in Theorem 2.4 with s=1, we get the following A-B fractional inequality of Simpson-Mercer type:
|16{Φ(d+D−ω1)+Φ(d+D−ω2)}+23Φ(d+D−ω1+ω22)+2δ(1−δ)Γ(δ)(ω2−ω1)δΦ(d+D−ω1+ω22)− 2δ−1M(δ)Γ(δ)(ω2−ω1)δ { AB(d+D−ω1)Iδ(d+D−ω1+ω22)Φ(d+D−ω1+ω22) +AB(d+D−ω2)Iδ(d+D−ω1+ω22)Φ(d+D−ω1+ω22)}|≤ω2−ω12(J5(δ)×{J6(δ)1q+J7(δ)1q}). |
Example 3.1. Let the function Jϱ:ℜ→[1,∞) be defined [54] as
Jϱ(u)=2ϱΓ(ϱ+1)u−ϱIϱ(u), u∈ℜ. |
Here, we consider the modified Bessel function of first kind given in
Jϱ(u)=∞∑n=0(u2)ϱ+2nn!Γ(ϱ+n+1). |
The first and second order derivative are given as
J′ϱ(u)=u2(ϱ+1)Jϱ+1(u). |
J′′ϱ(u)=14(ϱ+1)[u2(ϱ+1)Jϱ+2(u)+2Jϱ+1(u)]. |
If we use, Φ(u)=J′ϱ(u) and the above functions in Corollary 2.3, we have
|16{(d+D−ω1)2(ϱ+1)Jϱ+1(d+D−ω1)+4(d+D−ω1+ω222(ϱ+1)Jϱ+1(d+D−ω1+ω22))+d+D−ω22(ϱ+1)Jϱ+1(d+D−ω2)}−1ω2−ω1[Jϱ(d+D−ω1)−Jϱ(d+D−ω2)]|≤ω2−ω12[518{|14(ϱ+1)[d2(ϱ+1)Jϱ+2(d)+2Jϱ+1(d)]|+|14(ϱ+1)[D2(ϱ+1)Jϱ+2(D)+2Jϱ+1(D)]|}−12s[|14(ϱ+1)[ω21(ϱ+1)Jϱ+2(ω1)+2Jϱ+1(ω1)]|+|14(ϱ+1)[ω22(ϱ+1)Jϱ+2(ω2)+2Jϱ+1(ω2)]|]×(16(s+1)(5s+2−13s+1−6)−12(s+2)(5s+2+13s+2−2)+5s+2+12(s+1)(s+2)3s+2−2s(2s+7)3(s+1)(s+2))]. |
Example 3.2. If we use, Φ(u)=J′ϱ(u) and the above functions in Corollary 2.5, we have
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Consider the following two special means for 0<d<D.
The arithmetic mean:
A(d,D)=d+D2. |
The generalized logarithmic-mean:
Lk(d,D)=[Dk+1−dk+1(k+1)(D−d)]1k;k∈ℜ∖{−1,0}. |
Proposition 3.1. Suppose that d,D∈ℜ, s∈(0,1], 0<d<D, 0∉[d,D]. Then,
|16{2A((d+D−ω1)s,(d+D−ω2)s)}+4(2A(d,D)−A(ω1,ω2))s−Lss(d+D−ω1,d+D−ω2)|≤ s(ω2−ω1)2[(59A(ds−1,Ds−1)−12s−1A(ωs−11,ωs−12))(J3(1)+J4(1))]. |
Proof. The proof follows an immediate consequences from Corollary 2.3 by considering Φ(u)=us.
Proposition 3.2. Suppose that d,D∈ℜ, s∈(0,1], 0<d<D, 0∉[d,D]. Then for all q>1, the following inequality holds true:
|16{2A((d+D−ω1)−s,(d+D−ω2)−s)}+4(2A(d,D)−A(ω1,ω2))−s−L−s−s(d+D−ω1,d+D−ω2)|≤s(ω2−ω1)12(1+2p+1(3p+3))1p{(2A(d−(s+1)q,D−(s+1)q)−[ω−(s+1)q1+A(ω1,ω2)−(s+1)qs+1])1q+(2A(d−(s+1)q,D−(s+1)q)−[ω−(s+1)q2+A(ω1,ω2)−(s+1)qs+1])1q}. |
Proof. The proof follows an immediate consequence from Corollary 5 by considering Φ(u)=u−s.
Proposition 3.3. Suppose that d,D∈ℜ, s∈(0,1], 0<d<D, 0∉[d,D]. Then,
|16{2A((d+D−ω1)s,(d+D−ω2)s)}+4(2A(d,D)−A(ω1,ω2))s−Lss(d+D−ω1,d+D−ω2)|≤ s(ω2−ω1)12(1+2p+1(3p+3))1p{(2A(d−(s+1)q,D−(s+1)q)−[(2s+1−1)ω(s−1)q1+ω(s−1)q22s(s+1)])1q+(2A(d−(s+1)q,D−(s+1)q)−[(2s+1−1)ω(s−1)q2+ω(s−1)q1(s+1)2s])1q}. |
Proof. The proof follows an immediate consequences from Corollary 2.7 by considering Φ(u)=us.
The q-digamma(psi) function ϱρ, is the ρ-analogue of the digamma function ϱ (see [54,55]) given as:
ϱρ=−ln(1−ρ)+ lnρ∞∑k=0ρk+γ1−ρk+γ =−ln(1−ρ)+ lnρ∞∑k=0ρkγ1−ρkγ. |
For ρ>1 and γ>0, ρ-digamma function ϱρ can be given as:
ϱρ=−ln(ρ−1)+ lnρ[γ−12−∞∑k=0ρ−(k+γ)1−ρ−(k+γ)] =−ln(ρ−1)+ lnρ[γ−12−∞∑k=0ρ−kγ1−ρ−kγ]. |
Proposition 3.4. Suppose d,D,ρ be real numbers such that 0<d<D, 0<ρ<1, s∈(0,1] and 1p+1q=1,q>1. Then the following inequality holds true:
|16{ϱρ(d+D−ω1)+ ϱρ(d+D−ω2)}+23ϱρ(d+D−ω1+ω22)−1ω2−ω1∫d+D−ω1d+D−ω2ϱρ(γ)dγ| ≤ ω2−ω112(1+2p+1(3p+3))1p{(|ϱ′ρ(d)|q+|ϱ′ρ(D)|q−[2s+1−1(s+1)2s|ϱ′ρ(ω1)|q+ 1(s+1)2s |ϱ′ρ(ω2)|q])1q+ (|ϱ′ρ(d)|q+|ϱ′ρ(D)|q−[1(s+1)2s|ϱ′ρ(ω1)|q+ 2s+1−1(s+1)2s |ϱ′ρ(ω2)|q])1q}. |
Proof. By employing the definition of ρ-digamma function ϱρ(γ), it is easy to notice that ρ -digamma function Φ(γ)=ϱρ(γ) is completely monotonic on (0,∞). This ensures that the function Φ′(γ)=ϱ′ρ(γ) is s-convex on (0,∞) for each ρ∈(0,1). Now by applying Corollary 2.7, we get the desired result.
Proposition 3.5. Suppose d,D,q,ρ be real numbers such that 0<d<D, 1p+1q=1,q>1 and 0<ρ<1, s∈(0,1]. Then the following inequality holds true:
|16{ϱρ(d+D−ω1)+ ϱρ(d+D−ω2)}+23ϱρ(d+D−ω1+ω22)−1ω2−ω1∫d+D−ω1d+D−ω2ϱρ(γ)dγ| ≤ ω2−ω112(1+2p+13(p+1))1p{(|ϱ′ρ(d)|q+|ϱ′ρ(D)|q−[|ϱ′ρ(ω1)|q+|ϱ′ρ(ω1+ω22)|qs+1])1q+( |ϱ′ρ(d)|q+|ϱ′ρ(D)|q−[|ϱ′ρ(ω2)|q+|ϱ′ρ(ω1+ω22)|qs+1])1q}. |
Proof. By employing the definition of ρ-digamma function ϱρ(γ), it is easy to notice that ρ -digamma function Φ(γ)=ϱρ(γ) is completely monotonic on (0,∞). This ensures that the function Φ′(γ)=ϱ′ρ(γ) is s-convex on (0,∞) for each ρ∈(0,1). Now by applying Corollary 2.5, we get the desired result.
Proposition 3.6. Suppose d,D,ρ be real numbers such that 0<d<D, and 0<ρ<1, s∈(0,1]. Then the following inequality holds true:
|16{ϱρ(d+D−ω1)+ ϱρ(d+D−ω2)}+23ϱρ(d+D−ω1+ω22)−1ω2−ω1∫d+D−ω1d+D−ω2ϱρ(γ)dγ| ≤ω2−ω12[518(|ϱ′ρ(d)|+|ϱ′ρ(D)|)−12s(|ϱ′ρ(ω1)|+|ϱ′ρ(ω2)|){16(s+1)(5s+2−13s+1−6)−12(s+2)(5s+2+13s+2−2)+ 5s+2+12(s+1)(s+2)3s+2−2s(2s+7)3(s+1)(s+2)}]. |
Proof. By employing the definition of ρ-digamma function ϱρ(γ), it is easy to notice that ρ -digamma function Φ(γ)=ϱρ(γ) is completely monotonic on (0,∞). This ensures that the function Φ′(γ)=ϱ′ρ(γ) is s-convex on (0,∞) for each ρ∈(0,1). Now by applying Corollary 2.3, we get the desired result.
One can see that the recent developments in the field of inequalities are related to finding generalized versions and new bounds of various well-known inequalities employing different fractional integral operators. Researchers use new methodologies, new applications, and new operators to add originality to this subject. In this article, the incorporation of the Simpson-Mercer inequalities and the Atangana-Baleanu (A-B) fractional integral operator is studied for differentiable mappings, whose derivatives in absolute value at certain powers are s-convex in the second sense. Some novel cases of the established results are discussed as well. Moreover, the study also have applications to special means, modified Bessel functions, and q-digamma functions. An interesting scope is to check whether the methodology of this paper can be utilized to refine the Simpson-Mercer inequality via concepts such as interval analysis, quantum calculus, and coordinates. In future readers can work on modified A-B fractional operators as given in manucripts (see [49,50,51]) and modified Caputo-Fabrizio fractional operators (see [52,53]).
The authors received financial support from Taif University Researches Supporting Project number (TURSP-2020/031), Taif University, Taif, Saudi Arabia.
The authors declare no conflicts of interest.
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