# Some Simpson’s Riemann–Liouville Fractional Integral Inequalities with Applications to Special Functions

Authors: Jamshed Nasir; Shahid Qaisar; Saad Ihsan Butt; Khuram Ali Khan; Rostin Matendo Mabela
Journal: Journal of Function Spaces
 (2022)

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## Abstract

Based on the Riemann–Liouville fractional integral, a new form of generalized Simpson-type inequalities in terms of the first derivative is discussed. Here, some more inequalities for convexity as well as concavity are established. We expect that present outcomes are the generalization of already obtained results. Applications to beta,q-digamma, and Bessel functions are also provided.


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## Body

## 1. Introduction

Integral inequalities have been widely used in various sciences, including mathematical sciences, applied sciences, differential equations, and functional analysis. In most mathematical analysis areas, many types of integral inequalities are used. They are very important in approximation theory and numerical analysis, which estimate the error’s approximation [1–3]. Integral inequalities are useful tools in the study of different classes of differential equations and integral equations. Today, they are employed not only in mathematics but also in physics, computer science, and biology.In several zones of mathematics, convex functions show a vital role. Especially in optimization theory, they are magical due to the number of expedient properties. There is nice connection between the theory of convex functions and mathematical inequalities. There are several important inequalities due to their direct applications in applied sciences (see [4–8]).Several integral inequalities, such as Hölder’s inequality, Simpson’s inequality, Newton’s inequality, the Hermite–Hadamard inequality, Ostrowski’s inequality, Cauchy–Schwarz, and Chebyshev, are well known in classical analysis and have been proven and applied in the setup ofq-calculus using classical convexity [9–12]. Much work is being done in the domain of q-analysis, beginning with Euler, in order to achieve proficiency in mathematics that constructs quantum computing. q-Calculus is viewed as a link between mathematics, physics, and statistics. It has numerous applications in various areas of mathematics, including orthogonal polynomials, number theory, hypergeometric functions, and other mechanics, combinatorics, sciences, stochastic theory, quantum theory, and the theory of relativity. This important branch of mathematics appears to have been invented by Euler. In Newton’s work with infinite series, he used the q-parameter. Jackson later gave the q-calculus integral without the limit.During the period 1710–1761, Thomas Simpson developed critical methods for numerical integration and estimation of definite integrals, which became known as the Simpson rule. Nonetheless, Kepler used a similar approximation nearly last decades earlier, so it is also known as the Kepler rule. Simpson rule includes the three-point Newton–Cotes quadrature rule, so a three-step quadratic kernel estimation is sometimes referred to as a Newton-type result.An incredible reliance has been found between inequalities and the theory of convex functions. This relationship is the primary mental stability behind the vast data utilizing convex functions. The Simpson-type inequalities have been examined broadly in the course of recent decades. A functionℏ:J⊂ℜ⟶ℜ is said to be convex on σ1,σ2, with σ1<σ2 where σ1,σ2∈J:(1)ℏησ1+1−ησ2≤ηℏσ1+1−ηℏσ2,η∈0,1,an inequality which is notable as Simpson’s inequality in [13].Theorem 1.
Supposeℏ:σ1,σ2⟶ℜ is four times continuously differentiable function on σ1,σ2 and ℏ4∞=supθ∈σ1,σ2ℏ4θ<∞; then, the following inequality holds:(2)16ℏσ1+23ℏσ1+σ22+16ℏσ2−1σ2−σ1∫σ1σ2ℏθdθ≤σ2−σ142880ℏ4∞.Many scholars are interested in the Simpson-type inequality since it has been examined and studied for numerous classes of functions. Due to their efficacy and usefulness in pure and applied mathematics, Simpson-type and Newton-type results have been keenly interpolated for convex functions. The first striking result about the Simpson-type inequality along with its applications to the quadrature formula in numerical integration was given by Dragomir et al. in [14]. Later on, several new Simpson’s type inequalities with improved bounds were developed for s-convex functions by Alomari et al. [13] and Sarikaya et al. [15].Some researchers have demonstrated Simpson’s type inequalities and obtained various outcomes. A new generalization and extension of Simpson’s type inequalities were presented in [16, 17]. Qaisar et al. in [18, 19] gave Simpson’s type inequalities for twice differentiable convex mappings with applications. However, some fractional variants can be observed in [20–22]. There is massive literature regarding improvements and extensions of Simpson’s inequality in q-calculus. Recently, Ali et al. in [12] gave new quantum boundaries for quantum Simpson-type and Newton-type inequalities for preinvex functions. They also gave a brief literature review about the development of results connected to quantum Simpson’s inequality. The accompanying ongoing improvements for the fractional integral on Simpson’s inequality for ω>0 were demonstrated by Hwang et al. (see [20]).Theorem 2.
Supposeℏ:σ1,σ2⟶ℜ is a differentiable function whose derivative is continuous on σ1,σ2 and ℏ′∈ℒσ1,σ2; then, the following inequality holds:(3)Γω+12σ2−σ1ωJσ1+ωℏσ2+Jσ2−ωℏσ1−5ω−16ωℏσ1+σ22+6ω−5ω+16ωℏσ1+ℏσ22≤σ2−σ11ω+12ω+12ω+1−5ω+1+16ω+1+5ω−112.6ωℏ′σ1+ℏ′σ2.Throughout the text, the class of integrable functions onσ1,σ2 is denoted by ℒσ1,σ2. Another form utilizing Riemann–Liouville fractional integrals mentioned above for differentiable convex functions was presented by Iqbal et al. (see [22]).Theorem 3.
Supposeℏ:σ1,σ2⟶ℜ is a differentiable function and ℏ′∈ℒσ1,σ2 is integrable on σ1,σ2 with σ1<σ2 and 0<ω≤1. If ℏ′ is convex on σ1,σ2, then we have(4)16ℏσ1+23ℏσ1+σ22+16ℏσ2−Γω+12σ2−σ1ωJσ1+ωℏσ2+Jσ2−ωℏσ1≤σ2−σ12ω+1×2u−ω+26ω+1+2ω−13+122v−3−1ω+15v3−2ω+1+12ℏ′σ1+ℏ′σ2,where u=1/31/ω and vω=22ω−1+3/3.Proposition 1.
In above Theorem3, if ω=1, one can obtain(5)16ℏσ1+23ℏσ1+σ22+16ℏσ2−1σ2−σ1∫σ1σ2ℏθdθ≤5σ2−σ172ℏ′σ1+ℏ′σ2.Proposition 2.
In above Theorem3 with ℏσ1=ℏσ1+σ2/2=ℏσ2, one can obtain(6)1σ2−σ1∫σ1σ2ℏθdθ−ℏσ1+σ22≤5σ2−σ172ℏ′σ1+ℏ′σ2.Theorem 4.
Letℏ be defined as in Theorem 3, and if ℏ′ is convex on σ1,σ2, with q≥1, then we have(7)16ℏσ1+23ℏσ1+σ22+16ℏσ2−Γω+12σ2−σ1ωJσ1+ωℏσ2+Jσ2−ωℏσ1≤σ2−σ12ωS11−1/qϒ5ℏ′σ1q+ϒ6ℏ′σ2q21/q+ϒ5ℏ′σ1q+ϒ6ℏ′σ2q21/q+S21−1/qϒ7ℏ′σ1q+ϒ8ℏ′σ2q21/q+ϒ7ℏ′σ1q+ϒ8ℏ′σ2q21/q,ϒ1=∫01−u121−ηω−16dη=−161−u−12ω+1uω+1+12ω+1,ϒ2=∫1−u116−121−ηωdη=16−161−u−12ω+1uω+1,ϒ3=∫0v−1122ω−1−122ω−11+ηω+13dη=13+122ω−1v−1−vω+122ω−1ω+1+122ω−1ω+1,ϒ4=∫v−11122ω−11+ηω−122ω−1−13dη=2ω+122ω−1ω+1−13+122ω−1−vω+122ω−1ω+1+13+122ω−1v−1,ϒ5=∫0116−121−ηω1+ηdη=3ω+1+4ωω+2u−ωω+1u212ω+1ω+2−18,ϒ6=∫0116−121−ηω1−ηdη=2ωu2−ω+424ω+2,ϒ7=∫01122ω−1−122ω−11+ηω+131+ηdη=122ω−1v2−522ω−13+12−1ω+253v2−2ω+1+1213+122ω−1,ϒ8=∫01122ω−1−122ω−11+ηω+131−ηdη=122ω−112−2−v22ω−13+12+1ω+112−5v32−v+1ω+1ω+22ω+2+12−53v2.For the execution of differentiation and integration of real or complex number orders, fractional calculus proved as a helpful device which demonstrates its centrality. The theme had pulled in a lot of considerations from many authors who center around investigation of PDEs during the most recent couple of decades. For late outcomes identified with the current study, one can see [23–27]. Among a lot of fractional integrals which are grown up, the Riemann–Liouville fractional integral has been widely considered as a result of uses in numerous fields of sciences.Definition 1.
Supposeℏ∈ℒσ1,σ2. The left and right Riemann–Liouville fractional integral operator for ω>0 are(8)Jσ1+ωℏτ=1Γω∫σ1ττ−ηω−1ℏηdη,σ1<τ,Jσ2−ωℏτ=1Γω∫τσ2τ−ηω−1ℏηdη,τ<σ2.
Gamma function is defined asΓω=∫0∞e−uuω−1du. Note that Jσ1+0ℏτ=Jσ2−0ℏτ=ℏτ.
Ifω=1, the above integral becomes the classical integral.


## 2. Inequalities for Simpson’s Type

In this section, we give Simpson’s inequalities for the Riemann–Liouville integral operator for differentiable functions onσ1,σ2. For this, we give a new Riemann–Liouville integral operator auxiliary identity that will serve to produce subsequent results for improvements.Lemma 1.
Letℏ:σ1,σ2⟶ℜ be a differentiable mapping such that ℏ′∈ℒσ1,σ2 is integrable and 0<ω≤1 on σ1,σ2 with σ1<σ2; then, the following identity holds:(9)16ℏσ1+23ℏσ1+σ22+16ℏσ2−Γω+12σ2−σ1ωJσ1+ωℏσ2+Jσ2−ωℏσ1=σ2−σ12A1+A2+A3+A4,where(10)A1=∫01/2ηω−16ℏ′ησ2+1−ησ1dη,A2=∫1/21ηω−56ℏ′ησ2+1−ησ1dη,A3=∫01/216−ηωℏ′ησ1+1−ησ2dη,A4=∫1/2156−ηωℏ′ησ1+1−ησ2dη.Proof.
By integration by parts, we obtain(11)A1=∫01/2ηω−16ℏ′ησ2+1−ησ1dη=1σ2−σ112ω−16ℏσ1+σ22+16σ2−σ1ℏσ1−ωσ2−σ1∫01/2ηω−1ℏ′ησ2+1−ησ1dη,A1σ2−σ1=12ω−16ℏσ1+σ22+16ℏσ1−ω∫01/2ηω−1ℏ′ησ2+1−ησ1dη,A2σ2−σ1=16ℏσ2−12ω−56ℏσ1+σ22−ω∫1/21ηω−1ℏ′ησ2+1−ησ1dησ2−σ12A1+A2=412ℏσ1+σ22+112ℏσ1+ℏσ2−ω2∫01ηω−1ℏησ2+1−ησ1dη=412ℏσ1+σ22+112ℏσ1+ℏσ2−ω2∫σ1σ2θ−σ1ω−1σ2−σ1ωℏθdθ=412ℏσ1+σ22+112ℏσ1+ℏσ2−Γω+2σ2−σ1ωJσ2−ωℏσ1.
Similarly,(12)σ2−σ12A3+A4=412ℏσ1+σ22+112ℏσ1+ℏσ2−Γω+12σ2−σ1ωJσ1+ωℏσ2,which ends the proof.Theorem 5.
Letℏ be defined as in Lemma 1, and if ℏ′ is convex on σ1,σ2, then we have(13)16ℏσ1+23ℏσ1+σ22+16ℏσ2−Γω+12σ2−σ1ωJσ1+ωℏσ2+Jσ2−ωℏσ1≤σ2−σ12β+5γ−43+1+2ω1−2βω+1−2γω+12ωω+1ℏ′σ1+ℏ′σ2.Proof.
By using the properties of modulus on Lemma1, we have(14)16ℏσ1+23ℏσ1+σ22+16ℏσ2−Γω+12σ2−σ1ωJσ1+ωℏσ2+Jσ2−ωℏσ1≤σ2−σ12A1+A2+A3+A4,where β=1/61/ω and γ=5/61/ω.
By the convexity ofℏ′, ω∈0,1, and ∀η∈0,1, we have(15)16ℏσ1+23ℏσ1+σ22+16ℏσ2−Γω+12σ2−σ1ωJσ1+ωℏσ2+Jσ2−ωℏσ1≤σ2−σ12∫01/2ηω−16ℏ′ησ2+1−ησ1dη+σ2−σ12∫01/216−ηωℏ′ησ1+1−ησ2dη+σ2−σ12∫1/21ηω−56ℏ′ησ2+1−ησ1dη+σ2−σ12∫1/2156−ηωℏ′ησ1+1−ησ2dη=σ2−σ12∫01/2ηω−16ηℏ′σ2+1−ηℏ′σ1|dη+σ2−σ12∫01/216−ηωηℏ′σ1+1−ηℏ′σ2|dη+σ2−σ12∫1/21ηω−56ηℏ′σ2+1−ηℏ′σ1|dη+σ2−σ12∫1/2156−ηωηℏ′σ1+1−ηℏ′σ2|dη≤σ2−σ12β+5γ−43+1+2ω1−2βω+1−2γω+12ωω+1ℏ′σ1+ℏ′σ2.
Simple calculations yield that(16)S1=∫01/2ηω−16dη=4β−112+1−βω+1⋅2ω+12ω+1ω+1,S2=∫1/21ηω−56dη=54γ−312+2ω+1+1−γω+1⋅2ω+22ω+1ω+1,S3=∫01/2ηω−16ηdη=8β2−148+1−β3⋅2ω+32ω+2ω+2,S4=∫1/21ηω−56ηdη=40γ2−2548+2ω+2+1−γω+2⋅2ω+32ω+2ω+2,which completes the proof.Remark 1.
On lettingω=1 in Theorem 5, (13) reduces to inequality (5).Corollary 1.
By choosingℏσ1=ℏσ1+σ2/2=ℏσ2 in Theorem 5, inequality (13) becomes(17)ℏσ1+σ22−Γω+12σ2−σ1ωJσ1+ωℏσ2+Jσ2−ωℏσ1≤σ2−σ12β+5γ−43+1+2ω1−2βω+1−2γω+12ωω+1ℏ′σ1+ℏ′σ2.Remark 2.
By lettingω=1 in the above corollary, we get the following inequality, which looks better than the inequality presented by S. Kirmaci:(18)ℏσ1+σ22−1σ2−σ1∫σ1σ2ℏθdθ≤σ2−σ12536ℏ′σ1+ℏ′σ2.
The corresponding version for powers of the absolute value of the derivative is incorporated as follows.Theorem 6.
Letℏ be defined as in Lemma 1, and if ℏ′q is convex on σ1,σ2, with q≥1, then we have(19)​​​​16ℏσ1+23ℏσ1+σ22+16ℏσ2−Γω+12σ2−σ1ωJσ1+ωℏσ2+Jσ2−ωℏσ1≤σ2−σ12×1+2ωp+16ωp+1ωp+11/p12ω/q−1ℏ′σ1q+ℏ′σ1+σ22q1/q+ℏ′σ1+σ22q+ℏ′σ2q1/q.Proof.
Using Lemma1 and convexity, we obtain(20)16ℏσ1+23ℏσ1+σ22+16ℏσ2−Γω+12σ2−σ1ωJσ1+ωℏσ2+Jσ2−ωℏσ1≤σ2−σ12∫01/2ηω−16ℏ′ησ2+1−ησ1dη+∫1/21ηω−56ℏ′ησ2+1−ησ1dη+∫01/216−ηωℏ′ησ1+1−ησ2dη+∫1/2156−ηωℏ′ησ1+1−ησ2dη≤σ2−σ12∫01/2ηω−16pdη1/p∫01/2ℏ′ησ2+1−ησ1qdη1/q+σ2−σ12∫1/21ηω−56pdη1/p∫1/21ℏ′ησ2+1−ησ1qdη1/q+σ2−σ12∫01/2ηω−16pdη1/p∫01/2ℏ′ησ1+1−ησ2qdη1/q+σ2−σ12∫1/21ηω−56pdη1/p∫1/21ℏ′ησ1+1−ησ2qdη1/q=σ2−σ12∫01/616−ηωpdη+∫1/61/2ηω−16pdη1/p×∫01/2ℏ′ησ1+1−ησ2qdη1/q+σ2−σ12∫1/25/656−ηωpdη+∫5/61ηω−56pdη1/p×∫1/21ℏ′ησ2+1−ησ1qdη1/q+σ2−σ12∫01/616−ηωpdη+∫1/61/2ηω−16pdη1/p×∫01/2ℏ′ησ1+1−ησ2qdη1/q+σ2−σ12∫1/25/656−ηωpdη+∫5/61ηω−56pdη1/p×∫1/21ℏ′ησ1+1−ησ2qdη1/q≤σ2−σ121+2ωp+16ωp+1ωp+11/p×12ω/q−1ℏ′σ1q+ℏ′σ1+σ22q1/q+ℏ′σ1+σ22q+ℏ′σ2q1/q,which completes the proof.Theorem 7.
Letℏ be defined as in Lemma 1, and if ℏ′q is convex on σ1,σ2, with q≥1, then we have(21)16ℏσ1+23ℏσ1+σ22+16ℏσ2−Γω+12σ2−σ1ωJσ1+ωℏσ2+Jσ2−ωℏσ1≤σ2−σ12×S11−1/qS3ℏ′σ2q+S1−S3ℏ′σ1q1/q+S3ℏ′σ1q+S1−S3ℏ′σ2q1/q+S21−1/qS4ℏ′σ2q+S2−S4ℏ′σ1q1/q+S4ℏ′σ1q+S2−S4ℏ′σ2q1/q.Proof.
By the use of the power-mean integral inequality forq>1, we obtain(22)16ℏσ1+23ℏσ1+σ22+16ℏσ2−Γω+12σ2−σ1ωJσ1+ωℏσ2+Jσ2−ωℏσ1≤σ2−σ12∫01/2ηω−16ℏ′ησ2+1−ησ1dη+∫1/21ηω−56ℏ′ησ2+1−ησ1dη+∫01/2ηω−16ℏ′ησ1+1−ησ2dη+∫1/21ηω−56ℏ′ησ1+1−ησ2dη≤σ2−σ12∫01/2ηω−16dη1−1/q×∫01/2ℏ′ησ2+1−ησ1qdη1/q+∫1/21ηω−56dη1−1/q×∫1/21ℏ′ησ2+1−ησ1qdη1/q+∫01/2ηω−16dη1−1/q×∫01/2ℏ′ησ1+1−ησ2qdη1/q+∫1/21ηω−56dη1−1/q×∫1/21ℏ′ησ1+1−ησ2qdη1/q.
By using the convexity ofℏ′q,(23)∫01/2ηω−16ℏ′ησ2+1−ησ1qdη≤∫01/616−ηωηℏ′σ1q+1−ηℏ′σ2q|dη+∫1/61/2ηω−16ηℏ′σ1q+1−ηℏ′σ2q|dη∫1/21ηω−56ℏ′ησ2+1−ησ1qdη≤∫1/25/656−ηωηℏ′σ1q+1−ηℏ′σ2q|dη+∫5/61ηω−56ηℏ′σ2q+1−ηℏ′σ1q|dη.
By using the calculus tool, we get equation (21), which completes the proof.Remark 3.
Ifω=1 in Theorem 7, then it reduces to Corollary 2.7 in [19].
In the following theorem, we obtained the estimate of Simpson’s inequality (2) for concave functions.Theorem 8.
Supposeℏ:σ1,σ2⟶ℜ is a differentiable function on σ1,σ2 such that ℏ′∈ℒσ1,σ2. If ℏ′q is concave on σ1,σ2, for some fixed p>1 with q=p/p−1, then we have(24)16ℏσ1+23ℏσ1+σ22+16ℏσ2−Γω+12σ2−σ1ωJσ1+ωℏσ2+Jσ2−ωℏσ1≤σ2−σ12×S1ℏ′S3σ2+S1−S3σ1S1+ℏ′S3σ1+S1−S3σ2S1+S2ℏ′S4σ2+S2−S4σ1S2+ℏ′S4σ1+S2−S4σ2S2.Proof.
Utilizing concavity ofℏ′q and the power-mean inequality, we obtain(25)ℏ′ησ1+1−ησ2q≥ηℏ′σ1q+1−ηℏ′σ2q≥ηℏ′σ1+1−ηℏ′σ2q,ℏ′ησ1+1−ησ2≥ηℏ′σ1+1−ηℏ′σ2.
Therefore,ℏ′ is also concave. Jensen’s integral inequality follows that(26)16ℏσ1+23ℏσ1+σ22+16ℏσ2−Γω+12σ2−σ1ωJσ1+ωℏσ2+Jσ2−ωℏσ1≤σ2−σ12∫01/2ηω−16dηℏ′∫01/2ηω−1/6ησ2+1−ησ1dη∫01/2ηω−1/6dηq+σ2−σ12∫1/21ηω−56dηℏ′∫1/21ηω−5/6ησ2+1−ησ1dη∫1/21ηω−5/6dηq+σ2−σ12∫01/216−ηωdηℏ′∫01/21/6−ηωησ1+1−ησ2dη∫01/21/6−ηωdηq+σ2−σ12∫1/2156−ηωdηℏ′∫1/215/6−ηωησ1+1−ησ2dη∫1/215/6−ηωdηq=σ2−σ12S1ℏ′S3σ2+S1−S3σ1S1+σ2−σ12S2ℏ′S3σ1+S1−S3σ2S1+σ2−σ12S1ℏ′S4σ2+S2−S4σ1S2+σ2−σ12S2ℏ′S4σ1+S2−S4σ2S2,which completes the proof.Corollary 2.
By puttingω=1 in Theorem 8, then the following inequality becomes(27)16ℏσ1+23ℏσ1+σ22+16ℏσ2−1σ2−σ1∫σ1σ2ℏθdθ≤5σ2−σ172ℏ′29σ1+61σ290+ℏ′61σ1+29σ290.Remark 4.
Inequality (27) is a generalization of the obtained inequality as in Theorem 8 of [13].


## 3. Applications

### 3.1. Beta Function

In this section, letω>0, γ≥3, σ1=0, σ2=1, Γω be the gamma function, and ℏθ=θγ−1θ∈0,1. Then, ℏ′ is convex on 0,1.Let us recall that the beta function(28)Bp,q=∫01θp−11−θq−1dθp,q>0.Remark 5.
From Section2, we have(29)Γω+12σ2−σ1ωJσ1+ωℏσ2=ω2∫01θγ−11−θω−1dθ=ω2Bγ,ω,Γω+12σ2−σ1ωJσ2−ωℏσ1=ω2∫01θω+γ−2dθ=ω2γ+ω−1.Proposition 3.
In Theorem5, the following inequality holds:(30)23.2γ−1+16−ω2Bγ,ω+ω2γ+ω−1≤γ−12β+5γ−43+1+2ω1−2βω+1−2γω+12ωω+1.


### 3.2.q-Digamma Function

Suppose0<q<1; the q-digamma function φq is the q-analogue of the digamma function φ (see [28, 29]) given as(31)φq=−ln1−q+lnq∑k=0∞qk+ζ1−qk+ζ=−ln1−q+lnq∑k=0∞qkζ1−qkζ.Forq>1 and ζ>0, q-digamma function φq can be given as(32)φq=−lnq−1+lnqζ−12−∑k=0∞q−k+ζ1−q−k+ζ=−lnq−1+lnqζ−12−∑k=0∞q−kζ1−q−kζ.Proposition 4.
Assume thatσ1,σ2 are the real numbers such that 0<σ1<σ2 and 0<q<1. Then, the following inequality is valid:(33)φqσ1+σ22−1σ2−σ1∫σ1σ2φqεdε≤σ2−σ12536φq1σ1+φq1σ2.Proof.
The assertion can be obtained immediately by using inequality (33) which follows immediately from equation (18), when ℏε=φqε and ε>0 since ℏ′ε=φq′ε is convex on 0,+∞.


### 3.3. Modified Bessel Function

We recall the first kind of modified Bessel functionℑm, which has the series representation (see [28], p.77)(34)ℑmζ=∑n≥0ζ/2m+2nn!Γm+n+1,where ζ∈ℜ and m>−1. Consider the function Ωmζ:ℜ⟶1,∞ defined by(35)Ωmζ=2mΓm+1ζ−mℑmζ,where Γ is the gamma function.The first-order derivative formula ofΩmζ is given by [28](36)Ωm′ζ=ζ2m+1Ωm+1ζ,and the second derivative can be easily calculated from (36) to be(37)Ωm′′ζ=ζ2Ωm+2ζ4m+1m+2+Ωm+1ζ2m+1.Proposition 5.
Suppose thatm>−1 and 0<σ1<σ2. Then, we have(38)σ1+σ24m+1Ωm+1σ1+σ22−Ωmσ2−Ωmσ1σ2−σ1≤σ2−σ12536σ12Ωm+2σ14m+1m+2+Ωm+1σ12m+1+σ22Ωm+2σ24m+1m+2+Ωm+1σ22m+1.Proof.
Apply the inequality in (18) to the mapping ℏζ=Ωm′ζ, ζ>0 (note that all assumptions are satisfied), and identities (36) and (37).


## 3.1. Beta Function

In this section, letω>0, γ≥3, σ1=0, σ2=1, Γω be the gamma function, and ℏθ=θγ−1θ∈0,1. Then, ℏ′ is convex on 0,1.Let us recall that the beta function(28)Bp,q=∫01θp−11−θq−1dθp,q>0.Remark 5.
From Section2, we have(29)Γω+12σ2−σ1ωJσ1+ωℏσ2=ω2∫01θγ−11−θω−1dθ=ω2Bγ,ω,Γω+12σ2−σ1ωJσ2−ωℏσ1=ω2∫01θω+γ−2dθ=ω2γ+ω−1.Proposition 3.
In Theorem5, the following inequality holds:(30)23.2γ−1+16−ω2Bγ,ω+ω2γ+ω−1≤γ−12β+5γ−43+1+2ω1−2βω+1−2γω+12ωω+1.


## 3.2.q-Digamma Function

Suppose0<q<1; the q-digamma function φq is the q-analogue of the digamma function φ (see [28, 29]) given as(31)φq=−ln1−q+lnq∑k=0∞qk+ζ1−qk+ζ=−ln1−q+lnq∑k=0∞qkζ1−qkζ.Forq>1 and ζ>0, q-digamma function φq can be given as(32)φq=−lnq−1+lnqζ−12−∑k=0∞q−k+ζ1−q−k+ζ=−lnq−1+lnqζ−12−∑k=0∞q−kζ1−q−kζ.Proposition 4.
Assume thatσ1,σ2 are the real numbers such that 0<σ1<σ2 and 0<q<1. Then, the following inequality is valid:(33)φqσ1+σ22−1σ2−σ1∫σ1σ2φqεdε≤σ2−σ12536φq1σ1+φq1σ2.Proof.
The assertion can be obtained immediately by using inequality (33) which follows immediately from equation (18), when ℏε=φqε and ε>0 since ℏ′ε=φq′ε is convex on 0,+∞.


## 3.3. Modified Bessel Function

We recall the first kind of modified Bessel functionℑm, which has the series representation (see [28], p.77)(34)ℑmζ=∑n≥0ζ/2m+2nn!Γm+n+1,where ζ∈ℜ and m>−1. Consider the function Ωmζ:ℜ⟶1,∞ defined by(35)Ωmζ=2mΓm+1ζ−mℑmζ,where Γ is the gamma function.The first-order derivative formula ofΩmζ is given by [28](36)Ωm′ζ=ζ2m+1Ωm+1ζ,and the second derivative can be easily calculated from (36) to be(37)Ωm′′ζ=ζ2Ωm+2ζ4m+1m+2+Ωm+1ζ2m+1.Proposition 5.
Suppose thatm>−1 and 0<σ1<σ2. Then, we have(38)σ1+σ24m+1Ωm+1σ1+σ22−Ωmσ2−Ωmσ1σ2−σ1≤σ2−σ12536σ12Ωm+2σ14m+1m+2+Ωm+1σ12m+1+σ22Ωm+2σ24m+1m+2+Ωm+1σ22m+1.Proof.
Apply the inequality in (18) to the mapping ℏζ=Ωm′ζ, ζ>0 (note that all assumptions are satisfied), and identities (36) and (37).


## 4. Conclusion

The study dealt with investigating a new method to give fractional Simpson’s estimates for differentiable convex functions by taking into account the kernel given by Alomari et al. [13]. Several related estimations by employing Hölder’s inequality, bounds by using concave functions, and Jensen’s integral inequalities are presented. Finally, a representation of Simpson’s inequalities in terms of beta and q-digamma functions is depicted. Some estimation interims of modified Bessel functions are represented. Our technique is also plausible to give extensions for other fractional integral operators, e.g., k-Riemann–Liouville, Katugampola, conformable, and Atangana–Baleanu. Moreover, one can also extend these results in quantum calculus by exhibiting our method of arrangements of kernels for differentiable convex functions. All these results are open to be discussed for generalized convex functions and, in particular, can be given by using the method adopted in this article [12].



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