Novel Approaches to Dual Simpson-Type Inequalities via Caputo-Fabrizio Fractional Operators for Different Classes of Functions and Application

Novel Approaches to Dual Simpson-Type Inequalities via Caputo-Fabrizio Fractional Operators for Different Classes of Functions and Application Artion Kashuri1, Arslan Munir2, Hüseyin Budak3 1 Department of Mathematical Engineering, Polytechnic University of Tirana, 1001 Tirana, Albania 2 School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026 People’s Republic of China 3Department of Mathematics, Faculty of Science and Arts, Kocaeli University, Kocaeli 41001, Türkiye Corresponding author: m[email protected] ORCID IDs: First Author: 0000-0003-0115-3079 Second Author: 0009-0008-8148-5607 Third Author: 0000-0001-8843-955X DOI : 10.5281/zenodo.18011756 Abstract Integral inequalities are indeed a powerful tool in the context of numerical integration and quadrature methods. The study of fractional calculus continues to be an active area of research with practical implications for real-world applications. When combined with fractional operators, these integral inequalities become even more valuable in the analysis and estimation of errors in fractional calculus. The CaputoFabrizio fractional integral operator is one of the key concepts in fractional calculus. In the present study, we established a new identity through the fractional operator. By using this novel identity, some new error estimations of dual-Simpson type inequalities are obtained for the case of differentiable functions. Based on this identity, dualSimpson type inequalities are proved for bounded as well as Lipschitzian functions. In addition, we demonstrate an application of our findings to the quadrature formula. Keywords: Dual Simpson-type inequalities, Caputo-Fabrizio fractional integrals, s - M4-1 KOSC-2025 Proceedings convex function, Hölder’s inequality, power-mean inequality, Young’s inequality, bounded function, Lipschitzian function, quadrature formula. 1 Introduction and Preliminaries Convexity is indeed one of the most fundamental principles in analysis and mathematics. It plays a crucial role in various areas of pure and applied sciences, including optimization, functional analysis, numerical analysis, economics, physics, engineering, and the domain of inequalities. In all these fields, convexity allows for the development of important results. Convexity ensures that certain properties hold, making the analysis more tractable and providing powerful tools for solving problems. For interested researchers on convexity and integral inequalities, we refer to articles [ 1 , 2 , 3 ]. A function f : I→R is said to be convex if the inequality holds: f(ΥΨ + (1 −Υ) b)≤Υf(Ψ) + (1 −Υ) f(b), for all Ψ , b ∈I and Υ ∈[0,1] . In [ 4 ] Hudzik introduced the generalized convexity known as s -convexity. The function f : R+→R , where R+ = [0 , + ∞ ), is called to be s -convex in the second sense for a fixed real number s∈(0,1] for all Ψ , b ∈I and Υ∈[0,1], if f(ΥΨ + (1 −Υ) b)≤Υsf(Ψ) + (1 −Υ)sf(b). Inequality theory is well-known and still an interesting area of research with a wide range of applications in many fields of mathematics. In the last few decades, various authors have been interested in generalizing the different types of inequalities, the Simpson-type inequality is one of these. The classical Simpson-type inequality deals with the error term of Simpson’s Rule and provides an upper bound for the error between the true value of the integral and the approximation obtained through Simpson’s rule. We delineate three primary formulations as follows: (1) Simpson’s quadrature formula (Simpson’s 1/3rule) is stated as follows: Zb Ψ f(Φ) dΦ≈b−Ψ 6f(Ψ) + 4fΨ+b 2+f(b). (2) Simpson’s second formula or the Newton-cotes quadrature formula (Simpson’s M4-2 2nd Kocaeli Science Congress, November 19-21, 2025 3/8rule) is stated as follows: Zb Ψ f(Φ) dΦ≈b−Ψ 8f(Ψ) + 3f2Ψ+b 2+ 3fΨ+2b 2+f(b). The most important Newton-cotes quadrature that attained three-point Simpson-type inequality is described as follows: Theorem 1.1. Suppose f : [Ψ, b]→R be a four times differentiable function and continuous on (Ψ, b)and let   f(4)  ∞:= supΦ∈(Ψ,b)f(4) (Φ)<∞, then f(Ψ) 6+4 6fΨ+b 2+f(b) 6−1 b−ΨZb Ψ f(Φ) dΦ ≤1 2880   f(4)  ∞(b−Ψ)4. Overall, Simpson-type inequalities have been a subject of interest for researchers across different mathematical fields due to their practical relevance and theoretical significance. The study of Simpson-type inequalities for convex functions is an important area of research that contributes to the broader field of convex analysis. Simpson-type inequalities are a class of inequalities that involve certain functions and their properties, they are related to similar inequality to s -convex functions obtained [ 5 ]. Then, using differentiable convex mapping, Sarikaya et al. established new versions of Simpson-type inequalities [ 6 , 7 ]. Du et al. in [ 8 ] proved the Simpson’s inequalities using the extended ( s, m )-convex functions for differentiable mapping. Furthermore, Işcan et al. [ 9 ] established Simpsontype inequalities and giving new error bounds using the harmonically convex functions. Additionally, a few works [ 9 , 10 ] looked at the Simpson-type inequalities for different convex classes. Fractional calculus has gained significant attention and found applications in various fields over the last few decades. In pure and applied mathematics, fractional calculus is an active area of research, with mathematicians studying the properties of fractional derivatives, fractional differential equations, fractional integrals, fractional operators, and their connections to other areas of mathematics, such as fractional calculus on fractals. By incorporating fractional derivatives and integrals, researchers and practitioners can capture complex behaviors that cannot be adequately described by classical integer-order calculus. Many real dynamical systems exhibit behaviors that are better described and characterized by using non-integer order dynamical models based on fractional calculus. While fractional calculus has proven to be a valuable tool in various applications, it is important to note that it is not a replacement for classical integer-order calculus. Instead, it complements the traditional methods and provides additional flexibility to describe and 2nd Kocaeli Science Congress, November 19-21, 2025 M4-3 KOSC-2025 Proceedings understand more complex systems in nature. One can analysis many fractional integral inequalities in great detail because of the significance of fractional calculus as indicated in this paragraph. For instance, the Simpson inequality for differentiable functions to Riemann-Liouville fractional integrals were discussed by the authors in [ 11 ] and [ 12 ]. Abdeljawad et al. [ 13 ] obtained the new Simpson inequalities for generalized p -convex function on fractal sets with applications. Ertugral et al. [ 14 ] developed Simpson-type inequalities in the case of different using the generalized fractional integral. As a result, numerous studies are devoted to fractional Simpson inequalities in the case of different fractional integral operators [ 15 ]. Lei et al. [ 16 ] produced an estimate of the remainder for Simpson inequality via k -fractional integrals. Furthermore, several fractional Simpson type inequalities for functions with convex second derivatives in absolute value were demonstrated in the article [ 17 ]. For more on Simpson-type inequalities and other characteristics of Riemann-Liouville fractional integral, readers might see [ 18 , 19 ] and its references. For additional research on Simpson-type inequality, see these articles [ 20 , 21 ]. Sarikaya et al. [ 22 ] proved several Simpson-type inequalities for functions whose second derivatives are convex. Before we start to our main results, first we give the definitions of Riemann-Liouville fractional and Caputo-Fabrizio fractional integral operators as follows: Definition 1.1. [ 23 ] Suppose f∈L[Ψ, b] . The Riemann-Liouville fractional integrals of order ξ > 0are defined by: Iξ Ψ+f(Υ) = 1 Γ(ξ)ZΦ Ψ (Φ −Υ)ξ−1f(Υ) dΥ,Φ>Ψ, Iξ b−f(Υ) = 1 Γ(ξ)Zb Φ (Υ −Φ)ξ−1f(Υ) dΥ,Φ< b, where Γ (ξ)is the gamma function and I0 Ψ+f(Υ) = I0 b−f(Υ) = f(Υ). Definition 1.2. [24] Let H1(Ψ, b)be the Sobolev space of order one defined as: H1(Ψ, b) := ng∈L2(Ψ, b) : g′∈L2(Ψ, b)o, where L2(Ψ, b) :=    g(z) : Zb Ψ g2(z)dz!1 2 <∞   . Let f∈H1(Ψ, b) ,Ψ < b , ξ∈[0,1] , then the notion of left derivative in the sense of M4-4 2nd Kocaeli Science Congress, November 19-21, 2025 Caputo-Fabrizio is defined as: CF D ΨDξf(Φ) = β(ξ) 1−ξZΦ Ψ f′(Υ) e −ξ(Φ−Υ)ξ 1−ξdΥ and the associated integral operator is CF ΨIξf(Φ) = 1−ξ β(ξ)f(Φ) + ξ β(ξ)ZΦ Ψ f(Υ) dΥ, where β(ξ)> 0is the normalization function satisfying β (0) = β (1) = 1. For ξ = 0, ξ= 1, the left derivative is defined as follows, respectively CF D ΨD0f(Φ) = f′(Φ) CF D ΨD1f(Φ) = f(Φ) −f(Ψ) . For the right derivative operator CF D bDξf(Φ) = β(ξ) 1−ξZb Φ f′(Υ) e −ξ(b−Φ)ξ 1−ξdΥ and the associated integral operator is CF Iξ bf(Φ) = 1−ξ β(ξ)f(Φ) + ξ β(ξ)Zb Φ f(Υ) dΥ, where β(α)>0is a normalization function that satisfies β(0) = β(1) = 1. This research article aims to established an identity for Caputo-Fabrizio fractional integral operator. By utilizing this identity, we will show some dual Simpson type inequalities for differentiable functions whose derivatives are convex. Moreover, dual Simpson type inequalities will be obtained for different classes of functions, such as convex, bounded, and Lipschitzian. Furthermore, we will give an application to the quadrature formula. 2 Dual-Simpson type Inequalities for Differentiable Convex Functions In this section, we present several fractional inequalities of dual-Simpson type to a differentiable function. 2nd Kocaeli Science Congress, November 19-21, 2025 M4-5 KOSC-2025 Proceedings Lemma 2.1. Let f : [Ψ, mb]→R be a differentiable function on [Ψ, mb] with Ψ < mb and f′∈L[Ψ, mb], then the following equality holds true: 2 3f3Ψ+mb 4−1 3fΨ+mb 2+2 3fΨ+3mb 4 −β(ξ) ξ(mb −Ψ) hCF ΨIξf(k) + CF Iξ mbf(k)i+2 (1 −ξ) ξ(mb −Ψ)f(k) =b−Ψ 16 Z1 0 Υf′(1 −Υ) Ψ + Υ3Ψ+mb 4dΥ +Z1 0Υ−5 3f′(1 −Υ) 3Ψ+mb 4+ ΥΨ+mb 2dΥ +Z1 0Υ + 2 3f′(1 −Υ) Ψ+mb 2+ ΥΨ+3mb 4dΥ +Z1 0 (Υ −1) f′(1 −Υ) Ψ+3mb 4+mΥbdΥ. Proof. Let I=Z1 0 Υf′(1 −Υ) Ψ + Υ3Ψ+mb 4dΥ, +Z1 0Υ−5 3f′(1 −Υ) 3Ψ+mb 4+ ΥΨ+mb 2dΥ, +Z1 0Υ + 2 3f′(1 −Υ) Ψ+mb 2+ ΥΨ+3mb 4dΥ, +Z1 0 (Υ −1) f′(1 −Υ) Ψ+3mb 4+mΥbdΥ, I=I1+I2+I3+I4. By using the integration by parts, we get I1=Z1 0 Υf′(1 −Υ) Ψ + Υ3Ψ+mb 4dΥ =4Υ mb −Ψf(1 −Υ) Ψ + Υ3Ψ+mb 4 1 0 −4 mb −ΨZ1 0 f(1 −Υ) Ψ + Υ3Ψ+mb 4dΥ =4 mb −Ψf3Ψ+mb 4−4 mb −ΨZ1 0 f(1 −Υ) Ψ + Υ3Ψ+mb 4dΥ =4 mb −Ψf3Ψ+mb 4−16 (mb −Ψ)2Z3Ψ+mb 4 Ψ f(u)du. (1) M4-6 2nd Kocaeli Science Congress, November 19-21, 2025 Similarly, we have I2=−8 3 (mb −Ψ)fΨ+mb 2+20 3 (mb −Ψ)f3Ψ+mb 4−16 (b−Ψ)2ZΨ+mb 2 3Ψ+mb 4 f(u)du. (2) I3=20 3 (mb −Ψ)fΨ+3mb 4−8 3 (mb −Ψ)fΨ+mb 2−16 (mb −Ψ)2ZΨ+3mb 4 Ψ+mb 2 f(u)du, (3) and I4=4 mb −ΨfΨ+3mb 4−16 (mb −Ψ)2Zmb Ψ+3mb 4 f(u)du. (4) Adding the equalities (1)-(4), we get (I1+I2+I3+I4) =4 mb −Ψf3Ψ+mb 4−16 (mb −Ψ)2Z3Ψ+mb 4 Ψ f(u)du +−8 3 (mb −Ψ)fΨ+mb 2 +20 3 (mb −Ψ)f3Ψ+mb 4−16 (mb −Ψ)2ZΨ+mb 2 3Ψ+mb 4 f(u)du +20 3 (mb −Ψ)fΨ+3mb 4 −8 3 (mb −Ψ)fΨ+mb 2−16 (mb −Ψ)2ZΨ+3mb 4 Ψ+mb 2 f(u)du +4 mb −ΨfΨ+3mb 4 −16 (mb −Ψ)2Zmb Ψ+3mb 4 f(u)du =32 3 (mb −Ψ)f3Ψ+mb 4+32 3 (mb −Ψ)fΨ+3mb 4−16 3 (mb −Ψ)fΨ+mb 2 −16 (mb −Ψ)2Zmb Ψ f(u)du. (5) Multiplying the equality (5) with (mb−Ψ) 16 and subtracting 2(1−ξ) β(ξ)(mb−Ψ)f(k) , we obtain (mb −Ψ) 16 (I1+I2+I3+I4)−2 (1 −ξ) β(ξ) (mb −Ψ)f(k) =32 3 (mb −Ψ)f3Ψ+mb 4(mb −Ψ) 16 +32 3 (mb −Ψ)fΨ+3mb 4(mb −Ψ) 16 −16 3 (mb −Ψ)fΨ+mb 2(mb −Ψ) 16 −ξ (mb −Ψ) β(ξ)Zmb Ψ f(u)du −2 (1 −ξ) β(ξ) (mb −Ψ)f(k) =2 (mb −Ψ) 3f3Ψ+mb 4+2 (mb −Ψ) 3fΨ+3mb 4−1 3fΨ+mb 2 2nd Kocaeli Science Congress, November 19-21, 2025 M4-7 KOSC-2025 Proceedings −1 (mb −Ψ) ξ β(ξ)Zk Ψ f(u)du −(1 −ξ) β(ξ)f(k) + ξ β(ξ)Zmb k f(u)du −(1 −ξ) β(ξ)f(k)! =2 (mb −Ψ) 3f3Ψ+mb 4+2 (mb −Ψ) 3fΨ+3mb 4−1 3fΨ+mb 2 −1 (mb −Ψ) hCF ΨIξf(k) + CF Iξ mbf(k)i. Thus, we have mb −Ψ 16 Z1 0 Υf′(1 −Υ) Ψ + Υ3Ψ+mb 4dΥ +Z1 0Υ−5 3f′(1 −Υ) 3Ψ+mb 4+ ΥΨ+mb 2dΥ +Z1 0Υ + 2 3f′(1 −Υ) Ψ+mb 2+ ΥΨ+3mb 4dΥ +Z1 0 (Υ −1) f′(1 −Υ) Ψ+3mb 4+mΥbdΥ =2 3f3Ψ+mb 4−1 3fΨ+mb 2+2 3fΨ+3mb 4 −β(ξ) ξ(mb −Ψ) hCF ΨIξf(k) + CF Iξ mbf(k)i+2 (1 −ξ) ξ(mb −Ψ)f(k). The proof is completed. Corollary 2.1. If we put m= 1 in Lemma 2.1, then we have 2 3f3Ψ+b 4−1 3fΨ+b 2+2 3fΨ+3b 4 −β(ξ) ξ(b−Ψ) hCF ΨIξf(k) + CF Iξ bf(k)i+2 (1 −ξ) ξ(b−Ψ)f(k) =b−Ψ 16 Z1 0 Υf′(1 −Υ) Ψ + Υ3Ψ+b 4dΥ +Z1 0Υ−5 3f′(1 −Υ) 3Ψ+b 4+ ΥΨ+b 2dΥ +Z1 0Υ + 2 3f′(1 −Υ) Ψ+b 2+ ΥΨ+3b 4dΥ +Z1 0Υ + 2 3f′(1 −Υ) Ψ+b 2+ ΥΨ+3b 4dΥ +Z1 0 (Υ −1) f′(1 −Υ) Ψ+3b 4+ ΥbdΥ. Theorem 2.1. Assume that the assumptions of Lemma 2.1 are satisfied. If |f′| is M4-8 2nd Kocaeli Science Congress, November 19-21, 2025 s-convex on [Ψ, mb], then the following inequality holds true: 2 3f3Ψ+mb 4−1 3fΨ+mb 2+2 3fΨ+3mb 4 −β(ξ) ξ(mb −Ψ) hCF ΨIξf(k) + CF Iξ mbf(k)i+2 (1 −ξ) β(ξ) (mb −Ψ)f(k) ≤(mb −Ψ) 16 1 2+3s+s2f′(Ψ)+f′(mb)+1 3(7 + 2s) f′Ψ+mb 2 +1 3(10 + 8s) f′3Ψ+mb 4 + f′Ψ+3mb 4. Proof. By using the Lemma 2.1, properties of modulus and s-convexity of |f′|, we get 2 3f3Ψ+mb 4−1 3fΨ+mb 2+2 3fΨ+3mb 4 −β(ξ) ξ(mb −Ψ) hCF ΨIξf(k) + CF Iξ mbf(k)i+2 (1 −ξ) ξ(mb −Ψ)f(k) ≤mb −Ψ 16 Z1 0 Υ f′(1 −Υ) Ψ + Υ3Ψ+mb 4 dΥ +Z1 0 Υ−5 3 f′(1 −Υ) 3Ψ+mb 4+ ΥΨ+mb 2 dΥ +Z1 0 Υ + 2 3 f′(1 −Υ) Ψ+mb 2+ ΥΨ+3mb 4 dΥ +Z1 0 |Υ−1| f′(1 −Υ) Ψ+3mb 4+ Υmb dΥ ≤mb −Ψ 16 Z1 0 Υ(1 −Υ)sf′(Ψ)+ Υs f′3Ψ+mb 4dΥ +Z1 0 Υ−5 3(1 −Υ)s f′3Ψ+mb 4 + Υs f′Ψ+mb 2dΥ +Z1 0 Υ + 2 3(1 −Υ)s f′Ψ+b 2 + Υs f′Ψ+3mb 4dΥ +Z1 0 |Υ−1|(1 −Υ)s f′Ψ+3mb 4 + Υsf′(mb)dΥ =(mb −Ψ) 16 1 2+3s+s2f′(Ψ)+f′(mb)+1 3(7 + 2s) f′Ψ+mb 2 +1 3(10 + 8s) f′3Ψ+mb 4 + f′Ψ+3mb 4. This completes the proof. 2nd Kocaeli Science Congress, November 19-21, 2025 M4-9 KOSC-2025 Proceedings +Z1 0 Υ + 2 3 f′(1 −Υ) Ψ+mb 2+ ΥΨ+3mb 2 dΥ +Z1 0 |Υ−1| f′(1 −Υ) Ψ+3mb 2+ Υmb dΥ ≤mb −Ψ 16 1 pZ1 0 ΥpdΥ+1 qZ1 0 f′(1 −Υ) Ψ + Υ3Ψ+mb 4 qdΥ +1 pZ1 0 Υ−5 3 p dΥ+1 qZ1 0 f′(1 −Υ) 3Ψ+mb 4+ ΥΨ+mb 2 qdΥ +1 pZ1 0 Υ + 2 3 p dΥ+1 qZ1 0 f′(1 −Υ) Ψ+mb 2+ ΥΨ+3mb 2 qdΥ +1 pZ1 0 |Υ−1|pdΥ+1 qZ1 0 f′(1 −Υ) Ψ+3mb 4+ Υmb qdΥ ≤mb −Ψ 16 1 pZ1 0 ΥpdΥ +1 qZ1 0(1 −Υ)sf′(Ψ) q+ Υs f′3Ψ+mb 4 qdΥ +1 pZ1 0 Υ−5 3 p dΥ +1 qZ1 0(1 −Υ)s f′3Ψ+mb 4 q + Υs f′Ψ+mb 2 qdΥ +1 pZ1 0 Υ + 2 3 p dΥ +1 qZ1 0(1 −Υ)s f′Ψ+mb 2 q + Υs f′Ψ+3mb 4 qdΥ +1 pZ1 0 |Υ−1|pdΥ +1 qZ1 0(1 −Υ)s f′Ψ+3mb 4 q + Υsf′(mb) qdΥ =mb −Ψ 16 ×(s+ 1)1 q1 p(p+ 1) + 1 qf′(Ψ) q+ f′3Ψ+mb 4 q +1 q f′Ψ+3mb 4 q +f′(mb) q + 1 p 5p+1 −2p+1 3p+1 (p+ 1)!+1 q f′3Ψ+mb 4 q + f′Ψ+mb 2 q +1 q f′Ψ+mb 2 q + f′Ψ+3mb 4 q. This completes the proof. M4-16 2nd Kocaeli Science Congress, November 19-21, 2025 3 Dual-Simpson type Inequalities for Bounded Function In this section, we obtain several fractional inequalities of dual-Simpson type for bounded function. Theorem 3.1. Assume that the assumptions of Lemma 2.1 hold. If there exist constants −∞ < m◦< M < + ∞ such that m◦< f′(Φ) < M for all Φ ∈[Ψ, mb] , then the following inequality holds true: 2 3f3Ψ+mb 4−1 3fΨ+mb 2+2 3fΨ+3mb 4 −β(ξ) ξ(mb −Ψ) hCF ΨIξf(k) + CF Iξ mbf(k)i+2 (1 −ξ) ξ(mb −Ψ)f(k) ≤5 (mb −Ψ) M−m◦ 48 . Proof. From Lemma 2.1, we have 2 3f3Ψ+mb 4−1 3fΨ+mb 2+2 3fΨ+3mb 4 −β(ξ) ξ(mb −Ψ) hCF ΨIξf(k) + CF Iξ mbf(k)i+2 (1 −ξ) ξ(mb −Ψ)f(k) =mb −Ψ 16 Z1 0 Υf′(1 −Υ) Ψ + Υ3Ψ+mb 4dΥ +Z1 0Υ−5 3f′(1 −Υ) 3Ψ+mb 4+ ΥΨ+mb 2dΥ +Z1 0Υ + 2 3f′(1 −Υ) Ψ+mb 2+ ΥΨ+3mb 4dΥ +Z1 0 (Υ −1) f′(1 −Υ) Ψ+3mb 4+ ΥmbdΥ =mb −Ψ 16 "Z1 0 Υ f′(1 −Υ) Ψ + Υ3Ψ+mb 4−m◦+M 2!dΥ +Z1 0Υ−5 3 f′(1 −Υ) 3Ψ+mb 4+ ΥΨ+mb 2−m◦+M 2!dΥ +Z1 0Υ + 2 3 f′(1 −Υ) Ψ+mb 2+ ΥΨ+3mb 4−m◦+M 2!dΥ +Z1 0 (Υ −1) f′(1 −Υ) Ψ+3mb 4+ Υmb−m◦+M 2!dΥ#.(6) 2nd Kocaeli Science Congress, November 19-21, 2025 M4-17 KOSC-2025 Proceedings Taking the absolute value in both sides of equality (6), we get 2 3f3Ψ+mb 4−1 3fΨ+mb 2+2 3fΨ+3mb 4 −β(ξ) ξ(mb −Ψ) hCF ΨIξf(k) + CF Iξ mbf(k)i+2 (1 −ξ) β(ξ) (mb −Ψ)f(k) ≤b−Ψ 16 "Z1 0 Υ f′(1 −Υ) Ψ + Υ3Ψ+mb 4−m◦+M 2 dΥ +Z1 0 Υ−5 3 f′(1 −Υ) 3Ψ+mb 4+ ΥΨ+mb 2−m◦+M 2 dΥ +Z1 0 Υ + 2 3 f′(1 −Υ) Ψ+mb 2+ ΥΨ+3mb 2−m◦+M 2 dΥ +Z1 0 |Υ−1| f′(1 −Υ) Ψ+3mb 2+ Υmb−m◦+M 2 dΥ#.(7) Since m◦< f′(Φ) < M for all Φ∈[Ψ, mb], we obtain  f′(1 −Υ) Ψ + Υ3Ψ+mb 4−m◦+M 2 ≤M−m◦ 2, (8)  f′(1 −Υ) 3Ψ+mb 4+ ΥΨ+mb 2−m◦+M 2 ≤M−m◦ 2, (9)  f′(1 −Υ) Ψ+mb 2+ ΥΨ+3mb 4−m◦+M 2 ≤M−m◦ 2, (10) and  f′(1 −Υ) Ψ+3mb 4+ Υmb−m◦+M 2 ≤M−m◦ 2.(11) Applying inequalities (8)-(11) in (6), we have 2 3f3Ψ+mb 4−1 3fΨ+mb 2+2 3fΨ+3mb 4 −β(ξ) ξ(mb −Ψ) hCF ΨIξf(k) + CF Iξ mbf(k)i+2 (1 −ξ) ξ(mb −Ψ)f(k) ≤(mb −Ψ) (M−(mb −Ψ)) 32 ×Z1 0 ΥdΥ + Z1 0 Υ−5 3 dΥ + Z1 0 Υ + 2 3 dΥ + Z1 0 |Υ−1|dΥ M4-18 2nd Kocaeli Science Congress, November 19-21, 2025 =5 (mb −Ψ) M−m◦ 48 . The proof is completed. 4 Dual-Simpson type Inequalities for Lipschitzian Function In this section, we introduce Dual-Simpson type inequalities for Lipschitzian function. Theorem 4.1. Suppose that the assumptions of Lemma 2.1 are satisfied. If f′ is L-Lipschitzian function [Ψ, mb], then the following fractional holds true: 2 3f3Ψ+mb 4−1 3fΨ+mb 2+2 3fΨ+3mb 4 −β(ξ) ξ(mb −Ψ) hCF ΨIξf(k) + CF Iξ mbf(k)i+2 (1 −ξ) ξ(mb −Ψ)f(k) ≤13L(mb −Ψ)2 192 . Proof. From Lemma 2.1, we get 2 3f3Ψ+mb 4−1 3fΨ+mb 2+2 3fΨ+3mb 4 −β(ξ) ξ(mb −Ψ) hCF ΨIξf(k) + CF Iξ mbf(k)i+2 (1 −ξ) ξ(mb −Ψ)f(k) =mb −Ψ 16 Z1 0 Υf′(1 −Υ) Ψ + Υ3Ψ+mb 4dΥ +Z1 0Υ−5 3f′(1 −Υ) 3Ψ+mb 4+ ΥΨ+mb 2dΥ +Z1 0Υ + 2 3f′(1 −Υ) Ψ+mb 2+ ΥΨ+3mb 4dΥ +Z1 0 (Υ −1) f′(1 −Υ) Ψ+3mb 4+ ΥmbdΥ =b−Ψ 16 Z1 0 Υf′(1 −Υ) Ψ + Υ3Ψ+mb 4−f′(Ψ)dΥ +Z1 0Υ−5 3f′(1 −Υ) 3Ψ+mb 4+ ΥΨ+mb 2−f′3Ψ+mb 4dΥ +Z1 0Υ + 2 3f′(1 −Υ) Ψ+mb 2+ ΥΨ+3mb 4−f′Ψ+mb 2dΥ 2nd Kocaeli Science Congress, November 19-21, 2025 M4-19 KOSC-2025 Proceedings +Z1 0 (Υ −1) f′(1 −Υ) Ψ+3mb 4+ Υmb−f′Ψ+3mb 4dΥ +1 2f′(Ψ) −f′Ψ+3mb 4+7 6f′Ψ+mb 2−f′3Ψ+mb 4.(12) Taking the absolute value on both sides of equality (12) and utilizing the fact that f′ is L-Lipschitzian on [Ψ, mb], we have 2 3f3Ψ+mb 4−1 3fΨ+mb 2+2 3fΨ+3mb 4 −β(ξ) ξ(mb −Ψ) hCF ΨIξf(k) + CF Iξ mbf(k)i+2 (1 −ξ) ξ(mb −Ψ)f(k) ≤mb −Ψ 16 Z1 0 Υ f′(1 −Υ) Ψ + Υ3Ψ+mb 4−f′(Ψ) dΥ +Z1 0 Υ−5 3 f′(1 −Υ) 3Ψ+mb 4+ ΥΨ+mb 2−f′3Ψ+mb 4 dΥ +Z1 0 Υ + 2 3 f′(1 −Υ) Ψ+mb 2+ ΥΨ+3mb 4−f′Ψ+mb 2 dΥ +Z1 0 |Υ−1| f′(1 −Υ) Ψ+3mb 4+ Υmb−f′Ψ+3mb 4 dΥ +1 2 f′(Ψ) −f′Ψ+3mb 4 +7 6 f′Ψ+mb 2−f′3Ψ+mb 4 ≤mb −Ψ 16 mb −Ψ 4LZ1 0 Υ2dΥ + Z1 0 Υ−5 3 ΥdΥ + Z1 0 Υ + 2 3 ΥdΥ +Z1 0 |Υ−1|ΥdΥ+L 2 Ψ−Ψ+3mb 4 +7L 6 Ψ+mb 2−3Ψ+mb 4 =13L(mb −Ψ)2 192 . This completes the proof. 5 Dual Simpson’s Quadrature Formula Let d is the partition of the interval [Ψ, b] , d : Ψ = Φ 0< Φ 1< Φ 2< . . . < Φ n−1< Φ n = b and define the following quadrature formula: Zb Ψ f(Φ) dΦ = ζ(f, d)+R(f, d), M4-20 2nd Kocaeli Science Congress, November 19-21, 2025 with ζ(f, d) := n−1 X i=0 (Φi+1 −Φi) 32f3Φi+ Φi+1 4−fΦi+ Φi+1 2+ 2fΦi+ 3Φi+1 4, where R(f, d)is the approximation error. Proposition 5.1. Under the assumptions of Corollary 2.3 for every division d of [Ψ, b] , and ξ= 1,β(0) = β(1) = 1 the following inequality holds: |R(f, d)| ≤ n−1 X i=0 (Φi+1 −Φi)2 16 1 6f′(Φi)+f′(Φi+1)+ f′Φi+ Φi+1 2 + f′3Φi+ Φi+1 4 + f′Φi+ 3Φi+1 4. Proof. Applying the Theorem 2.1 on the subinterval [Φi,Φi+1] , (i= 0,1,2, . . . , n −1) , we have  1 32f3Φi+ Φi+1 4−fΦi+ Φi+1 2+ 2fΦi+ 3Φi+1 4−ZΦi+1 Φi f(Φ) dΦ (13) ≤(Φi+1 −Φi) 16 1 6f′(Φi)+f′(Φi+1)+ f′Φi+ Φi+1 2 + f′3Φi+ Φi+1 4 + f′Φi+ 3Φi+1 4. Summing over i from 0to n− 1and multiplying on both sides inequality (13) by (Φi+1 −Φi), we deduce by the triangle inequality the desired result. 6 Conclusion In this paper, the main aim is to derive dual-Simpson-type inequalities for various function classes using the Caputo-Fabrizio fractional integral operator. First of all, we give an integral identity that is essential in order to establish the main findings of the article. Using Caputo-Fabrizio fractional integrals, a few dual-Simpson-type inequalities are investigated for differentiable functions. Employing the new approach, we extended the study of dual-Simpson type inequalities using Hölder’s, bounded, Lipschitzian and power-mean integral inequalities. Furthermore, an application to the quadrature formula is discussed. In the future, it would be fascinating to apply these findings to other 2nd Kocaeli Science Congress, November 19-21, 2025 M4-21 KOSC-2025 Proceedings convexities and fractional integral operators. We believe that much research in this intriguing area of inequalities and numerical analysis will focus on the future. 7 Acknowledgments The authors wish to thank the editors and reviewers for their valuable comments and suggestions for the betterment of this article. References [1] N. A. Alqahtani., S. Qaisar, A. Munir, M. Naeem, & H. Budak. Error bounds for fractional integral inequalities with applications. Fractal and Fractional, 8(4), 208, (2024). [2] A. Munir., A. Qayyum., S. S. A. Supadi., H. Budak. I. Faiz. (2024). 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