A Note on Fractional Simpson-like Type Inequality for Functions whose Second Derivatives are of Bounded Variation

A Note on Fractional Simpson-like Type Inequality for Functions whose Second Derivatives are of Bounded Variation Hüseyin Budak1, İrem Çay1, İrem Bağlan1 1Department of Mathematics, Kocaeli University, Kocaeli, Türkiye Corresponding author: [email protected] ORCID IDs: First Author: 0000-0001-8843-955X Second Author: 0000-0001-9234-2523 Third Author: 0000-0003-4900-6027 DOI : 10.5281/zenodo.18023597 Abstract In this study, a Simpson-type equality for the Riemann–Liouville fractional integral is established. Using this equality, several fractional Simpson-type inequalities are proved for functions whose second derivatives are of bounded variation. Keywords: Fractional integrals, Simpson-type inequality, bounded variation 1 Introduction Fractional calculus has become an essential tool in the analysis of various mathematical models due to its ability to capture memory and hereditary properties. In particular, fractional integral inequalities play a significant role in establishing bounds and estimating errors in numerical approximation methods. Among these, Simpson-type inequalities provide effective estimates for integrals and have recently been extended to the fractional setting. Motivated by these developments, we investigate a Simpson-type equality involving the Riemann–Liouville fractional integral. Based on this equality, we derive several fractional Simpson-like inequalities for functions whose second derivatives are of bounded variation. Now, let us give the definition of a function of bounded variation and introduce the basic concepts of fractional calculus that will be needed in subsequent analysis. Definition 1.1. Let P : a = x0< x1< ... < xn = b be any partition of [ a, b ]and let ∆F(xi) = F(xi+1)−F(xi). Then Fis said to be of bounded variation if the sum n X i=1 |∆F(xi)| is bounded for all such partitions. M23-1 KOSC-2025 Proceedings Let F be of bounded variation on [ a, b ], and P ∆ F ( P )denotes the sum Pn i=1 |∆F(xi)| corresponding to the partition Pof [a, b]. The number b _ a (F) := sup nX∆F(P) : P∈(P[a, b])o is called the total variation of F on [ a, b ]. Here P([a, b]) denotes the family of partitions of [ a, b ] . Definition 1.2. (See [ 4 , 7 ]) If f∈L1[a, b] is a function, then the integrals Iα a+f(x) and Iα b−f(x) of order α > 0are defined by Iα a+f(x) = 1 Γ (a)Zx a (x−t)α−1f(t)dt, x > a and Iα b−f(x) = 1 Γ (α)Zb x (t−x)α−1f(t)dt, x < b, respectively. These integrals are called Riemann-Liouville fractional integrals in the literature. Here, Γis the Gamma function defined by Γ(α) = Z∞ 0 e−uuα−1du. Fractional integral inequalities and their applications have been established using the RiemannLiouville fractional integral. For example, Sarikaya and colleagues have established some Simpsonlike inequalities for functions with convex second derivatives [ 8 ]. Hezenci and Budak have also proven some fractional Simpson-like inequalities for functions with convex third derivatives in absolute values [ 5 ]. Furthermore, some fractional Simpson-like inequalities for functions with convex second derivatives in absolute values have been proven in [ 6 ]. For more information on various properties of Riemann-Liouville fractional integrals and various fractional integral operators, the reader is referred to [3,2,1]. The aim of this paper is to establish a new Simpson-like equation based on the RiemannLiouville fractional integral and, using this equation, to obtain various fractional Simpson-like inequalities for functions with bounded variations in second derivatives. The overall design of this paper consists of four sections, including the introduction. A new Simpson-type equation, which serves as the main tool of the study, is established in Section 2. In Section 3, various fractional Simpson-type inequalities are derived using the equation obtained in Section 2. Finally, in Section 4, the results are discussed. 2 A New and Essential Equality In this section, we derive a key integral identity that forms the foundation for the principal results presented in this paper. Lemma 2.1. Let F : [ a, b ] →R be a a three times differentiable function on [ a, b ]such that M23-2 2nd Kocaeli Science Congress, November 19-21, 2025 F′′′L∈[a, b]. Then, the following equality holds 1 (α+ 1)(α+ 2) F(a)+(α2+ 3α)Fa+b 2+F(b)−2α−1Γ(α+ 1) (b−a)αJα a+b 2 −F(a) + Jα a+b 2 +F(b) =(b−a)2 8(α+ 1)(α+ 2) Zb a p(x)d(F′′(x)), where p(x) =     2(x−a) b−aα+2 −2(x−a) b−a2, a < x ≤a+b 2, 2(b−x) b−a2−2(b−x) b−aα+2 ,a+b 2< x < b. (1) Proof. Zb a p(x)d(F′′(x)) = Za+b 2 a"2(x−a) b−aα+2 −2(x−a) b−a2#d(F′′(x)) +Zb a+b 2"2(b−x) b−a2 −2(b−x) b−aα+2#d(F′′(x)) By applying integration by parts, we obtain I1=Za+b 2 a"2(x−a) b−aα+2 −2(x−a) b−a2#d(F′′(x)) ="2(x−a) b−aα+2 −2(x−a) b−a2#F′′(x) a+b 2 a −Za+b 2 a"2 b−aα+2 (α+ 2)(x−a)α+1 −8 (b−a)2(x−a)#d(F′(x)) =−"2 b−aα+2 (α+ 2)(x−a)α+1 −8 (b−a)2(x−a)#F′(x) a+b 2 a +Za+b 2 a"2 b−aα+2 (α+ 2)(α+ 1)(x−a)α−8 (b−a)2#d(F(x)) =4 b−a−2(α+ 2) b−aF′a+b 2+"(α+ 1)(α+ 2) 2 b−aα+2 (x−a)α−8 (b−a)2#F(x) a+b 2 a −α(α+ 1)(α+ 2)2α+2Γ(α) (b−a)α+2 Za+b 2 a (x−a)αF(x)dx Similarly, we obtain I2=Zb a+b 2"2(b−x) b−a2 −2(b−x) b−aα+2#d(F′′(x)) =2(α+ 2) b−a−4 b−aF′a+b 2+"8 (b−a)2−(α+ 1)(α+ 2) 2 b−aα+2 (b−x)α#F(x) b a+b 2 −α(α+ 1)(α+ 2)2α+2Γ(α) (b−a)α+2 Zb a+b 2 (b−x)αF(x)dx 2nd Kocaeli Science Congress, November 19-21, 2025 M23-3 KOSC-2025 Proceedings If we add I1and I2, we obtain the following equation I1+I2= 8 (α+ 1)(α+ 2) (b−a)2−2 (b−a)2Fa+b 2+8 (b−a)2[F(a) + F(b)] −α(α+ 1)(α+ 2)2α+2Γ(α) (b−a)α+2 Jα a+b 2−F(a) + Jα a+b 2+F(b) Thus the proof is completed. 3 Simpson-Like Inequalities Theorem 3.1. Let F : [ a, b ] →R be such that F′′ is a continuous function of bounded variation on [a, b]. Then we have the inequality  1 (α+ 1)(α+ 2) F(a)+(α2+ 3α)Fa+b 2+F(b)−2α−1Γ(α+ 1) (b−a)αJα a+b 2 −F(a) + Jα a+b 2 +F(b) ≤α22/α−3(b−a)2 (α+ 1)(α+ 2)2/α+2 b _ a (F′′). Proof. From the equality (1), we can write  1 (α+ 1)(α+ 2) F(a)+(α2+ 3α)Fa+b 2+F(b)−2α−1Γ(α+ 1) (b−a)αJα a+b 2 −F(a) + Jα a+b 2 +F(b) =(b−a)2 8(α+ 1)(α+ 2) Zb a p(x)dF′′(x) =1 2(α+ 1)(α+ 2) ×Za+b 2 a 2 b−aα (x−a)α+2 −(x−a)2dF′′(x)+Zb a+b 2(b−x)2−2 b−aα (b−x)α+2dF′′(x) ≤1 2(α+ 1)(α+ 2) Za+b 2 a 2 b−aα (x−a)α+2 −(x−a)2dF′′(x) +1 2(α+ 1)(α+ 2) Zb a+b 2(b−x)2−2 b−aα (b−x)α+2dF′′(x) ≤1 2(α+ 1)(α+ 2)      supx∈[a, a+b 2]2 b−aα (x−a)α+2 −(x−a)2 a+b 2 _ a (F′′) +supx∈[a+b 2,b] (b−x)2−2 b−aα (b−x)α+2 b _ a+b 2 (F′′)     ≤α22/α−3(b−a)2 (α+ 1)(α+ 2)2/α+2 b _ a (F′′) Thus the proof is completed. Corollary 3.1. If we choose α = 1 in Theorem 3.1, then the following Simpson type inequality M23-4 2nd Kocaeli Science Congress, November 19-21, 2025 holds:  1 6F(a)+4Fa+b 2+F(b)−1 b−aZb a F(x)dx ≤(b−a)2 324 b _ a (F′′). 4 Conclusions In this study, a new Simpson-type equation is derived in the framework of the Riemann–Liouville fractional integral, and various fractional Simpson-like inequalities are derived using this equation for functions whose second derivatives have a bounded total variation. The results generalize the classical Simpson inequality to the fractional integral context and provide new evaluations valid for a broader class of functions. The findings here are expected to provide a foundation for applications of fractional calculus associated with fractional integral operators. References [1] H. Budak, F. Hezenci, and H. Kara. On generalized ostrowski, simpson and trapezoidal type inequalities for co–ordinated convex functions via generalized fractional integrals. Advances in Difference Equations, 2021(312):1–32, 2021. [2] H. Budak, F. Hezenci, and H. Kara. On parameterized inequalities of ostrowski and simpson type for convex functions via generalized fractional integrals. Mathematical Methods in the Applied Sciences, 44(17):12522– 12536, 2021. [3] H. Budak, H. Kara, M. Z. Sarikaya, and M. E. Kiris. New extensions of the hermite-hadamard inequalities involving riemannliouville fractional integrals. Miskolc Mathematical Notes, 21(2):665–678, 2020. [4] R. Gorenflo and F. Mainardi. Fractional Calculus: Integral and differential equations of fractional order. Springer Verlag, Wien, 1997. [5] F. Hezenci and H. Budak. A note on fractional simpson-like type inequalities for functions whose third derivatives are convex. Filomat, 37(12):3715–3724, 2023. [6] F. Hezenci, H. Budak, and H. Kara. New version of fractional simpson type inequalities for twice differentiable functions. Advances in Difference Equations, 2021(460):1–10, 2021. [7] A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo. Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam, 2006. [8] M. Z. Sarikaya and E. Set. On newinequalities of simpson’s type for functionswhose second derivatives absolute values are convex. Journal of Applied Mathematics, 9(1):37–45, 2013. 2nd Kocaeli Science Congress, November 19-21, 2025 M23-5