Euler-Maclaurin-type Inequalities for Generalized Fractional Integrals on Functions of Bounded Variation

Euler-Maclaurin-type Inequalities for Generalized Fractional Integrals on Functions of Bounded Variation Tuğba Acar1, Fatih Hezenci1, Arslan Munir2 1 Department of Mathematics, Faculty of Science and Arts, Duzce University, Duzce 81620, Türkiye 2School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, People’s Republic of China Corresponding author: [email protected] ORCID IDs: First Author: 0009-0006-4055-8182 Second Author: 0000-0003-1008-5856 Third Author: 0009-0008-8148-5607 DOI : 10.5281/zenodo.18033277 Abstract This paper presents a comprehensive approach that utilizes generalized fractional integrals to derive Euler-Maclaurin-type inequalities for functions of bounded variation. Moreover, by choosing particular forms of functions and suitable parameters, we obtain several special cases. These cases clearly show the generality and practical applicability of the obtained results. These results not only extend previously known findings but also provide a more general setting for studying such inequalities. Keywords: Quadrature formulae, convex functions, generalized fractional integrals. 1 Introduction The theory of inequalities is a well-established and fundamental area of mathematics, with deep connections to various mathematical fields and a wide range of applications. Among the key concepts in this theory are convex functions, which play a crucial role in formulating and proving many important inequalities. In recent years, fractional calculus has also attracted significant interest due to its powerful theoretical framework and numerous practical applications in science and engineering. Because of its importance, many researchers have focused on developing and studying fractional integral inequalities. The bounds of such inequalities can be derived not only from Hermite-Hadamard-type results but also from Simpson, Newton, and Euler-Maclaurin-type inequalities. Simpson’s quadratic formula for functions of bounded variation, along with its applications in the theory of special means, is investigated by Dragomir in [ 1 ]. Furthermore, Budak et al. [ 2 ] are established various forms of Simpson-type inequalities involving generalized fractional M33-1 KOSC-2025 Proceedings integrals within the framework of differentiable convex functions. For additional information on Simpson-type inequalities and further properties of generalized fractional integrals, readers are encouraged to consult [3] and the references therein. Since the three-point Newton-Cotes quadrature formula is a special case of Simpson’s second rule, evaluations involving three-step quadratic kernels are often referred to as Newton-type results in the literature. A substantial body of research has focused on developing and analyzing such inequalities. For instance, in [ 4 ], several Newton-type integral inequalities are derived for functions whose first derivatives are arithmetically-harmonically convex in absolute value at a given power. Similarly, in [ 5 ], fractional Newton-type inequalities are established for functions of bounded variation. Furthermore, Gao and Shi [ 6 ] presented specific applications for certain classes of real functions and introduced a new Newton-type inequality based on convexity. For further details on Newton-type inequalities involving convex and differentiable functions, readers may refer to [7] and the references therein. Dedic et al. [ 8 ] develop a set of inequalities, and their results are used to derive error estimates for the Maclaurin quadrature rules. In addition, the results are applied to provide some error estimates for the Simpson 3 / 8quadrature rule in [ 9 ]. Next, in [ 10 ], some Euler-Maclaurin-type inequalities are considered for the case of differentiable convex functions. Furthermore, in [ 11 ], some corrected Euler-Maclaurin-type inequalities are established using the Riemann-Liouville fractional integrals. For further details on these types of inequalities, readers are encouraged to consult [12,13,14] and the references therein. 2 Preliminaries In this section, we present several definitions and notations that are frequently used throughout the main part of the paper. Simpson’s quadrature formula, also known as Simpson’s 1 / 3rule, is expressed as follows: Zb a f(x)dx ≈b−a 6f(a) + 4fa+b 2+f(b).(1) Theorem 2.1. Let us consider that f : [a, b]→R is a four times differentiable and continuous function on (a, b)and   f(4)  ∞= sup x∈(a,b)f(4)(x)<∞.Then, the following inequality holds:  1 6f(a)+4fa+b 2+f(b)−1 b−aZb a f(x)dx ≤1 2880   f(4)  ∞(b−a)4. Simpson’s second formula, also known as the Newton-Cotes quadrature formula (Simpson’s 3/8rule; cf. [13]), is expressed as follows: Zb a f(x)dx ≈b−a 8f(a) + 3f2a+b 3+ 3fa+ 2b 3+f(b).(2) Theorem 2.2. Let us note that f : [a, b]→R is a four times differentiable and continuous M33-2 2nd Kocaeli Science Congress, November 19-21, 2025 function on (a, b)and   f(4)  ∞= sup x∈(a,b)f(4)(x)<∞.Then, one has the inequality  1 8f(a) + 3f2a+b 3+ 3fa+ 2b 3+f(b)−1 b−aZb a f(x)dx ≤1 6480   f(4)  ∞(b−a)4. The corresponding dual Simpson’s 3 / 8formula, known as the Maclaurin rule and based on the Maclaurin formula (cf. [13]), is given as follows: Zb a f(x)dx ≈b−a 83f5a+b 6+ 2fa+b 2+ 3fa+ 5b 6.(3) Theorem 2.3. If f : [a, b]→R is a four times differentiable and continuous function on (a, b) and   f(4)  ∞= sup x∈(a,b)f(4)(x)<∞,then the inequality  1 83f5a+b 6+ 2fa+b 2+ 3fa+ 5b 6−1 b−aZb a f(x)dx ≤7 51840   f(4)  ∞(b−a)4 is valid. The formulae (1) , (2) , and (3) are valid for any function f whose fourth derivative is continuous on the interval [a, b]. In the literature, numerous papers discuss inequalities related to generalized fractional integrals. For example, Sarikaya and Ertugral establish Hermite-Hadamard-type inequalities for generalized fractional integrals [ 15 ]. In addition, Ertugral and Sarikaya [ 16 ] present some Simpson-type inequalities for these fractional integral operators. Definition 2.1. [ 15 ] Note that φ : [0 ,∞ ) → [0 ,∞ )the condition R1 0 φ(t) tdt < ∞. Then, the following left-sided and right-sided generalized fractional integral operators are described as a+Iφf(x) = Zx a φ(x−t) x−tf(t)dt, x > a (4) and b−Iφf(x) = Zb x φ(t−x) t−xf(t)dt, x < b, (5) respectively. A key feature of generalized fractional integrals is that they encompass several important types of fractional integrals, such as the Riemann-Liouville fractional integral, the k -RiemannLiouville fractional integral, Katugampola fractional integrals, the conformable fractional integral, Hadamard fractional integrals, and others. These notable special cases of the integral operators (4) and (5) are summarized as follows: i. If we choose φ(t) = t, then the operators (4) and (5) become to the Riemann integral. ii. If we consider φ(t) = tα Γ(α) and α > 0 , then the operators (4) and (5) reduce to the 2nd Kocaeli Science Congress, November 19-21, 2025 M33-3 KOSC-2025 Proceedings Riemann-Liouville fractional integrals Jα a+f(x)and Jα b−f(x), respectively [17,18]. Here, Jα a+f(x) = 1 Γ(α)Zx a (x−t)α−1f(t)dt, x > a and Jα b−f(x) = 1 Γ(α)Zb x (t−x)α−1f(t)dt, x < b. Γis Gamma function described by the integral formula Γ (x) := ∞ Z 0 tx−1e−tdt, x ∈R+. iii. For φ(t) = 1 kΓk(α)tα k and α, k > 0 , the operators (4) and (5) become to the k -RiemannLiouville fractional integrals Jα a+,k f(x)and Jα b−,kf(x), respectively. Here, Jα a+,kf(x) = 1 Γk(α)Zx a (x−t)α k−1f(t)dt, x > a and Jα b−,kf(x) = 1 Γk(α)Zb x (t−x)α k−1f(t)dt, x < b. Γkis k-Gamma function defined by Γk(α) = Z∞ 0 tα−1e−tk kdt, R(α)>0 and Γk(α) = kα k−1Γα k,R(α)>0; k > 0. 3 Main results Lemma 3.1. Let us consider that f : [ a, b ] →R is an absolutely continuous function ( a, b )so that f′∈L1[a, b]. Then, the following equality holds: 1 83f5a+b 6+ 2fa+b 2+ 3fa+ 5b 6−1 2Λ (1) ha+b 2−Iφf(a) +a+b 2+Iφf(b)i(6) =b−a 2Λ (1) b Z a K(x)df(x). M33-4 2nd Kocaeli Science Congress, November 19-21, 2025 Here, Λ (t) = t R0 φ((b−a 2)y) ydy and K(x) =                                2 b−aΛ2(x−a) b−a, a ≤x < 5a+b 6, 2 b−ahΛ2(x−a) b−a−3 4Λ (1)i,5a+b 6≤x < a+b 2, −2 b−ahΛ2(b−x) b−a−3 4Λ (1)i,a+b 2≤x < a+5b 6, −2 b−aΛ2(b−x) b−a,a+5b 6≤x≤b. Proof. Let us first consider the function K ( x ) . Then, with the help of the integrating by parts, it follows from that b Z a K(x)df(x)(7) =2 b−a       5a+b 6 Z a Λ2 (x−a) b−adf(x) + a+b 2 Z 5a+b 6 Λ2 (x−a) b−a−3 4Λ (1)df(x) − a+5b 6 Z a+b 2 Λ2 (b−x) b−a−3 4Λ (1)df(x)− b Z a+5b 6 Λ2 (b−x) b−adf(x)       =2 b−a       Λ2 (x−a) b−af(x) 5a+b 6 a − 5a+b 6 Z a φ(x−a) x−af(x)dx +Λ2 (x−a) b−a−3 4Λ (1)f(x) a+b 2 5a+b 6 − a+b 2 Z 5a+b 6 φ(x−a) x−af(x)dx −Λ2 (b−x) b−a−3 4Λ (1)f(x) a+5b 6 a+b 2 − a+5b 6 Z a+b 2 φ(b−x) b−xf(x)dx −Λ2 (b−x) b−af(x) b a+5b 6 − b Z a+5b 6 φ(b−x) b−xf(x)dx       =2 b−aΛ (1) 43f5a+b 6+ 2fa+b 2+ 3fa+ 5b 6 − a+b 2 Z a φ(x−a) x−af(x)dx − b Z a+b 2 φ(b−x) b−xf(x)dx       =2 b−aΛ (1) 43f5a+b 6+ 2fa+b 2+ 3fa+ 5b 6 −ha+b 2−Iφf(a) +a+b 2+Iφf(b)io. 2nd Kocaeli Science Congress, November 19-21, 2025 M33-5 KOSC-2025 Proceedings Hence, we obtain readily 1 32f3a+b 4−fa+b 2+ 2fa+ 3b 4−1 2Λ (1) [b−Iφf(a) +a+Iφf(b)] =b−a 2Λ (1) Zb a K(x)df(x) Theorem 3.1. If f: [a, b]→Ris a function of bounded variation on [a, b],then we have  1 83f5a+b 6+ 2fa+b 2+ 3fa+ 5b 6−1 2Λ (1) ha+b 2−Iφf(a) +a+b 2+Iφf(b)i ≤1 2Λ (1) max Λ1 3, Λ1 3−3 4Λ (1) ,1 4Λ (1)b _ a (f). Here, b W a(f)denotes the total variation of fon [a, b]. Proof. It is known that if g, f : [a, b]→R are such that g is continuous on [a, b] and f is of bounded variation on [a, b], then b R a g(t)df(t)exist and  b Z a g(t)df(t) ≤sup t∈[a,b] |g(t)| b _ a (f).(8) By using (8), it follows  1 83f5a+b 6+ 2fa+b 2+ 3fa+ 5b 6−1 2Λ (1) ha+b 2−Iφf(a) +a+b 2+Iφf(b)i =b−a 4Λ (1)  b Z a K(x)df(x) ≤1 2Λ (1)         5a+b 6 Z a Λ2 (x−a) b−adf(x) + a+b 2 Z 5a+b 6 Λ2 (x−a) b−a−3 4Λ (1)df(x) + a+5b 6 Z a+b 2 3 4Λ (1) −Λ2 (b−x) b−adf(x) + b Z a+5b 6 Λ2 (b−x) b−adf(x)        ≤1 2Λ (1)      sup x∈[a, 5a+b 6] Λ2 (x−a) b−a 5a+b 6 _ a (f) M33-6 2nd Kocaeli Science Congress, November 19-21, 2025 + sup x∈[5a+b 6,a+b 2] Λ2 (x−a) b−a−3 4Λ (1) a+b 2 _ 5a+b 6 (f) + sup x∈[a+b 2,a+5b 6] 3 4Λ (1) −Λ2 (b−x) b−a a+5b 6 _ a+b 2 (f) + sup x∈[a+5b 6,b] Λ2 (b−x) b−a b _ a+5b 6 (f)     =1 2Λ (1)      Λ1 3 5a+b 6 _ a (f) + max 1 4Λ (1) , Λ1 3−3 4Λ (1) a+b 2 _ 5a+b 6 (f) + max  Λ1 3−3 4Λ (1) ,1 4Λ (1) a+5b 6 _ a+b 2 (f) +Λ 1 3b _ a+5b 6 (f)     ≤1 2Λ (1) max Λ1 3, Λ1 3−3 4Λ (1) ,1 4Λ (1)b _ a (f). This finishes the proof of Theorem 3.1. Remark 3.1. Let us consider φ(t) = t in Theorem 3.1. Then, the following Euler-Maclaurin-type inequality holds:  1 83f5a+b 6+ 2fa+b 2+ 3fa+ 5b 6−1 b−aZb a f(t)dt ≤5 24 b _ a (f). This is proved by Gumus et al. in paper [19, Corollary 6]. Remark 3.2. Consider φ(t) = tα Γ(α) in Theorem 3.1. Then, Theorem 3.1 becomes to[ 19 , Theorem 9]. Corollary 3.1. Note that φ(t) = 1 kΓk(α)tα kin Theorem 3.1. Then, we obtain  1 83f5a+b 6+ 2fa+b 2+ 3fa+ 5b 6−2α k−1Γk(α+k) (b−a)α kJα a+b 2−,kf(a) + Jα a+b 2+,kf(b) ≤1 2max (1 3α k,1 3α k−3 4 ,1 4)b _ a (f), 2nd Kocaeli Science Congress, November 19-21, 2025 M33-7 KOSC-2025 Proceedings References [1] Dragomir S.S. On Simpson’s quadrature formula for mappings of bounded variation and applications, Tamkang J. of Math. 30, 53–58, 1999. [2] Budak H.; Hezenci F.; and Kara H. 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