Milne Formula for Functions of Bounded Variation via Generalized Fractional Integrals

Milne Formula for Functions of Bounded Variation via Generalized Fractional Integrals Fatih Hezenci Department of Mathematics, Faculty of Science and Arts, Duzce University, Duzce 81620, Türkiye Corresponding author: [email protected] ORCID IDs: First Author: 0000-0003-1008-5856 DOI : 10.5281/zenodo.18017714 Abstract In this paper, some Milne formulas are established by generalized fractional integrals for functions of bounded variation. By choosing particular functions and suitable parameter values, several important special cases can be derived, which clearly illustrate the wide applicability of the proposed approach. This allows us to see how the general results reduce to simpler and more familiar formulas. The results improve the understanding of fractional inequalities and offer a solid mathematical basis for future research in fractional analysis and related fields. Namely, one may consider generalizing our findings by exploring alternative classes of convex functions or different types of fractional integral operators. Keywords: Milne formula, functions of bounded variation, generalized fractional integrals. 1 Introduction Many mathematicians have studied numerical integration formulas and their error bounds using different methods. To find these error bounds, they have also explored many mathematical inequalities for various kinds of functions, such as convex, bounded, and Lipschitz functions. For example, in papers [ 1 , 2 ], some error bounds for the midpoint and trapezoidal inequalities in numerical integration are obtained by using convex functions. In [ 3 ], Simpson-type inequalities are introduced, and their applications to quadrature inequalities within the context of numerical analysis are investigated. Furthermore, Budak et al. [ 4 ] are studied several forms of Simpson-type inequalities for differentiable convex functions within the setting of generalized fractional integrals. The three-point Newton-Cotes quadrature rule, commonly known as Simpson’s second rule, has a significant role in numerical integration. Since it employs three-point quadratic kernels, these formulas are often referred to as Newton-type results. Milne-type inequalities can be viewed as extensions of Newton-type formulas, using higher-order or generalized kernels to achieve better approximation accuracy. Recently, several studies have focused on deriving and extending Milne-type inequalities by using convexity properties and fractional integral operators, which provide wider applications in numerical analysis. For instance, in [ 5 ], some error bounds for Newton-type inequalities in numerical integration have been established by employing convex M18-1 KOSC-2025 Proceedings functions. Likewise, certain Newton-type inequalities based on convexity are presented in [ 6 ], along with applications to specific cases of real functions. Several new estimates of Milne’s quadrature rule have been derived by Djenaoui and Meftah [ 7 ], particularly for functions whose first derivative is s -convex. The paper [ 8 ] presents error bounds for Milne-type inequalities applied to functions of bounded variation. Moreover, in [ 9 ], fractional Milne-type inequalities are derived using differentiable convex functions. The study [ 10 ] establishes error bounds for Milne’s formula, an open Newton-Cotes quadrature rule, for differentiable convex functions using both classical and fractional calculus approaches. For further details, one may refer to [ 11 , 12 , 13 ], as well as the references cited therein. 2 Preliminaries Generalized fractional integrals are extensions of classical fractional integrals that allow integration of functions with respect to more general kernels or parameters. For example, Sarikaya and Ertugral established Hermite–Hadamard-type inequalities for generalized fractional integrals [ 14 ]. Definition 2.1. [ 14 ] Let 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 (1) and b−Iφf(x) = Zb x φ(t−x) t−xf(t)dt, x < b, (2) respectively. The most important feature of generalized fractional integrals is that they generalize some important types of fractional integrals such as Riemann-Liouville fractional integral, k -RiemannLiouville fractional integral, conformable fractional integral, Katugampola fractional integrals, Hadamard fractional integrals etc. The important special cases of the integral operators (1) and (2) are as follows: i. If we choose φ(t) = t, then the operators (1) and (2) become to the Riemann integral. ii. Let us consider φ(t) = tα Γ(α) and α > 0 . Then, the operators (1) and (2) coincide with the Riemann-Liouville fractional integrals Jα a+f(x)and Jα b−f(x), respectively [15,16]. 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. The symbol Γrepresents the Gamma function, which is defined by the integral Γ (x) := ∞ Z 0 tx−1e−tdt, x ∈R+. M18-2 2nd Kocaeli Science Congress, November 19-21, 2025 iii. When φ ( t ) = 1 kΓk(α)tα k and α, k > 0, the integral operators (1) and (2) specialize to the k-Riemann–Liouville 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 32f3a+b 4−fa+b 2+ 2fa+ 3b 4−1 2Λ (1) [b−Iφf(a) +a+Iφf(b)] (3) =b−a 2Λ (1) Zb a ψ(x)df(x) Here, Λ (t) = t R0 φ((b−a)y) ydy and ψ(x) =                                1 b−ahΛx−a b−a−Λb−x b−a+ Λ (1)i, a ≤x≤3a+b 4, 1 b−ahΛx−a b−a−Λb−x b−a−1 3Λ (1)i,3a+b 4<x≤a+b 2, 1 b−ahΛx−a b−a−Λb−x b−a+1 3Λ (1)i,a+b 2<x≤a+3b 4, 1 b−ahΛx−a b−a−Λb−x b−a−Λ (1)i,a+3b 4<x≤b. Proof. Let us first consider the function ψ(x). Then, with the help of the integrating by parts, it follows from that Zb a ψ(x)df(x)(4) =1 b−a       3a+b 4 Z aΛx−a b−a−Λb−x b−a+ Λ (1)df(x) 2nd Kocaeli Science Congress, November 19-21, 2025 M18-3 KOSC-2025 Proceedings + a+b 2 Z 3a+b 4 Λx−a b−a−Λb−x b−a−1 3Λ (1)df(x) + a+3b 4 Z a+b 2 Λx−a b−a−Λb−x b−a+1 3Λ (1)df(x) + b Z a+3b 4 Λx−a b−a−Λb−x b−a−Λ (1)df(x)       In a similar way, we get 3a+b 4 Z aΛx−a b−a−Λb−x b−a+ Λ (1)df(x)(5) =Λx−a b−a−Λb−x b−a+ Λ (1)f(x) 3a+b 4 a − 3a+b 4 Z aφ(x−a) x−a+φ(b−x) b−xf(x)dx =Λ1 4−Λ3 4+ Λ (1)f3a+b 4 − 3a+b 4 Z aφ(x−a) x−a+φ(b−x) b−xf(x)dx. Similarly, we have Za+b 2 3a+b 4Λx−a b−a−Λb−x b−a−1 3Λ (1)df(x)(6) =Λx−a b−a−Λb−x b−a−1 3Λ (1)f(x) a+b 2 3a+b 4 − a+b 2 Z 3a+b 4 φ(x−a) x−a+φ(b−x) b−xf(x)dx =−1 3Λ (1) fa+b 2−Λ1 4−Λ3 4−1 3Λ (1)f3a+b 4 − a+b 2 Z 3a+b 4 φ(x−a) x−a+φ(b−x) b−xf(x)dx, Za+3b 4 a+b 2Λx−a b−a−Λb−x b−a+1 3Λ (1)df(x)(7) =Λx−a b−a−Λb−x b−a+1 3Λ (1)f(x) a+3b 4 a+b 2 − a+3b 4 Z a+b 2 φ(x−a) x−a+φ(b−x) b−xf(x)dx =Λ3 4−Λ1 4+1 3Λ (1)fa+ 3b 4−1 3Λ (1) fa+b 2 M18-4 2nd Kocaeli Science Congress, November 19-21, 2025 − a+3b 4 Z a+b 2 φ(x−a) x−a+φ(b−x) b−xf(x)dx, and b Z a+3b 4 Λx−a b−a−Λb−x b−a−Λ (1)df(x)(8) =Λx−a b−a−Λb−x b−a−Λ (1)f(x) b a+3b 4 − b Z a+3b 4 φ(x−a) x−a+φ(b−x) b−xf(x)dx =−Λ3 4−Λ1 4−Λ (1)fa+ 3b 4 − b Z a+3b 4 φ(x−a) x−a+φ(b−x) b−xf(x)dx. By putting the equalities (5)-(8) in (4), we have Zb a ψ(x)df(x) =2Λ (1) (b−a) 1 32f3a+b 4−fa+b 2+ 2fa+ 3b 4 −1 b−a b Z aφ(x−a) x−a+φ(b−x) b−xf(x)dx =2Λ (1) (b−a) 1 32f3a+b 4−fa+b 2+ 2fa+ 3b 4−1 b−a[b−Iφf(a) +a+Iφf(b)] . 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 ψ(x)df(x) Theorem 3.1. Note that f : [a, b]→R is a function of bounded variation on [a, b]. Then, one can obtain  1 32f3a+b 4−fa+b 2+ 2fa+ 3b 4−1 2Λ (1) [b−Iφf(a) +a+Iφf(b)] ≤1 2Λ (1) max  Λ1 4−Λ3 4+ Λ (1)  Λ3 4−Λ1 4+1 3Λ (1)b _ a (f). 2nd Kocaeli Science Congress, November 19-21, 2025 M18-5 KOSC-2025 Proceedings Here, d W c(f)denotes the total variation of fon [c, d]. Proof. If we take modules of equality (3), we have  1 32f3a+b 4−fa+b 2+ 2fa+ 3b 4−1 2Λ (1) [b−Iφf(a) +a+Iφf(b)] (9) =b−a 2Λ (1) Zb a ψ(x)df(x) . It is well known that if two functions g, f : [ a, b ] →R satisfy that g is continuous on [ a, b ] and fis of bounded variation on [a, b], then the integral Zb a g(t)df(t) exists. Then, it yields Zb a g(t)df(t) ≤sup t∈[a,b] |g(t)| b _ a (f).(10) With the help of the inequality (10), we obtain Zb a ψ(x)df(x) (11) ≤1 b−a(Z3a+b 4 aΛx−a b−a−Λb−x b−a+ Λ (1)df(x) +Za+b 2 3a+b 4Λx−a b−a−Λb−x b−a−1 3Λ (1)df(x) +Za+3b 4 a+b 2Λx−a b−a−Λb−x b−a+1 3Λ (1)df(x) +Zb a+3b 4Λx−a b−a−Λb−x b−a−Λ (1)df(x)) ≤1 b−a     sup x∈[a, 3a+b 4] Λx−a b−a−Λb−x b−a+ Λ (1) 3a+b 4 _ a (f) + sup x∈[3a+b 4,a+b 2] Λx−a b−a−Λb−x b−a−1 3Λ (1) a+b 2 _ 3a+b 4 (f) + sup x∈[a+b 2 a+3b 4] Λx−a b−a−Λb−x b−a+1 3Λ (1) a+3b 4 _ a+b 2 (f) + sup x∈[a+3b 4,b] Λx−a b−a−Λb−x b−a−Λ (1) b _ a+3b 4 (f)     =1 b−a     Λ1 4−Λ3 4+ Λ (1) 3a+b 4 _ a (f) M18-6 2nd Kocaeli Science Congress, November 19-21, 2025 REFERENCES + max 1 3Λ (1) , Λ1 4−Λ3 4−1 3Λ (1) a+b 2 _ 3a+b 4 (f) + max  Λ3 4−Λ1 4+1 3Λ (1) ,1 3Λ (1) a+3b 4 _ a+b 2 (f) + Λ3 4−Λ1 4−Λ (1) b _ a+3b 4 (f)     ≤1 b−amax  Λ1 4−Λ3 4+ Λ (1)  Λ3 4−Λ1 4+1 3Λ (1)b _ a (f). Then, we get the following inequality  1 32f3a+b 4−fa+b 2+ 2fa+ 3b 4−1 2Λ (1) [b−Iφf(a) +a+Iφf(b)] ≤1 2Λ (1) max  Λ1 4−Λ3 4+ Λ (1)  Λ3 4−Λ1 4+1 3Λ (1)b _ a (f). which is complete the desired results of Theorem 3.1. Remark 3.1. Consider φ(t) = tin Theorem 3.1. Then, the following Milne formula satisfies:  1 32fa+ 3b 4−fa+b 2+ 2f3a+b 4−1 b−aZb a f(t)dt ≤5 12 b _ a (f). This is established by Unal et al. in [13, Corollary 6]. Remark 3.2. Let us consider φ(t) = tα Γ(α) in Theorem 3.1. Then, Theorem 3.1 reduces to [ 12 , Theorem 2.2]. Corollary 3.1. For φ(t) = 1 kΓk(α)tα kin Theorem 3.1, we have  1 32f3a+b 4−fa+b 2+ 2fa+ 3b 4−Γk(α+k) 2 (b−a)α khJα a+,kf(b) + Jα b−,kf(a)i ≤1 2max (1−3α k+ 4α k 4α k ,3α k−1 4α k +1 3)b _ a (f). References [1] Dragomir, S.S.; Agarwal, R. Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula. Appl. Math. 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