Some Error Bounds for Milne's Formula via Fractional Integral Operators

Some Error Bounds for Milne’s Formula via Fractional Integral Operators Fatih Hezenci1, Hüseyin Budak2, Hasan Kara1 1 Department of Mathematics, Faculty of Science and Arts, Duzce University, Duzce 81620, Türkiye 2Department of Mathematics, Faculty of Science and Arts, Kocaeli University, Kocaeli 41001, Türkiye Corresponding author: [email protected] ORCID IDs: First Author: 0000-0003-1008-5856 Second Author: 0000-0001-8843-955X Third Author: 0000-0002-2075-944X DOI : 10.5281/zenodo.18017772 Abstract This study investigates some Milne-type inequalities for differentiable convex functions. Our main goal is to show how such inequalities can be established by using the power mean inequality, and in doing so, we naturally extend several well-known inequalities in the existing literature to more general forms. Moreover, we study special cases of the main theorem by choosing particular functions and parameters. This allows us to see how the general results reduce to simpler and more familiar inequalities. We also provide an illustrative example in order to clarify the use of our results. Furthermore, graphs are included to make the inequalities more understandable and to demonstrate their validity in practice. Overall, this study presents a useful contribution to the theory of inequalities for differentiable convex functions and may serve as a basis for further research. Keywords: Milne’s formula, quadrature formulae, open Newton-Cotes formulas. 1 Introduction Fractional calculus is a branch of mathematics that extends the concept of derivatives and integrals to non-integer (fractional) orders. This generalization of classical calculus has proven to be highly effective in various fields such as physics, engineering, and applied sciences. Common definitions of fractional integrals include the Riemann-Liouville, conformable, and tempered fractional integrals, among others. New bounds and integral identities can be derived using a variety of inequality techniques, including Hermite-Hadamard, Simpson-type, as well as Newton and Milne-type inequalities. One of the most widely used Newton-Cotes quadrature methods, incorporating a three-point Simpson-type inequality, is given as follows: M19-1 KOSC-2025 Proceedings Theorem 1.1. Let us consider f : [a, b]→R is a four times continuously differentiable function on (a, b),and   f(4)  ∞= sup x∈(a,b)f(4)(x)<∞.Then, one has the following inequality  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 inequalities for differentiable convex functions and their fractional versions have been studied extensively. Simpson-type inequalities and their application to quadrature inequalities in numerical analysis are proved in paper [ 1 ]. Moreover, in paper [ 2 ], fractional Simpson-type inequalities are investigated for the case of function whose second derivatives in absolute value are convex. Furthermore, in paper [ 3 ], some fractional Simpson-type inequalities are proved by functions whose second derivatives in absolute value are convex. One of the classical closed-type quadrature rules is the Simpson 3 / 8rule, which is based on the Simpson 3/8inequality, formulated as follows: Theorem 1.2 (See [ 4 ]).Let us consider that f : [a, b]→R is a four times continuously differentiable function on (a, b), and   f(4)  ∞ = sup x∈(a,b)f(4)(x)<∞. Then, the following inequality satisfies:  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. Simpson’s second rule is a consequence of the three-point Newton-Cotes quadrature formula, which often leads to the classification of evaluations involving three-step quadratic kernels as Newton-type results. In the literature, such results are commonly referred to as Newton-type inequalities. A considerable number of mathematicians have studied and contributed to the development of these inequalities. For instance, some Newton-type inequalities are presented for the case of functions whose first derivative in absolute value at certain power are arithmeticallyharmonically convex in paper [ 5 ]. In addition, in paper [ 6 ], some Riemann-Liouville fractional Newton-type inequalities were established for functions of bounded variation. Newton-type inequalities are further discussed in papers [ 7 , 8 ] and the referenced works within those papers provide additional information on this topic. The open-type Milne formula, derived from Newton-Cotes formulas, is similar to the closedtype Simpson formula in that it holds under the same conditions. Theorem 1.3 (See [ 9 ]).If f : [a, b]→R is a four times continuously differentiable mapping on (a, b)and   f(4)  ∞= sup x∈(a,b)f(4)(x)<∞.Then, one has the inequality  1 32f(a)−fa+b 2+ 2f(b)−1 b−aZb a f(x)dx ≤7 (b−a)4 23040   f(4)  ∞. The well-known Riemann-Liouville fractional integrals that are given as follows: Definition 1.1 (See [ 10 ]).The Riemann-Liouville integrals Jα a+f ( x )and Jα b−f ( x )of order α > 0 M19-2 2nd Kocaeli Science Congress, November 19-21, 2025 are given by 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, respectively for f∈L1 [ a, b ]. The gamma function is defined Γ (x) := ∞ R0 tx−1e−tdt for x∈R+ . The Riemann-Liouville integrals are equals to the classical integrals for the case of α= 1. Djenaoui and Meftah [ 11 ] established several estimates for Milne’s quadrature rule in the case of functions whose first derivative is s -convex. Additionally, Alomari and Liu [ 12 ] provided error estimates for Milne’s rule applicable to functions of bounded variation and absolutely continuous functions. Moreover, Budak et al. [ 13 ] developed fractional versions of Milne’s formula by employing differentiable convex functions. For more information about these type of inequalities, one can refer to [14]. Ali et al. [ 15 ] computed error bounds using one of the open Newton-Cotes formulas, namely, Milne’s formula for differentiable convex functions within both classical and fractional calculus frameworks. To support their main findings, they established the following integral identity: Lemma 1.1 (See [ 15 ]).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a+ 3b 4−fa+b 2+ 2f3a+b 4−Γ (α+ 1) 2 (b−a)αJα a+f(b) + Jα b−f(a) =b−a 2 4 X i=1 Ii. Here,                                    I1= 1 4 R0 tα[f′(tb + (1 −t)a)−f′(ta + (1 −t)b)] dt, I2= 1 2 R1 4tα−2 3[f′(tb + (1 −t)a)−f′(ta + (1 −t)b)] dt, I3= 3 4 R1 2tα−1 3[f′(tb + (1 −t)a)−f′(ta + (1 −t)b)] dt, I4= 1 R3 4 (tα−1) [f′(tb + (1 −t)a)−f′(ta + (1 −t)b)] dt. 2 Main Result Theorem 2.1. Under the assumptions that Lemma 1.1 holds and the function |f′|q , q≥ 1is convex on [a, b].Then, we have the following Milne’s formula  1 32fa+ 3b 4−fa+b 2+ 2f3a+b 4−Γ (α+ 1) 2 (b−a)αJα a+f(b) + Jα b−f(a) (1) 2nd Kocaeli Science Congress, November 19-21, 2025 M19-3 KOSC-2025 Proceedings ≤b−a 2(Ω1(α))1−1 qΩ5(α)f′(b) q+ (Ω1(α)−Ω5(α)) f′(a) q1 q +Ω5(α)f′(a) q+ (Ω1(α)−Ω5(α)) f′(b) q1 q + (Ω2(α))1−1 qΩ6(α)f′(b) q+ (Ω2(α)−Ω6(α)) f′(a) q1 q +Ω6(α)f′(a) q+ (Ω2(α)−Ω6(α)) f′(b) q1 q + (Ω3(α))1−1 qΩ7(α)f′(b) q+ (Ω3(α)−Ω7(α)) f′(a) q1 q +Ω7(α)f′(a) q+ (Ω3(α)−Ω7(α)) f′(b) q1 q + (Ω4(α))1−1 qΩ8(α)f′(b) q+ (Ω4(α)−Ω8(α)) f′(a) q1 q +Ω8(α)f′(a) q+ (Ω4(α)−Ω8(α)) f′(b) q1 q. Here, Ω1(α) = 1 4 Z 0 tαdt =1 4α+1 (α+ 1), Ω2(α) = 1 2 Z 1 4  tα−2 3 dt =                                      −((α+1)2α+1−6)4α+2α3 12(α+1)2α4α,0< α ≤ln(2 3) ln(1 4), −2(α+1)3 1 α−α21 α+34α−31 α+1 12(α+1)3 1 α4α −2(α+1)3 1 α−α21 α+22α−31 α+1 6(α+1)3 1 α2α, ln(2 3) ln(1 4)< α ≤ln(2 3) ln(1 2), ((α+1)2α+1−6)4α+2α3 12(α+1)2α4α, α > ln(2 3) ln(1 2), M19-4 2nd Kocaeli Science Congress, November 19-21, 2025 Ω3(α) = 3 4 Z 1 2  tα−1 3 dt =                                      3α+1 4α+1(α+1) −1 2α+1(α+1) −1 12,0< α ≤ln(1 3) ln(1 2), 31 α+1−(α+1)3 1 α−2α2α 6(α+1)3 1 α2α −3(α+1)3 1 α−4α4α−3α+1 α+2 12(α+1)3 1 α4α, ln(1 3) ln(1 2)< α ≤ln(1 3) ln(3 4), 1 2α+1(α+1) −3α+1 4α+1(α+1) +1 12, α > ln(1 3) ln(3 4), Ω4(α) = 1 Z 3 4 |tα−1|dt =α−3 4 (α+ 1) +1 α+ 1 3 4α+1 . Ω5(α) = 1 4 Z 0 tα+1dt =1 4α+2 (α+ 2), Ω6(α) = 1 2 Z 1 4 t tα−2 3 dt =                                1 α+2 1 2α+2 −1 α+2 1 4α+2 −1 16,0< α ≤ln(2 3) ln(1 4), α α+2 2 31+ 2 α+1 α+2 1 4α+2 +1 α+2 1 2α+2 −5 48, ln(2 3) ln(1 4)< α ≤ln(2 3) ln(1 2), −1 α+2 1 2α+2 +1 α+2 1 4α+2 +1 16, α > ln(2 3) ln(1 2), Ω7(α) = 3 4 Z 1 2 t tα−1 3 dt =                                1 α+2 3 4α+2 −1 α+2 1 2α+2 −5 96,0< α ≤ln(1 3) ln(1 2), α α+2 1 31+ 2 α+1 α+2 1 2α+2 +1 α+2 3 4α+2 −13 96, ln(1 3) ln(1 2)< α ≤ln(1 3) ln(3 4), −1 α+2 3 4α+2 +1 α+2 1 2α+2 +5 96, α > ln(1 3) ln(3 4), and Ω8(α) = 1 Z 3 4 t|tα−1|dt =7 32 −1 α+ 2 +1 α+ 2 3 4α+2 . Proof. If we initially apply the power-mean inequality to (1), then we have  1 32fa+ 3b 4−fa+b 2+ 2f3a+b 4−Γ (α+ 1) 2 (b−a)αJα a+f(b) + Jα b−f(a) ≤b−a 2             1 4 Z 0 |tα|dt    1−1 q    1 4 Z 0 |tα|f′(tb + (1 −t)a) qdt    1 q 2nd Kocaeli Science Congress, November 19-21, 2025 M19-5 KOSC-2025 Proceedings +    1 4 Z 0 |tα|dt    1−1 q    1 4 Z 0 |tα|f′(ta + (1 −t)b) qdt    1 q +    1 2 Z 1 4  tα−2 3 dt    1−1 q    1 2 Z 1 4  tα−2 3f′(tb + (1 −t)a) qdt    1 q +    1 2 Z 1 4  tα−2 3 dt    1−1 q    1 2 Z 1 4  tα−2 3f′(ta + (1 −t)b) qdt    1 q +    3 4 Z 1 2  tα−1 3 dt    1−1 q    3 4 Z 1 2  tα−1 3f′(tb + (1 −t)a) qdt    1 q +    3 4 Z 1 2  tα−1 3 dt    1−1 q    3 4 Z 1 2  tα−1 3f′(ta + (1 −t)b) qdt    1 q +    1 Z 3 4 |tα−1|dt    1−1 q    1 Z 3 4 |tα−1|f′(tb + (1 −t)a) qdt    1 q +    1 Z 3 4 |tα−1|dt    1−1 q    1 Z 3 4 |tα−1|f′(ta + (1 −t)b) qdt    1 q         . Using the fact that |f′|qis convex, it leads to  1 32fa+ 3b 4−fa+b 2+ 2f3a+b 4−Γ (α+ 1) 2 (b−a)αJα a+f(b) + Jα b−f(a) ≤b−a 2             1 4 Z 0 tαdt    1−1 q    1 4 Z 0 tαtf′(b) q+(1−t)f′(a) qdt    1 q M19-6 2nd Kocaeli Science Congress, November 19-21, 2025 +    1 4 Z 0 tαdt    1−1 q    1 4 Z 0 tαtf′(a) q+(1−t)f′(b) qdt    1 q +    1 2 Z 1 4  tα−2 3 dt    1−1 q    1 2 Z 1 4  tα−2 3tf′(b) q+(1−t)f′(a) qdt    1 q +    1 2 Z 1 4  tα−2 3 dt    1−1 q    1 2 Z 1 4  tα−2 3tf′(a) q+(1−t)f′(b) qdt    1 q +    3 4 Z 1 2  tα−1 3 dt    1−1 q    3 4 Z 1 2  tα−1 3tf′(b) q+(1−t)f′(a) qdt    1 q +    3 4 Z 1 2  tα−1 3 dt    1−1 q    3 4 Z 1 2  tα−1 3tf′(a) q+(1−t)f′(b) qdt    1 q +    1 Z 3 4 |tα−1|dt    1−1 q    1 Z 3 4 |tα−1|tf′(b) q+(1−t)f′(a) qdt    1 q +    1 Z 3 4 |tα−1|dt    1−1 q    1 Z 3 4 |tα−1|tf′(a) q+(1−t)f′(b) qdt    1 q         =b−a 2(Ω1(α))1−1 qΩ5(α)f′(b) q+ (Ω1(α)−Ω5(α)) f′(a) q1 q +Ω5(α)f′(a) q+ (Ω1(α)−Ω5(α)) f′(b) q1 q + (Ω2(α))1−1 qΩ6(α)f′(b) q+ (Ω2(α)−Ω6(α)) f′(a) q1 q 2nd Kocaeli Science Congress, November 19-21, 2025 M19-7 KOSC-2025 Proceedings +Ω6(α)f′(a) q+ (Ω2(α)−Ω6(α)) f′(b) q1 q + (Ω3(α))1−1 qΩ7(α)f′(b) q+ (Ω3(α)−Ω7(α)) f′(a) q1 q +Ω7(α)f′(a) q+ (Ω3(α)−Ω7(α)) f′(b) q1 q + (Ω4(α))1−1 qΩ8(α)f′(b) q+ (Ω4(α)−Ω8(α)) f′(a) q1 q +Ω8(α)f′(a) q+ (Ω4(α)−Ω8(α)) f′(b) q1 q. Corollary 2.1. If we assign α= 1 in Theorem 2.1, then the following Milne’s formula holds:  1 32fa+ 3b 4−fa+b 2+ 2f3a+b 4−1 b−a b Z a f(t)dt ≤b−a 4   1 8  |f′(b)|q+ 5 |f′(a)|q 6!1 q + |f′(a)|q+ 5 |f′(b)|q 6!1 q  +7 241−1 q  5|f′(b)|q+ 9 |f′(a)|q 48 !1 q + 5|f′(a)|q+ 9 |f′(b)|q 48 !1 q    . This inequality helps us find the error bound of Milne’s rule. Example 2.1. Consider a function f : [ a, b ] = [0 , 2] →R given by f ( x ) = x2 . From Theorem 2.1 with α∈(0,10] and q= 2, the left-hand side of (1) reduces to  4α (α+ 1) (α+ 2) −2 3 and the right hand-side of (1) coincides with 4n(Ω1(α))1 2h[Ω5(α)]1 2+ [Ω1(α)−Ω5(α)]1 2i + (Ω2(α))1 2h[Ω6(α)]1 2+ [Ω2(α)−Ω6(α)]1 2i + (Ω3(α))1 2h[Ω7(α)]1 2+ [Ω3(α)−Ω7(α)]1 2i + (Ω4(α))1 2h[Ω8(α)]1 2+ [Ω4(α)−Ω8(α)]1 2io. M19-8 2nd Kocaeli Science Congress, November 19-21, 2025 Finally, we get the inequality  4α (α+ 1) (α+ 2) −2 3 (2) ≤4n(Ω1(α))1 2h[Ω5(α)]1 2+ [Ω1(α)−Ω5(α)]1 2i + (Ω2(α))1 2h[Ω6(α)]1 2+ [Ω2(α)−Ω6(α)]1 2i + (Ω3(α))1 2h[Ω7(α)]1 2+ [Ω3(α)−Ω7(α)]1 2i + (Ω4(α))1 2h[Ω8(α)]1 2+ [Ω4(α)−Ω8(α)]1 2io. 0 0.05 0.1 0.15 0.2 0.25 0.3 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 The left terms The right terms (a) A graph plotted over the interval 0 < α ≤ln(2 3) ln(1 4) . 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0 0.2 0.4 0.6 0.8 1 1.2 The left terms The right terms (b) The graph of the function defined on the interval ln(2 3) ln(1 4)< α ≤ln(2 3) ln(1 2). Figure 1: The left-hand side of (2) is consistently below the right-hand side of this inequality for all values of α∈0,ln(2 3) ln(1 2)in Example 2.1. 0.4 0.6 0.8 1 1.2 1.4 1.6 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 The left terms The right terms (a) A graphical representation corresponding to the interval ln(2 3) ln(1 2)< α ≤ln(1 3) ln(1 2). 1.5 2 2.5 3 3.5 4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 The left terms The right terms (b) A plot over the domain from ln(1 3) ln(1 2)to ln(1 3) ln(3 4). Figure 2: The left-hand side of (2) remains consistently below the right-hand side of the inequality for all values of α∈ln(2 3) ln(1 2),ln(1 3) ln(3 4)in Example 2.1. 2nd Kocaeli Science Congress, November 19-21, 2025 M19-9