Fractional Calculus Approach to Perturbed Trapezoid-type Inequalities: Novel Error Estimates and Applications
Abstract
2nd Kocaeli Science Congress (KOSC-2025), 19-21 November 2025, Kocaeli, TÜRKİYE https://fefkongre.kocaeli.edu.tr/en
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Fractional Calculus Approach to Perturbed Trapezoid-type Inequalities: Novel Error Estimates and Applications Arslan Munir1, Shumin Li1, Hüseyin Budak3 1 School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026 People’s Republic of China 2Department of Mathematics, Faculty of Science and Arts, Kocaeli University, Kocaeli 41001, Türkiye Corresponding author: m[email protected] ORCID IDs: First Author: 0009-0008-8148-5607 Second Author: 0000-0002-6236-6273 Third Author: 0000-0001-8843-955X DOI : 10.5281/zenodo.18011658 Abstract Inequalities involving fractional operators have also been an active area of research. These inequalities play a crucial role in establishing bounds, estimates, and stability conditions for solutions to fractional integrals. The main aim in the current article is to establish a identity through the Caputo-Fabrizio integral operator. By using this identity, we are obtained the new fractional bounds and error estimated for the perturbed trapezoid-type inequalities. With the aid of this identity, it is proved that new fractional perturbed trapezoid-type inequalities for functions whose third derivatives in absolute value are convex. In addition, the research has acquired fractional perturbed trapezoid-type inequalities for (α, m) -convex function. This is the first article for fractional version of perturbed trapezoid-type inequalities. Finally, we present several corollaries to demonstrate how new results are better than previous results. Moreover, we have given application to the quadrature formula. Keywords: Perturbed trapezoid type inequality, (α, m) -convex function, Fractional integrals, Young’s inequality. M3-1
KOSC-2025 Proceedings 1 Introduction Convexity simplifies problem-solving and often leads to computationally efficient solutions. Convex functions and sets have well-defined mathematical properties, which allow for the development of efficient algorithms and analytical solutions. This makes convexity a valuable tool for researchers and practitioners in physics, statistics, engineering, and among other disciplines. Convexity theory provides a unifying framework to understand and prove many of these inequalities. It allows mathematicians to explore and generalize these inequalities, leading to the development of new results and techniques in various mathematical disciplines. The interested researchers on convexity and integral inequalities we refer to see these articles [ 2 , 3 , 4 ]. The original and generalized version is defined as: Definition 1.1. [1]A function f:I→Ris said to be convex if the inequality holds: f(ψπ + (1 −ψ)ν)≤ψf (π)+(1−ψ)f(ν), for all π, ν ∈Iand ψ∈[0,1] . Definition 1.2. [5]Let f: [0, y]→Rand (α, m)∈[0,1]2is said to be (α, m)-convex if: f(ψπ + (1 −ψ)ν)≤ψαf(π) + m(1 −ψα)f(ν), for all π, ν ∈[0, y]and ψ∈[0,1] . The Hermite-Hadamard inequality is a well-known result in mathematical analysis and can be attributed to two mathematicians Charles Hermite and Jacques Hadamard is indeed known as the Hermite-Hadamard inequality. The Hermite-Hadamard inequality is a result in the theory of convex functions and states as follows: fπ+ν 2≤Zν π f(Ω)dΩ≤f(π) + f(ν) 2. This inequality provides bounds on the integral of a convex function over a closed interval and has applications in various branches of mathematics, including calculus, analysis, and optimization. The Hermite-Hadamard inequality has garnered significant attention from mathematicians and researchers since its introduction. This inequality is not only elegant but also versatile, making it a valuable tool in mathematical analysis and its applications. Mathematicians have explored various aspects of the inequality, including its extensions, generalizations, have been published based on the definition of convexity [6,7,8,9,10,11]. M3-2 2nd Kocaeli Science Congress, November 19-21, 2025
Fractional calculus, a branch of mathematics that generalized the concept of differentiation and integration to non-integer orders, has indeed gained remarkable popularity and importance in recent decades. Its applications span a wide range of fields, making it a valuable tool in various scientific and engineering disciplines. The use of fractional integral operators to popularize and extend well-known integral inequalities reflects the evolving nature of mathematics and its applications. It allows for a deeper exploration of mathematical concepts and provides valuable insights into the behavior of systems with fractional order dynamics. As a result, this topic continues to attract the attention of mathematicians and researchers interested in both theoretical and applied mathematics. The Riemann-Liouville fractional integral is just one of several fractional integral operators used in fractional calculus. Other operators, such as the Caputo fractional integral, are also commonly used depending on the specific problem and boundary conditions. Prior to using the Caputo-Fabrizio fractional integral defined by Caputo, let’s review its definition. Definition 1.3. [ 12 ] Suppose f∈L[π, ν] . The Riemann-Liouville fractional integrals of order α > 0, defined on the left and right sides are given by: Iα π+f(ψ) = 1 Γ (α)ZΩ π (Ω −ψ)α−1f(ψ)dψ,Ω> π, Iα ν−f(ψ) = 1 Γ (α)Zν Ω (ψ−Ω)α−1f(ψ)dψ,Ω< ν, where Γ (.)is the gamma function and I0 π+f(ψ)=I0 ν−f(ψ)=f(ψ). Definition 1.4. [ 13 ]Let f∈H1(π, ν) , π < ν , for all ψ∈[0,1] , where β(α)> 0is a normalizer satisfying β(0) = β(1) = 1, then the left and right fractional integrals are defined by: CF πIψf(Ω) = 1−α β(α)f(Ω) + α β(α)ZΩ π f(Ω) dΩ, CF Iψ νf(Ω) = 1−α β(α)f(Ω) + α β(α)Zν Ω f(Ω) dΩ. The study of fractional H-H (Hermite-Hadamard type) inequalities has indeed garnered significant attention in recent years within the realm of fractional calculus and mathematical analysis. These inequalities are extensions or generalizations of the classical H-H 2nd Kocaeli Science Congress, November 19-21, 2025 M3-3
KOSC-2025 Proceedings (Hermite-Hadamard type) inequality to the framework of fractional calculus. Therefore, it is crucial to create and investigate the H-H (Hermite-Hadamard type) inequality for a variety of convex functions using different fractional operators. Fractional calculus continues to evolve, and researchers explore new operators, for example (Caputo, CaputoFabrizio, Riemann-Liouville, k -Riemann-Liouville, Katugampola, Conformable) and their applications to address complex problems in various domains. Agarwal et. al [ 14 ] derived H-H (Hermite-Hadamard type) inequalities using generalized k -fractional integrals. By utilizing harmonic convex functions, Awan et al. [ 15 ] obtained some conformable fractional estimates for H-H (Hermite-Hadamard type) inequalities. Zhao et. al [ 16 ], where s -convex functions were employed to establish H-H (Hermite-Hadamard type) inequalities using ψ -Riemann-Liouville fractional integrals. For further inequalities that can be solved using fractional integrals, and the references therein [ 17 , 18 , 19 ]. H-H (Hermite-Hadamard type) inequalities are a class of inequalities, and choosing the most suitable fractional integral operator for studying these inequalities can depend on the specific problem and the properties of the functions involved. It is difficult to generalized and extend H-H (Hermite-Hadamard type) inequalities to the various fractional integral operators. Therefore, the concept of using general fractional integral operators has been proposed by some researchers to meet the needs of modern mathematics. Set et. al [ 20 ] established H-H (Hermite-Hadamard type) inequalities involving generalized fractional integral operators. Furthermore, Ahmad et. al [ 21 ] proved several H-H (Hermite-Hadamard type) inequalities for convex functions using fractional integral operators with exponential kernel. Aljaaidi et.al [ 22 ] studied some novel fractional integral H-H (Hermite-Hadamard type) inequalities based on the generalized proportional fractional integral operators. For more details, we refer the readers to [23,24,25]. The aim of the current research article is to establish new perturbed trapezoidtype inequalities for ( α, m )-convex functions in the setting of Caputo-Fabrizio integral operators. We are using the Caputo-Fabrizio integral to obtain the new fractional bounds and error estimates for perturbed trapezoid-type inequalities. In addition, the research has acquired fractional perturbed trapezoid-type inequalities for functions whose third derivatives in absolute value are convex. Moreover, we have given application to the quadrature formula. 2 Fractional Perturbed Trapezoid-Type Inequality In this section, we introduce new perturbed trapezoid-type inequalities by applying Caputo-Fabrizio fractional operators. M3-4 2nd Kocaeli Science Congress, November 19-21, 2025
Lemma 2.1. A function f : [π, ν]→R is an absolutely continuous function (π, ν) such that f′′′ ∈L1[π, ν]. Then the following equality holds: (Ω −π)2f′(π)−(Ω −ν)2f′(ν)+2f(Ω) (ν−π) + 4 [(Ω −π)f(π) + (Ω −ν)f(ν)] 6 (ν−π) −β(α) α(ν−π)hCF πIαf(k) + CF Iα νf(k)i+2 (1 −α) α(ν−π)f(k) =1 6 (ν−π)Z1 0 ψ(1 −ψ)2(Ω −π)4f′′′ (ψΩ + (1 −ψ)π) −Z1 0 ψ(1 −ψ)2(Ω −ν)4f′′′ (ψΩ + (1 −ψ)ν)dψ. Proof. Let Z1 0 ψ(1 −ψ)2(Ω −π)4f′′′ (ψΩ + (1 −ψ)π)dψ −Z1 0 ψ(1 −ψ)2(Ω −ν)4f′′′ (ψΩ+(1−ψ)ν)dψ =I1−I2. Integration by parts, we get I1=Z1 0 ψ(1 −ψ)2(Ω −π)4f′′′ (ψΩ + (1 −ψ)π)dψ =ψ(1 −ψ)2(Ω −π)4 Ω−πf′′ (ψΩ + (1 −ψ)π) 1 0 −(Ω −π)4 Ω−πZ1 01−4ψ+ 3ψ2f′′ (ψΩ + (1 −ψ)π)dψ =−(Ω −π)3Z1 01−4ψ+ 3ψ2f′′ (ψΩ + (1 −ψ)π)dψ =−(Ω −π)3 1−4ψ+ 3ψ2 Ω−πf′′ (ψΩ + (1 −ψ)π) 1 0 −1 Ω−πZ1 0 (6ψ−4) f′(ψΩ+(1−ψ)π)dψ = (Ω −π)2f′(π) + (Ω −π)2Z1 0 (6ψ−4) f′(ψΩ + (1 −ψ)π)dψ = (Ω −π)2f′(π) + (Ω −π)2"(6ψ−4) Ω−πf′(ψΩ + (1 −ψ)π) 1 0 −6 Ω−πZ1 0 f(ψΩ + (1 −ψ)π)dψ# = (Ω −π)2f′(π)+2f(Ω) (Ω −π)+4f(π) (Ω −π)−6 (Ω −π)Z1 0 f(ψΩ + (1 −ψ)π)dψ = (Ω −π)2f′(π)+2f(Ω) (Ω −π)+4f(π) (Ω −π)−6ZΩ π f(u)du. (1) 2nd Kocaeli Science Congress, November 19-21, 2025 M3-5
KOSC-2025 Proceedings Similarly, we have I2=Z1 0 ψ(1 −ψ)2(Ω −ν)4f′′′ (ψΩ + (1 −ψ)ν)dψ =ψ(1 −ψ)2(Ω −ν)4 Ω−νf′′ (ψΩ + (1 −ψ)ν) 1 0 −(Ω −ν)4 Ω−νZ1 01−4ψ+ 3ψ2f′′ (ψΩ + (1 −ψ)ν)dψ =−(Ω −ν)3Z1 01−4ψ+ 3ψ2f′′ (ψΩ+(1−ψ)ν)dψ =−(Ω −ν)3 1−4ψ+ 3ψ2 Ω−νf′′ (ψΩ + (1 −ψ)ν) 1 0 −1 Ω−νZ1 0 (6ψ−4) f′(ψΩ + (1 −ψ)ν)dψ = (Ω −ν)2f′(ν)+(Ω−ν)2Z1 0 (6ψ−4) f′(ψΩ + (1 −ψ)ν)dψ = (Ω −ν)2f′(ν)+(Ω−ν)2"(6ψ−4) Ω−νf′(ψΩ + (1 −ψ)ν) 1 0 −6 Ω−νZ1 0 f(ψΩ + (1 −ψ)ν)dψ# = (Ω −ν)2f′(ν)+2f(Ω) (Ω −ν) + 4f(ν) (Ω −ν)−6 (Ω −ν)Z1 0 f(ψΩ + (1 −ψ)ν)dψ = (Ω −ν)2f′(ν)+2f(Ω) (Ω −ν) + 4f(ν) (Ω −ν) + 6 Zν Ω f(u)du. (2) Subtracting the equalities (1) and (2), we get I1−I2= (Ω −π)2f′(π)+2f(Ω) (Ω −π)+4f(π) (Ω −π)−6ZΩ π f(u)du −(Ω −ν)2f′(ν) −2f(Ω) (Ω −ν)−4f(ν) (Ω −ν)−6Zν Ω f(u)du = (Ω −π)2f′(π)−(Ω −ν)2f′(ν)+2f(Ω) (ν−π) + 4 [(Ω −π)f(π)+(Ω−ν)f(ν)] −6Zν π f(u)du. (3) Multiplying by the equality (3) with 1 6and subtracting 2(1−α) β(α)f(k), we obtain (I1−I2)1 6−2 (1 −α) β(α)f(k) =(Ω −π)2f′(π) 6−(Ω −ν)2f′(ν) 6+2f(Ω) (ν−π) 6 +4 [(Ω −π)f(π)+(Ω−ν)f(ν)] 6−Zν π f(u)du −2 (1 −α) β(α)f(k) =(Ω −π)2f′(π) + (Ω −ν)2f′(ν)+2f(Ω) (ν−π) + 4 [(Ω −π)f(π) + (Ω −ν)f(ν)] 6 M3-6 2nd Kocaeli Science Congress, November 19-21, 2025
− α β(α)Zk π f(u)du +(1 −α) β(α)f(k) + α β(α)Zν k f(u)du +(1 −α) β(α)f(k)! =(Ω −π)2f′(π) + (Ω −ν)2f′(ν)+2f(Ω) (ν−π) + 4 [(Ω −π)f(π) + (Ω −ν)f(ν)] 6 −hCF πIαf(k) + CF Iα νf(k)i. Thus, we have (Ω −π)2f′(π)−(Ω −ν)2f′(ν)+2f(Ω) (ν−π) + 4 [(Ω −π)f(π) + (Ω −ν)f(ν)] 6 (ν−π) −β(α) α(ν−π)hCF πIαf(k) + CF Iα νf(k)i+2 (1 −α) α(ν−π)f(k) =1 6 (ν−π)Z1 0 ψ(1 −ψ)2(Ω −π)4f′′′ (ψΩ + (1 −ψ)π) −(Ω −ν)4f′′′ (ψΩ+(1−ψ)ν)dψi. The proof of Lemma 2.1 is completed. Theorem 2.1. Let assumptions of Lemma 2.1 hold. If |f′′′| is (α, m) -convex on [π, ν] and (α, m)∈(0,1]2, then the following fractional inequality holds: (Ω −π)2f′(π)−(Ω −ν)2f′(ν)+2f(Ω) (ν−π) + 4 [(Ω −π)f(π) + (Ω −ν)f(ν)] 6 (ν−π) −β(α) α(ν−π)hCF πIαf(k) + CF Iα νf(k)i+2 (1 −α) α(ν−π)f(k) ≤(Ω −π)4 6(ν−π)"2 α3+ 9α2+ 26α+ 24 f′′′ (Ω)+m α3+ 9α2+ 26α 12 (α3+ 9α2+ 26α+ 24)!f′′′ (π)# +(Ω −ν)4 6(ν−π)"2 α3+ 9α2+ 26α+ 24 f′′′ (Ω)+m α3+ 9α2+ 26α 12 (α3+ 9α2+ 26α+ 24)!f′′′ (ν)#. Proof. By using the Lemma 2.1, since (α, m)-convexity of |f′′′|, we get (Ω −π)2f′(π)−(Ω −ν)2f′(ν)+2f(Ω) (ν−π) + 4 [(Ω −π)f(π) + (Ω −ν)f(ν)] 6 (ν−π) −β(α) α(ν−π)hCF πIψf(k) + CF Iψ νf(k)i+2 (1 −α) α(ν−π)f(k) ≤(Ω −π)4 6 (ν−π)Z1 0 ψ(1 −ψ)2f′′′ (ψΩ+(1−ψ)π)dψ 2nd Kocaeli Science Congress, November 19-21, 2025 M3-7
KOSC-2025 Proceedings +(Ω −ν)4 6 (ν−π)Z1 0 ψ(1 −ψ)2f′′′ (ψΩ+(1−ψ)ν)dψ ≤(Ω −π)4 6 (ν−π)Z1 0 ψ(1 −ψ)2ψαf′′′ (Ω)+m(1 −ψα)f′′′ (π)dψ +(Ω −ν)4 6 (ν−π)Z1 0 ψ(1 −ψ)2ψαf′′′ (Ω)+m(1 −ψα)f′′′ (ν)dψ ≤(Ω −π)4 6(ν−π)"2 α3+ 9α2+ 26α+ 24 f′′′ (Ω)+m α3+ 9α2+ 26α 12 (α3+ 9α2+ 26α+ 24)!f′′′ (π)# +(Ω −ν)4 6(ν−π)"2 α3+ 9α2+ 26α+ 24 f′′′ (Ω)+m α3+ 9α2+ 26α 12 (α3+ 9α2+ 26α+ 24)!f′′′ (ν)#. The proof of Theorem 2.1 is completed. Corollary 2.1. If we put Ω = π+ν 2in Theorem 2.1, then we have ν−π 24 f′(π)−f′(ν)+1 3f(π) + f(ν)+fπ+ν 2 −β(α) α(ν−π)hCF πIαf(k) + CF Iα νf(k)i+2 (1 −α) α(ν−π)f(k) ≤(ν−π)3 96 "2 α3+ 9α2+ 26α+ 24 f′′′ π+ν 2 +m α3+ 9α2+ 26α 12 (α3+ 9α2+ 26α+ 24)!f′′′ (π) +2 α3+ 9α2+ 26α+ 24 f′′′ π+ν 2 +m α3+ 9α2+ 26α 12 (α3+ 9α2+ 26α+ 24)!f′′′ (ν)#. Corollary 2.2. If we put m= 1 in Corollary 2.1, then we have ν−π 24 f′(π)−f′(ν)+1 3f(π) + f(ν)+fπ+ν 2 −β(α) α(ν−π)hCF πIαf(k) + CF Iα νf(k)i+2 (1 −α) α(ν−π)f(k) ≤(ν−π)3 96 "2 α3+ 9α2+ 26α+ 24 f′′′ π+ν 2 + α3+ 9α2+ 26α 12 (α3+ 9α2+ 26α+ 24)!f′′′ (π) +2 α3+ 9α2+ 26α+ 24 f′′′ π+ν 2 + α3+ 9α2+ 26α 12 (α3+ 9α2+ 26α+ 24)!f′′′ (ν)#. Corollary 2.3. If we put α= 1 in Corollary 2.2, then we have ν−π 24 f′(π)−f′(ν)+1 3f(π) + f(ν)+fπ+ν 2 M3-8 2nd Kocaeli Science Congress, November 19-21, 2025
−β(α) α(ν−π)hCF πIαf(k) + CF Iα νf(k)i ≤(ν−π)3 96 1 2f′′′ (π)+f′′′ (ν)+2 3 f′′′ π+ν 2. Corollary 2.4. Applying again convexity in Corollary 2.3, then we have ν−π 24 f′(π)−f′(ν)+1 3f(π) + f(ν)+fπ+ν 2 −β(α) α(ν−π)hCF πIαf(k) + CF Iα νf(k)i ≤(ν−π)3 1152 f′′′ (π)+f′′′ (ν). Theorem 2.2. Suppose assumptions of Lemma 2.1 hold . If |f′′′| is (α, m) -convex on [π, ν],(α, m)∈(0,1]2and q > 1, then the following fractional inequality holds: (Ω −π)2f′(π)−(Ω −ν)2f′(ν)+2f(Ω) (ν−π) + 4 [(Ω −π)f(π) + (Ω −ν)f(ν)] 6 (ν−π) −β(α) α(ν−π)hCF πIαf(k) + CF Iα νf(k)i+2 (1 −α) α(ν−π)f(k) ≤(Ω −π)4 6(ν−π) (Γ (1 + p),Γ (1 + 2p)) Γ (2 + 3p) 1 p1 α+ 1 f′′′ (Ω) q+mα α+ 1f′′′ (π) q1 q +(Ω −ν)4 6(ν−π) (Γ (1 + p),Γ (1 + 2p)) Γ (2 + 3p) 1 p1 α+ 1 f′′′ (Ω) q+mα α+ 1f′′′ (ν) q1 q . Proof. By employing the Hölder inequality since (α, m)-convexity of |f′′′|q, we get (Ω −π)2f′(π)−(Ω −ν)2f′(ν)+2f(Ω) (ν−π) + 4 [(Ω −π)f(π) + (Ω −ν)f(ν)] 6 (ν−π) −β(α) α(ν−π)hCF πIαf(k) + CF Iα νf(k)i+2 (1 −α) α(ν−π)f(k) ≤(Ω −π)4 6 (ν−π)Z1 0 ψ(1 −ψ)2f′′′ (ψΩ+(1−ψ)π)dψ +(Ω −ν)4 6 (ν−π)Z1 0 ψ(1 −ψ)2f′′′ (ψΩ + (1 −ψ)ν)dψ ≤(Ω −π)4 6 (ν−π) Z1 0ψ(1 −ψ)2pdψ 1 pZ1 0f′′′ (ψΩ + (1 −ψ)π) q 1 q dψ 2nd Kocaeli Science Congress, November 19-21, 2025 M3-9
KOSC-2025 Proceedings (Ω −π)2f′(π)−(Ω −ν)2f′(ν)+2f(Ω) (ν−π) + 4 [(Ω −π)f(π) + (Ω −ν)f(ν)] 6 (ν−π) −β(α) α(ν−π)hCF πIαf(k) + CF Iα νf(k)i+2 (1 −α) α(ν−π)f(k) ≤(Ω −π)4 60(ν−π) f′′′ 5Ω+π 6+(Ω −ν)4 60(ν−π) f′′′ 5Ω+ν 6. Proof. From Lemma 2.1, we get (Ω −π)2f′(π)−(Ω −ν)2f′(ν)+2f(Ω) (ν−π) + 4 [(Ω −π)f(π) + (Ω −ν)f(ν)] 6 (ν−π) −β(α) α(ν−π)hCF πIαf(k) + CF Iα νf(k)i+2 (1 −α) α(ν−π)f(k) ≤(Ω −π)4 6 (ν−π)Z1 0 ψ(1 −ψ)2f′′′ (ψΩ + (1 −ψ)ν)dψ +(Ω −ν)4 6 (ν−π)Z1 0 ψ(1 −ψ)2f′′′ (ψΩ + (1 −ψ)π)dψ.(4) By using the Jensen inequality, we have Z1 0f′′′ (ψΩ + (1 −ψ)ν)dψ ≤"Z1 0 ψ(1 −ψ)2dψ f′′′ R1 0ψ(1 −ψ)2(ψΩ+(1−ψ)π)dψ R1 0ψ(1 −ψ)2dψ !# ≤ f′′′ 5Ω+π 6 .(5) Similarly, we have Z1 0f′′′ (ψΩ + (1 −ψ)ν)dψ ≤"Z1 0 ψ(1 −ψ)2dψ f′′′ R1 0ψ(1 −ψ)2(ψΩ+(1−ψ)ν)dψ R1 0ψ(1 −ψ)2dψ !# ≤ f′′′ 5Ω+ν 6 .(6) Using the equalities (5) and (6) in (4), we get (Ω −π)2f′(π)−(Ω −ν)2f′(ν)+2f(Ω) (ν−π) + 4 [(Ω −π)f(π) + (Ω −ν)f(ν)] 6 (ν−π) −β(α) α(ν−π)hCF πIαf(k) + CF Iα νf(k)i+2 (1 −α) α(ν−π)f(k) M3-16 2nd Kocaeli Science Congress, November 19-21, 2025
≤(Ω −π)4 60(ν−π) f′′′ 5Ω+π 6+(Ω −ν)4 60(ν−π) f′′′ 5Ω+ν 6. This completes the proof. Corollary 2.15. If we choice Ω = π+ν 2in Theorem 2.5, we have ν−π 24 f′(π)−f′(ν)+1 3f(π) + f(ν)+fπ+ν 2 −β(α) α(ν−π)hCF πIαf(k) + CF Iα νf(k)i+2 (1 −α) α(ν−π)f(k) ≤(ν−π)3 960 f′′′ 7π+ 5ν 12 + f′′′ 5π+ 7ν 12 . 3 Quadrature formula Let d is the partition of the interval [π, ν] , d : π = Ω 0< Ω 1< Ω 2< ..... < Ω n−1< Ω n = ν and Let the Zν π f(Ω) dΩ = Z(f, d) + R(f, d), where Z(f, d) = n−1 X i=0 Ωi+1 −Ωi 24 f′(Ωi)−f′(Ωi+1)+1 3f′(Ωi)+f′(Ωi+1)+fΩi+ Ωi+1 2, where the approximation error R(f, d)of the interval I. Proposition 3.1. Under the assumptions of Lemma 2.1, then in Corollary 2.3 for every division dof [π, ν], then the following inequality holds: |R(f, d)| ≤ n−1 X i=0 (Ωi+1 −Ωi)3 96 1 2f′′′ (Ωi)+f′′′ (Ωi+1)+2 3 f′′′ Ωi+ Ωi+1 2. Proof. Applying the Corollary 2.3 on the subinterval [Ωi,Ωi+1] , (i= 0,1,2,3....n −1) and α= 1,β(0) = β(1) = 1, we have Ωi+1 −Ωi 24 f′(Ωi)−f′(Ωi+1)+1 3f′(Ωi)+f′(Ωi+1)+fΩi+ Ωi+1 2−ZΩi+1 Ωi f(Ω) dΩ (7) 2nd Kocaeli Science Congress, November 19-21, 2025 M3-17
KOSC-2025 Proceedings ≤(Ωi+1 −Ωi)3 96 1 2f′′′ (Ωi)+f′′′ (Ωi+1)+2 3 f′′′ Ωi+ Ωi+1 2. Summing over i from 0to n− 1, we deduce by the triangle inequality, we attained the result. This completes the proof. 4 Conclusion Convexity and fractional integral operators are indeed important mathematical tools used to deal with integral inequalities and related problems in mathematical analysis. This paper is to derive a new perturbed trapezoid-type inequalities by using the CaputoFabrizio fractional integral. First of all, we give an integral identity that is essential in order to establish the main finding of the article. Using the Caputo-Fabrizio fractional integral, some perturbed trapezoid-type inequalities are investigated for three times differentiable convex functions. In addition, the research has acquired fractional perturbed trapezoidtype inequalities for (α, m) -convex function. Furthermore, we have given application to the quadrature formula. In future work, it would be fascinating to apply these findings to other convexities and fractional operators for example modified Atangana Baleanu, k-Riemann-Liouville, Katugampola and Conformable. 5 Funding No Funding 6 Competing interests The authors declare that they have no competing interests. 7 Authors contributions This work was carried out in collaboration between all authors. All authors defined the research theme, read and approved the manuscript. M3-18 2nd Kocaeli Science Congress, November 19-21, 2025
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