New Results on Weddle's Type Inequalities Involving Tempered Fractional Integrals

New Results on Weddle’s Type Inequalities Involving Tempered Fractional Integrals Asia Shehzadi1, Hüseyin Budak2, Mehmet Zeki Sarikaya3, Wali Haider1 1 School of Mathematics and Statistics, Central South University, Changsha 410083, China 2Department of Mathematics, Faculty of Science and Arts, Kocaeli University, Kocaeli 41001, Türkiye 3 Department of Mathematics Faculty of Science and Arts, Düzce University Düzce 81620, Turkey Corresponding author: [email protected] ORCID IDs: First Author: 0009-0005-1101-5536 Second Author: 0000-0001-8843-955X Third Author: 0000-0002-6165-9242 Fourth Author: 0009-0001-7065-2755 DOI : 10.5281/zenodo.18011864 Abstract Integral inequalities represent a central component of modern mathematical analysis, providing essential connections between theory and various applied disciplines. This study develops new inequalities for convex functions through the use of tempered fractional integrals, employing a known integral identity as the basis of the investigation. In particular, we establish Weddle’s type inequalities for differentiable convex functions and extend the results to critical functional classes, such as Lipschitzian functions, bounded functions, and functions with bounded variation. These developments not only enhance the theoretical framework of fractional calculus but also provide potential applications in computational analysis and numerical integration. Keywords: Convexity, Weddles formula, Tempered fractional integrals. 1 Introduction and Motivation The study of convex functions has a well-established history in the field of science and continued an active area of research for over a hundred years. Formally, the definition of a convex function can be stated as: A mapping F : [ σ, υ ] →R⊂R is said to be a convex function if the given inequality is satisfied: F(tσ + (1 −t)υ)≤tF(σ)+(1−t)F(υ)(1) for all σ, υ ∈I, t ∈[0,1].Also, we say that Fis concave, if the inequality (1) is reversed. Significant research demonstrated a strong link between convexity theory and integral inequalities, highlighting their importance in differential equations and applied mathematics. The M5-1 KOSC-2025 Proceedings convexity principle is utilized by the Jensen, Gruss, Hölder, Hermite Hadamard, Simpson, EulerMaclaurin, and Milne type of inequalities to attain more accurate conclusions. Additionally, it aids us with approximation theories, differential equations, and optimization methods. Researchers examine these relationships to advance theory and enhance problem-solving in various scientific domains [29,33,31]. The Hermite–Hadamard inequality is one of the most notable of several significant inequalities derived from the characteristics of convex functions. It asserts that for any convex function defined on a closed interval, its value at the interval’s midpoints provides an upper and lower bound for its integral. More precisely, the Hermite-Hadamard inequality can be outlined as follows: If F:I→Ris a convex function on Iand σ, υ ∈Iwith σ < υ, then Fσ+υ 2≤1 υ−σZυ σ F(t)dt ≤F(σ) + F(υ) 2(2) holds. Both the inequalities in (2) hold in the reversed direction if F′is concave. For differentiable convex functions, Dragomir and Agarwal [ 6 ] have reported two new inequalities connected with inequalities (2) . Additionally, they offer some applications of special means of real numbers. Zhao et al. [ 35 ] have derived Bullen-type inequalities for differentiable convex functions by involving generalized fractional integrals. In 2010, Sarikaya et al. [ 28 ] explored several novel Simpson-type inequalities employing s -convexity and offered various applications to the special mean of real numbers. Chen and Huang extended these newly established Simpsontype inequalities by leveraging Riemann-Liouville fractional integrals [ 5 ]. Considering generalized Riemann-Liouville fractional integrals, Budak et al. [ 3 ] obtained several approximations for Simpson’s 1 / 3formula for differentiable convex functions. Toseef et al. [ 34 ] established a novel identity by generalizing Weddle’s rule for the first time. They investigated several results related to differentiable convex functions and studied different function classes. Mateen et al. established Weddle’s type equality using Riemann-Liouville fractional integrals. Also, they generalized results for the Riemann-Liouville fractional integrals reported in [ 22 ]. Readers may consult [ 7 , 10 ] and the associated references for additional information. Fractional calculus extends the concepts of integration and differentiation to non-integer orders, serving as an effective tool for modeling memory-dependent and anomalous processes. To address certain limitations of standard fractional operators, tempered fractional calculus introduces an exponential tempering parameter into the fractional kernel, which mitigates the singular and unbounded behavior of classical fractional operators. This change enables us to exert better control over long-range interactions. It offers greater freedom to replicate physical phenomena, such as anomalous diffusion and viscoelasticity, while preserving the essential features of fractional dynamics. The incorporation of fractional and tempered fractional calculus into the theory of inequalities has created novel opportunities for mathematical analysis. Recent studies [ 23 , 14 , 31 ] have demonstrated that these advanced frameworks facilitate the formulation of innovative Hermite–Hadamard, Simpson, and Weddle-type inequalities for convex functions. The mathematical preliminaries and concepts presented here will be widely employed throughout the research process. M5-2 2nd Kocaeli Science Congress, November 19-21, 2025 Definition 1.1. The terminology of the gamma, incomplete gamma function and the λ -incomplete gamma function are explained by ⋎(α) := ∞ Z 0 tα−1etdt, ⋎(α, ϖ) := ϖ Z 0 tα−1e−tdt, and ⋎λ(α, ϖ) := ϖ Z 0 tα−1e−λtdt, respectively. Here, 0<α<∞and λ≥0. Some features of the λ-incomplete gamma function are listed the following: Remark 1.1. [21] Under the conditions α > 0;x, λ ≥0and σ < υ, we achieve i. ⋎λ(υ−σ)(α, 1) = 1 R0 tα−1e−λ(υ−σ)tdt =1 (υ−σ)α⋎λ(α, υ −σ), ii. 1 R0 ⋎λ(υ−σ)(α, ϖ)dϖ =⋎λ(α,υ−σ) (υ−σ)α−⋎λ(α+1,υ−σ) (υ−σ)α+1 . Definition 1.2 (See [ 15 , 12 ]).The Riemann-Liouville fractional integral of order α > 0, is expressed as follows: Jα σ+F(ϖ) = 1 Γ(α)Zϖ σ (ϖ−t)α−1F(t)dt, ϖ > σ and Jα υ−F(ϖ) = 1 Γ(α)Zυ ϖ (t−ϖ)α−1F(t)dt, ϖ < υ. Here, Γ(α)is the well-konown Gamma function and J0 σ+F(ϖ)=J0 υ−F(ϖ)=F(ϖ). Definition 1.3 (See [ 30 , 19 ]).Let F ∈ L [ σ, υ ], where 0 ≤σ < υ . For α≥ 0and λ≥ 0, the tempered fractional integral operators Z(α,λ) σ+Fand Z(α,λ) υ−Fare expressed as Z(α,λ) σ+F(ϖ) = 1 Γ(α)Zϖ σ (ϖ−t)α−1e−λ(ϖ−t)F(t)dt, ϖ ∈[σ, υ] and Z(α,λ) υ−F(ϖ) = 1 Γ(α)Zυ ϖ (t−ϖ)α−1e−λ(t−ϖ)F(t)dt, ϖ ∈[σ, υ] respectively. By substituting λ = 0 in Definition 1.3, Definition 1.2 can be quickly produced. To go deeper into the many instances of tempered fractional integrals, it is recommended to refer to the following books: [27,20]. The groundbreaking research of Buschman [ 2 ] introduced the notion of fractional integration with weak singular and exponential kernels, serving as the foundation for tempering fractional 2nd Kocaeli Science Congress, November 19-21, 2025 M5-3 KOSC-2025 Proceedings calculus. Mohammad et al. [ 21 ] have generalized results based on Riemann and Riemann–Liouville integrals by introducing the λ -incomplete gamma function and establishing Hermite–Hadamardtype inequalities involving tempered fractional integrals. In [ 1 ], Simpson-type inequalities for convex functions have obtained using tempered fractional integral operators, based on Holder and power mean inequalities. This extended previous work employing Riemann-Liouville fractional integrals. In tempered fractional calculus, Kucche et al. [ 16 ] studied a Hilfer-type operator. They examined its major aspects, including compositional properties, mappings in function spaces, and other features of functional analysis. New integral inequalities for subadditive functions and their products have been introduced, with Kashuri et al. [ 17 ] formulating an identity and demonstrating various Hermite–Hadamard-type inequalities for subadditive functions that incorporate tempered fractional integrals. Many researchers have made significant contributions to the development of tempered fractional integral theories; refer to [ 4 , 13 , 32 ] and the cited references. Inspired by earlier research, this work utilizes tempered fractional integrals to prove several inequalities of the type of Weddle’s formula. The analysis examines important categories of functions in the context of tempered fractional calculus, encompassing convex, bounded, Lipschitzian functions, and functions with bounded variation. The proposed findings offer an expanded view on fractional inequalities and illustrate the adaptability of the tempered fractional methodology. Additionally, the results build upon and expand existing frameworks, producing the Riemann–Liouville case for λ= 0 and traditional Weddle’s inequality for α= 1. The study is divided into three sections, beginning with the introduction and preliminaries, which provide a concise summary of appropriate research in the field and provide fundamental definitions of tempered fractional calculus. In Section 2, we investigate a variety of functional classes, such as Lipschitzian, bounded, and functions with bounded variation. In Section 3, we present a summary of our findings and propose potential directions for future research. Lemma 1.1. Let F : [ σ, υ ] →R is an absolutely continuous function on the interval ( σ, υ )with F′∈L1[σ, υ], then the following holds: 1 20 F(σ) + 5Fυ+ 5σ 6+Fυ+ 2σ 3+ 6Fσ+υ 2+F2υ+σ 3+ 5F5υ+σ 6+F(υ) −Γ(α) 2⋎λ(α, υ −σ)hZ(α,λ) υ−F(σ) + Z(α,λ) σ+F(υ)i=(υ−σ)α+1 2⋎λ(α, υ −σ) 6 X i=1 Ii,(3) where                      I1=Z1 6 0⋎λ(υ−σ)(α, t)−1 20 ⋎λ(υ−σ)(α, 1)F′(tυ +(1−t)σ)− F′(tσ + (1 −t)υ)dt, I2=Z1 3 1 6⋎λ(υ−σ)(α, t)−6 20 ⋎λ(υ−σ)(α, 1)F′(tυ +(1−t)σ)− F′(tσ + (1 −t)υ)dt, I3=Z1 2 1 3⋎λ(υ−σ)(α, t)−7 20 ⋎λ(υ−σ)(α, 1)F′(tυ +(1−t)σ)− F′(tσ + (1 −t)υ)dt, M5-4 2nd Kocaeli Science Congress, November 19-21, 2025                        I4=Z2 3 1 2⋎λ(υ−σ)(α, t)−13 20 ⋎λ(υ−σ)(α, 1)F′(tυ +(1−t)σ)− F′(tσ + (1 −t)υ)dt, I5=Z5 6 2 3⋎λ(υ−σ)(α, t)−14 20 ⋎λ(υ−σ)(α, 1)F′(tυ +(1−t)σ)− F′(tσ + (1 −t)υ)dt, I6=Z1 5 6⋎λ(υ−σ)(α, t)−19 20 ⋎λ(υ−σ)(α, 1)F′(tυ + (1 −t)σ)− F′(tσ + (1 −t)υ)dt. 2 Main Results Theorem 2.1. Suppose the conditions stated in Lemma 1.1 satisfied. If there exist m, M ∈R such that m≤ F′(t)≤Mfor t∈[σ, υ], then we have the subsequent inequality  1 20 F(σ) + 5Fυ+ 5σ 6+Fυ+ 2σ 3+ 6Fσ+υ 2+F2υ+σ 3+ 5F5υ+σ 6+F(υ) −Γ(α) 2⋎λ(α, υ −σ)hZ(α,λ) υ−F(σ) + Z(α,λ) σ+F(υ)i ≤(υ−σ)α+1 2⋎λ(α, υ −σ)[K1(α, λ) + K2(α, λ) + K3(α, λ) + K4(α, λ) + K5(α, λ) + K6(α, λ)] (M−m), (4) where                                                        K1(α, λ) = Z1 6 0 ⋎λ(υ−σ)(α, t)−1 20 ⋎λ(υ−σ)(α, 1) dt, K2(α, λ) = Z1 3 1 6 ⋎λ(υ−σ)(α, t)−6 20 ⋎λ(υ−σ)(α, 1) dt, K3(α, λ) = Z1 2 1 3 ⋎λ(υ−σ)(α, t)−7 20 ⋎λ(υ−σ)(α, 1) dt, K4(α, λ) = Z2 3 1 2 ⋎λ(υ−σ)(α, t)−13 20 ⋎λ(υ−σ)(α, 1) dt, K5(α, λ) = Z5 6 2 3 ⋎λ(υ−σ)(α, t)−14 20 ⋎λ(υ−σ)(α, 1) dt, K6(α, λ) = Z1 5 6 ⋎λ(υ−σ)(α, t)−19 20 ⋎λ(υ−σ)(α, 1) dt. Proof. Considering Lemma 1.1, we can conclude 1 20 F(σ) + 5Fυ+ 5σ 6+Fυ+ 2σ 3+ 6Fσ+υ 2+F2υ+σ 3+ 5F5υ+σ 6+F(υ) −Γ(α) 2⋎λ(α, υ −σ)hZ(α,λ) υ−F(σ) + Z(α,λ) σ+F(υ)i =(υ−σ)α+1 2⋎λ(α, υ −σ)"Z1 6 0⋎λ(υ−σ)(α, t)−1 20 ⋎λ(υ−σ)(α, 1)F′(tυ + (1 −t)σ)−m+M 2dt 2nd Kocaeli Science Congress, November 19-21, 2025 M5-5 KOSC-2025 Proceedings +Z1 3 1 6⋎λ(υ−σ)(α, t)−6 20 ⋎λ(υ−σ)(α, 1)F′(tυ +(1−t)σ)−m+M 2dt +Z1 2 1 3⋎λ(υ−σ)(α, t)−7 20 ⋎λ(υ−σ)(α, 1)F′(tυ +(1−t)σ)−m+M 2dt +Z2 3 1 2⋎λ(υ−σ)(α, t)−13 20 ⋎λ(υ−σ)(α, 1)F′(tυ +(1−t)σ)−m+M 2dt +Z5 6 2 3⋎λ(υ−σ)(α, t)−14 20 ⋎λ(υ−σ)(α, 1)F′(tυ +(1−t)σ)−m+M 2dt +Z1 5 6⋎λ(υ−σ)(α, t)−19 20 ⋎λ(υ−σ)(α, 1)F′(tυ + (1 −t)σ)−m+M 2dt +Z1 6 0⋎λ(υ−σ)(α, t)−1 20 ⋎λ(υ−σ)(α, 1)m+M 2− F′(tσ + (1 −t)υ)dt +Z1 3 1 6⋎λ(υ−σ)(α, t)−6 20 ⋎λ(υ−σ)(α, 1)m+M 2− F′(tσ + (1 −t)υ)dt +Z1 2 1 3⋎λ(υ−σ)(α, t)−7 20 ⋎λ(υ−σ)(α, 1)m+M 2− F′(tσ + (1 −t)υ)dt +Z2 3 1 2⋎λ(υ−σ)(α, t)−13 20 ⋎λ(υ−σ)(α, 1)m+M 2− F′(tσ + (1 −t)υ)dt +Z5 6 2 3⋎λ(υ−σ)(α, t)−14 20 ⋎λ(υ−σ)(α, 1)m+M 2− F′(tσ + (1 −t)υ)dt +Z1 5 6⋎λ(υ−σ)(α, t)−19 20 ⋎λ(υ−σ)(α, 1)m+M 2− F′(tσ + (1 −t)υ)dt#.(5) Utilizing the modulus properties in (5), we can deduce  1 20 F(σ) + 5Fυ+ 5σ 6+Fυ+ 2σ 3+ 6Fσ+υ 2+F2υ+σ 3+ 5F5υ+σ 6+F(υ) −Γ(α) 2⋎λ(α, υ −σ)hZ(α,λ) υ−F(σ) + Z(α,λ) σ+F(υ)i ≤(υ−σ)α+1 2⋎λ(α, υ −σ)"Z1 6 0 ⋎λ(υ−σ)(α, t)−1 20 ⋎λ(υ−σ)(α, 1) F′(tυ + (1 −t)σ)−m+M 2 dt +Z1 3 1 6 ⋎λ(υ−σ)(α, t)−6 20 ⋎λ(υ−σ)(α, 1) F′(tυ + (1 −t)σ)−m+M 2 dt +Z1 2 1 3 ⋎λ(υ−σ)(α, t)−7 20 ⋎λ(υ−σ)(α, 1) F′(tυ + (1 −t)σ)−m+M 2 dt +Z2 3 1 2 ⋎λ(υ−σ)(α, t)−13 20 ⋎λ(υ−σ)(α, 1) F′(tυ + (1 −t)σ)−m+M 2 dt +Z5 6 2 3 ⋎λ(υ−σ)(α, t)−14 20 ⋎λ(υ−σ)(α, 1) F′(tυ + (1 −t)σ)−m+M 2 dt +Z1 5 6 ⋎λ(υ−σ)(α, t)−19 20 ⋎λ(υ−σ)(α, 1) F′(tυ + (1 −t)σ)−m+M 2 dt M5-6 2nd Kocaeli Science Congress, November 19-21, 2025 +Z1 6 0 ⋎λ(υ−σ)(α, t)−1 20 ⋎λ(υ−σ)(α, 1) m+M 2− F′(tσ + (1 −t)υ) dt +Z1 3 1 6 ⋎λ(υ−σ)(α, t)−6 20 ⋎λ(υ−σ)(α, 1) m+M 2− F′(tσ + (1 −t)υ) dt +Z1 2 1 3 ⋎λ(υ−σ)(α, t)−7 20 ⋎λ(υ−σ)(α, 1) m+M 2− F′(tσ + (1 −t)υ) dt +Z2 3 1 2 ⋎λ(υ−σ)(α, t)−13 20 ⋎λ(υ−σ)(α, 1) m+M 2− F′(tσ + (1 −t)υ) dt +Z5 6 2 3 ⋎λ(υ−σ)(α, t)−14 20 ⋎λ(υ−σ)(α, 1) m+M 2− F′(tσ + (1 −t)υ) dt +Z1 5 6 ⋎λ(υ−σ)(α, t)−19 20 ⋎λ(υ−σ)(α, 1) m+M 2− F′(tσ + (1 −t)υ) dt#. Assuming the given conditions m≤ F′(t)≤Mfor t∈[σ, υ], it becomes  F′(tυ + (1 −t)σ)−m+M 2 ≤M−m 2,(6) and  m+M 2− F′(tσ + (1 −t)υ) ≤M−m 2.(7) By employing inequalities (6) and (7), we gain  1 20 F(σ) + 5Fυ+ 5σ 6+Fυ+ 2σ 3+ 6Fσ+υ 2+F2υ+σ 3+ 5F5υ+σ 6+F(υ) −Γ(α) 2⋎λ(α, υ −σ)hZ(α,λ) υ−F(σ) + Z(α,λ) σ+F(υ)i ≤(υ−σ)α+1 2⋎λ(α, υ −σ)"Z1 6 0 ⋎λ(υ−σ)(α, t)−1 20 ⋎λ(υ−σ)(α, 1) dt +Z1 3 1 6 ⋎λ(υ−σ)(α, t)−6 20 ⋎λ(υ−σ)(α, 1) dt +Z1 2 1 3 ⋎λ(υ−σ)(α, t)−7 20 ⋎λ(υ−σ)(α, 1) dt +Z2 3 1 2 ⋎λ(υ−σ)(α, t)−13 20 ⋎λ(υ−σ)(α, 1) dt +Z5 6 2 3 ⋎λ(υ−σ)(α, t)−14 20 ⋎λ(υ−σ)(α, 1) dt +Z1 5 6 ⋎λ(υ−σ)(α, t)−19 20 ⋎λ(υ−σ)(α, 1) dt#(M−m) =(υ−σ)α+1 2⋎λ(α, υ −σ)[K1(α, λ) + K2(α, λ) + K3(α, λ) + K4(α, λ) + K5(α, λ) + K6(α, λ)] (M−m). Hence, the proof is complete. Remark 2.1. Let us consider λ= 0 in Theorem 2.1, we obtain  1 20 F(σ) + 5Fυ+ 5σ 6+Fυ+ 2σ 3+ 6Fσ+υ 2+F2υ+σ 3+ 5F5υ+σ 6+F(υ) −Γ(α+ 1) 2(υ−σ)αJα υ−F(σ) + Jα σ+F(υ) 2nd Kocaeli Science Congress, November 19-21, 2025 M5-7 KOSC-2025 Proceedings ≤α(υ−σ) 2[K1(α, 0) + K2(α, 0) + K3(α, 0) + K4(α, 0) + K5(α, 0) + K6(α, 0)] (M−m), which is reported in [22]. Remark 2.2. If we assign λ= 0 and α= 1 in Theorem 2.1, then we get  1 20 F(σ) + 5Fυ+ 5σ 6+Fυ+ 2σ 3+ 6Fσ+υ 2+F2υ+σ 3+ 5F5υ+σ 6+F(υ) −1 υ−σZυ σ F(t)dt ≤13(υ−σ) 450 |F′(σ)|+|F′(υ)|(M−m), which is defined in [22, Theorem 5]. Corollary 2.1. Suppose the conditions stated in Theorem 2.1, for any t within the range [ σ, υ ], assuming the existence of a positive constant Msatisfying |F′(t)|≤M, it gives  1 20 F(σ) + 5Fυ+ 5σ 6+Fυ+ 2σ 3+ 6Fσ+υ 2+F2υ+σ 3+ 5F5υ+σ 6+F(υ) −Γ(α) 2⋎λ(α, υ −σ)hZ(α,λ) υ−F(σ) + Z(α,λ) σ+F(υ)i ≤(υ−σ)α+1 ⋎λ(α, υ −σ)[K1(α, λ) + K2(α, λ) + K3(α, λ) + K4(α, λ) + K5(α, λ) + K6(α, λ)] M. Corollary 2.2. If we consider λ= 0,in Corollary 2.1, then we obtain  1 20 F(σ) + 5Fυ+ 5σ 6+Fυ+ 2σ 3+ 6Fσ+υ 2+F2υ+σ 3+ 5F5υ+σ 6+F(υ) −Γ(α+ 1) 2(υ−σ)αJα υ−F(σ) + Jα σ+F(υ) ≤α(υ−σ) [K1(α, 0) + K2(α, 0) + K3(α, 0) + K4(α, 0) + K5(α, 0) + K6(α, 0)] M. Corollary 2.3. Let λ= 0 and α= 1 in Corollary 2.1. Then, it yields  1 20 F(σ) + 5Fυ+ 5σ 6+Fυ+ 2σ 3+ 6Fσ+υ 2+F2υ+σ 3+ 5F5υ+σ 6+F(υ) −1 υ−σZυ σ F(t)dt ≤13(υ−σ) 225 M. Theorem 2.2. Suppose the conditions stated in Lemma 1.1 hold. If F′ is an Lipschitzian functions on the interval [σ, υ], then we have the subsequent inequality  1 20 F(σ) + 5Fυ+ 5σ 6+Fυ+ 2σ 3+ 6Fσ+υ 2+F2υ+σ 3+ 5F5υ+σ 6+F(υ) −Γ(α) 2⋎λ(α, υ −σ)hZ(α,λ) υ−F(σ) + Z(α,λ) σ+F(υ)i ≤(υ−σ)α+2 2⋎λ(α, υ −σ)[K1(α, λ) + K2(α, λ) + K3(α, λ)−K4(α, λ)−K5(α, λ)−K6(α, λ) −2 (K7(α, λ) + K8(α, λ) + K9(α, λ)) + 2 (K10(α, λ) + K11(α, λ) + K12(α, λ))] L.(8) M5-8 2nd Kocaeli Science Congress, November 19-21, 2025 Here, K1(α, λ)−K12(α, λ)are defined as in Theorem 2.1 and                                                        K7(α, λ) = Z1 6 0 t ⋎λ(υ−σ)(α, t)−1 20 ⋎λ(υ−σ)(α, 1) dt, K8(α, λ) = Z1 3 1 6 t ⋎λ(υ−σ)(α, t)−6 20 ⋎λ(υ−σ)(α, 1) dt, K9(α, λ) = Z1 2 1 3 t ⋎λ(υ−σ)(α, t)−7 20 ⋎λ(υ−σ)(α, 1) dt, K10(α, λ) = Z2 3 1 2 t ⋎λ(υ−σ)(α, t)−13 20 ⋎λ(υ−σ)(α, 1) dt, K11(α, λ) = Z5 6 2 3 t ⋎λ(υ−σ)(α, t)−14 20 ⋎λ(υ−σ)(α, 1) dt, K12(α, λ) = Z1 5 6 t ⋎λ(υ−σ)(α, t)−19 20 ⋎λ(υ−σ)(α, 1) dt. Proof. Using the modulus in Lemma 1.1, it follows that  1 20 F(σ) + 5Fυ+ 5σ 6+Fυ+ 2σ 3+ 6Fσ+υ 2+F2υ+σ 3+ 5F5υ+σ 6+F(υ) −Γ(α) 2⋎λ(α, υ −σ)hZ(α,λ) υ−F(σ) + Z(α,λ) σ+F(υ)i ≤(υ−σ)α+1 2⋎λ(α, υ −σ)"Z1 6 0 ⋎λ(υ−σ)(α, t)−1 20 ⋎λ(υ−σ)(α, 1)F′(tυ + (1 −t)σ)− F′(tσ + (1 −t)υ)dt +Z1 3 1 6 ⋎λ(υ−σ)(α, t)−6 20 ⋎λ(υ−σ)(α, 1)F′(tυ + (1 −t)σ)− F′(tσ + (1 −t)υ)dt +Z1 2 1 3 ⋎λ(υ−σ)(α, t)−7 20 ⋎λ(υ−σ)(α, 1)F′(tυ + (1 −t)σ)− F′(tσ + (1 −t)υ)dt +Z2 3 1 2 ⋎λ(υ−σ)(α, t)−13 20 ⋎λ(υ−σ)(α, 1)F′(tυ + (1 −t)σ)− F′(tσ + (1 −t)υ)dt +Z5 6 2 3 ⋎λ(υ−σ)(α, t)−14 20 ⋎λ(υ−σ)(α, 1)F′(tυ + (1 −t)σ)− F′(tσ + (1 −t)υ)dt +Z1 5 6 ⋎λ(υ−σ)(α, t)−19 20 ⋎λ(υ−σ)(α, 1)F′(tυ + (1 −t)σ)− F′(tσ + (1 −t)υ)dt#. Given that F′is L-Lipschitzian function, we acquire  1 20 F(σ) + 5Fυ+ 5σ 6+Fυ+ 2σ 3+ 6Fσ+υ 2+F2υ+σ 3+ 5F5υ+σ 6+F(υ) −Γ(α) 2⋎λ(α, υ −σ)hZ(α,λ) υ−F(σ) + Z(α,λ) σ+F(υ)i ≤(υ−σ)α+1 2⋎λ(α, υ −σ)" Z1 6 0 ⋎λ(υ−σ)(α, t)−1 20 ⋎λ(υ−σ)(α, 1) dt +Z1 3 1 6 ⋎λ(υ−σ)(α, t)−6 20 ⋎λ(υ−σ)(α, 1) dt +Z1 2 1 3 ⋎λ(υ−σ)(α, t)−7 20 ⋎λ(υ−σ)(α, 1) dt! 2nd Kocaeli Science Congress, November 19-21, 2025 M5-9 KOSC-2025 Proceedings Informed consent statement: N/A. Data availability statement: N/A. Competing interest: The authors declare no conflicts of interest References [1] Almoneef, A. A., Hyder, A. A., Hezenci, F., & Budak, H. (2023). Simpson–type inequalities by means of tempered fractional integrals. Aims Math, 8, 29411-29423. [2] Buschman, R. G. (1972). Decomposition of an integral operator by use of Mikusiński calculus. SIAM Journal on Mathematical Analysis. 3(1), 83-85. [3] Budak, H., Hezenci, F., & Kara, H. (2021). 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