Novel Results on Fractional Weddle's Type Inequalities for Several Functional Classes via Tempered Fractional Integrals

Novel Results on Fractional Weddle’s Type Inequalities for Several Functional Classes via Tempered Fractional Integrals Bilal Ahmad1, Hüseyin Budak2, Wali Haider3, Asia Shehzadi4, Asra Ahmad5 1School of Mathematics and Computational Science, Xiangtan University, Xiangtan, 411105, China 2Department of Mathematics, Kocaeli University, Kocaeli 41001, Türkiye 3School of Mathematics and Statistics, Shaanxi Normal University, Xi’an, Shaanxi, 710119, China 4 School of Mathematics and Statistics, Central South University, Changsha 410083, China 5 Department of Mathematics and Statistics,University of Agriculture Faisalabad, Pakistan Corresponding author: asrac[email protected] ORCID IDs: First Author: 0009-0002-8603-0421 Second Author: 0000-0001-8843-955X Third Author: 0009-0001-7065-2755 Fourth Author: 0009-0005-1101-5536 Fifth Author: 0009-0008-7864-3635 DOI : 10.5281/zenodo.18012139 Abstract The theory of inequalities forms the backbone of analysis by refining essential estimates between functions and their means. Weddle’s type inequalities yield sharper error estimates and enhance the precision of the accuracy of integral approximation techniques. It serves as a cornerstone for improvements in mathematical modeling and integral calculus. In this investigation, we derive unique inequalities for convex functions employing tempered fractional integrals, concentrating on Weddle’s type inequalities. A significant equality is formulated within the framework of tempered fractional integrals, and this provides the cornerstone for our study. Based on this equality, we develop various Weddle-type inequalities for differentiable convex mappings connected with tempered fractional integrals. These mappings extend the applicability of fractional calculus by extending classical notions and presenting further insight into the behavior of convex functions. We attain distinctive types of Weddle’s type inequalities by involving key mathematical tools such as Lipschitzian, bounded functions, and functions with bounded variation, enhancing the implementation of these inequalities in diverse mathematical domains. Keywords: Weddle’s formula type inequality, Lipschitzian functions, bounded functions, bounded variation, tempered fractional integrals. M10-1 KOSC-2025 Proceedings 1 Introduction and Motivation A key concept in mathematics, especially in calculus, optimization and mathematical analysis is a convexity.A function F( ξ )defined on [Θ , ρ ]is convex if it satisfies the following inequality for any pair of points (Θ,F(Θ)) and (ρ, F(ρ)) lying within the interval: F(ξΘ + (1 −ξ)ρ)≤ξF(Θ) + (1 −ξ)F(ρ),(1) for every ξ value between 0and 1. Numerous fields, such as economics, biology, and optimization, depend on the convexity of functions and their extended versions. The idea of classical convexity has attracted a lot of attention lately. To learn more about convex functions, see [ 13 , 27 ] and its references. Because it creates constraints and provides crucial insights into how functions behave in various domains, inequality theory is key to mathematical analysis. Inequalities are important in the context of optimization because they provide an environment that is conducive to ideal solutions. Inequality theory also has a significant impact on numerical techniques, leading to novel error estimations and more accurate integral approximations. It improves the precision and consistency of models and algorithms in a number of fields, such as computer science, engineering, and economics. Recently, there has been a lot of attention in classical inequalities concerning integral operators connected with different convexities. Jensen, Gruss, Hölder, Hermite–Hadamard, Simpson, Euler–Maclaurin, and Milne-type inequalities [18,2] are some significant inequalities. Among the significant mathematical inequalities involving convex maps are the Hermite– Hadamard type inequalities, which have the following mathematical definition. When a function F: [Θ, ρ]→Ris convex, we obtain FΘ+ρ 2≤1 ρ−ΘZρ Θ F(ϖ)dϖ ≤F(Θ) + F(ρ) 2.(2) Numerous mathematical fields, including analysis, optimization theory, and economics, have numerous applications incorporating inequality (2) . A new version of the Hermite–Hadamard– Mercer inequalities for harmonically convex functions has been provided by You et al. [ 32 ], who also looked at particular ways to solve recently discovered inequalities. Abbas et al. [ 1 ] discovered Hermite–Hadamard type inequalities for exponentially subadditive functions by utilizing Riemann–Liouville fractional integrals. Hwang et al. [ ? ] demonstrated particular applications in the Beta function and discovered numerous improvements and related generalizations of the Hermite–Hadamard inequality for fractional integrals. By introducing derivatives and integrals of fractional/non-integer orders, fractional calculus (FC), first presented in 1695 through the correspondence between Leibniz and Marquis de l’Hospital, generalizes classical calculus. The theory was subsequently developed by mathematicians such as Euler, Liouville, and Riemann, and it gained recognition as a critical instrument in both theoretical and applied mathematics during the 19th century. The non-local aspect of FC sets it apart and is especially useful for describing memory and inherited effects in materials and systems. In many domains, including geometry, dynamics, and fractional differential equations, FC has grown in importance as a tool for modeling non-standard events [ 26 , 15 , 12 ]. These days, it is employed in fields including viscoelasticity, M10-2 2nd Kocaeli Science Congress, November 19-21, 2025 bioengineering, control theory, and acoustics [ 3 ].Its position in science and engineering has been cemented by foundational publications like those by Samko, Kilbas, and Marichev (1993) and Podlubny (1999), which enable the study of the intricate and unpredictable ways that natural systems function. The following is a list of the well-known Riemann-Liouville (RL) fractional integrals: Definition 1.1 (See [ 19 , 14 ]).Presume F ∈L1 [Θ , ρ ] , the RL fractional integral of order α > 0, expressed as: Jα Θ+F(κ) = 1 Γ(α)Zκ Θ (κ−κ)α−1F(κ)dκ, κ>Θ and Jα ρ−F(κ) = 1 Γ(α)Zρ κ (κ−κ)α−1F(κ)dκ, κ< ρ. Definition 1.2. The Gamma function, as described in [ 9 ], is described using the subsequent integral expression. Γ(χ) = Z∞ 0 e−κκχ−1dκ, Re(χ)>0. For a more detailed description of RL fractional integrals, see references [24,4,10]. However, in practical mathematics and differential equation analysis, fractional integral inequalities are crucial. Since ancient times, mathematical inequalities—which compare two numbers or expressions to show their relationships or relative magnitudes—have been an essential concept in mathematics. Convex functions and the theory of inequalities are closely related; several famous inequalities, such as Hermite–Hadamard, Simpson, Newton, Euler-Maclaurin, and Boole’s type inequalities, were developed for convex functions. Convexity serves as a reliable tool for studying inequalities, providing a basis for using geometric and analytical techniques to confirm these conclusions.Convex function analysis and inequality analysis are closely related, with each impacting the development of the other. kindly see [ 16 , 5 , 29 ] and the sources listed therein for more details on convexity and Newton-type inequalities associated with convex differentiable functions. Buschman [ 6 ] was the first to investigate the tempered fractional integral in this field. Liu et al. [ 20 ] and Meerschaert et al. [ 23 ] made more thorough contributions after this. Tempered fractional calculus, a significant subfield of fractional calculus, identifies a unique intersection between fractional calculus with analytic kernels and weighted fractional calculus [ 11 ]. Given its broad use and significance, it may be viewed as an extension of fractional calculus. Its rediscovery under different names, such as significant fractional calculus [ 11 , 7 ] and generalized proportional fractional calculus [17], highlights its importance. The important preliminary steps that serve as the foundation for our main conclusions are reviewed in the next section. Definition 1.3 (See [ 25 ]).For real values α > 0and κ, λ ≥ 0, the λ -incomplete gamma function 2nd Kocaeli Science Congress, November 19-21, 2025 M10-3 KOSC-2025 Proceedings is identified as: Υλ(α, κ) := Zκ 0 κα−1e−λκdκ. λ= 1 corresponds to the incomplete gamma function [8]: Υ(α, κ) := Zκ 0 κα−1e−κdκ. Where, 0<α<∞and λ≥0. Remark 1.1 (See [25]).For the real values α > 0and κ, λ ≥0, we gain I. Υλ(ρ−Θ)(α, 1) = R1 0κα−1e−λ(ρ−Θ)κdκ =1 (ρ−Θ)αΥλ(α, ρ −Θ). II. R1 0Υλ(ρ−Θ)(α, κ)dκ=Υλ(α,ρ−Θ) (ρ−Θ)α−Υλ(α+1,ρ−Θ) (ρ−Θ)α+1 . Definition 1.4 (See [ 21 , 23 ]).Tempered fractional integral operators of order α > 0and λ≥ 0 are presented as: Z(α,λ) Θ+ F(κ) = 1 Γ(α)Zκ Θ (κ−κ)α−1e−λ(κ−κ)F(κ)dκ, κ∈[Θ, ρ] and Z(α,λ) ρ−F(κ) = 1 Γ(α)Zρ κ (κ−κ)α−1e−λ(κ−κ)F(κ)dκ, κ∈[Θ, ρ], respectively for F∈L1[Θ, ρ]. The RL fractional integral equations in Definition 1.1 are obviously comparable if we assume λ= 0 in Definition 1.4. A number of examples of integral inequality for tempered fractional integrals from earlier works were presented by Gul and Yalcin [ 14 ], who developed Minkowski and Hermite–Hadamard integral inequalities using tempered fractional integral operators. Interested readers are referred to [26,28,30] for more details. Motivated by earlier studies, we identified some of Weddle’s formula-type inequalities using tempered fractional integrals. The main types of function classes in tempered fractional integrals—bounded, Lipschitzian, and functions with bounded variation—are examined in this study. The most significant benefit of these inequalities is that they became Riemann–Liouville fractional Weddle’s type inequalities when λ = 0. The modified inequalities become traditional Weddle’s type inequalities when α= 1. Section 1.1 will establish some conclusions regarding Wwddle’s type inequalities for differentiable convex functions. We also evaluate various functional classes, such as Lipschitzian, bounded, and functions with bounded variation. Finally, we summarize our findings and research opportunities for future work. The study is organized into three sections, starting with an introduction and preliminaries that include basic definitions of fractional calculus and a brief summary of important studies in the field. M10-4 2nd Kocaeli Science Congress, November 19-21, 2025 1.1 Main Contributions 1.1 Main Contributions Lemma 1.1. If F: [Θ , ρ ] →R be a function that is absolutely continuous on (Θ , ρ )such that F′∈L1[Θ, ρ].Then, the following equality is valid: 1 20 F(Θ) + 5F5Θ+ρ 6+F2Θ+ρ 3+ 6FΘ+ρ 2+FΘ+2ρ 3+ 5FΘ+5ρ 6+F(ρ) −Γ(α) 2⋎λα, ρ−Θ 2"Z(α,λ) (Θ+ρ 2)+F(ρ)+Z(α,λ) (Θ+ρ 2)−F(Θ)#=(ρ−Θ)α+1 2α+2 ⋎λα, ρ−Θ 2 ×"Z1 3 0⋎λ(ρ−Θ 2)(α, ξ)−1 10 ⋎λ(ρ−Θ 2)(α, 1)F′ξ 2ρ+2−ξ 2Θ−F′ξ 2Θ + 2−ξ 2ρdξ +Z2 3 1 3⋎λ(ρ−Θ 2)(α, ξ)−3 5⋎λ(ρ−Θ 2)(α, 1)F′ξ 2ρ+2−ξ 2Θ−F′ξ 2Θ + 2−ξ 2ρdξ +Z1 2 3⋎λ(ρ−Θ 2)(α, ξ)−7 10 ⋎λ(ρ−Θ 2)(α, 1)F′ξ 2ρ+2−ξ 2Θ−F′ξ 2Θ + 2−ξ 2ρdξ#. (3) Proof. By acknowledging the essential rules of integration, it is enough to state that I1=Z1 3 0⋎λ(ρ−Θ 2)(α, ξ)−1 10 ⋎λ(ρ−Θ 2)(α, 1)F′ξ 2ρ+2−ξ 2Θdξ (4) =2 ρ−Θ⋎λ(ρ−Θ 2)(α, ξ)−1 10 ⋎λ(ρ−Θ 2)(α, 1)Fξ 2ρ+2−ξ 2Θ 1 3 0 −2 ρ−ΘZ1 3 0 ξα−1e−λ(ρ−Θ 2)ξFξ 2ρ+2−ξ 2Θdξ =2 ρ−Θ⋎λ(ρ−Θ 2)α, 1 3−1 10 ⋎λ(ρ−Θ 2)(α, 1)F5Θ+ρ 6+ ⋎λ(ρ−Θ 2)(α, 1) 5(ρ−Θ) F(Θ) −2 ρ−ΘZ1 3 0 ξα−1e−λ(ρ−Θ 2)ξFξ 2ρ+2−ξ 2Θdξ, I2=2 ρ−Θ⋎λ(ρ−Θ 2)α, 2 3−3 5⋎λ(ρ−Θ 2)(α, 1)F2Θ+ρ 3(5) −⋎λ(ρ−Θ 2)α, 1 3−3 5⋎λ(ρ−Θ 2)(α, 1)F5Θ+ρ 6 −2 ρ−ΘZ2 3 1 3 ξα−1e−λ(ρ−Θ 2)ξFξ 2ρ+2−ξ 2Θdξ. In the same way, we find I3=2 ρ−Θ⋎λ(ρ−Θ 2)(α, 1) −7 10 ⋎λ(ρ−Θ 2)(α, 1)FΘ+ρ 2 −⋎λ(ρ−Θ 2)α, 2 3−7 10 ⋎λ(ρ−Θ 2)(α, 1)F2Θ+ρ 3 −2 ρ−ΘZ1 2 3 ξα−1e−λ(ρ−Θ 2)ξFξ 2ρ+2−ξ 2Θdξ. (6) 2nd Kocaeli Science Congress, November 19-21, 2025 M10-5 KOSC-2025 Proceedings Consequently, we arrive at the subsequent equality by merging (4)-(6) I1+I2+I3=2 10(ρ−Θ) ⋎λ(ρ−Θ 2)(α, 1)F(Θ) + 5 ⋎λ(ρ−Θ 2)(α, 1)F5Θ+ρ 6 +⋎λ(ρ−Θ 2)(α, 1)F2Θ+ρ 3+3⋎λ(ρ−Θ 2)(α, 1)FΘ+ρ 2 −2 ρ−ΘZ1 0 ξα−1e−λ(ρ−Θ 2)ξFξ 2ρ+2−ξ 2Θdξ.(7) Similarly I4+I5+I6=−2 10(ρ−Θ) ⋎λ(ρ−Θ 2)(α, 1)F(ρ) + ⋎λ(ρ−Θ 2)(α, 1)FΘ+2ρ 3 +5 ⋎λ(ρ−Θ 2)(α, 1)FΘ+5ρ 6+3⋎λ(ρ−Θ 2)(α, 1)FΘ+ρ 2 +2 ρ−ΘZ1 0 ξα−1e−λ(ρ−Θ 2)ξFξ 2Θ + 2−ξ 2ρdξ.(8) By subtracting equalities (7) and (8) , then we make the substitutions ϖ = ξ 2ρ + 2−ξ 2 Θand ϖ=ξ 2Θ + 2−ξ 2ρfor ξ∈[0,1], it becomes [I1+I2+I3−(I4+I5+I6)] = ⋎λ(ρ−Θ 2)(α, 1) 5(ρ−Θ) F(Θ) + 5F5Θ+ρ 6+F2Θ+ρ 3+ 6FΘ+ρ 2+FΘ+2ρ 3 +5FΘ+5ρ 6+F(ρ)−2α+1Γ(α) (ρ−Θ)α+1 "Z(α,λ) (Θ+ρ 2)+F(ρ) + Z(α,λ) (Θ+ρ 2)−F(Θ)#.(9) Multiplying both sides of (9) by ρ−Θ 4⋎ λ(ρ−Θ 2)(α,1) , the equality (3) is obtained. 1.2 Weddle’s type Inequality for Bounded and Lipschitzian Functions Through Fractional Integral This segment focuses on presenting fractional Weddle’s-type inequalities specifically for bounded and Lipschitzian functions. Theorem 1.1. If all conditions of Lemma 1.1 are accomplished. If there exist m, M ∈R such that m≤F′(ξ)≤Mfor ξ∈[Θ, ρ],then we have the following Wedlde’s type inequality:  1 20 F(Θ) + 5F5Θ+ρ 6+F2Θ+ρ 3+ 6FΘ+ρ 2+FΘ+2ρ 3+ 5FΘ+5ρ 6+F(ρ) −Γ(α) 2⋎λα, ρ−Θ 2"Z(α,λ) (Θ+ρ 2)+F(ρ) + Z(α,λ) (Θ+ρ 2)−F(Θ)# ≤(ρ−Θ)α+1 2α+2 ⋎λα, ρ−Θ 2[∆1(α, λ) + ∆2(α, λ) + ∆3(α, λ)](M−m),(10) M10-6 2nd Kocaeli Science Congress, November 19-21, 2025 1.2 Weddle’s type Inequality for Bounded and Lipschitzian Functions Through Fractional Integral where ∆1(α, λ) = Z1 3 0 ⋎λ(ρ−Θ 2)(α, ξ)−1 10 ⋎λ(ρ−Θ 2)(α, 1) dξ, ∆2(α, λ) = Z2 3 1 3 ⋎λ(ρ−Θ 2)(α, ξ)−3 5⋎λ(ρ−Θ 2)(α, 1) dξ, ∆3(α, λ) = Z1 2 3 ⋎λ(ρ−Θ 2)(α, ξ)−7 10 ⋎λ(ρ−Θ 2)(α, 1) dξ. Proof. With the assistance of Lemma 1.1, we establish  1 20 F(Θ) + 5F5Θ+ρ 6+F2Θ+ρ 3+ 6FΘ+ρ 2+FΘ+2ρ 3+ 5FΘ+5ρ 6+F(ρ) −Γ(α) 2⋎λα, ρ−Θ 2"Z(α,λ) (Θ+ρ 2)+F(ρ) + Z(α,λ) (Θ+ρ 2)−F(Θ)# ≤(ρ−Θ)α+1 2α+2 ⋎λα, ρ−Θ 2"Z1 3 0⋎λ(ρ−Θ 2)(α, ξ)−1 10 ⋎λ(ρ−Θ 2)(α, 1)F′ξ 2ρ+2−ξ 2Θ−m+M 2dξ +Z2 3 1 3⋎λ(ρ−Θ 2)(α, ξ)−3 5⋎λ(ρ−Θ 2)(α, 1)F′ξ 2ρ+2−ξ 2Θ−m+M 2dξ +Z1 2 3⋎λ(ρ−Θ 2)(α, ξ)−7 10 ⋎λ(ρ−Θ 2)(α, 1)F′ξ 2ρ+2−ξ 2Θ−m+M 2dξ +Z1 3 0⋎λ(ρ−Θ 2)(α, ξ)−1 10 ⋎λ(ρ−Θ 2)(α, 1)m+M 2−F′ξ 2Θ + 2−ξ 2ρdξ +Z2 3 1 3⋎λ(ρ−Θ 2)(α, ξ)−3 5⋎λ(ρ−Θ 2)(α, 1)m+M 2−F′ξ 2Θ + 2−ξ 2ρdξ +Z1 2 3⋎λ(ρ−Θ 2)(α, ξ)−7 10 ⋎λ(ρ−Θ 2)(α, 1)m+M 2−F′ξ 2Θ + 2−ξ 2ρdξ#.(11) Through the use of modulus properties in (11), we achieve  1 20 F(Θ) + 5F5Θ+ρ 6+F2Θ+ρ 3+ 6FΘ+ρ 2+FΘ+2ρ 3+ 5FΘ+5ρ 6+F(ρ) −Γ(α) 2⋎λα, ρ−Θ 2"Z(α,λ) (Θ+ρ 2)+F(ρ) + Z(α,λ) (Θ+ρ 2)−F(Θ)# ≤(ρ−Θ)α+1 2α+2 ⋎λα, ρ−Θ 2"Z1 3 0 ⋎λ(ρ−Θ 2)(α, ξ)−1 10 ⋎λ(ρ−Θ 2)(α, 1) F′ξ 2ρ+2−ξ 2Θ−m+M 2 dξ +Z2 3 1 3 ⋎λ(ρ−Θ 2)(α, ξ)−3 5⋎λ(ρ−Θ 2)(α, 1) F′ξ 2ρ+2−ξ 2Θ−m+M 2 dξ +Z1 2 3 ⋎λ(ρ−Θ 2)(α, ξ)−7 10 ⋎λ(ρ−Θ 2)(α, 1) F′ξ 2ρ+2−ξ 2Θ−m+M 2 dξ 2nd Kocaeli Science Congress, November 19-21, 2025 M10-7 KOSC-2025 Proceedings +Z1 3 0 ⋎λ(ρ−Θ 2)(α, ξ)−1 10 ⋎λ(ρ−Θ 2)(α, 1) m+M 2−F′ξ 2Θ + 2−ξ 2ρ dξ +Z2 3 1 3 ⋎λ(ρ−Θ 2)(α, ξ)−3 5⋎λ(ρ−Θ 2)(α, 1) m+M 2−F′ξ 2Θ + 2−ξ 2ρ dξ +Z1 2 3 ⋎λ(ρ−Θ 2)(α, ξ)−7 10 ⋎λ(ρ−Θ 2)(α, 1) m+M 2−F′ξ 2Θ + 2−ξ 2ρ dξ#. From the statements m≤F′(ξ)≤Mfor ξ∈[Θ, ρ],we get  F′ξ 2ρ+2−ξ 2Θ−m+M 2 ≤M−m 2,(12) and  m+M 2−F′ξ 2Θ + 2−ξ 2ρ ≤M−m 2.(13) Through the application of inequalities (12) and (13), we reach  1 20 F(Θ) + 5F5Θ+ρ 6+F2Θ+ρ 3+ 6FΘ+ρ 2+FΘ+2ρ 3+ 5FΘ+5ρ 6+F(ρ) −Γ(α) 2⋎λα, ρ−Θ 2"Z(α,λ) (Θ+ρ 2)+F(ρ) + Z(α,λ) (Θ+ρ 2)−F(Θ)# ≤(ρ−Θ)α+1 2α+2 ⋎λα, ρ−Θ 2"Z1 3 0 ⋎λ(ρ−Θ 2)(α, ξ)−1 10 ⋎λ(ρ−Θ 2)(α, 1) dξ +Z2 3 1 3 ⋎λ(ρ−Θ 2)(α, ξ)−3 5⋎λ(ρ−Θ 2)(α, 1) dξ +Z1 2 3 ⋎λ(ρ−Θ 2)(α, ξ)−7 10 ⋎λ(ρ−Θ 2)(α, 1) dξ#(M−m) ≤(ρ−Θ)α+1 2α+2 ⋎λα, ρ−Θ 2[∆1(α, λ) + ∆2(α, λ) + ∆3(α, λ)](M−m). This proof has been finalized. Remark 1.2. If λ= 0 in Theorem 1.1, we conclude  1 20 F(Θ) + 5F5Θ+ρ 6+F2Θ+ρ 3+ 6FΘ+ρ 2+FΘ+2ρ 3+ 5FΘ+5ρ 6+F(ρ) −2α−1Γ(α+ 1) (ρ−Θ)αJα (Θ+ρ 2)+F(ρ)+Jα (Θ+ρ 2)−F(Θ) ≤α(ρ−Θ) 4(∆1(α, 0) + ∆2(α, 0) + ∆3(α, 0)) (M−m), which is derived in [22]. Remark 1.3. If we assume λ= 0 and α= 1 in Theorem 1.1, then we gain  1 20 F(Θ) + 5F5Θ+ρ 6+F2Θ+ρ 3+ 6FΘ+ρ 2+FΘ+2ρ 3+ 5FΘ+5ρ 6+F(ρ) M10-8 2nd Kocaeli Science Congress, November 19-21, 2025 1.2 Weddle’s type Inequality for Bounded and Lipschitzian Functions Through Fractional Integral −1 ρ−ΘZρ Θ F(ϖ)dϖ ≤13(ρ−Θ) 450 (M−m), which is established in [31]. Corollary 1.1. Under conditions of Theorem 1.1, if there exist M∈R+ such that | F ′ ( ξ ) |≤M for all ξ∈[Θ, ρ],then we attain  1 20 F(Θ) + 5F5Θ+ρ 6+F2Θ+ρ 3+ 6FΘ+ρ 2+FΘ+2ρ 3+ 5FΘ+5ρ 6+F(ρ) −Γ(α) 2⋎λα, ρ−Θ 2"Z(α,λ) (Θ+ρ 2)+F(ρ) + Z(α,λ) (Θ+ρ 2)−F(Θ)# ≤(ρ−Θ)α+1 2α+1 ⋎λα, ρ−Θ 2[∆1(α, λ) + ∆2(α, λ) + ∆3(α, λ)]M. Corollary 1.2. If λ= 0 in Theorem 1.1, we conclude  1 20 F(Θ) + 5F5Θ+ρ 6+F2Θ+ρ 3+ 6FΘ+ρ 2+FΘ+2ρ 3+ 5FΘ+5ρ 6+F(ρ) −2α−1Γ(α+ 1) (ρ−Θ)αJα (Θ+ρ 2)+F(ρ)+Jα (Θ+ρ 2)−F(Θ) ≤α(ρ−Θ) 4[∆1(α, 0) + ∆2(α, 0) + ∆3(α, 0)]M. Corollary 1.3. If we choose λ= 0 and α= 1 in Corollary 1.1, then we get inequality  1 20 F(Θ) + 5F5Θ+ρ 6+F2Θ+ρ 3+ 6FΘ+ρ 2+FΘ+2ρ 3+ 5FΘ+5ρ 6+F(ρ) −1 ρ−ΘZρ Θ F(ϖ)dϖ ≤13(ρ−Θ) 225 M. Theorem 1.2. If all conditions of Lemma 1.1 are accomplished. If F ′ is a L -Lipschitzian on [Θ, ρ],then we have the subsequent inequality:  1 20 F(Θ) + 5F5Θ+ρ 6+F2Θ+ρ 3+ 6FΘ+ρ 2+FΘ+2ρ 3+ 5FΘ+5ρ 6+F(ρ) −Γ(α) 2⋎λα, ρ−Θ 2"Z(α,λ) (Θ+ρ 2)+F(ρ)+Z(α,λ) (Θ+ρ 2)−F(Θ)# ≤(ρ−Θ)α+2 2α+2 ⋎λα, ρ−Θ 2[ψ1(α, λ) + ψ2(α, λ) + ψ3(α, λ)−[ψ4(α, λ) + ψ5(α, λ) + ψ6(α, λ)]]L, (14) where ψ1(α, λ) = Z1 3 0 ⋎λ(ρ−Θ 2)(α, ξ)−1 10 ⋎λ(ρ−Θ 2)(α, 1) |(1 −ξ)dξ, ψ2(α, λ) = Z2 3 1 3 ⋎λ(ρ−Θ 2)(α, ξ)−3 5⋎λ(ρ−Θ 2)(α, 1) |(1 −ξ)dξ, 2nd Kocaeli Science Congress, November 19-21, 2025 M10-9 KOSC-2025 Proceedings which is demonstrated by Mateen et al. in [22, Corrollary 7]. 1.4 Future Research Directions In this work, we have established several new Weddle-type inequalities within the framework of tempered fractional integrals for differentiable convex functions. By formulating a fundamental equality associated with tempered fractional operators, we developed a unified approach that serves as the basis for deriving these refined inequalities. 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