EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6556 ISSN 1307-5543 – ejpam.com Published by New York Business Global Novel Fractional Hermite–Hadamard and Product-Type Inequalities via Raina Function and Preinvex Mappings with Entropy Applications Muhammad Tariq1,7, Waqar Afzal2, Muhammad Nadeem3, Angel E. Muñoz-Zavala4, Jorge E. Maćıas-Dı́az5,6,∗ 1 Mathematics Research Center, Near East University, Near East Boulevard, PC: 99138, Nicosia /Mersin 10, Turkey 2 Abdus Salam School of Mathematical Sciences, Government College University, 68-B, New Muslim Town, Lahore 54600, Pakistan 3 Department of Mathematics, Virtual University of Pakistan, Multan Campus 4 Faculty of Basic Sciences, Autonomous University of Aguascalientes, Mexico 5 Department of Mathematics and Didactics of Mathematics, Tallinn University, Tallinn 10120, Estonia 6 Department of Mathematics and Physics, Autonomous University of Aguascalientes, Aguascalientes 20100, Mexico 7 Department of Mathematics, Balochistan Residential College, Loralai, Balochistan, Pakistan Abstract. In this paper, we employ the Atangana-Baleanu fractional integral operator to develop new versions of Hermite–Hadamard and Pachpatte-type integral inequalities within the frame- work of generalized convexity involving Raina’s function. By this approach, we derive a novel fractional integral identity associated with Raina’s functions. Furthermore, leveraging Young’s in- equality, the power mean inequality, and Hölder’s inequality, we establish several new extensions of Hermite–Hadamard-type inequalities via the Atangana–Baleanu fractional operator. Our results significantly improve upon existing findings, both in terms of generality and special cases. To validate our results, we provide remarks that recover various earlier inequalities. Additionally, we present applications related to entropy measures that demonstrate the practical utility of our main findings. 2020 Mathematics Subject Classifications: 11B73, 11B83 Key Words and Phrases: Preinvex functions, Hermite–Hadamard-type inequalities, AB-fractional operators, Raina special functions, Pachpatte-type integral inequalities, fractional calculus ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6556 Email address: jemacias@correo.uaa.mx (J. E. Maćıas-Dı́az) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) M. Tariq et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6556 2 of 28 1. Introduction Convex analysis offers a strong mathematical foundation for researching and resolving issues in a wide range of fields. Convexity is a widely developed concept that is funda- mentally introduced in the widely acclaimed book Inequalities by G. Pólya, G.H. Hardy, and J.E. Littlewood [1]. This foundational work, which focused solely on the study of inequalities, swiftly established itself as a standard reference for mathematicians and is still a must-read for anyone interested in learning more about this fascinating subject. Engineering [2], finance [3], economics [4], and optimization [5] are just a few of the fields in which the theory of convex functions finds extensive use. Furthermore, convex analysis serves as the foundation for the creation of strong numerical techniques, which offer the fundamental resources required to successfully explore and resolve challenging mathemat- ical issues. Mathematical inequalities offers a key basis for comprehending the perform of func- tions under integration, leading to crucial applications in both theoretical and applied mathematics. Fractional integrals inequalities involving convex functions, which establish connections between the integrals of convex functions and their values at particular points, are an effective tool in mathematical analysis. They are used in a number of areas, such as optimization, information theory, and probability theory. For numerical methods, such as the trapezoidal rule [6], Simpson’s rule [7], and others, these inequalities are essential, particularly when estimating the error bounds. Growing exponentially in popularity, fractional calculus allows one to define fractional derivatives and fractional integrals in several ways. Notably, the first concept of frac- tional calculus was put forth by Leibniz and L’Hospital in 1695. Particularly in view of the flaws of traditional calculus, the roots and ideas of fractional calculus have lately attracted great attention. Fractional calculus is the study of fractional order integrals and derivatives together with their applications in real and complex domains. Fractional integral inequalities allow us to determine the exact stability and uniqueness of fractional differential equations. Almost every nonlinear discipline or area of study in the modern world is impacted by fractional methods and tools. The subject fractional calculus has many applications in control systems [8], transform theory [9], nanotechnology [10], mod- eling [11, 12], fluid flow [13], mathematical biology [14], epidemiology [15], optimal control [16, 17] and physics [18]. The aforementioned widespread viewpoints and significance made the discussion of fractional operators intriguing to readers and academics. When defining and assessing statistical issues and formulas that resemble quadratures, this the- ory is helpful. An important development in fractional calculus is the Atangana-Baleanu fractional integral operator (ABFIO), which provides improved modeling capabilities for complex systems. For other relevant inequalities involving different fractional operators, we refer the reader to the following sources [19–23] Because of their strong applications in analysis, differential equations, and the applied sciences, fractional convex integral inequalities have attracted a lot of attention lately. By using generalized proportional fractional integral operators defined with respect to another function, Rashid et al. [24] introduced new estimates of integral inequalities and offered M. Tariq et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6556 3 of 28 a versatile framework for extending classical results like Hermite–Hadamard inequalities. Similarly, a larger class of functions could be analyzed under fractional integral operators thanks to the development of some mean-type fractional integral inequalities by Samraiz et al. [25] using various convexity concepts. However, Akın [26] extended the use of convex integral inequalities in dynamic systems and brought continuous and discrete fractional analysis together by establishing fractional maximal delta integral inequalities on time scales. To gain greater control over memory effects, Mohammed et al. [27] derived gener- alized Hermite–Hadamard inequalities using tempering fractional integrals. In the context of weighted inequalities, Rahman et al. [28] investigated weighted fractional integral in- equalities for Chebyshev functionals, offering useful resources for weighting structures and monotonicity in fractional calculus. By developing generalized reverse Minkowski inequali- ties using conformable fractional integrals, Rashid et al. [29] expanded on classical inequal- ities and provided insight into fractional variational problems. Rahman et al. [30] recently introduced hattaf fractional operators, which are a more general type of fractional opera- tor. They also make it possible to use new types of integral inequalities in many different fields. Tunç et al. [31] enhanced the resources for fractional integral analysis by proposing novel generalized fractional integral operators that resulted in new Hermite–Hadamard and Ostrowski-type inequalities. Sahoo et al. [32] presented generalized exponential-type convex functions and employed them to formulate novel Ostrowski-type fractional integral inequalities, thereby encompassing a broader spectrum of convexity behavior. Kashuri et al. [33] provided a unifying approach and illustrated the flexibility of the fractional in- tegral framework by constructing integral inequalities using general fractional operators. Through their work with fractional calculus involving general analytic kernels, Mohammed and Fernandez [34] expanded on this idea, allowing for a highly flexible approach to solving integral inequalities suitable for a range of applications. Finally, Rahman et al. [35] stud- ied fractional integral inequalities for monotone weighted Chebyshev functionals, which are crucial for fractional analysis that involves monotonicity and weighted conditions. For additional results on inequalities derived via alternative fractional operators, the reader is referred to the following literature [36–40]. The primary contribution of this study lies in the introduction of a novel class of gen- eralized convex mappings, developed primarily based on Condition A, which is formally defined below. This newly established class incorporates the Raina function in a manner that, to the best of our knowledge, has not previously been explored in the context of the related inequalities. Moreover, while prior work on inequalities associated with general- ized convexity has predominantly utilized classical integrals, Riemann–Liouville fractional integrals, and Caputo–Fabrizio operators, our approach leverages the Atangana–Baleanu fractional operators to generalize these inequalities within the framework of the newly introduced convexity concept. To substantiate and validate our findings, we also present several remarks and special cases demonstrating that many known results from the litera- ture can be recovered as particular instances of our general framework. This comparative analysis underscores the broader applicability and enhanced generality of the results de- veloped in this work. The organization of this paper is as follows. In Section 2, we revisit several fundamental M. Tariq et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6556 4 of 28 concepts and definitions that form the foundation for our subsequent analysis. Section 3.1 introduces the notion of generalized convex functions (GCF ) and explores their key al- gebraic properties. In Section 4, we present a novel Hermite–Hadamard-type inequality based on the Atangana–Baleanu fractional integral operator (ABFIO), accompanied by several interesting corollaries and remarks. Section 5 is devoted to deriving a new integral identity, which serves as the basis for refined versions of the Hermite–Hadamard inequality. In Section 6, we investigate a new class of Pachpatte-type inequalities via ABFIO, along with related corollaries and remarks. Finally, Section 8 summarizes the main findings and suggests potential directions for future research. 2. Preliminaries This section reviews key definitions and concepts essential for our subsequent analysis, including convexity, the Hermite–Hadamard inequality, Mittag–Leffler functions, general- ized convex sets, and generalized convex functions. Additionally, we discuss Condition C, Hölder’s inequality, and the power mean inequality. The section concludes with a brief overview of the Caputo–Fabrizio derivative and the Atangana–Baleanu fractional integral operator (ABFIO), which are fundamental to our study. Definition 1 ([2]). A real-valued function Λ is said to be convex, if Λ (xla + (1− x) lb) ≤ xΛ (la) + (1− x) Λ (lb) , (2.1) holds for all la, lb ∈ I and x ∈ [0, 1]. The most famous inequality involving convex functions is the Hermite-Hadamard in- equality [41–43] stated as: Theorem 1. If Λ : [la, lb] → R is a convex function, then Λ ( la + lb 2 ) ≤ 1 lb − la ∫ lb la Λ(x)dx ≤ Λ(la) + Λ(lb) 2 . (2.2) The inequality has been used to establish limits and estimates in information theory, particularly in connection with quantum calculus and quantum integral inequalities. Ge- ometric settings of this inequality help to establish relationships between the average over an interval and the value of a function at its midpoint. Hermite-Hadamard inequality is a useful tool for studying a range of economic phenomena involving convex functions, including asset pricing and optimization, income distribution, and production. Raina [44] introduced a generalized class of functions defined as Dκ µ,δ(z) = D κ(0), κ(1),... µ,δ (z) = ∞∑ k=0 κ(k) Γ(µk + δ) zk, (2.3) where κ = (κ(0), κ(1), . . . , κ(k), . . .), µ, δ > 0, and |z| < R. Equation (2.3) serves as a generalization of the classical Mittag–Leffler function. M. Tariq et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6556 5 of 28 If µ = 1, δ = 0, and κ(k) = (η)k(θ)k (ξ)k for k = 0, 1, 2, . . ., where η, θ, ξ are complex parameters such that ξ /∈ {0,−1,−2, . . .}, and where the Pochhammer symbol is defined by (η)k = Γ(η + k) Γ(η) = η(η + 1) · · · (η + k − 1), k = 0, 1, 2, . . . , and the domain is restricted to |z| ≤ 1 with z ∈ C, then equation (2.3) reduces to the classical hypergeometric function: D(η, θ; ξ; z) = ∞∑ k=0 (η)k(θ)k k!(ξ)k zk. Moreover, if κ = (1, 1, . . .), µ = α, δ = 1, with ℜ(α) > 0, then equation (2.3) reduces to the one-parameter Mittag–Leffler function: Eα(z) = ∞∑ k=0 zk Γ(1 + αk) . (2.4) Equation (2.4) is referred to as a classical Mittag–Leffler function. The Mittag–Leffler function appears usually in the study of fractional calculus and especially in the studies of fractional conjecture of the kinetic equation, super diffusive transport, random walks, Lévy flights, and in the studies of complicated structures. Cortez presented the generalized convex set and the convex function pertaining to Raina’s function in [45, 46]. Definition 2 (See [46]). Let ϱ = (ϱ(0), . . . , ϱ(v), . . .) and ϵ, σ > 0. A set X ̸= ∅ is said to be generalized convex, if la + x ∆ϱ ϵ,σ(lb − la) ∈ X, (2.5) for all la, lb ∈ X and x ∈ [0, 1]. Definition 3 (See [46]). Let ϱ represent a bounded sequence then ϱ = (ϱ(0), . . . , ϱ(v), . . .) and ϵ, σ > 0. If real-valued Λ holds the following inequality Λ ( la + x ∆ϱ ϵ,σ(lb − la) ) ≤ xΛ(lb) + (1− x)Λ(la), (2.6) for all la, lb ∈ X, where la < lb and x ∈ [0, 1], then Λ is said to be generalized convex function. Remark 1. If ∆ϱ ϵ,σ(lb − la) = lb − la > 0, then achieve Definition 1. M. Tariq et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6556 6 of 28 The following Condition-A first time explored by Ahmad et.al [47]. Condition C: Let X be generalized convex subset w.r.t. ∆ϱ ϵ,σ(·). For any la, lb ∈ X and x ∈ [0, 1], ∆ϱ ϵ,σ ( la − (la + x ∆ϱ ϵ,σ(lb − la)) ) = −x ∆ϱ ϵ,σ(lb − la), ∆ϱ ϵ,σ ( lb − ( la + x ∆ϱ ϵ,σ(lb − la) ) ) = (1− x) ∆ϱ ϵ,σ(lb − la). Note that, for every la, lb ∈ X and for all x1, x2 ∈ [0, 1] from Condition-A, we have ∆ϱ ϵ,σ ( la + x2 ∆ ϱ ϵ,σ(lb − la)− (la + x1 ∆ ϱ ϵ,σ(lb − la)) ) = (x2 − x1) ∆ ϱ ϵ,σ(lb − la). (2.7) Some well-known integral inequalities such as Hölder inequality and power-mean in- equality will be used. Theorem 2. [48] Assume that p > 1 and 1 p + 1 q = 1. Assume thats Λ1,Λ2 : [x1, x2] → R are such that |Λ1|p and |Λ2|q are integrable on [x1, x2]. Then∫ 1 0 |Λ1(x)Λ2(x)|dx ≤ (∫ 1 0 |Λ1(x)|pdx ) 1 p (∫ 1 0 |Λ2(x)|qdx ) 1 q . If we get |Λ1||Λ2| = (|Λ1| 1 p )(|Λ1| 1 q |Λ2|) in the Hölder inequality, then we obtain the following power mean integral inequality as a simple result of the Hölder integral inequality. Theorem 3. [48] Assume that Λ ≥ 1 and 1 p + 1 q = 1. Assume thats Λ1,Λ2 : [x1, x2] → R are such that |Λ1|p and |Λ2|q are integrable on [x1, x2]. Then∫ 1 0 |Λ1(x)Λ2(x)|dx ≤ (∫ 1 0 |Λ1(x)|dx )1− 1 q (∫ 1 0 |Λ1(x)|dx ∫ 1 0 |Λ2(x)|qdx ) 1 q . With the advancement of fractional calculus, numerous mathematicians have intro- duced a variety of fractional derivative and integral operators to address complex phe- nomena arising in real-world applications. These operators aim to capture memory effects and hereditary properties inherent in many physical, biological, and engineering systems. Over time, several notable formulations have emerged in the literature, some of which are listed below. In Caputo-Fabrizio (C-F) derivative operator, Atangana and Baleanu utilizing the Mittag-Leffler function and investigate the new derivative operators as follows. Definition 4. [49] Let Λ ∈ H1(la, lb), lb > la, γ ∈ [0, 1) then, the definition of the Caputo- Fabrizio derivative is given by ABC la Dγ t [Λ(t)] = B(γ) 1− γ ∫ t la Λ′(x)Eγ [ −γ (t− x)γ (1− γ) ] dx. (2.8) Definition 5. [49] Let Λ ∈ H1(la, lb), lb > la, γ ∈ [0, 1) then, the definition of the Caputo- Fabrizio derivative is given by ABR la Dγ t [Λ(t)] = B(γ) 1− γ d dt ∫ t la Λ(x)Eγ [ −γ (t− x)γ (1− γ) ] dx. (2.9) M. Tariq et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6556 7 of 28 Equations (2.8) and (2.9) have a non-local kernel. Also in equation (2.8) when the function is constant we get zero. The related fractional integral operator has been defined by Atangana-Baleanu as follows. Definition 6. [49] The fractional integral associated to the new fractional derivative with non-local kernel of a function Λ ∈ H1(la, lb) is defined: AB la Iγ{Λ(t)} = 1− γ B(γ) Λ(t) + γ B(γ)Γ(γ) ∫ t la Λ(y)(t− y)γ−1dy, where b > a, γ ∈ (0, 1]. In [50], Abdeljawad and Baleanu introduced right hand side of the integral operator as follows: The right fractional new integral with Mittag-Leffler kernel of order γ ∈ (0, 1] is defined by ABIγlb{Λ(t)} = 1− γ B(γ) Λ(t) + γ B(γ)Γ(γ) ∫ lb t Λ(y)(y − t)γ−1dy. 3. The major results In this section, we present our main results. 3.1. Generalized Convex Function and Its Properties In this section, we utilize the Definition 3 and examine some of its algebraic properties. Theorem 4. If Λ1, Λ2 are two GCF , then (Λ1 + Λ2) is also an GCF . Proof. Since given that Λ1 and Λ2 be two GCF , then (Λ1 + Λ2) (la + x∆ϱ ϵ,σ(lb − la)) = Λ1(la + x∆ϱ ϵ,σ(lb − la)) + Λ2(la + x∆ϱ ϵ,σ(lb − la)) ≤ (1− x) Λ1 (la) + xΛ1 (lb) + (1− x) Λ2 (la) + xΛ2 (lb) = (1− x) [Λ1 (la) + Λ2 (la)] + x [Λ1 (lb) + Λ2 (lb)] = (1− x) (Λ1 + Λ2)(la) + x(Λ1 + Λ2)(lb). This completes the proof. Theorem 5. If Λ is GCF , then (cΛ) is also an GCF . Proof. Since Λ is GCF , and c is any constant number, then (cΛ) (la + x∆ϱ ϵ,σ(lb − la)) ≤ c ( (1− x) Λ (la) + xΛ (lb) ) M. Tariq et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6556 8 of 28 = (1− x) cΛ (la) + xcΛ (lb) = (1− x) (cΛ) (la) + x(cΛ) (lb) . This completes the proof. Theorem 6. Composition of two GCF is also an GCF . Proof. (Λ2 ◦ Λ1) (la + x∆ϱ ϵ,σ(lb − la)) = Λ2(Λ1(la + x∆ϱ ϵ,σ(lb − la))) ≤ Λ2 ( (1− x) Λ1 (la) + xΛ1 (lb) ) ≤ (1− x) Λ2(Λ1 (la)) + xΛ2(Λ1 (lb)) = (1− x) (Λ2 ◦ Λ1) (la) + x(Λ2 ◦ Λ1) (lb) . This completes the desired proof. Theorem 7. Let 0 < la < lb, Λj : X = [la, lb] → [0,+∞) be a family of GCF and Λ(u) = supj Λj(u). Then, Λ is an GCF for m ∈ (0, 1], x ∈ [0, 1], and U = {Λ ∈ [la, lb] : Λ(Λx) <∞} is an interval. Proof. Let la, lb ∈ U , and x ∈ [0, 1], then Λ(la + x∆ϱ ϵ,σ(lb − la)) = sup j Λj(la + x∆ϱ ϵ,σ(lb − la)) ≤ (1− x) sup j Λj (la) + x sup j Λj (lb) = (1− x) Λ (la) + xΛ (lb) <∞. This completes the proof. 4. Hermite–Hadamard Inequality Pertaining to AB Fractional Integral Operator The main goal of this portion is to provide a new sort of the Hermite-Hadamard-type inequality for a GCF via ABFIO. Theorem 8. Let I ⊆ R be an open and non-empty convex subset and la, lb ∈ I with la < mla+∆ϱ ϵ,σ(lb− la). If Λ : [la, la +∆ϱ ϵ,σ(lb − la)] → R is a GCF , Λ ∈ L [la, la +∆ϱ ϵ,σ(lb − la)] and ∆ϱ ϵ,σ satisfies Condition C, the following inequalities for ABFIO hold Λ ( 2la +∆ϱ ϵ,σ(lb − la) 2 ) ≤ B(γ)Γ(γ) 2 [∆ϱ ϵ,σ(lb − la)] γ [ AB la Iγ { Λ ( la +∆ϱ ϵ,σ(lb − la) )} + ABIγ la+∆ϱ ϵ,σ(lb−la) {Λ (la)} ] M. Tariq et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6556 9 of 28 − (1− γ)Γ(γ) 2 [∆ϱ ϵ,σ(lb − la)] γ [ Λ (la) + Λ ( la +∆ϱ ϵ,σ(lb − la) )] ≤ Λ (la) + Λ (lb) 2 , (4.1) where γ ∈ (0, 1], B(γ) > 0 is a normalization function, and Γ(·) denotes the Gamma function. Proof. Since Λ is GCF on [la, la +∆ϱ ϵ,σ(lb − la)], we can write 2Λ ( 2la +∆ϱ ϵ,σ(lb − la) 2 ) ≤ Λ ( la + x∆ϱ ϵ,σ(lb − la) ) + Λ ( la + (1− x)∆ϱ ϵ,σ(lb − la) ) . (4.2) Multiplying both sides of inequality (4.2) by γ B(γ) Γ(γ) xγ−1 and integrating the re- sulting expression with respect to x over the interval [0, 1], we obtain 2 B(γ)Γ(γ) Λ ( 2la +∆ϱ ϵ,σ(lb − la) 2 ) ≤ γ B(γ)Γ(γ) ∫ 1 0 xγ−1Λ ( la + x∆ϱ ϵ,σ(lb − la) ) dx + γ B(γ)Γ(γ) ∫ 1 0 tγ−1Λ ( la + (1− x)∆ϱ ϵ,σ(lb − la) ) dx = γ B(γ)Γ(γ) [∆ϱ ϵ,σ(lb − la)] γ ∫ la+∆ϱ ϵ,σ(lb−mla) la (x− la) γ−1 Λ(x)dx + γ B(γ)Γ(γ) [∆ϱ ϵ,σ(lb − la)] γ ∫ la+∆ϱ ϵ,σ(lb−la) la ( la +∆ϱ ϵ,σ(lb − la)− y )γ−1 Λ(y)dy. Then we can write 2 B(γ)Γ(γ) Λ ( 2la +∆ϱ ϵ,σ(lb − la) 2 ) ≤ 1 [∆ϱ ϵ,σ(lb − la)] γ [ γ B(γ)Γ(γ) ∫ la+∆ϱ ϵ,σ(lb−la) la (x− la) γ−1 Λ(x)dx+ (1− γ) B(γ) Λ (la) ] − (1− γ) B(γ) [∆ϱ ϵ,σ(lb − la)] γΛ (la) + 1 [∆ϱ ϵ,σ(lb − la)] γ [ γ B(γ)Γ(γ) ∫ la+∆ϱ ϵ,σ(lb−mla) la ( la +∆ϱ ϵ,σ(lb − la)− y )γ−1 Λ(y)dy + (1− γ) B(γ) Λ ( la +∆ϱ ϵ,σ(lb − la) )] − (1− γ) B(γ) [∆ϱ ϵ,σ(lb − la)] γΛ ( la +∆ϱ ϵ,σ(lb − la) ) . M. Tariq et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6556 10 of 28 So, using ABFIO, we get 2 B(γ)Γ(γ) Λ ( 2la +∆ϱ ϵ,σ(lb − la) 2 ) ≤ 1 [∆ϱ ϵ,σ(lb − la)] γ [ AB la Iγ { Λ ( la +∆ϱ ϵ,σ(lb − la) )} + ABIγ la+∆ϱ ϵ,σ(lb−la) {Λ (la)} ] − (1− γ) B(γ) [∆ϱ ϵ,σ(lb − la)] γ [ Λ (la) + Λ ( la +∆ϱ ϵ,σ(lb − la) )] and the first inequality is proved. For the proof of the second inequality in the above inequality (4.1), we first note that if Λ is a GCF , then we can write Λ ( la + x∆ϱ ϵ,σ(lb − la) ) ≤ (1− x)Λ (la) + xΛ (lb) and Λ ( la + (1− x)∆ϱ ϵ,σ(lb − la) ) ≤ mxΛ (la) + (1− x)Λ (lb) . By summing the above inequalities term by term, we arrive at Λ ( la + x∆ϱ ϵ,σ(lb − la) ) + Λ ( la + (1− x)∆ϱ ϵ,σ(lb − la) ) ≤ Λ (la) + Λ (lb) . (4.3) Next, we multiply both sides of inequality (4.3) by γ B(γ)Γ(γ) xγ−1 and integrate the re- sulting expression with respect to x over the interval [0, 1]. This yields γ B(γ)Γ(γ) ∫ 1 0 xγ−1Λ ( la + x∆ϱ ϵ,σ(lb − la) ) dx + γ B(γ)Γ(γ) ∫ 1 0 xγ−1Λ ( la + (1− x)∆ϱ ϵ,σ(lb − la) ) dx ≤ γ B(γ)Γ(γ) [Λ (la) + Λ (lb)] ∫ 1 0 xγ−1dx. Then, we can write 1 [∆ϱ ϵ,σ(lb − la)] γ [ AB la Iγ { Λ ( la +∆ϱ ϵ,σ(lb − la) )} + ABIγ la+∆ϱ ϵ,σ(lb−la) {Λ (la)} ] − (1− γ) B(γ) [∆ϱ ϵ,σ(lb − la)] γ [ Λ (la) + Λ ( la +∆ϱ ϵ,σ(lb − la) )] ≤ Λ (la) + Λ (lb) B(γ)Γ(γ) . So, the proof of this theorem is completed. M. Tariq et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6556 11 of 28 Remark 2. Choosing ∆ϱ ϵ,σ(lb − la) = lb − la in the above theorem, we have the result in [51, Proposition 2.1], inequality (13). Remark 3. Considering Theorem 8, we establish the following new mathematical ap- proach of Hermite-Hadamard inequality pertaining to the classical Mittag-Leffler function via ABFIO if we pick ϱ = (1, 1, ...) with ϵ =α and σ = 1: Λ ( 2la + Eα(lb − la) 2 ) ≤ B(γ)Γ(γ) 2 [Eα(lb − la)] γ [ AB mlaI γ {Λ (la + Eα(lb − la))}+ ABIγla+Eα(lb−mla) {Λ (la)} ] − (1− γ)Γ(γ) 2 [Eα(lb − la)] γ [Λ (la) + Λ (la + Eα(lb − la))] ≤ Λ (la) + Λ (lb) 2 . 5. Refinements of Hermite-Hadamard Type Inequality via AB Fractional Integral Operator This section’s goal is to explore and offer a novel equality. We derive some novel enhancements of Hermite-Hadamard-type inequalities using an ABFIO based on this re- cently studied equality. To improve the content and grab readers’ attention, we include a few remarks. First, we prove a lemma in the frame of ABFIO. Throughout in this section, B(γ) represents the normalization function and Γ(.) rep- resents the Gamma function. Lemma 1. Let I ⊆ R be an open, non-empty convex set, and let la, lb ∈ I with la < la +∆ϱ ϵ,σ(lb − la). Suppose that Λ : I → R is a differentiable function such that Λ′ ∈ L [ la, la +∆ϱ ϵ,σ(lb − la) ] . Then, the following identity involving the Atangana–Baleanu fractional integral operator holds: B(γ)Γ(γ) [∆ϱ ϵ,σ(lb − la)] γ+1 [ AB la Iγ { Λ ( la +∆ϱ ϵ,σ(lb − la) )} + ABIγ la+∆ϱ ϵ,σ(lb−la) {Λ (la)} ] − ( [∆ϱ ϵ,σ(lb − la)] γ + (1− γ)Γ(γ) [∆ϱ ϵ,σ(lb − la)] γ+1 )[ Λ (la) + Λ ( la +∆ϱ ϵ,σ(lb − la) )] = ∫ 1 0 (1− x)γΛ′ (la + x∆ϱ ϵ,σ(lb − la) ) dx− ∫ 1 0 xγΛ′ (la + x∆ϱ ϵ,σ(lb − la) ) dx where γ ∈ (0, 1], x ∈ [0, 1]. M. Tariq et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6556 12 of 28 Proof. By using integration, we have∫ 1 0 (1− x)γΛ′ (la + x∆ϱ ϵ,σ(lb − la) ) dx = (1− x)γΛ (la + x∆ϱ ϵ,σ(lb − la)) ∆ϱ ϵ,σ(lb − la) ∣∣∣∣1 0 + γ ∆ϱ ϵ,σ(lb − la) ∫ 1 0 Λ ( la + x∆ϱ ϵ,σ(lb − la) ) (1− x)γ−1dx =− Λ (la) ∆ϱ ϵ,σ(lb − la) + γ ∆ϱ ϵ,σ(lb − la) ∫ 1 0 (1− x)γ−1Λ ( la + x∆ϱ ϵ,σ(lb − la) ) dx =− Λ (la) ∆ϱ ϵ,σ(lb − la) + γ [∆ϱ ϵ,σ(lb − la)] γ+1 ∫ la+∆ϱ ϵ,σ(lb−la) la ( la +∆ϱ ϵ,σ(lb − la)− x )γ−1 Λ(x)dx. (5.1) Multiplying both sides of inequality (5.1) by 1 B(γ)Γ(γ) , we obtain 1 B(γ)Γ(γ) ∫ 1 0 (1− x)γΛ′ (la + x∆ϱ ϵ,σ(lb − la) ) dx = − Λ (la) B(γ)Γ(γ)∆ϱ ϵ,σ(lb −mla) + γ B(γ)Γ(γ) [∆ϱ ϵ,σ(lb − la)] γ+1 ∫ la+∆ϱ ϵ,σ(lb−la) la ( la +∆ϱ ϵ,σ(lb − la)− x )γ−1 Λ(x)dx. Then, we can write 1 B(γ)Γ(γ) ∫ 1 0 (1− x)γΛ′ (la + x∆ϱ ϵ,σ(lb − la) ) dx = − Λ (la) B(γ)Γ(γ)∆ϱ ϵ,σ(lb − la) + 1 [∆ϱ ϵ,σ(lb − la)] γ+1 [ γ B(γ)Γ(γ) ∫ la+∆ϱ ϵ,σ(lb−la) la ( la +∆ϱ ϵ,σ(lb − la)− x )γ−1 Λ(x)dx + (1− γ) B(γ) Λ ( la +∆ϱ ϵ,σ(lb − la) )] − (1− γ) B(γ) [∆ϱ ϵ,σ(lb − la)] γ+1Λ ( la +∆ϱ ϵ,σ(lb − la) ) . Using ABFIO, we have 1 B(γ)Γ(γ) ∫ 1 0 (1− x)γΛ′ (la + x∆ϱ ϵ,σ(lb − la) ) dx =− Λ (la) B(γ)Γ(γ)∆ϱ ϵ,σ(lb − la) + 1 [∆ϱ ϵ,σ(lb − la)] γ+1 [ AB la Iγ { Λ ( la +∆ϱ ϵ,σ(lb − la) )}] − (1− γ) B(γ) [∆ϱ ϵ,σ(lb − la)] γ+1Λ ( mla +∆ϱ ϵ,σ(lb − la) ) . (5.2) M. Tariq et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6556 13 of 28 Similarly, using integration, we get∫ 1 0 xγΛ′ (la + x∆ϱ ϵ,σ(lb − la) ) dx = xγΛ (la + x∆ϱ ϵ,σ(lb − la)) ∆ϱ ϵ,σ(lb − la) ∣∣∣∣1 0 − γ ∆ϱ ϵ,σ(lb − la) ∫ 1 0 Λ ( la + x∆ϱ ϵ,σ(lb − la) ) xγ−1dx = Λ(la +∆ϱ ϵ,σ(lb − la)) Ω (lb, la) − γ ∆ϱ ϵ,σ(lb − la) ∫ 1 0 xγ−1Λ ( la + x∆ϱ ϵ,σ(lb − la) ) dx = Λ(la +∆ϱ ϵ,σ(lb − la)) ∆ϱ ϵ,σ(lb − la) − γ [∆ϱ ϵ,σ(lb − la)] γ+1 ∫ la+∆ϱ ϵ,σ(lb−la) la (u− la) γ−1 Λ(u)du. (5.3) Multiplying both sides of inequality (5.3) by − 1 B(γ)Γ(γ) , we have − 1 B(γ)Γ(γ) ∫ 1 0 xγΛ′ (la + x∆ϱ ϵ,σ(lb − la) ) dx =− Λ (la +∆ϱ ϵ,σ(lb − la)) B(γ)Γ(γ)∆ϱ ϵ,σ(lb − la) + γ B(γ)Γ(γ) [∆ϱ ϵ,σ(lb − la)] γ+1 ∫ la+∆ϱ ϵ,σ(lb−la) la (u− la) γ−1 Λ(u)du. Then we can write − 1 B(γ)Γ(γ) ∫ 1 0 xγΛ′ (la + x∆ϱ ϵ,σ(lb − la) ) dx = − Λ (la +∆ϱ ϵ,σ(lb − la)) B(γ)Γ(γ)∆ϱ ϵ,σ(lb − la) + 1 [∆ϱ ϵ,σ(lb − la)] γ+1 [ γ B(γ)Γ(γ) ∫ la+∆ϱ ϵ,σ(lb−la) la (u− la) γ−1 Λ(u)du + (1− γ) B(γ) Λ (la) ] − (1− γ) B(γ) [∆ϱ ϵ,σ(lb − la)] γ+1Λ (la) . Using ABFIO, we have − 1 B(γ)Γ(γ) ∫ 1 0 xγΛ′ (la + x∆ϱ ϵ,σ(lb − la) ) dx =− Λ (la +∆ϱ ϵ,σ(lb − la)) B(γ)Γ(γ)∆ϱ ϵ,σ(lb − la) + 1 [∆ϱ ϵ,σ(lb − la)] γ+1 [ ABIγ la+∆ϱ ϵ,σ(lb−la) {Λ (la)} ] − (1− γ) B(γ) [∆ϱ ϵ,σ(lb − la)] γ+1Λ (la) . (5.4) By adding identities (5.2) and (5.4), we obtain the proof of Lemma 1. M. Tariq et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6556 14 of 28 Remark 4. If we put ∆ϱ ϵ,σ(lb − la) = lb − la in Lemma 1 then we have the result in [51, Theorem 3.1], equality (29). Theorem 9. Let I ⊆ R be an open and non-empty convex set and la, lb ∈ I with la < la + ∆ϱ ϵ,σ(lb − la). Suppose that Λ : I → R is a differentiable function and Λ′ ∈ L [la, la +∆ϱ ϵ,σ(lb − la)]. If |Λ′| is GCRF , then we have the following inequality for AB- FIO ∣∣∣∣∣ B(γ)Γ(γ) [∆ϱ ϵ,σ(lb − la)] γ+1 [ AB mlaI γ { Λ ( la +∆ϱ ϵ,σ(lb − la) )} + ABIγ la+∆ϱ ϵ,σ(lb−la) {Λ (la)} ] − ( [∆ϱ ϵ,σ(lb − la)] γ + (1− γ)Γ(γ) [∆ϱ ϵ,σ(lb − la)] γ+1 )[ Λ (la) + Λ ( la +∆ϱ ϵ,σ(lb − la) )]∣∣∣∣∣ ≤ |Λ′ (la)|+ |Λ′ (lb)| γ + 1 , where γ ∈ (0, 1]. Proof. Using the identity given in Lemma 1 and the properties of the modulus, we can write ∣∣∣∣∣ B(γ)Γ(γ) [∆ϱ ϵ,σ(lb − la)] γ+1 [ AB la Iγ { Λ ( la +∆ϱ ϵ,σ(lb − la) )} + ABIγ la+∆ϱ ϵ,σ(lb−la) {Λ (la)} ] − ( [∆ϱ ϵ,σ(lb − la)] γ + (1− γ)Γ(γ) [∆ϱ ϵ,σ(lb − la)] γ+1 )[ Λ (la) + Λ ( la +∆ϱ ϵ,σ(lb − la) )]∣∣∣∣∣ = ∣∣∣∣∫ 1 0 (1− x)γΛ′ (la + x∆ϱ ϵ,σ(lb − la) ) dx− ∫ 1 0 xγΛ′ (la + x∆ϱ ϵ,σ(lb − la) ) dx ∣∣∣∣ ≤ ∫ 1 0 (1− x)γ ∣∣Λ′ (la + x∆ϱ ϵ,σ(lb − la) )∣∣ dx+ ∫ 1 0 xγ ∣∣Λ′ (la + x∆ϱ ϵ,σ(lb − la) )∣∣ dx. Since |Λ′| is GCRF , we obtain∣∣∣∣∣ B(γ)Γ(γ) [∆ϱ ϵ,σ(lb − la)] γ+1 [ AB la Iγ { Λ ( la +∆ϱ ϵ,σ(lb − la) )} + ABIγ la+∆ϱ ϵ,σ(lb−la) {Λ (la)} ] − ( [∆ϱ ϵ,σ(lb − la)] γ + (1− γ)Γ(γ) [∆ϱ ϵ,σ(lb − la)] γ+1 )[ Λ (la) + Λ ( la +∆ϱ ϵ,σ(lb − la) )]∣∣∣∣∣ ≤ ∫ 1 0 (1− x)γ [ (1− x) ∣∣Λ′ (la) ∣∣+ x ∣∣Λ′ (lb) ∣∣] dx+ ∫ 1 0 xγ [ (1− x) ∣∣Λ′ (la) ∣∣+ x ∣∣Λ′ (lb) ∣∣] dx = |Λ′ (la)|+ |Λ′ (lb)| γ + 1 . So, the proof is completed. M. Tariq et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6556 15 of 28 Remark 5. Considering Theorem 9, we establish the following new mathematical ap- proach of Hermite-Hadamard inequality pertaining to the classical Mittag-Leffler function via ABFIO if we pick ϱ = (1, 1, . . .) with ϵ =α and σ = 1:∣∣∣∣ B(γ)Γ(γ) [Eα(lb − la)] γ+1 [ AB la Iγ {Λ (la + Eα(lb − la))}+ ABIγla+Eα(lb−la) {Λ (la)} ] − ( [Eα(lb − la)] γ + (1− γ)Γ(γ) [Eα(lb − la)] γ+1 ) [Λ (la) + Λ (la + Eα(lb − la))] ∣∣∣∣ ≤ |Λ′ (la)|+ |Λ′ (lb)| γ + 1 , Corollary 1. In the above Theorem 9, if we choose ∆ϱ ϵ,σ(lb − la) = lb − la, then we obtain∣∣∣∣ B(γ)Γ(γ) (lb − la) γ+1 [ AB la Iγ {Λ (lb)}+ ABIγlb {Λ (la)} ] − ( (lb − la) γ + (1− γ)Γ(γ) (lb − la) γ+1 ) [Λ (la) + Λ (lb)] ∣∣∣∣ ≤ |Λ′ (la)|+ |Λ′ (lb)| γ + 1 . Theorem 10. Let I ⊆ R be an open and non-empty covex set and la, lb ∈ I with la < la +∆ϱ ϵ,σ(lb −mla). Suppose that Λ : I → R is a differentiable function and Λ′ ∈ L [ la, la +∆ϱ ϵ,σ(lb − la) ] . If |Λ′|q is a GCF , then we have the following inequality for ABFIO∣∣∣∣∣ B(γ)Γ(γ) [∆ϱ ϵ,σ(lb − la)] γ+1 [ AB la Iγ { Λ ( la +∆ϱ ϵ,σ(lb − la) )} + ABIγ la+∆ϱ ϵ,σ(lb−la) {Λ (mla)} ] − ( [∆ϱ ϵ,σ(lb − la)] γ + (1− γ)Γ(γ) [∆ϱ ϵ,σ(lb − la)] γ+1 )[ Λ (la) + Λ ( la +∆ϱ ϵ,σ(lb − la) )]∣∣∣∣∣ ≤ 2 ( 1 γp+ 1 ) 1 p ( |Λ′ (la)|q + |Λ′ (lb)|q 2 ) 1 q , where p−1 + q−1 = 1, q > 1, γ ∈ (0, 1]. Proof. By using Lemma 1, we get∣∣∣∣∣ B(γ)Γ(γ) [∆ϱ ϵ,σ(lb − la)] γ+1 [ AB la Iγ { Λ ( la +∆ϱ ϵ,σ(lb − la) )} + ABIγ la+∆ϱ ϵ,σ(lb−la) {Λ (la)} ] − ( [∆ϱ ϵ,σ(lb − la)] γ + (1− γ)Γ(γ) [∆ϱ ϵ,σ(lb − la)] γ+1 )[ Λ (la) + Λ ( la +∆ϱ ϵ,σ(lb − la) )]∣∣∣∣∣ ≤ ∫ 1 0 (1− x)γ ∣∣Λ′ (la + x∆ϱ ϵ,σ(lb − la) )∣∣ dx+ ∫ 1 0 xγ ∣∣Λ′ (la + x∆ϱ ϵ,σ(lb − la) )∣∣ dx. M. Tariq et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6556 16 of 28 By applying Hölder inequality, we get∣∣∣∣∣ B(γ)Γ(γ) [∆ϱ ϵ,σ(lb − la)] γ+1 [ AB la Iγ { Λ ( la +∆ϱ ϵ,σ(lb − la) )} + ABIγ la+∆ϱ ϵ,σ(lb−la) {Λ (la)} ] − ( [∆ϱ ϵ,σ(lb − la)] γ + (1− γ)Γ(γ) [∆ϱ ϵ,σ(lb − la)] γ+1 )[ Λ (la) + Λ ( la +∆ϱ ϵ,σ(lb − la) )]∣∣∣∣∣ ≤ (∫ 1 0 (1− x)γpdx ) 1 p (∫ 1 0 ∣∣Λ′ (la + x∆ϱ ϵ,σ(lb − la) )∣∣q dx) 1 q + (∫ 1 0 xγpdx ) 1 p (∫ 1 0 ∣∣Λ′ (la + x∆ϱ ϵ,σ(lb − la) )∣∣q dx) 1 q . By using GmCF of |Λ′|q, we obtain∣∣∣∣ B(γ)Γ(γ) [Ω (lb, la)] γ+1 [ AB la Iγ { Λ ( la +∆ϱ ϵ,σ(lb − la) )} + ABIγ la+∆ϱ ϵ,σ(lb−la) {Λ (la)} ] − ( [∆ϱ ϵ,σ(lb − la)] γ + (1− γ)Γ(γ) [∆ϱ ϵ,σ(lb − la)] γ+1 )[ Λ (la) + Λ ( la +∆ϱ ϵ,σ(lb − la) )]∣∣∣∣∣ ≤ (∫ 1 0 (1− x)γpdx ) 1 p (∫ 1 0 [ (1− x) ∣∣Λ′ (la) ∣∣q + x ∣∣Λ′ (lb) ∣∣q] dx) 1 q + (∫ 1 0 xγpdx ) 1 p (∫ 1 0 [ (1− x) ∣∣Λ′ (la) ∣∣q + x ∣∣Λ′ (lb) ∣∣q] dx) 1 q . By calculating the integrals in the above inequality, we get the desired result. Remark 6. Considering Theorem 10, we establish the following new mathematical ap- proach of Hermite-Hadamard inequality pertaining to the classical Mittag-Leffler function via ABFIO if we pick ϱ = (1, 1, ...) with ϵ =α and σ = 1:∣∣∣∣ B(γ)Γ(γ) [Eα(lb − la)] γ+1 [ AB la Iγ {Λ (la + Eα(lb − la))}+ ABIγla+Eα(lb−la) {Λ (la)} ] − ( [Eα(lb − la)] γ + (1− γ)Γ(γ) [Eα(lb − la)] γ+1 ) [Λ (la) + Λ (la + Eα(lb − la))] ∣∣∣∣ ≤ 2 ( 1 γp+ 1 ) 1 p ( |Λ′ (la)|q + |Λ′ (lb)|q 2 ) 1 q . Corollary 2. In the above Theorem, if we choose ∆ϱ ϵ,σ(lb − la) = lb − la, then we obtain∣∣∣∣ B(γ)Γ(γ) (lb − la) γ+1 [ AB la Iγ {Λ (lb)}+ ABIγlb {Λ (la)} ] M. Tariq et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6556 17 of 28 − ( (lb − la) γ + (1− γ)Γ(γ) (lb − la) γ+1 ) [Λ (la) + Λ (lb)] ∣∣∣∣ ≤ 2 ( 1 γp+ 1 ) 1 p ( |Λ′ (la)|q + |Λ′ (lb)|q 2 ) 1 q . Theorem 11. Let I ⊆ R be an open and non-empty convex set and la, lb ∈ I with la < la + ∆ϱ ϵ,σ(lb − la). Suppose that Λ : I → R is a differentiable function and Λ′ ∈ L [la, la +∆ϱ ϵ,σ(lb − la)]. If |Λ′|q is a GCF , then we have the following inequality for AB- FIO ∣∣∣∣∣ B(γ)Γ(γ) [∆ϱ ϵ,σ(lb − la)] γ+1 [ AB la Iγ { Λ ( la +∆ϱ ϵ,σ(lb − la) )} + ABIγ la+∆ϱ ϵ,σ(lb−la) {Λ (la)} ] − ( [∆ϱ ϵ,σ(lb − la)] γ + (1− γ)Γ(γ) [∆ϱ ϵ,σ(lb − la)] γ+1 )[ Λ (la) + Λ ( la +∆ϱ ϵ,σ(lb − la) )]∣∣∣∣∣ ≤ ( 1 γ + 1 )1− 1 q [( |Λ′ (la)|q γ + 2 + |Λ′ (lb)|q (γ + 1)(γ + 2) ) 1 q + ( |Λ′ (la)|q (γ + 1)(γ + 2) + |Λ′ (lb)|q γ + 2 ) 1 q ] , where γ ∈ (0, 1], q ≥ 1. Proof. Employing Lemma 1 and utilizing the power mean inequality, we get∣∣∣∣∣ B(γ)Γ(γ) [∆ϱ ϵ,σ(lb − la)] γ+1 [ AB la Iγ { Λ ( la +∆ϱ ϵ,σ(lb − la) )} + ABIγ la+∆ϱ ϵ,σ(lb−la) {Λ (la)} ] − ( [∆ϱ ϵ,σ(lb − la)] γ + (1− γ)Γ(γ) [∆ϱ ϵ,σ(lb − la)] γ+1 )[ Λ (la) + Λ ( la +∆ϱ ϵ,σ(lb − la) )]∣∣∣∣∣ ≤ ∫ 1 0 (1− x)γ ∣∣Λ′ (la + x∆ϱ ϵ,σ(lb − la) )∣∣ dx+ ∫ 1 0 xγ ∣∣Λ′ (la + x∆ϱ ϵ,σ(lb − la) )∣∣ dx ≤ (∫ 1 0 (1− x)γdx )1− 1 q (∫ 1 0 (1− x)γ ∣∣Λ′ (la + x∆ϱ ϵ,σ(lb − la) )∣∣q dx) 1 q + (∫ 1 0 xγdx )1− 1 q (∫ 1 0 xγ ∣∣Λ′ (la + x∆ϱ ϵ,σ(lb − la) )∣∣q dx) 1 q · By using GCF of |Λ′|q, we have∣∣∣∣∣ B(γ)Γ(γ) [∆ϱ ϵ,σ(lb − la)] γ+1 [ AB la Iγ { Λ ( la +∆ϱ ϵ,σ(lb − la) )} + ABIγ la+∆ϱ ϵ,σ(lb−la) {Λ (la)} ] − ( [∆ϱ ϵ,σ(lb − la)] γ + (1− γ)Γ(γ) [∆ϱ ϵ,σ(lb − la)] γ+1 )[ Λ (la) + Λ ( la +∆ϱ ϵ,σ(lb − la) )]∣∣∣∣∣ ≤ (∫ 1 0 (1− x)γdx )1− 1 q (∫ 1 0 (1− x)γ [ (1− x) ∣∣Λ′ (la) ∣∣q + x ∣∣Λ′ (lb) ∣∣q] dx) 1 q M. Tariq et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6556 18 of 28 + (∫ 1 0 xγdx )1− 1 q (∫ 1 0 xγ [ (1− x) ∣∣Λ′ (la) ∣∣q + x ∣∣Λ′ (lb) ∣∣q] dx) 1 q = ( 1 γ + 1 )1− 1 q [( |Λ′ (la)|q γ + 2 + |Λ′ (lb)|q (γ + 1)(γ + 2) ) 1 q + ( |Λ′ (la)|q (γ + 1)(γ + 2) + |Λ′ (lb)|q γ + 2 ) 1 q ] . Th proof is completed. Remark 7. Considering Theorem 11, we establish the following new mathematical ap- proach of Hermite-Hadamard inequality pertaining to the classical Mittag-Leffler function via ABFIO if we pick ϱ = (1, 1, . . .) with ϵ =α and σ = 1:∣∣∣∣ B(γ)Γ(γ) [Eα(lb − la)] γ+1 [ AB la Iγ {Λ (la + Eα(lb − la))}+ ABIγla+Eα(lb−la) {Λ (mla)} ] − ( [Eα(lb − la)] γ + (1− γ)Γ(γ) [Eα(lb − la)] γ+1 ) [Λ (la) + Λ (la + Eα(lb − la))] ∣∣∣∣ ≤ ( 1 γ + 1 )1− 1 q [( |Λ′ (la)|q γ + 2 + |Λ′ (lb)|q (γ + 1)(γ + 2) ) 1 q + ( |Λ′ (la)|q (γ + 1)(γ + 2) + |Λ′ (lb)|q γ + 2 ) 1 q ] , Corollary 3. In the above Theorem, if we choose ∆ϱ ϵ,σ(lb − la) = lb − la, we obtain∣∣∣∣ B(γ)Γ(γ) (lb − la) γ+1 [ AB la Iγ {Λ (lb)}+ ABIγlb {Λ (la)} ] − ( (lb − la) γ + (1− γ)Γ(γ) (lb − la) γ+1 ) [Λ (la) + Λ (lb)] ∣∣∣∣ ≤ ( 1 γ + 1 )1− 1 q [( |Λ′ (la)|q γ + 2 + |Λ′ (lb)|q (γ + 1)(γ + 2) ) 1 q + ( |Λ′ (la)|q (γ + 1)(γ + 2) + |Λ′ (lb)|q γ + 2 ) 1 q ] . Theorem 12. Let I ⊆ R be an open and non-empty convex set and la, lb ∈ I with la < la + ∆ϱ ϵ,σ(lb − la). Suppose that Λ : I → R is a differentiable function and Λ′ ∈ L [la, la +∆ϱ ϵ,σ(lb − la)]. If |Λ′|q is a GCF , then we have the following inequality for AB- FIO ∣∣∣∣∣ B(γ)Γ(γ) [∆ϱ ϵ,σ(lb − la)] γ+1 [ AB la Iγ { Λ ( la +∆ϱ ϵ,σ(lb − la) )} + ABIγ la+∆ϱ ϵ,σ(lb−la) {Λ (la)} ] − ( [∆ϱ ϵ,σ(lb − la)] γ + (1− γ)Γ(γ) [∆ϱ ϵ,σ(lb − la)] γ+1 )[ Λ (la) + Λ ( la +∆ϱ ϵ,σ(lb − la) )]∣∣∣∣∣ ≤ 2 p(γp+ 1) + |Λ′ (la)|q + |Λ′ (lb)|q q , where p−1 + q−1 = 1, q > 1, γ ∈ (0, 1]. M. Tariq et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6556 19 of 28 Proof. By using the identity given in Lemma 1 and applying the Young inequality xy ≤ 1 px p + 1 qy q, we get∣∣∣∣∣ B(γ)Γ(γ) [∆ϱ ϵ,σ(lb − la)] γ+1 [ AB la Iγ { Λ ( la +∆ϱ ϵ,σ(lb − la) )} + ABIγ la+∆ϱ ϵ,σ(lb−la) {Λ (la)} ] − ( [∆ϱ ϵ,σ(lb − la)] γ + (1− γ)Γ(γ) [∆ϱ ϵ,σ(lb − la)] γ+1 )[ Λ (la) + Λ ( la +∆ϱ ϵ,σ(lb − la) )]∣∣∣∣∣ ≤ ∫ 1 0 (1− x)γ ∣∣Λ′ (la + x∆ϱ ϵ,σ(lb − la) )∣∣ dx+ ∫ 1 0 xγ ∣∣Λ′ (la + x∆ϱ ϵ,σ(lb − la) )∣∣ dx. ≤ 1 p ∫ 1 0 (1− x)γpdx+ 1 q ∫ 1 0 ∣∣Λ′ (la + x∆ϱ ϵ,σ(lb − la) )∣∣q dx + 1 p ∫ 1 0 xγpdx+ 1 q ∫ 1 0 ∣∣Λ′ (la + x∆ϱ ϵ,σ(lb − la) )∣∣q dx. By using GCF of |Λ′|q and by a simple computation, we have the desired result. Remark 8. Considering Theorem 12, we establish the following new mathematical ap- proach of Hermite-Hadamard inequality pertaining to the classical Mittag-Leffler function via ABFIO if we pick ϱ = (1, 1, . . .) with ϵ =α and σ = 1:∣∣∣∣ B(γ)Γ(γ) [Eα(lb − la)] γ+1 [ AB la Iγ {Λ (la + Eα(lb − la))}+ ABIγla+Ω(lb,la) {Λ (la)} ] − ( [Eα(lb − la)] γ + (1− γ)Γ(γ) [Eα(lb − la)] γ+1 )[ Λ (la) + Λ ( la +∆ϱ ϵ,σ(lb − la) )]∣∣∣∣ ≤ 2 p(γp+ 1) + |Λ′ (la)|q + |Λ′ (lb)|q q , Corollary 4. In the above Theorem 12, if we choose ∆ϱ ϵ,σ(lb − la) = lb − la, we obtain∣∣∣∣ B(γ)Γ(γ) (lb − la) γ+1 [ AB la Iγ {Λ (lb)}+ ABIγlb {Λ (la)} ] − ( (lb − la) γ + (1− γ)Γ(γ) (lb − la) γ+1 ) [Λ (la) + Λ (lb)] ∣∣∣∣ ≤ 2 p(γp+ 1) + |Λ′ (la)|q + |Λ′ (lb)|q q . 6. Pachpatte-Type Inequality via AB Fractional Integral Operator In this section, We study and explore the Pachpatte-type inequality via ABFIO. We enhance this section’s utility through the notes that are provided. M. Tariq et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6556 20 of 28 Theorem 13. Let I ⊆ R be an open and non-empty convex set and la, lb ∈ I with la < la + ∆ϱ ϵ,σ(lb − la). If Λ1,Λ2 : [la, la +∆ϱ ϵ,σ(lb − la)] → R are GCF , Λ1,Λ2 ∈ L [la, la +∆ϱ ϵ,σ(lb − la)], then the following inequality for ABFIO holds 1 [∆ϱ ϵ,σ(lb − la)] γ [ AB la Iγ { Λ1Λ2 ( la +∆ϱ ϵ,σ(lb − la) )} + ABIγ la+∆ϱ ϵ,σ(lb−la) {Λ1Λ2 (la)} ] ≤ γ B(γ)Γ(γ) [ [Λ1 (la) Λ2 (la) + Λ1 (lb) Λ2 (lb)] ( 2 γ(γ + 1)(γ + 2) + 1 γ + 2 ) +2 [Λ1 (la) Λ2 (lb) + Λ1 (lb) Λ2 (la)] (γ + 1)(γ + 2) ] + (1− γ) B(γ) [∆ϱ ϵ,σ(lb − la)] γ [ Λ1 (la) Λ2 (la) +Λ1 ( la +∆ϱ ϵ,σ(lb − la) ) Λ2 ( la +∆ϱ ϵ,σ(lb − la) ) ] , where γ ∈ (0, 1]. Proof. Since Λ1 and Λ2 are GCF on [la, la +∆ϱ ϵ,σ(lb − la)], we get Λ1 ( la + x∆ϱ ϵ,σ(lb − la) ) ≤ (1− x)Λ1 (la) + xΛ1 (lb) (6.1) and Λ2 ( la + x∆ϱ ϵ,σ(lb − la) ) ≤ (1− x)Λ2 (la) + xΛ2 (lb) . (6.2) By multiplying both inequalities 6.1 and 6.2 side by side, we get Λ1 ( la + x∆ϱ ϵ,σ(lb − la) ) Λ2 ( la + x∆ϱ ϵ,σ(lb − la) ) ≤ (1− x)2Λ1 (la) Λ2 (la) + x2Λ1 (lb) Λ2 (lb) + x(1− x) [Λ1 (la) Λ2 (lb) + Λ1 (lb) Λ2 (la)] . (6.3) By multiplying both sides of (6.3) with (1 − x)γ−1 and integrating the resulting in- equality w.r.t. x over [0, 1], we obtain∫ 1 0 (1− x)γ−1Λ1 ( la + x∆ϱ ϵ,σ(lb − la) ) Λ2 ( la + x∆ϱ ϵ,σ(lb − la) ) dx ≤ ∫ 1 0 (1− x)γ−1 [ (1− x)2Λ1 (la) Λ2 (la) + x2Λ1 (lb) Λ2 (lb) +x(1− x) [Λ1 (la) Λ2 (lb) + Λ1 (lb) Λ2 (la)]] dx = Λ1 (la) Λ2 (la) γ + 2 + 2 Λ1 (lb) Λ2 (lb) γ(γ + 1)(γ + 2) + [Λ1 (la) Λ2 (lb) + Λ1 (lb) Λ2 (la)] (γ + 1)(γ + 2) . By changing the variable la+ x∆ϱ ϵ,σ(lb− la) = x, we can write the inequality in (6.4) as 1 [∆ϱ ϵ,σ(lb − la)] γ ∫ la+∆ϱ ϵ,σ(lb−la) la ( la +∆ϱ ϵ,σ(lb − la)− x )γ−1 Λ1(x)Λ2(x)dx M. Tariq et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6556 21 of 28 ≤ Λ1 (la) Λ2 (la) γ + 2 + 2 Λ1 (lb) Λ2 (lb) γ(γ + 1)(γ + 2) + [Λ1 (la) Λ2 (lb) + Λ1 (lb) Λ2 (la)] (γ + 1)(γ + 2) . (6.4) By multiplying the both sides of (6.4) by γ B(γ)Γ(γ) and then adding the term (1−γ) B(γ)[∆ϱ ϵ,σ(lb−la)] γΛ1 (la +∆ϱ ϵ,σ(lb − la)) Λ2 (la +∆ϱ ϵ,σ(lb − la)) to both sides of (6.4) and fi- nally using ABFIO, we get 1 [∆ϱ ϵ,σ(lb − la)] γ [ AB la Iγ { Λ1Λ2 ( la +∆ϱ ϵ,σ(lb − la) )}] ≤ γ B(γ)Γ(γ) [ Λ1 (la) Λ2 (la) γ + 2 + 2 Λ1 (lb) Λ2 (lb) γ(γ + 1)(γ + 2) + [Λ1 (la) Λ2 (lb) + Λ1 (lb) Λ2 (la)] (γ + 1)(γ + 2) ] + (1− γ) B(γ) [∆ϱ ϵ,σ(lb − la)] γΛ1 ( la +∆ϱ ϵ,σ(lb − la) ) Λ2 ( la +∆ϱ ϵ,σ(lb − la) ) . (6.5) Similarly, by multiplying both sides of (6.3) with xγ−1 and integrating the resulting inequality w.r.t. x over [0, 1], we obtain∫ 1 0 xγ−1Λ1 ( la + x∆ϱ ϵ,σ(lb − la) ) Λ2 ( la + x∆ϱ ϵ,σ(lb − la) ) dx ≤ ∫ 1 0 xγ−1 [ (1− x)2Λ1 (la) Λ2 (la) + x2Λ1 (lb) Λ2 (lb) +x(1− x) [Λ1 (la) Λ2 (lb) + Λ1 (lb) Λ2 (la)]] dx = 2 Λ1 (la) Λ2 (la) γ(γ + 1)(γ + 2) + Λ1 (lb) Λ2 (lb) γ + 2 + [Λ1 (la) Λ2 (lb) + Λ1 (lb) Λ2 (la)] (γ + 1)(γ + 2) . By making calculations similar to those in the proof of (6.5), we obtain 1 [∆ϱ ϵ,σ(lb − la)] γ [ ABIγ la+∆ϱ ϵ,σ(lb−la) {Λ1Λ2 (la)} ] ≤ γ B(γ)Γ(γ) [ 2 Λ1 (la) Λ2 (la) γ(γ + 1)(γ + 2) + 2 Λ1 (lb) Λ2 (lb) γ + 2 + [Λ1 (la) Λ2 (lb) + Λ1 (lb) Λ2 (la)] (γ + 1)(γ + 2) ] + (1− γ) B(γ) [∆ϱ ϵ,σ(lb − la)] γΛ1 (la) Λ2 (la) . (6.6) Adding (6.5) and (6.6) side by side, we get 1 [∆ϱ ϵ,σ(lb − la)] γ [ AB la Iγ { Λ1Λ2 ( la +∆ϱ ϵ,σ(lb − la) )} +ABIγ la+∆ϱ ϵ,σ(lb−la) {Λ1Λ2 (la)} ] ≤ γ B(γ)Γ(γ) [ [Λ1 (la) Λ2 (la) + Λ1 (lb) Λ2 (lb)] ( 2 γ(γ + 1)(γ + 2) + 1 γ + 2 ) +2 [Λ1 (la) Λ2 (lb) + Λ1 (lb) Λ2 (la)] (γ + 1)(γ + 2) ] M. Tariq et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6556 22 of 28 + (1− γ) B(γ) [∆ϱ ϵ,σ(lb − la)] γ [ Λ1 (la) Λ2 (la) +Λ1 ( la +∆ϱ ϵ,σ(lb − la) ) Λ2 ( la +∆ϱ ϵ,σ(lb − la) ) ] . The proof is completed. Remark 9. Considering Theorem 13, we establish the following new mathematical ap- proach of Pachpatte-type inequality pertaining to the classical Mittag-Leffler function via ABFIO if we pick ϱ = (1, 1, ...) with ϵ =α and σ = 1: 1 [Ω (lb, la, )] γ [ AB la Iγ {Λ1Λ2 (la + Eα(lb − la))}+ ABIγla+Eα(lb−la) {Λ1Λ2 (la)} ] ≤ γ B(γ)Γ(γ) [ [Λ1 (la) Λ2 (la) + Λ1 (lb) Λ2 (lb)] ( 2 γ(γ + 1)(γ + 2) + 1 γ + 2 ) +2 [Λ1 (la) Λ2 (lb) + Λ1 (lb) Λ2 (la)] (γ + 1)(γ + 2) ] + (1− γ) B(γ) [Eα(lb − la)] γ [ Λ1 (la) Λ2 (la) +Λ1 (la + Eα(lb − la)) Λ2 (la + Eα(lb − la)) ] . Also, in the above Theorem 13, if we put ∆ϱ ϵ,σ(lb − la) = lb − la, then we get the new variant of Pachpatte-type integral inequality involving convexity via ABFIO. Theorem 14. If Λ1,Λ2 : [la, lb] → R are convex functions, Λ1,Λ2 ∈ L [la, lb], then the following inequality for ABFIO holds 1 (lb − la) γ [ AB la Iγ {Λ1Λ2 (lb)}+ ABIγlb {Λ1Λ2 (la)} ] ≤ γ B(γ)Γ(γ) [ [Λ1 (la) Λ2 (la) + Λ1 (lb) Λ2 (lb)] ( 2 γ(γ + 1)(γ + 2) + 1 γ + 2 ) +2 [Λ1 (la) Λ2 (lb) + Λ1 (lb) Λ2 (la)] (γ + 1)(γ + 2) ] + (1− γ) B(γ) (lb − la) γ [Λ1 (la) Λ2 (la) + Λ1 (lb) Λ2 (lb)] . 7. Applications to entropy Entropy is important because it provides a fundamental measure of uncertainty, dis- order, or information content across various scientific disciplines. It also has a strong relation with integral inequalities, particularly in the study of convex functions and func- tional analysis. For example, many entropy-based inequalities, such as the Gibbs inequal- ity or logarithmic Sobolev inequalities, are special cases or extensions of classical integral M. Tariq et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6556 23 of 28 inequalities. These relationships allow researchers to analyze the behavior of complex systems by connecting probabilistic concepts with analytical tools. This interplay be- tween entropy and integral inequalities is crucial in various fields, including mathematical physics, optimization, statistical mechanics, and information geometry, where it helps to derive bounds, characterize stability, and study the distribution of measures. Consider the exponential random variable Eλ with parameter λ > 0, whose probability density function (PDF) is f(x) = λe−λx, x ≥ 0. The Shannon entropy of Eλ is defined by H(Eλ) = − ∫ ∞ 0 f(x) log f(x) dx = − ∫ ∞ 0 λe−λx log ( λe−λx ) dx. Compute the Shannon entropy explicitly using standard integration, H(Eλ) = − ∫ ∞ 0 λe−λx ( log λ− λx ) dx = − log λ ∫ ∞ 0 λe−λx dx+ λ ∫ ∞ 0 λxe−λx dx = − log λ · 1 + λ · 1 λ = 1− log λ. Define the function Λ related to the integrand and consider Λ(x) := − log f(x) = − log ( λe−λx ) = λx− log λ, which is linear (hence convex) on [0,∞). Choose the interval and Mittag-Leffler deformation and restrict the domain to the compact interval [la, lb] := [ 0, 1 λ ] . Define the Mittag-Leffler type deformation ∆ϱϵ,σ(lb) := ∞∑ k=0 ϱ(k) Γ(ϵk + σ) ( 1 λ )k , with parameters ϱ(k) = 1, ϵ = α ∈ (0, 1), σ = 1, so that ∆ (1) α,1 ( 1 λ ) = ∞∑ k=0 ( 1 λ )k Γ(1 + αk) =: Eα ( 1 λ ) , the classical Mittag-Leffler function. M. Tariq et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6556 24 of 28 Applying Theorem 8 to Λ. From the integral inequality theorem for generalized-convex functions and AB-fractional integrals, we have the double inequality: Λ ( 2la + ∆ ϱ ϵ,σ(lb) 2 ) ≤ B(γ)Γ(γ) 2 [∆ϱϵ,σ(lb)] γ [ 0∆ϱϵ,σ(lb)Λ ( ∆ϱϵ,σ(lb) ) + ∆ϱϵ,σ(lb)0Λ(0) ] (7.1) − (1− γ)Γ(γ) 2 [∆ϱϵ,σ(lb)] γ [ Λ(0) + Λ ( ∆ϱϵ,σ(lb) )] ≤ Λ(0) + Λ(lb) 2 . Substitute the values: la = 0, lb = 1 λ , ∆ϱϵ,σ(lb) = Eα ( 1 λ ) , and recall Λ(x) = λx− log λ. Thus, Λ ( Eα ( 1 λ ) 2 ) = λ · Eα ( 1 λ ) 2 − log λ, and Λ(0) + Λ ( 1 λ ) 2 = (− log λ) + (1− log λ) 2 = 1− 2 log λ 2 . Therefore, inequality (7.1) becomes λ · Eα ( 1 λ ) 2 − log λ ≤ (fractional AB integral expression) ≤ 1− 2 log λ 2 . This inequality provides fractional integral bounds on the entropy-related function Λ, linking fractional calculus, Mittag-Leffler functions, and information-theoretic measures. 8. Conclusions Fractional calculus has attracted considerable attention from researchers and scholars across a wide range of disciplines. At the same time, convexity theory has emerged as a powerful analytical framework for constructing novel numerical models that address com- plex problems in both pure and applied sciences. The growing interest in convex analysis and its related inequalities is driven by ongoing theoretical advancements, generalizations, and diverse applications. In this work, we first present a novel approach to the Hermite– Hadamard inequality via generalized convex functions (GCF) over the Atangana–Baleanu fractional integral operator (ABFIO), accompanied by several remarks and corollaries. Secondly, we establish a new identity involving the Raina function and derive refined ver- sions of the Hermite–Hadamard inequality using classical tools such as Hölder’s inequality, the power mean inequality, and Young’s inequality. Thirdly, we propose a new modifi- cation of a Pachpatte-type inequality based on a recently introduced concept within the M. Tariq et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6556 25 of 28 ABFIO framework. Moreover, the established inequalities have potential implications in the contexts of interval analysis and quantum calculus. Integral inequalities, in particular, represent a rapidly evolving area of research. 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