EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 17, No. 4, 2024, 4014-4049 ISSN 1307-5543 – ejpam.com Published by New York Business Global Some New Fractional Hermite-Hadamard type Inequalities For Generalized Class of Godunova-Levin Functions By Means of Interval Center-Radius Order Relation with Applications Waqar Afzal1, Mehreen S. Khan2, Mutum Zico Meetei3, Mujahid Abbas4,5, Jorge E. Maćıas-Dı́az6,7,∗, Hector Vargas-Rodŕıguez8 1 Department of Mathematics, Government College University, Katchery Road, Lahore 54000, Pakistan 2 Department of Mathematics, Faculty of Science, Jazan University, Jazan 45142, Saudi Arabia 3 Department of Mathematics, College of Science, Jazan University, P.O. Box 114, Jazan 45142, Saudi Arabia 4 Department of Mechanical Engineering Sciences, Faculty of Engineering and the Built Environment, Doornfontein Campus, University of Johannesburg, South Africa 5 Department of Medical Research, China Medical University, Taichung 406040, Taiwan 6 Department of Mathematics, School of Digital Technologies, Tallinn University, Narva Rd. 25, 10120 Tallinn, Estonia 7 Department of Mathematics and Physics, Autonomous University of Aguascalientes, Ave. Universidad 940, Ciudad Universitaria, Aguascalientes 20100, Mexico 8 Department of Exact Sciences and Technology, Los Lagos University Center, University of Guadalajara, Jalisco, Mexico Abstract. The purpose of this article is to establish several new forms of Hermite-Hadamard inequalities by utilizing fractional integral operators via a totally interval midpoint-radius order relation for differentiable Godunova-Levin mappings. Moreover, in order to verify our main results, we construct some non-trivial examples and remarks that lead to other generalized convex mappings with different settings. Furthermore, we exploit special cases of Hölder’s, Young’s, and Minkowski- type inequalities in order to develop new bounds of Hermite-Hadamard inequality. Finally, we relate our key results with special means and demonstrate some of their applications. 2020 Mathematics Subject Classifications: 11B73, 11B83 Key Words and Phrases: Hermite-Hadamard, cr-order, Fractional Operators, Interval mappings ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v17i4.5594 Email addresses: waqar2989@gmail.com (W. Afzal), mskhan@jazanu.edu.sa (M. S. Khan), mmeetei@jazanu.edu.sa (M. Z. Meetei), abbas.mujahid@gmail.com (M. Abbas), jemacias@correo.uaa.mx (J. E. Maćıas-Dı́az), hvargas@culagos.udg.mx (H. Vargas-Rodriguez) https://www.ejpam.com 4014 Copyright: © 2024 The Author(s). (CC BY-NC 4.0) J. E. Maćıas-Dı́az et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 4014-4049 4015 1. Introduction Convex analysis provides a powerful mathematical framework for analyzing problems in various fields, especially due to the well-behaved properties of convex sets and functions. Its uses span a variety of fields, including control theory [56], economics [54], machine learning [38], and optimization [53]. In control theory [55], systems are often formulated as convex problems, where the system needs to minimize energy or error subject to dynamic constraints; in signal processing, it aids in the design of codes that minimize transmission errors, enhancing communication reliability [24]. Convex analysis is closely related to economic theory, particularly in the study of utility functions [14], which represent rational consumer preferences where utility increases with consumption, but at a diminishing rate. For more recent applications in diverse disciplines of applied sciences, we refer to [22, 26, 27, 59, 64] and the references therein. Interval analysis is a mathematical methodology that allows numerical algorithms to address uncertainty more rigorously. It has applications in a variety of domains, including numerical computation, global optimization, control systems, engineering, and computer graphics. Borwein et al. [17] initially defined convex interval-valued functions (IVFs) in 1981, and since then, several researchers have extended and promoted different types of convexity by using IVFs. For example, include preinvex [48], harmonic convex [41], Godunova-Levin [5], (h1, h2)-convex [9], log-convex [49], coordinated convex [62], and var- ious others [21, 31, 34, 39, 40, 43, 51, 52] and the references therein. It’s important to remember that the partial order relation defines these convex IVFs, meaning that any two intervals may not be comparable. This indicates that the maximum-minimum problem cannot be solved since it is impossible to determine which of them is the greatest or small- est interval using these orderings. Hu and Wang [23] introduced the cr-order, which takes into account the midpoint and radius of two intervals to address this limitation. This order is total, meaning that any two interval numbers are comparable. In [61] authors provided the appropriate optimization conditions for the constrained optimization issue of interval-valued objective function and provided a novel definition of convex IVF using cr-order. Among the several different types of inequalities, the Hermite-Hadamard type inequal- ity is a fundamental component of convex analysis, offering crucial perspectives and in- struments for theoretical investigation and real-world applications in numerous scientific domains. The inequality is defined as follows: Consider Φ : Ω ⊆ R → R a convex mapping on the interval Ω with ε1, ε2 ∈ Ω. Then, the inequality listed below is true: Φ ( ε1 + ε2 2 ) ≤ 1 ε2 − ε1 ∫ ε2 ε1 Φ(θ) dθ ≤ Φ(ε1) + Φ(ε2) 2 . (1) The Hermite-Hadamard inequality is frequently employed in optimization problems involv- ing convex functions. It helps establish bounds on integrals of convex functions, which can be crucial for finding optimal solutions in various mathematical models. Various authors study this inequality using different methodologies, including different kinds of interval- J. E. Maćıas-Dı́az et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 4014-4049 4016 valued order relations, stochastic and fuzzy-valued mappings, and various kinds of frac- tional operators. For example, in [50], authors used generalized convex mappings, also known as preinvex functions, and developed various variations of Hermite and Hadamard inequalities for interval-valued functions; in [63], authors used h-convex mappings on coor- dinates in the sense of interval-valued functions and developed Hadamard and Bullen type inclusions; in [10], authors used (h1, h2)-Godunova and Levin functions and developed Hermite-Hadamard and Jensen type inequalities; in [35], authors used harmonical con- vex mappings and developed double inequalities using inclusion relation; in [18] Dragomir developed Hermite–Hadamard’s type inequalities for operator convex functions; in [19] au- thors show some new generalization of Hermite–Hadamard and Mercer forms of inequal- ities for geometric–arithmetic convexity by using interval maps. Some other important results and inequalities connected to these employing different types of fractional oper- ators, including Hadamard, Atangana–Baleanu, Caputo–Fabrizio and Riemann–Liouville fractional integrals (see refs. [1, 2, 6, 28, 30, 42, 60]). As the primary focus of this paper is on cr-interval order relations, recent advances in center-radius order relations should be recalled using a different type of convex mapping. In [58], the authors initially presented the idea of cr-order in covex sense. In comparison to other order relations, this relation is more compatible and possesses a various additional characteristics that other interval order relations lack. In [25] authors defined a new class of convex mapping for convex optimizing problems in the context of cr-order based on their work. In response to these discoveries, Liu et al. [36, 37] derived discrete versions of Jensen and Hermite-Hadamard inequalities based on two different types of generalized convex mappings by using cr-order. As a result of using superquadratic functions in a fractional frame of reference via cr-order relations, Khan and Saad [33] developed sev- eral novel bounds for various kinds of double inequalities. To explore entropy and mean characteristics, Fahad et al. [20] used geometric and arithmetic-cr-convex functions. Afzal et al. [4, 8] created different types of discrete Jensen type and Hermite-Hadamard inequality utilizing the conventional Riemann integral operator by using the cr-h-Godunova- Levin function in convex and harmonic convex sense. Theorem 1 (see [4]). Let h : (0, 1) → R+ such that h ( 1 2 ) ̸= 0, and Φ : [ε1, ε2] → R+ I be an cr-h-Godunova-Levin mapping, ε1, ε2 ∈ R+, then the inequality stated below holds true: h ( 1 2 ) 2 Φ ( ε2 + ε1 2 ) ⪯cr 1 ε2 − ε1 ∫ ε2 ε1 Φ(v)dv ⪯cr [Φ(ε1) + Φ(ε2)] ∫ 1 0 d♭ h(♭) . Sahoo et al. [45] employed fractional integral operators to construct the following double relation for cr-convex functions: Theorem 2 (see [45]). Let Φ : [ε1, ε2] → R+ I be an cr-convex function on [ε1, ε2], then the inequality stated below holds true: J. E. Maćıas-Dı́az et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 4014-4049 4017 Φ ( ε2 + ε1 2 ) ⪯cr 2α−1Γ(α+ 1) (ε2 − ε1)α ( J( ε1+ε2 2 )+Φ(ε2) + Jα( ε1+ε2 2 )−Φ(ε1) ) ⪯cr Φ(ε1) + Φ(ε2) 2 Shah et al. [47] employed cr-convex stochastic processes to establish different variations of Hermite-Hadamard type relations in the Mercer sense. Theorem 3 (see [47]). Let Φ : [ε1, ε2]×Ω → R+ I be a γ-convex cr-interval-valued stochastic processes then the inequality stated below holds true:( 1− e−ζ ) γ ( 1 2 ) Φ ( η1 + η2 − ε1 + ε2 2 , . ) ⪯cr 1− β 2 [ Jβ η1+η2−ε−1 [Φ] (η1 + η2 − ε2) + Jβ η1+η2−ε+2 [Φ] (η1 + η2 − ε1) ] ⪯cr [ Φ (η1, .) + Φ (η2, .)− Φ(ε1, .) + Φ(ε2, .) 2 ] ∆. For some additional results and inequalities obtained using other types of generalized convex mappings under center radius order relations connected to developed results, please see the following publications [7, 12, 44, 46] and their references. This study is regarded fresh and significant since it presents new and original conclu- sions using center-radius interval order relations. Furthermore, this is the first time that Hermite-Hadamard and its numerous versions, including product form, weighted form, and employing symmetric mappings, are produced by Atangana-Baleanu fractional inte- gral operators under cr-order relation. Additionally, we utilize a number of additional known results, such as Minkowski, Holder, and Young, in the development of these re- sults. Furthermore, we provide bounds of these inequalities in terms of special functions and many applications in terms of special means. We are especially inspired by the works of these authors [4, 11, 13, 20, 33] to introduced a new and improved form of several inequalities, which undergo multiple improvements and reverses in various circumstances. This note is structured into five parts, starting with an introduction and foundational discussion of the topic related to preliminary. In Section 3, we develop numerous innovative versions of the double inequality, including its product and weighted forms, using various other well-known inequalities. In Section 4, we tie our conclusions to special means and demonstrate their applications. Finally, in Section 5, we provide a precise conclusion and possible future work. 2. Preliminaries In this section, we present some well-known definitions and outcomes that can be uti- lized to support the paper’s core findings. Furthermore, we will go over some fundamental ideas relating to fractional and interval calculus. Some fundamental topics are not fully J. E. Maćıas-Dı́az et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 4014-4049 4018 covered here; thus, we refer to [20]. Prior to proceeding, we correct a few notations that are utilized in the article. • RI : space of intervals in R; • Φ = Φ: interval maps become dysfunctional; • ⊆: inclusion interval order relation; • ⪯cr: cr-interval order relation; • ≤: standard order relation; • IVF: interval-valued function; 2.1. Set-valued Analysis The space containing all subsets of R in n-dimensional interval space RI . RI = {[ε1, ε2] : ε1, ε2 ∈ R and ε1 ≤ ε2}, To define the Hausdorff metric in RI , use this formula: H(ε1, ε2) = max{d(ε1, ε2), d(ε2, ε1)}, (2) where d(ε1, ε2) = supν∈ε1 d(ν, ε2), and d(ν, ε2) = minµ∈ε2 d(ν, µ) = minµ∈ε2 |ν − µ|. Remark 1. The Hausdorff metric (2) can also be expressed as follows: H([ε1, ε1], [ε2, ε2]) = max{|ε1 − ε2|, |ε1 − ε2|}. In interval space, we call this the Moore metric. For instance, if ζ1 = [ε1, ε1] and ζ2 = [ε2, ε2] are two closed intervals, then the Minkowski sum, scalar multiplication, and difference are defined as follows: ζ1 + ζ2 = {ε1 + ε2 | ε1 ∈ ζ1, ε2 ∈ ζ2} and Γζ1 = {Γε1 | ε1 ∈ ζ1}. and ζ1 − ζ2 = [ε1 − ε2, ε1 − ε2], with the product ζ1 · ζ2 = [min{ε1ε2, ε1ε2, ε1ε2, ε1ε2}, sup{ε1ε2, ε1ε2, ε1ε2, ε1ε2}], and the division ζ1 ζ2 = [ min { ε1 ε2 , ε1 ε2 , ε1 ε2 , ε1 ε2 } ,max { ε1 ε2 , ε1 ε2 , ε1 ε2 , ε1 ε2 }] , where 0 /∈ ζ2. J. E. Maćıas-Dı́az et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 4014-4049 4019 Definition 1 (see [4]). For any two intervals the center-radius order relation is defined as ζ1 = [ε1, ε2] = ⟨ωc, ωr⟩ = 〈 ε1+ε2 2 , ε2−ε1 2 〉 , ζ2 = [ε1, ε2] = ⟨Ωc,Ωr⟩ = 〈 ε1+ε2 2 , ε2−ε1 2 〉 , where ζ1 ⪯cr ζ2 ⇐⇒ { ωc < Ωc, if ωc ̸= Ωc; ωr ≤ Ωr, if ωr = Ωr. The relation ⪯cr satisfies the following relational properties for any three intervals ζ1 = [ε1, ε2] = ⟨ωc, ωr⟩, ζ2 = [ε1, ε2] = ⟨Ωc,Ωr⟩ and ζ3 = [η1, η2] = ⟨ηc, ηr⟩ : Reflexivity: ζ1 ⪯cr ζ1. Anti-symmetry: ζ1 ⪯cr ζ2 and ζ2 ⪯cr ζ1. Transitivity: ζ1 ⪯cr ζ2 and ζ2 ⪯cr ζ3 then ζ1 ⪯cr ζ3 . Comparability: ζ2 ⪯cr ζ3 or ζ3 ⪯cr ζ2. Theorem 4 (see [4]). Let Φ : [ε1, ε2] → R+ I be an interval set-valued map given by Φ = [Φ, Φ̄]. Then the Φ is Riemann integrable on [ε1, ε2] iff Φ and Φ̄ are Riemann integrable on [ε1, ε2] and∫ ε2 ε1 Φ(e)de = [∫ ε2 ε1 Φ(e)de, ∫ ε2 ε1 Φ̄(e)de ] We shall refer to the set of all Riemann integrable interval-valued maps on [ε1, ε2] as IR([ε1,ε2]). Theorem 5 (see [4]). Let Φ,H : [ε1, ε2] → R+ I given by Φ = [Φ, Φ̄], and H = [H , H̄ ]. If Φ,H ∈ IR([ε1,ε2]), and Φ(e) ⪯cr H (e), ∀ e ∈ [ε1, ε2], then∫ ε2 ε1 Φ(e)de ⪯cr ∫ ε2 ε1 H (e)de. Example 1. Consider Φ = [v + 1, 2v + 2] and H = [v2 + 2, 3v + 2], ∀ v ∈ [0, 1]. Φc = 3v + 3 2 ,Φr = v + 1 2 ,Hc = v2 + 3v + 4 2 and Hr = 3v − v2 2 . From Definition 1, we have Φ(v) ⪯cr H (v), ∀ v ∈ [0, 1]. Since, ∫ 1 0 [v + 1, 2v + 2]dv = [ 3 2 , 3 ] . and ∫ 1 0 [v2 + 2, 2v + 2]dv = [ 7 3 , 7 2 ] From Theorem 5, we have ∫ 1 0 Φ(v)dv ⪯cr ∫ 1 0 H (v)dv. J. E. Maćıas-Dı́az et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 4014-4049 4020 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 1 v v a lu es v2 2 + v v2 + 2v v3 3 + 2v 3v2 2 + 2v Figure 1: Graphical validation of Theorem 5 . Definition 2 (see [4]). Let Φ : [ε1, ε2] → R+ I be an cr set-valued map given by Φ = [Φ, Φ̄]; then, Φ is said to be cr-convex if Φ(♭ε1 + (1− ♭)ε2) ⪯cr ♭Φ(ε1) + (1− ♭)Φ(ε2), holds for all ε1, ε2 ∈ B ⊂ R and ♭ ∈ [0, 1]. Definition 3 (see [4]). Let Φ : [ε1, ε2] → R+ I be an cr set-valued map given by Φ = [Φ, Φ̄] and h : (0, 1) → R be non-negative function; then, Φ is said to be cr-h-convex if Φ(♭ε1 + (1− ♭)ε2) ⪯cr h(♭)Φ(ε1) + h(1− ♭)Φ(ε2), holds for all ε1, ε2 ∈ B ⊂ R and ♭ ∈ (0, 1). Definition 4 (see [4]). Let Φ : [ε1, ε2] → R+ I be an cr set-valued map given by Φ = [Φ, Φ̄] and h : (0, 1) → R be non-negative function; then, Φ is said to be cr-h-Godunova-Levin if Φ(♭ε1 + (1− ♭)ε2) ⪯cr Φ(ε1) h(♭) + Φ(ε2) h(1− ♭) , holds for all ε1, ε2 ∈ B ⊂ R and ♭ ∈ (0, 1). The class of all cr-h-Godunova-Levin convex mappings are denoted by SGX(h, [ε1, ε2],R + I ). Remark 2. • If h(♭) = 1 ♭s , then Definition 4 recovers cr-s-convex functions in [4]. • If h(♭) = 1, then Definition 4 recovers cr-p-functions in [4]. • If h(♭) = 1 ♭ , then Definition 4 recovers cr-convex functions in [45] Definition 5 (see [15]). Let Φ : [ε1, ε2] → R+ I be an set-valued map given by Φ = [Φ, Φ̄]. The interval-valued left-sided and right-sided Atangana-Baleanu fractional integral of function Φ and order ς > 0 is defined by J. E. Maćıas-Dı́az et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 4014-4049 4021 AB ε1I ς t{Φ(t)} = 1− ς B(ς) Φ(t) + ς B(ς)Γ(ς) ∫ t ε1 Φ(♭)(t− ♭)ς−1 d♭, ABIςε2{Φ(t)} = 1− ς B(ς) Φ(t) + ς B(ς)Γ(ς) ∫ ε2 t Φ(♭)(♭− t)ς−1 d♭, where ε1 < ε2, ς ∈ (0, 1],Γ(♭) = ∫∞ 0 t♭−1e−t dt is the special function, B(ς) > 0 such that B(0) = B(1) = 1, ∥B(ς)∥ = 1, and βa = βa(p, q) = ∫ a 0 ♭p−1(1− ♭)q−1 d♭ is the beta integral in incomplete sense. Theorem 6 (see [4]). Let Φ : [ε1, ε2] → R+ I be an interval set-valued map given by Φ = [Φ, Φ̄], then we have AB ε1I ς t{Φ(t)} = [ AB ε1I ς t{Φ(t)}, ABε1I ς t{Φ(t)} ] and ABIςε2{Φ(t)} = [ ABIςε2{Φ(t)}, ABIςε2{Φ(t)} ] . The following inequalities are frequently used to produce our major results. Theorem 7 (see [3]). (Hölder inequality). Let 1 < p and 1 p + 1 q = 1. Consider two real-valued functions Φ and ג on [ε1, ε2] with |Φ|p, q|ג| are also integrable on [ε1, ε2], then one has ∫ ε2 ε1 |Φ(♭)ג(♭)|d♭ ≤ (∫ ε2 ε1 |Φ(♭)|pd♭ ) 1 p (∫ ε2 ε1 ♭qd|(♭)ג| ) 1 q Another generalized variant of Hölder’s inequality is defined as follows. Theorem 8 (see [16]). Let two real-valued functions Φ and ג on [ε1, ε2] and |Φ|, |Φ||ג|q are also integrable on [ε1, ε2], then one has∫ ε2 ε1 |Φ(♭)ג(♭)|d♭ ≤ (∫ ε2 ε1 |Φ(♭)|d♭ )1− 1 q (∫ ε2 ε1 |Φ(♭)||ג(♭)|qd♭ ) 1 q . Theorem 9 (see [16]). (Young’s inequality). Consider p, q be positive real numbers satisfying 1 p + 1 q = 1. Then if Φ,H are nonnegative functions then we have, G H ≤ Φp p + H q q , and equality holds iff Φp = H q. The following two below Lemmas [29] also play a very crucial role in creating our main findings. [see [29]] J. E. Maćıas-Dı́az et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 4014-4049 4022 Let Φ : B◦ ⊂ R → R is a differentiable mapping on B◦, where ε1, ε2 ∈ B◦, with ε1 < ε2. If Φ ′ ∈ L[ε1, ε2] (space of all measurable function), then one has Bk(Φ, ε1, ε2) = k−1∑ ȷ=0 1 2k [ Φ ( (k−ȷ)ε1 + ȷε2 k ) +Φ ( (k−ȷ− 1)ε1 + (ȷ+ 1)ε2 k )] − 1 ε2 − ε1 ∫ ε2 ε1 Φ(ς)dς = k−1∑ ȷ=0 ε2 − ε1 2k2 [∫ 1 0 (1− 2♭)Φ′ ( ♭ (k−ȷ)ε1 + ȷε2 k +(1− ♭) (k−ȷ− 1)ε1 + (ȷ+ 1)ε2 k ) d♭ ] . holds. [see [32]] Let ε1 < ε2, ε1, ε2 ∈ R+,Φ : R+ → R+ is a differentiable mapping. If Φ′′ ∈ L[ε1, ε2], for each ς ∈ (0, 1], then one has 1 ε2 − ε1 [ ABIςε2+ε1 2 {Φ(ε1)}+ AB ε2+ε1 2 Iςε2{Φ(ε2)} ] − 1 (ε2 − ε1)B(ς) [Φ(ε1) + Φ(ε2)]− (ε2 − ε1) ς−1 2ς−1B(ς)Γ(ς) Φ ( ε2 + ε1 2 ) = (ε2 − ε1) ς−1 2(ς + 1)B(ς)Γ(ς) ∫ 1 0 wς(♭) [ Φ′′(♭ε1 + (1− ♭)ε2) + Φ′′(♭ε2 + (1− ♭)ε1) ] d♭, where wς(♭) = { ♭ς+1, ♭ ∈ [ 0, 12 ) , (1− ♭)ς+1, ♭ ∈ [ 1 2 , 1 ] . In [57], the authors introduced these type of inequalities that utilize the s-convexity with the help of Lemma 2.1. Theorem 10. Let Φ : B ⊂ R → R is a differentiable mapping on B◦, where ε1, ε2 ∈ B◦, with ε1 < ε2. If |Φ′|q is s-convex on [ε1, ε2] for some q > 1, then one has |Bk(Φ, ε1, ε2)| ≤ k−1∑ ȷ=0 ε2 − ε1 2k2 ( 1 p+ 1 ) 1 p ( 1 s+ 1 ) 1 q × [∣∣∣∣Φ′ ( (k−ȷ)ε1 + ȷε2 k )∣∣∣∣q + ∣∣∣∣Φ′ ( (k−ȷ− 1)ε1 + (ȷ+ 1)ε2 k )∣∣∣∣q] 1 q holds, where 1 p + 1 q = 1. J. E. Maćıas-Dı́az et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 4014-4049 4023 3. The main results This section uses the cr-h-Godunova-Levin function to build multiple forms of Hermite- Hadamard inequality, with several particular cases. Theorem 11. Let h : (0, 1) → R+ and h ̸= 0. Let Φ : [ε1, ε2] → R+ I is cr-h-Godunova- Levin mapping, ε1, ε2 ∈ R+, ε1 < ε2. If Φ ∈ L[ε1, ε2], then the following relation holds true: h ( 1 2 ) (ε2 − ε1) ς B(ς)Γ(ς) Φ ( ε2 + ε1 2 ) + 1− ς B(ς) [ Φ(ε1) + Φ(ε2) ] ⪯cr AB ε1I ς ε2{Φ(ε2)}+ ABIςε2{Φ(ε1)} ⪯cr [ Φ(ε1) + Φ(ε2) B(ς) ] [ 1− ς + ς(ε2 − ε1) ς Γ(ς) × ∫ 1 0 ♭ς−1 ( 1 h(♭) + 1 h(1− ♭) ) d♭ ] , (3) where ς ∈ (0, 1). Proof. As Φ ∈ SGX(h, [ε1, ε2],R + I ), we have Φ ( ν1 + ν2 2 ) ⪯cr 1[ h ( 1 2 )] [Φ(ν1) + Φ(ν2)] , let ν1 = ♭ε1 + (1− ♭)ε2, ν2 = ♭ε2 + (1− ♭)ε1, the above relation becomes as h ( 1 2 ) Φ ( ε2 + ε1 2 ) ⪯cr [ Φ(♭ε1 + (1− ♭)ε2) + Φ(♭ε2 + (1− ♭)ε1) ] . (4) Multiplying by ♭ς−1 in (4) and integrating, we have 1 ς Φ ( ε2 + ε1 2 ) ⪯cr 1 h ( 1 2 ) [∫ 1 0 ♭ς−1Φ(♭ε1 + (1− ♭)ε2)d♭ + ∫ 1 0 ♭ς−1Φ(♭ε2 + (1− ♭)ε1)d♭ ] , that is h ( 1 2 ) ς Φ ( ε2 + ε1 2 ) ⪯cr ∫ 1 0 ♭ς−1Φ(♭ε1 + (1− ♭)ε2)d♭ + ∫ 1 0 ♭ς−1Φ(♭ε2 + (1− ♭)ε1)d♭. J. E. Maćıas-Dı́az et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 4014-4049 4024 Multiplying the above relation with ς(ε2−ε1)ς B(ς)Γ(ς) and adding the expression 1−ς B(ς) [ Φ(ε1)+Φ(ε2) ] , we get that h ( 1 2 ) (ε2 − ε1) ς B(ς)Γ(ς) Φ ( ε2 + ε1 2 ) + 1− ς B(ς) [ Φ(ε1) + Φ(ε2) ] ⪯cr ς(ε2 − ε1) ς B(ς)Γ(ς) ∫ 1 0 ♭ς−1Φ(♭ε1 + (1− ♭)ε2)d♭ + ς(ε2 − ε1) ς B(ς)Γ(ς) ∫ 1 0 ♭ς−1Φ(♭ε2 + (1− ♭)ε1)d♭+ 1− ς B(ς) [ Φ(ε1) + Φ(ε2) ] . In the final two integrals of the preceding relation, let a = ♭ε1 + (1 − ♭)ε2 and B = ♭ε2 + (1− ♭)ε1 respectively, we have h ( 1 2 ) (ε2 − ε1) ς B(ς)Γ(ς) Φ ( ε2 + ε1 2 ) + 1− ς B(ς) [ Φ(ε1) + Φ(ε2) ] ≤ AB ε1I ς ε2{Φ(ε2)}+ ABIςε2{Φ(ε1)}, so the first relation of (3) holds. Taking into account Definition 4, we have Φ(♭ε1 + (1− ♭)ε2) ⪯cr Φ(ε1) h(♭) + Φ(ε2) h(1− ♭) . . Multiplying aforementioned result with ♭ς−1, and integrating, we have∫ 1 0 ♭ς−1Φ(♭ε1 + (1− ♭)ε2)d♭ ⪯cr Φ(ε1) ∫ 1 0 ♭ς−1d♭ h(♭) + Φ(ε2) ∫ 1 0 ♭ς−1d♭ h(1− ♭) . (5) Multiplying both sides of (5) by ς(ε2−ε1)ς B(ς)Γ(ς) and adding the expression 1−ς B(ς)Φ(ε2) to both sides of the desired relation, we get ς(ε2 − ε1) ς B(ς)Γ(ς) ∫ 1 0 ♭ς−1Φ(♭ε1 + (1− ♭)ε2)d♭+ 1− ς B(ς) Φ(ε2) ⪯cr ς(ε2 − ε1) ς B(ς)Γ(ς) [ Φ(ε1) ∫ 1 0 ♭ς−1d♭ h(♭) +Φ(ε2) ∫ 1 0 ♭ς−1d♭ h(1− ♭) ] + 1− ς B(ς) Φ(ε2). (6) Making a modification in the previous integral of the preceding relation, a = ♭ε1+(1−♭)ε2, then the above relation become as AB ε1I ς ε2{Φ(ε2)} ⪯cr ς(ε2 − ε1) ς B(ς)Γ(ς) [ Φ(ε1) ∫ 1 0 ♭ς−1d♭ h(♭) +Φ(ε2) ∫ 1 0 ♭ς−1d♭ h(1− ♭) ] + 1− ς B(ς) Φ(ε2). (7) J. E. Maćıas-Dı́az et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 4014-4049 4025 Again by Definition 4, we have Φ(♭ε1 + (1− ♭)ε2) ⪯cr Φ(ε1) h(♭) + Φ(ε2) h(1− ♭) . Multiplying aforementioned relation with ♭ς−1, and integrating, we have ς(ε2 − ε1) ς B(ς)Γ(ς) ∫ 1 0 ♭ς−1Φ(♭ε2 + (1− ♭)ε1)d♭+ 1− ς B(ς) Φ(ε1) ⪯cr ς(ε2 − ε1) ς B(ς)Γ(ς) [ Φ(ε2) ∫ 1 0 ♭ς−1d♭ h(♭) +Φ(ε1) ∫ 1 0 ♭ς−1d♭ h(1− ♭) ] + 1− ς B(ς) Φ(ε1). Making a modification in the previous integral of the preceding relation with some dummy variable , b = ♭ε2 + (1− ♭)ε1, then the above relation becomes ABIςε2{Φ(ε1)} ⪯cr ς(ε2 − ε1) ς B(ς)Γ(ς) [ Φ(ε2) ∫ 1 0 ♭ς−1d♭ h(♭) +Φ(ε1) ∫ 1 0 ♭ς−1d♭ h(1− ♭) ] + 1− ς B(ς) Φ(ε1). (8) Adding (7) and (8), we can get that the second relation of (3). This finishes the proof. Example 2. Let Φ : [1, 4] → R+ I defined as Φ(µ) = [ 2eµ + 1, 3eµ + √ µ 3 ] with h(♭) = 1 ♭ , ς = 1 2 , then we have h ( 1 2 ) (ε2 − ε1) ς B(ς)Γ(ς) Φ ( ε2 + ε1 2 ) + 1− ς B(ς) [ Φ(ε1) + Φ(ε2) ] ≈ [81.77390, 103.32659], AB ε1I ς ε2{Φ(ε2)}+ ABIςε2{Φ(ε1)} ≈ [90.33565, 131.54364]. and [ Φ(ε1) + Φ(ε2) B(ς) ] [ 1− ς + ς(ε2 − ε1) ς Γ(ς) × ∫ 1 0 ♭ς−1 ( 1 h(♭) + 1 h(1− ♭) ) d♭ ] = [ e4 + e+ √ 3 √ π ( 2e4 + 2e+ 2 ) 3π + 1, ( 3e4 + 3e+ 1 )(1 2 + √ 3 √ π 3π )] ≈ [96.30783, 142.81028]. J. E. Maćıas-Dı́az et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 4014-4049 4026 Thus, we have [81.77390, 103.32659] ⪯cr [90.33565, 131.54364] ⪯cr [96.30783, 142.81028]. Consequently, Theorem 11 is correct. The different types of settings allow us to get results for other types of generalized convex mappings, as described in the remark below. Remark 3. (i) If h(♭) = 1 ♭s , then Theorem 11 yields an outcome for the cr-s-convex function for AB integral operators: 2s (ε2 − ε1) ς B(ς)Γ(ς) Φ ( ε2 + ε1 2 ) + 1− ς B(ς) [ Φ(ε1) + Φ(ε2) ] ≤ AB ε1I ς ε2{Φ(ε2)}+ ABIςε2{Φ(ε1)} ≤ [ Φ(ε1) + Φ(ε2) B(ς) ] [ 1− ς + ς(ε2 − ε1) ς Γ(ς)(s+ς) + ς(ε2 − ε1) ς Γ(ς) Γ(ς)Γ(s+1) Γ(ς + s+2) ] . (9) (ii) If h(♭) = 1, then Theorem 11 yields an outcome for the cr-p-convex function for AB integral operators: (ε2 − ε1) ς B(ς)Γ(ς) Φ ( ε2 + ε1 2 ) + 1− ς B(ς) [ Φ(ε1) + Φ(ε2) ] ≤ AB ε1I ς ε2{Φ(ε2)}+ ABIςε2{Φ(ε1)} ≤ [ Φ(ε1) + Φ(ε2) B(ς) ] [ 1− ς + 2(ε2 − ε1) ς Γ(ς) ] . (10) Theorem 12. Let h : (0, 1) → R+ and h ̸= 0. Let Φ : [ε1, ε2] → R+ I is cr-h-Godunova- Levin mapping, ε1, ε2 ∈ R+, ε1 < ε2 and ג : [ε1, ε2] → R+ is symmetric about ε1+ε2 2 . If Φ ∈ L[ε1, ε2], then the following relation holds true: h ( 1 2 ) 2 Φ ( ε2 + ε1 2 )[ AB ε1I ς ε2{ג(ε2)}+ ABIςε2{ג(ε1)} ] − h ( 1 2 ) 2 Φ ( ε2 + ε1 2 ) 1− ς B(ς) [ (ε1)ג + (ε2)ג ] + 1− ς B(ς) [ Φ(ε1)ג(ε1) + Φ(ε2)ג(ε2) ] ≤ AB ε1I ς ε2{(Φג(ε2)}+ ABIςε2{(Φג(ε1)} ⪯cr ς(ε2 − ε1) ς B(ς)Γ(ς) [ Φ(ε1) + Φ(ε2) ] × ∫ 1 0 ♭ς−1 [ 1 h(♭) + 1 h(1− ♭) ] ε2♭)ג + (1− ♭)ε1)d♭ + 1− ς B(ς) [ Φ(ε1)ג(ε1) + Φ(ε2)ג(ε2) ] , (11) where ς ∈ (0, 1]. J. E. Maćıas-Dı́az et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 4014-4049 4027 Proof. As Φ ∈ SGX(h, [ε1, ε2],R + I ), we have Φ ( ε2 + ε1 2 ) ≤ 1 h ( 1 2 )[Φ(♭ε1 + (1− ♭)ε2) + Φ(♭ε2 + (1− ♭)ε1) ] . (12) Multiplying above relation with h ( 1 2 ) ♭ς−1ג(♭ε2 + (1 − ♭)ε1), and integrating the desired relation over (0, 1), we have h ( 1 2 ) Φ ( ε2 + ε1 2 )∫ 1 0 ♭ς−1ג(♭ε2 + (1− ♭)ε1)d♭ ≤ ∫ 1 0 ♭ς−1 [ Φ(♭ε1 + (1− ♭)ε2) + Φ(♭ε2 + (1− ♭)ε1) ] ε2♭)ג + (1− ♭)ε1)d♭. Let u = ♭ε2 + (1− ♭)ε1, then the above relation becomes h ( 1 2 ) 1 (ε2 − ε1)ς Φ ( ε2 + ε1 2 )∫ ε2 ε1 (u− ε1) ς−1ג(u)du ⪯cr 1 (ε2 − ε1)ς [∫ ε2 ε1 (u− ε1) ς−1Φ(ε2 + ε1 − u)ג(u)du + ∫ ε2 ε1 (u− ε1) ς−1Φ(u)ג(u)du ] . Making a modification in the previous integral of the preceding relation, v = ε2+ε1−u, from ε2)ג + ε1 − v) = ,(v)ג one has h ( 1 2 ) 1 (ε2 − ε1)ς Φ ( ε2 + ε1 2 )∫ ε2 ε1 (u− ε1) ς−1ג(u)du ≤ 1 (ε2 − ε1)ς [∫ ε2 ε1 (ε2 − v)ς−1Φ(v)ג(v)dv+ ∫ ε2 ε1 (u− ε1) ς−1Φ(u)ג(u)du ] . Multiplying above relation with ς(ε2−ε1)ς B(ς)Γ(ς) and adding the expression 1−ς B(ς) [ Φ(ε1) + Φ(ε2) ] to both sides of the desired results, we get that h ( 1 2 ) ς B(ς)Γ(ς) Φ ( ε2 + ε1 2 )∫ ε2 ε1 (u− ε1) ς−1ג(u)du+ 1− ς B(ς) [ Φ(ε1)ג(ε1) + Φ(ε2)ג(ε2) ] ⪯cr ς B(ς)Γ(ς) [∫ ε2 ε1 (ε2 − v)ς−1Φ(v)ג(v)dv+ ∫ ε2 ε1 (u− ε1) ς−1Φ(u)ג(u)du ] + 1− ς B(ς) [ Φ(ε1)ג(ε1) + Φ(ε2)ג(ε2) ] . From this, it can be follows as h ( 1 2 ) Φ ( ε2 + ε1 2 ) ABIςε2{ג(ε1)} − h ( 1 2 ) Φ ( ε2 + ε1 2 ) 1− ς B(ς) (ε1)ג + 1− ς B(ς) [ Φ(ε1)ג(ε1) + Φ(ε2)ג(ε2) ] ⪯cr AB ε1I ς ε2{(Φג(ε2)}+ ABIςε2{(Φג(ε1)}. (13) J. E. Maćıas-Dı́az et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 4014-4049 4028 Similarly, multiplying h ( 1 2 ) ♭ς−1ג(♭ε1+(1− ♭)ε2) on both sides of (12) and integrating, we have h ( 1 2 ) Φ ( ε2 + ε1 2 )∫ 1 0 ♭ς−1ג(♭ε1 + (1− ♭)ε2)d♭ ≤ ∫ 1 0 ♭ς−1 [ Φ(♭ε1 + (1− ♭)ε2) + Φ(♭ε2 + (1− ♭)ε1) ] ε1♭)ג + (1− ♭)ε2)d♭. Let u = ♭ε1 + (1− ♭)ε2, then the above result becomes h ( 1 2 ) 1 (ε2 − ε1)ς Φ ( ε2 + ε1 2 )∫ ε2 ε1 (ε2 − u)ς−1ג(u)du ⪯cr 1 (ε2 − ε1)ς [∫ ε2 ε1 (ε2 − u)ς−1Φ(u)ג(u)du + ∫ ε2 ε1 (ε2 − u)ς−1Φ(ε2 + ε1 − u)ג(u)du ] . Making a modification in the previous integral of the preceding relation, v = ε2 + ε1 − u, ε2)ג + ε1 − v) = ,(v)ג we have h ( 1 2 ) 1 (ε2 − ε1)ς Φ ( ε2 + ε1 2 )∫ ε2 ε1 (ε2 − u)ς−1ג(u)du ≤ 1 (ε2 − ε1)ς [∫ ε2 ε1 (ε2 − u)ς−1Φ(u)ג(u)du+ ∫ ε2 ε1 (v− ε1) ς−1Φ(v)ג(v)dv ] . Multiplying above relation with ς(ε2−ε1)ς B(ς)Γ(ς) and adding the expression 1−ς B(ς) [ Φ(ε1) + Φ(ε2) ] to both sides of the desired result, we get that h ( 1 2 ) ς B(ς)Γ(ς) Φ ( ε2 + ε1 2 )∫ ε2 ε1 (ε2 − u)ς−1ג(u)du + 1− ς B(ς) [ Φ(ε1)ג(ε1) + Φ(ε2)ג(ε2) ] ⪯cr ς B(ς)Γ(ς) [∫ ε2 ε1 (ε2 − u)ς−1Φ(u)ג(u)du+ ∫ ε2 ε1 (v− ε1) ς−1Φ(v)ג(v)dv ] + 1− ς B(ς) [ Φ(ε1)ג(ε1) + Φ(ε2)ג(ε2) ] . From this, it can be follows that h ( 1 2 ) Φ ( ε2 + ε1 2 ) AB ε1I ς ε2{ג(ε2)} − h ( 1 2 ) Φ ( ε2 + ε1 2 ) 1− ς B(ς) (ε2)ג + 1− ς B(ς) [ Φ(ε1)ג(ε1) + Φ(ε2)ג(ε2) ] ⪯cr AB ε1I ς ε2{(Φג(ε2)}+ ABIςε2{(Φג(ε1)}. (14) J. E. Maćıas-Dı́az et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 4014-4049 4029 Adding (13) and (14), we can get the first relation in (11). Now again taking into account Definition 4, we have Φ(♭ε1 + (1− ♭)ε2) ⪯cr Φ(ε1) h(♭) + Φ(ε2) h(1− ♭) , Φ(♭ε2 + (1− ♭)ε1) ⪯cr Φ(ε2) h(♭) + Φ(ε1) h(1− ♭) , adding the above two relations yields that Φ(♭ε1 + (1− ♭)ε2) + Φ(♭ε2 + (1− ♭)ε1) ≤ [ 1 h(♭) + 1 h(1− ♭) ] [ Φ(ε1) + Φ(ε2) ] . Multiplying aforementioned result with ♭ς−1ג(♭ε2+(1− ♭)ε1) and integrating, we have∫ 1 0 ♭ς−1 [ Φ(♭ε1 + (1− ♭)ε2) + Φ(♭ε2 + (1− ♭)ε1) ] ε2♭)ג + (1− ♭)ε1)d♭ ≤ [ Φ(ε1) + Φ(ε2) ] ∫ 1 0 ♭ς−1 [ 1 h(♭) + 1 h(1− ♭) ] ε2♭)ג + (1− ♭)ε1)d♭. Making a modification in the previous integral of the preceding relation, v = ε2 + ε1 − u, from ε2)ג + ε1 − v) = ,(v)ג we have 1 (ε2 − ε1)ς [∫ ε2 ε1 (ε2 − v)ς−1Φ(v)ג(v)dv+ ∫ ε2 ε1 (u− ε1) ς−1Φ(u)ג(u)du ] ≤ [ Φ(ε1) + Φ(ε2) ] ∫ 1 0 ♭ς−1 [ 1 h(♭) + 1 h(1− ♭) ] ε2♭)ג + (1− ♭)ε1)d♭. Multiplying above relation with ς(ε2−ε1)ς B(ς)Γ(ς) and adding the expression 1−ς B(ς) [ Φ(ε1) + Φ(ε2) ] , we have ς B(ς)Γ(ς) [∫ ε2 ε1 (ε2 − v)ς−1Φ(v)ג(v)dv+ ∫ ε2 ε1 (u− ε1) ς−1Φ(u)ג(u)du ] + 1− ς B(ς) [ Φ(ε1)ג(ε1) + Φ(ε2)ג(ε2) ] ⪯cr ς(ε2 − ε1) ς B(ς)Γ(ς) [ Φ(ε1) + Φ(ε2) ] × ∫ 1 0 ♭ς−1 [ 1 h(♭) + 1 h(1− ♭) ] ε2♭)ג + (1− ♭)ε1)d♭ + 1− ς B(ς) [ Φ(ε1)ג(ε1) + Φ(ε2)ג(ε2) ] , that is AB ε1I ς ε2{(Φג(ε2)}+ ABIςε2{(Φג(ε1)} J. E. Maćıas-Dı́az et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 4014-4049 4030 ⪯cr ς(ε2 − ε1) ς B(ς)Γ(ς) [ Φ(ε1) + Φ(ε2) ] × ∫ 1 0 ♭ς−1 [ 1 h(♭) + 1 h(1− ♭) ] ε2♭)ג + (1− ♭)ε1)d♭ + 1− ς B(ς) [ Φ(ε1)ג(ε1) + Φ(ε2)ג(ε2) ] , so the second relation holds true in (11). This concludes the proof. Example 3. Let Φ : [1, 4] → R+ I defined as Φ(µ) = [ 2eµ + 1, 3eµ + √ µ 3 ] with h1(♭) = 1 ♭ , η = 1 2 and a real-valued symmetric functions are defined as (ð)ג = ð− 1 for ð ∈ [ 1, 52 ] and (ð)ג = −ð+ 4 for ð ∈ [ 5 2 , 4 ] , then we consider h ( 1 2 ) 2 Φ ( ε2 + ε1 2 )[ AB ε1I ς ε2{ג(ε2)}+ ABIςε2{ג(ε1)} ] − h ( 1 2 ) 2 Φ ( ε2 + ε1 2 ) 1− ς B(ς) [ (ε1)ג + (ε2)ג ] + 1− ς B(ς) [ Φ(ε1)ג(ε1) + Φ(ε2)ג(ε2) ] = Φ ( 5 2 )[ 1 2 ג ( 5 2 ) + 1 2 √ π ∫ 5 2 1 (µ− 1) ( 5 2 − µ )−1 2 + 1 2 ג ( 5 2 ) + 1 2 √ π ∫ 4 5 2 (−µ+ 4) ( µ− 5 2 )−1 2 ] dµ ≈ [73.10130, 106.84792], and AB ε1I ς ε2{(Φג(ε2)}+ ABIςε2{(Φג(ε1)} = [ 1 2 Φ ( 3 2 ) + 1 2 √ π ∫ 5 2 1 [ 2eµ + 1, 3eµ + √ µ 3 ] (µ− 1) ( 5 2 − µ )−1 2 + 1 2 Φ ( 3 2 ) + 1 2 √ π ∫ 4 5 2 [ 2eµ + 1, 3eµ + √ µ 3 ] (−µ+ 4) ( µ− 5 2 )−1 2 ] dµ ≈ [81.16120, 111.35182] . Finally, we have ς(ε2 − ε1) ς B(ς)Γ(ς) [ Φ(ε1) + Φ(ε2) ] × ∫ 1 0 ♭ς−1 [ 1 h(♭) + 1 h(1− ♭) ] ε2♭)ג + (1− ♭)ε1)d♭ + 1− ς B(ς) [ Φ(ε1)ג(ε1) + Φ(ε2)ג(ε2) ] ≈ [85.14621, 115.36241] . J. E. Maćıas-Dı́az et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 4014-4049 4031 This implies that [73.10130, 106.84792] ⪯cr [81.16120, 111.35182] ⪯cr [85.14621, 115.36241] . Consequently, Theorem 12 is valid. Remark 4. (i) If h(♭) = 1 ♭s , then Theorem 12 yields an outcome for the cr-s-convex function for AB integral operators: 2s−1Φ ( ε2 + ε1 2 )[ AB ε1I ς ε2{ג(ε2)}+ ABIςε2{ג(ε1)} ] − 2s−1Φ ( ε2 + ε1 2 ) 1− ς B(ς) [ (ε1)ג + (ε2)ג ] + 1− ς B(ς) [ Φ(ε1)ג(ε1) + Φ(ε2)ג(ε2) ] ⪯cr AB ε1I ς ε2{(Φג(ε2)}+ ABIςε2{(Φג(ε1)} ⪯cr ς(ε2 − ε1) ς B(ς)Γ(ς) [ Φ(ε1) + Φ(ε2) ] × ∫ 1 0 ♭ς−1 [ ♭s + (1− ♭)s ] ε2♭)ג + (1− ♭)ε1)d♭ + 1− ς B(ς) [ Φ(ε1)ג(ε1) + Φ(ε2)ג(ε2) ] . (15) (ii) If h(♭) = 1, then Theorem 12 yields an outcome for the cr-p-convex function for AB integral operators: 1 2 f ( ε2 + ε1 2 )[ AB ε1I ς ε2{ג(ε2)}+ ABIςε2{ג(ε1)} ] − 1 2 Φ ( ε2 + ε1 2 ) 1− ς B(ς) [ (ε1)ג + (ε2)ג ] + 1− ς B(ς) [ Φ(ε1)ג(ε1) + Φ(ε2)ג(ε2) ] ⪯cr AB ε1I ς ε2{(Φג(ε2)}+ ABIςε2{(Φג(ε1)} ⪯cr 2ς(ε2 − ε1) ς B(ς)Γ(ς) [ Φ(ε1) + Φ(ε2) ] × ∫ 1 0 ♭ς−1ג(♭ε2 + (1− ♭)ε1)d♭ + 1− ς B(ς) [ Φ(ε1)ג(ε1) + Φ(ε2)ג(ε2) ] . Using Holder and Young inequalities, we present a novel refinement of (H-H) fractional integral inequalities when the function Φ is twice differentiable and belongs to the class of cr-Godunova-Levin mappings, based on the identity in Lemma 2.1. J. E. Maćıas-Dı́az et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 4014-4049 4032 Theorem 13. Let h : (0, 1) → R+ and h ̸= 0. Let Φ : [ε1, ε2] → R+ I is cr-h-Godunova- Levin mapping, ε1, ε2 ∈ R+, ε1 < ε2. If Φ′′ ∈ L[ε1, ε2] and |Φ′′| is also cr-h-Godunova- Levin function, then the following double relation hold true: 1 ε2 − ε1 [ ABIςε2+ε1 2 {Φ(ε1)}+ AB ε2+ε1 2 Iςε2{Φ(ε2)} ] − 1 (ε2 − ε1)B(ς) [ Φ(ε1) + Φ(ε2) ] − (ε2 − ε1) ς−1 2ς−1B(ς)Γ(ς) Φ ( ε2 + ε1 2 ) ⪯cr (ε2 − ε1) ς−1 (ς + 1)B(ς)Γ(ς) [ |Φ′′(ε1)|+|Φ′′(ε2)| ] × ∫ 1 2 0 ♭ς+1 [ 1 h(♭) + 1 h(1− ♭) ] d♭, (16) where ς ∈ (0, 1]. Proof. Firstly, from Lemma 2.1, we have 1 ε2 − ε1 [ ABIςε2+ε1 2 {Φ(ε1)}+ AB ε2+ε1 2 Iςε2{Φ(ε2)} ] − 1 (ε2 − ε1)B(ς) [ Φ(ε1) + Φ(ε2) ] − (ε2 − ε1) ς−1 2ς−1B(ς)Γ(ς) Φ ( ε2 + ε1 2 ) ⪯cr (ε2 − ε1) ς−1 2(ς + 1)B(ς)Γ(ς) × ∫ 1 0 |wς(♭)| [ |Φ′′(♭ε1 + (1− ♭)ε2)|+|Φ′′(♭ε2 + (1− ♭)ε1)| ] d♭. (17) As |Φ′′| is cr-h-Godunova-Levin function, we have 1 ε2 − ε1 [ ABIςε2+ε1 2 {Φ(ε1)}+ AB ε2+ε1 2 Iςε2{Φ(ε2)} ] − 1 (ε2 − ε1)B(ς) [ Φ(ε1) + Φ(ε2) ] − (ε2 − ε1) ς−1 2ς−1B(ς)Γ(ς) Φ ( ε2 + ε1 2 ) ≤ (ε2 − ε1) ς−1 2(ς + 1)B(ς)Γ(ς) × {∫ 1 2 0 ♭ς+1 [ |Φ′′(ε1)| h(♭) + |Φ′′(ε2)| h(1− ♭) ] d♭ + ∫ 1 1 2 (1− ♭)ς+1 [ |Φ′′(ε1)| h(♭) + |Φ′′(ε2)| h(1− ♭) ] d♭ + ∫ 1 2 0 ♭ς+1 [ |Φ′′(ε2)| h(♭) + |Φ′′(ε1)| h(1− ♭) ] d♭ + ∫ 1 1 2 (1− ♭)ς+1 [ |Φ′′(ε2)| h(♭) + |Φ′′(ε1)| h(1− ♭) ] d♭ } = (ε2 − ε1) ς−1 (ς + 1)B(ς)Γ(ς) J. E. Maćıas-Dı́az et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 4014-4049 4033 × {∫ 1 2 0 ♭ς+1 [ |Φ′′(ε1)| h(♭) + |Φ′′(ε2)| h(1− ♭) ] d♭ + ∫ 1 2 0 ♭ς+1 [ |Φ′′(ε2)| h(♭) + |Φ′′(ε1)| h(1− ♭) ] d♭ } = (ε2 − ε1) ς−1 (ς + 1)B(ς)Γ(ς) [ |Φ′′(ε1)|+|Φ′′(ε2)| ] ∫ 1 2 0 ♭ς+1 [ 1 h(♭) + 1 h(1− ♭) ] d♭. Remark 5. (i) If h(♭) = 1 ♭s , then Theorem 13 yields an outcome for the cr-s-convex function for AB integral operators: 1 ε2 − ε1 [ ABIςε2+ε1 2 {Φ(ε1)}+ AB ε2+ε1 2 Iςε2{Φ(ε2)} ] − 1 (ε2 − ε1)B(ς) [ Φ(ε1) + Φ(ε2) ] − (ε2 − ε1) ς−1 2ς−1B(ς)Γ(ς) Φ ( ε2 + ε1 2 ) ≤ (ε2 − ε1) ς−1 (ς + 1)B(ς)Γ(ς) [ |Φ′′(ε1)|+|Φ′′(ε2)| ] [(12)ς+s+2 ς + s+2 + β 1 2 (ς + 2, s+1) ] . (ii) If h(♭) = 1, then Theorem 13 yields an outcome for the cr-p-convex function for AB integral operators: 1 ε2 − ε1 [ ABIςε2+ε1 2 {Φ(ε1)}+ AB ε2+ε1 2 Iςε2{Φ(ε2)} ] − 1 (ε2 − ε1)B(ς) [ Φ(ε1) + Φ(ε2) ] − (ε2 − ε1) ς−1 2ς−1B(ς)Γ(ς) Φ ( ε2 + ε1 2 ) ≤ ( 1 2 )ς+1 (ς + 1)(ς + 2) (ε2 − ε1) ς−1 B(ς)Γ(ς) [ |Φ′′(ε1)|+|Φ′′(ε2)| ] . Theorem 14. Let h : (0, 1) → R+ and h ̸= 0. Let Φ : [ε1, ε2] → R+ I is cr-h-Godunova- Levin mapping, ε1, ε2 ∈ R+, ε1 < ε2. If Φ′′ ∈ L[ε1, ε2] and |Φ′′| is also cr-h-Godunova- Levin function, then the following double relation holds true: 1 ε2 − ε1 [ ABIςε2+ε1 2 {Φ(ε1)}+ AB ε2+ε1 2 Iςε2{Φ(ε2)} ] − 1 (ε2 − ε1)B(ς) [ Φ(ε1) + Φ(ε2) ] − (ε2 − ε1) ς−1 2ς−1B(ς)Γ(ς) Φ ( ε2 + ε1 2 ) ⪯cr (ε2 − ε1) ς−1 (ς + 1)B(ς)Γ(ς) ( ( 1 2 )ςp+2p ςp+ p+ 1 ) 1 p [ |Φ′′(ε1)|+|Φ′′(ε2)| ] × [(∫ 1 0 d♭ h(♭) ) 1 q + (∫ 1 0 d♭ h(1− ♭) ) 1 q ] , (18) J. E. Maćıas-Dı́az et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 4014-4049 4034 where ς ∈ (0, 1], 1 p + 1 q = 1. Proof. According to Lemma 2.1 and taking into account Hölder’s inequality and apply it to relation (17), we have 1 ε2 − ε1 [ ABIςε2+ε1 2 {Φ(ε1)}+ AB ε2+ε1 2 Iςε2{Φ(ε2)} ] − 1 (ε2 − ε1)B(ς) [ Φ(ε1) + Φ(ε2) ] − (ε2 − ε1) ς−1 2ς−1B(ς)Γ(ς) Φ ( ε2 + ε1 2 ) ⪯cr (ε2 − ε1) ς−1 2(ς + 1)B(ς)Γ(ς) (∫ 1 0 |wς(♭)|pd♭ ) 1 p [(∫ 1 0 |Φ′′(♭ε1 + (1− ♭)ε2)|qd♭ ) 1 q + (∫ 1 0 |Φ′′(♭ε2 + (1− ♭)ε1)|qd♭ ) 1 q ] . (19) As |Φ′′|q is an cr-h-Godunova-Levin mapping, we have 1 ε2 − ε1 [ ABIςε2+ε1 2 {Φ(ε1)}+ AB ε2+ε1 2 Iςε2{Φ(ε2)} ] − 1 (ε2 − ε1)B(ς) [ Φ(ε1) + Φ(ε2) ] − (ε2 − ε1) ς−1 2ς−1B(ς)Γ(ς) Φ ( ε2 + ε1 2 ) ⪯cr (ε2 − ε1) ς−1 (ς + 1)B(ς)Γ(ς) ( ( 1 2 )ςp+2p ςp+ p+ 1 ) 1 p × {[∫ 1 0 ( |Φ′′(ε1)|q h(♭) + |Φ′′(ε2)|q h(1− ♭) ) d♭ ] 1 q + [∫ 1 0 ( |Φ′′(ε2)|q h(♭) + |Φ′′(ε1)|q h(1− ♭) ) d♭ ] 1 q } . Then, we apply the fact that ε2∑ k=1 (uk + vk) ε1 ≤ ε2∑ k=1 uk ε1 + ε2∑ k=1 vk ε1 , for 0 < ε1 < 1, u1, u2, · · · , uε2 ≥ 0, v1, v2, · · · , vε2 ≥ 0. This further implies as follows: 1 ε2 − ε1 [ ABIςε2+ε1 2 {Φ(ε1)}+ AB ε2+ε1 2 Iςε2{Φ(ε2)} ] − 1 (ε2 − ε1)B(ς) [ Φ(ε1) + Φ(ε2) ] − (ε2 − ε1) ς−1 2ς−1B(ς)Γ(ς) Φ ( ε2 + ε1 2 ) ⪯cr (ε2 − ε1) ς−1 (ς + 1)B(ς)Γ(ς) ( ( 1 2 )ςp+2p ςp+ p+ 1 ) 1 p J. E. Maćıas-Dı́az et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 4014-4049 4035 × [(∫ 1 0 |Φ′′(ε1)|q h(♭) d♭ ) 1 q + (∫ 1 0 |Φ′′(ε2)|q h(1− ♭) d♭ ) 1 q + (∫ 1 0 |Φ′′(ε2)|q h(♭) d♭ ) 1 q + (∫ 1 0 |Φ′′(ε1)|q h(1− ♭) d♭ ) 1 q ] = (ε2 − ε1) ς−1 (ς + 1)B(ς)Γ(ς) ( ( 1 2 )ςp+2p ςp+ p+ 1 ) 1 p [ |Φ′′(ε1)|+|Φ′′(ε2)| ] × [(∫ 1 0 d♭ h(♭) ) 1 q + (∫ 1 0 d♭ h(1− ♭) ) 1 q ] . The finishes the proof. Remark 6. (i) If h(♭) = 1 ♭s , then Theorem 14 yields an outcome for the cr-s-convex function for AB integral operators: 1 ε2 − ε1 [ ABIςε2+ε1 2 {Φ(ε1)}+ AB ε2+ε1 2 Iςε2{Φ(ε2)} ] − 1 (ε2 − ε1)B(ς) [ Φ(ε1) + Φ(ε2) ] − (ε2 − ε1) ς−1 2ς−1B(ς)Γ(ς) Φ ( ε2 + ε1 2 ) ⪯cr (ε2 − ε1) ς−1 (ς + 1)B(ς)Γ(ς) (( 1 2 )ς+1 ς + 2 ) 1 p [ |Φ′′(ε1)|q+|Φ′′(ε2)|q ] 1 q × [( 1 2 )ς+s+2 ς + s+2 + β 1 2 (ς + 2, s+1) ] 1 q . (20) (ii) If h(♭) =, then Theorem 14 yields an outcome for the cr-p-convex function for AB integral operators: 1 ε2 − ε1 [ ABIςε2+ε1 2 {Φ(ε1)}+ AB ε2+ε1 2 Iςε2{Φ(ε2)} ] − 1 (ε2 − ε1)B(ς) [ Φ(ε1) + Φ(ε2) ] − (ε2 − ε1) ς−1 2ς−1B(ς)Γ(ς) Φ ( ε2 + ε1 2 ) ⪯cr (ε2 − ε1) ς−1 (ς + 1)B(ς)Γ(ς) ( 1 2 )ς+1 ς + 2 [ |Φ′′(ε1)|q+|Φ′′(ε2)|q ] 1 q . (21) Theorem 15. Let h : (0, 1) → R+ and h ̸= 0. Let Φ : [ε1, ε2] → R+ I is cr-h-Godunova- Levin mapping, ε1, ε2 ∈ R+, ε1 < ε2. If Φ′′ ∈ L[ε1, ε2] and |Φ′′| is also cr-h-Godunova- Levin function, then the following double relation holds true: 1 ε2 − ε1 [ ABIςε2+ε1 2 {Φ(ε1)}+ AB ε2+ε1 2 Iςε2{Φ(ε2)} ] J. E. Maćıas-Dı́az et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 4014-4049 4036 − 1 (ε2 − ε1)B(ς) [ Φ(ε1) + Φ(ε2) ] − (ε2 − ε1) ς−1 2ς−1B(ς)Γ(ς) Φ ( ε2 + ε1 2 ) ⪯cr (ε2 − ε1) ς−1 (ς + 1)B(ς)Γ(ς)  ( 1 2 )(ς+1) ( q−p q−1 ) (q− 1) (ς + 1)(q− p) + q− 1 1− 1 q × [ |Φ′′(ε1)|q+|Φ′′(ε2)|q ] 1 q [∫ 1 2 0 ♭ςp+pd♭ h(♭) + ∫ 1 2 0 ♭ςp+pd♭ h(1− ♭) ] 1 q , (22) where ς ∈ (0, 1], q ≥ p > 1. Proof. By using the Holder’s inequality and taking into account relation (17), we have 1 ε2 − ε1 [ ABIςε2+ε1 2 {Φ(ε1)}+ AB ε2+ε1 2 Iςε2{Φ(ε2)} ] − 1 (ε2 − ε1)B(ς) [ Φ(ε1) + Φ(ε2) ] − (ε2 − ε1) ς−1 2ς−1B(ς)Γ(ς) Φ ( ε2 + ε1 2 ) = (ε2 − ε1) ς−1 2(ς + 1)B(ς)Γ(ς) × ∫ 1 0 |wς(♭)| q−p q ·|wς(♭)| p q [ |Φ′′(♭ε1 + (1− ♭)ε2)|+|Φ′′(♭ε2 + (1− ♭)ε1)| ] d♭ ⪯cr (ε2 − ε1) ς−1 2(ς + 1)B(ς)Γ(ς) (∫ 1 0 |wς(♭)| q−p q−1d♭ )1− 1 q × [(∫ 1 0 |wς(♭)|p|Φ′′(♭ε1 + (1− ♭)ε2)|qd♭ ) 1 q + (∫ 1 0 |wς(♭)|p|Φ′′(♭ε2 + (1− ♭)ε1)|qd♭ ) 1 q ] . As |Φ′′|q is cr-h-Godunova-Levin mapping, one has 1 ε2 − ε1 [ ABIςε2+ε1 2 {Φ(ε1)}+ AB ε2+ε1 2 Iςε2{Φ(ε2)} ] − 1 (ε2 − ε1)B(ς) [ Φ(ε1) + Φ(ε2) ] − (ε2 − ε1) ς−1 2ς−1B(ς)Γ(ς) Φ ( ε2 + ε1 2 ) = (ε2 − ε1) ς−1 2(ς + 1)B(ς)Γ(ς) × ∫ 1 0 |wς(♭)| q−p q ·|wς(♭)| p q [ |Φ′′(♭ε1 + (1− ♭)ε2)|+|Φ′′(♭ε2 + (1− ♭)ε1)| ] d♭ ⪯cr (ε2 − ε1) ς−1 2(ς + 1)B(ς)Γ(ς) (∫ 1 0 |wς(♭)| q−p q−1d♭ )1− 1 q × [(∫ 1 0 |wς(♭)|p|Φ′′(ε1)|qd♭ h(♭) + ∫ 1 0 |wς(♭)|p|Φ′′(ε2)|qd♭ h(1− ♭) ) 1 q + (∫ 1 0 |wς(♭)|p|Φ′′(ε2)|qd♭ h(♭) + ∫ 1 0 |wς(♭)|p|Φ′′(ε1)|qd♭ h(1− ♭) ) 1 q ] J. E. Maćıas-Dı́az et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 4014-4049 4037 = (ε2 − ε1) ς−1 2(ς + 1)B(ς)Γ(ς)  ( 1 2 )(ς+1) ( q−p q−1 ) (q− 1) (ς + 1)(q− p) + q− 1 1− 1 q {[ |Φ′′(ε1)|q (∫ 1 2 0 ♭ςp+pd♭ h(♭) + ∫ 1 1 2 (1− ♭)ςp+pd♭ h(♭) ) +|Φ′′(ε2)|q (∫ 1 2 0 ♭ςp+pd♭ h(1− ♭) + ∫ 1 1 2 (1− ♭)ςp+pd♭ h(1− ♭) )] 1 q + [ |Φ′′(ε2)|q (∫ 1 2 0 ♭ςp+pd♭ h(♭) + ∫ 1 1 2 (1− ♭)ςp+pd♭ h(♭) ) +|Φ′′(ε1)|q (∫ 1 2 0 ♭ςp+pd♭ h(1− ♭) + ∫ 1 1 2 (1− ♭)ςp+pd♭ h(1− ♭) )] 1 q  = (ε2 − ε1) ς−1 (ς + 1)B(ς)Γ(ς)  ( 1 2 )(ς+1) ( q−p q−1 ) (q− 1) (ς + 1)(q− p) + q− 1 1− 1 q × [ |Φ′′(ε1)|q+|Φ′′(ε2)|q ] 1 q [∫ 1 2 0 ♭ςp+pd♭ h(♭) + ∫ 1 2 0 ♭ςp+pd♭ h(1− ♭) ] 1 q . Remark 7. (i) If h(♭) = 1 ♭s , then Theorem 15 yields an outcome for the cr-s-convex function for AB integral operators: 1 ε2 − ε1 [ ABIςε2+ε1 2 {Φ(ε1)}+ AB ε2+ε1 2 Iςε2{Φ(ε2)} ] − 1 (ε2 − ε1)B(ς) [ Φ(ε1) + Φ(ε2) ] − (ε2 − ε1) ς−1 2ς−1B(ς)Γ(ς) Φ ( ε2 + ε1 2 ) ⪯cr (ε2 − ε1) ς−1 (ς + 1)B(ς)Γ(ς)  ( 1 2 )(ς+1) ( q−p q−1 ) (q− 1) (ς + 1)(q− p) + q− 1 1− 1 q × [ |Φ′′(ε1)|q+|Φ′′(ε2)|q ] 1 q [ ( 1 2 )ςp+p+s+1 ςp+ p+ s+1 + β 1 2 (ςp+ p+ 1, s+1) ] 1 q . (ii) If h(♭) = 1, then Theorem 15 yields an outcome for the cr-p-convex function for AB integral operators: 1 ε2 − ε1 [ ABIςε2+ε1 2 {Φ(ε1)}+ AB ε2+ε1 2 Iςε2{Φ(ε2)} ] − 1 (ε2 − ε1)B(ς) [ Φ(ε1) + Φ(ε2) ] − (ε2 − ε1) ς−1 2ς−1B(ς)Γ(ς) Φ ( ε2 + ε1 2 ) ⪯cr (ε2 − ε1) ς−1 (ς + 1)B(ς)Γ(ς)  ( 1 2 )(ς+1) ( q−p q−1 ) (q− 1) (ς + 1)(q− p) + q− 1 1− 1 q × ( ( 1 2 )ςp+p ςp+ p+ 1 ) 1 q [ |Φ′′(ε1)|q+|Φ′′(ε2)|q ] 1 q . J. E. Maćıas-Dı́az et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 4014-4049 4038 Theorem 16. Let h : (0, 1) → R+ and h ̸= 0. Let Φ : [ε1, ε2] → R+ I is cr-h-Godunova- Levin mapping, ε1, ε2 ∈ R+, ε1 < ε2. If |Φ′′|∈ L[ε1, ε2] and |Φ′′| is also cr-h-Godunova- Levin function, then the following double relation holds true: 1 ε2 − ε1 [ ABIςε2+ε1 2 {Φ(ε1)}+ AB ε2+ε1 2 Iςε2{Φ(ε2)} ] − 1 (ε2 − ε1)B(ς) [ Φ(ε1) + Φ(ε2) ] − (ε2 − ε1) ς−1 2ς−1B(ς)Γ(ς) Φ ( ε2 + ε1 2 ) ⪯cr (ε2 − ε1) ς−1 2(ς + 1)B(ς)Γ(ς) { ( 1 2 )ςp+p−1 (ςp+ p+ 1)p + 1 q [ |Φ′′(ε1)|q+|Φ′′(ε2)|q ] ∫ 1 0 ( 1 h(♭) + 1 h(1− ♭) ) d♭ } , where ς ∈ (0, 1]. Proof. By using the Holder’s inequality and taking into account result (17), based on the Young’s result: ab ≤ 1 p ap + 1 q bq, we obtain 1 ε2 − ε1 [ ABIςε2+ε1 2 {Φ(ε1)}+ AB ε2+ε1 2 Iςε2{Φ(ε2)} ] − 1 (ε2 − ε1)B(ς) [ Φ(ε1) + Φ(ε2) ] − (ε2 − ε1) ς−1 2ς−1B(ς)Γ(ς) Φ ( ε2 + ε1 2 ) ⪯cr (ε2 − ε1) ς−1 2(ς + 1)B(ς)Γ(ς) [ 2 p ∫ 1 0 |wς(♭)|pd♭ + 1 q (∫ 1 0 |Φ′′(♭ε1 + (1− ♭)ε2)|qd♭+ ∫ 1 0 |Φ′′(♭ε2 + (1− ♭)ε1)|qd♭ )] . As |Φ′′|q is cr-h-Godunova-Levin, one has 1 ε2 − ε1 [ ABIςε2+ε1 2 {Φ(ε1)}+ AB ε2+ε1 2 Iςε2{Φ(ε2)} ] − 1 (ε2 − ε1)B(ς) [ Φ(ε1) + Φ(ε2) ] − (ε2 − ε1) ς−1 2ς−1B(ς)Γ(ς) Φ ( ε2 + ε1 2 ) ⪯cr (ε2 − ε1) ς−1 2(ς + 1)B(ς)Γ(ς) { 2 p ∫ 1 0 |wς(♭)|pd♭ + 1 q [∫ 1 0 |Φ′′(ε1)|qd♭ h(♭) + ∫ 1 0 |Φ′′(ε2)|qd♭ h(1− ♭) + ∫ 1 0 |Φ′′(ε2)|qd♭ h(♭) + ∫ 1 0 |Φ′′(ε1)|qd♭ h(1− ♭) ]} = (ε2 − ε1) ς−1 2(ς + 1)B(ς)Γ(ς) { ( 1 2 )ςp+p−1 (ςp+ p+ 1)p + 1 q [ |Φ′′(ε1)|q+|Φ′′(ε2)|q ] J. E. Maćıas-Dı́az et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 4014-4049 4039 × ∫ 1 0 ( 1 h(♭) + 1 h(1− ♭) ) d♭ } . The proof is completed. Remark 8. (i) If h(♭) = 1 ♭s , then Theorem 16 yields an outcome for the cr-s-convex function for AB integral operators: 1 ε2 − ε1 [ ABIςε2+ε1 2 {Φ(ε1)}+ AB ε2+ε1 2 Iςε2{Φ(ε2)} ] − 1 (ε2 − ε1)B(ς) [ Φ(ε1) + Φ(ε2) ] − (ε2 − ε1) ς−1 2ς−1B(ς)Γ(ς) Φ ( ε2 + ε1 2 ) ⪯cr (ε2 − ε1) ς−1 (ς + 1)B(ς)Γ(ς) { ( 1 2 )ςp+p (ςp+ p+ 1)p + 1 q(s+1) [ |Φ′′(ε1)|q+|Φ′′(ε2)|q ]} . (ii) If h(♭) = 1, then Theorem 16 yields an outcome for the cr-p-convex function for AB integral operators: 1 ε2 − ε1 [ ABIςε2+ε1 2 {Φ(ε1)}+ AB ε2+ε1 2 Iςε2{Φ(ε2)} ] − 1 (ε2 − ε1)B(ς) [ Φ(ε1) + Φ(ε2) ] − (ε2 − ε1) ς−1 2ς−1B(ς)Γ(ς) Φ ( ε2 + ε1 2 ) ⪯cr (ε2 − ε1) ς−1 (ς + 1)B(ς)Γ(ς) { ( 1 2 )ςp+p (ςp+ p+ 1)p + 1 q [ |Φ′′(ε1)|q+|Φ′′(ε2)|q ]} . Theorem 17. Let h : (0, 1) → R+ and h ̸= 0. Let Φ : [ε1, ε2] → R+ I is cr-h-Godunova- Levin mapping, ε1, ε2 ∈ R+, ε1 < ε2 and [h(♭)]q ∈ L1[0, 1],Φ ∈ L1[ε1, ε2]. If |Φ′| is an cr-h-Godunova-Levin mapping on [ε1, ε2], then the following relation |Bk(Φ, ε1, ε2)| = k−1∑ ȷ=0 ε2 − ε1 2k2 [( 1 2 ) q−1 q (∫ 1 0 |1− 2♭| h(♭) ∣∣∣∣Φ′ ( (k−ȷ)ε1 + ȷε2 k )∣∣∣∣q d♭ + ∫ 1 0 |1− 2♭| h(1− ♭) ∣∣∣∣Φ′ ( (k−ȷ− 1)ε1 + (ȷ+ 1)ε2 k )∣∣∣∣q d♭) 1 q ] holds, where 1 < p and 1 p + 1 q = 1. Proof. Let q ≥ 1 and using identity from Lemma 2.1 and taking into account Power- J. E. Maćıas-Dı́az et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 4014-4049 4040 mean inequality, then we have |Bk(Φ, ε1, ε2)| ⪯cr k−1∑ ȷ=0 ε2 − ε1 2k2 (∫ 1 0 ∣∣∣∣(1− 2♭)Φ′ ( ♭ (k−ȷ)ε1 + ȷε2 k + (1− ♭) (k−ȷ− 1)ε1 + (ȷ+ 1)ε2 k )∣∣∣∣d♭) ⪯cr k−1∑ ȷ=0 ε2 − ε1 2k2 (∫ 1 0 |1− 2♭|d♭ )1− 1 q × (∫ 1 0 |1− 2♭| ∣∣∣∣Φ′ ( ♭ (k−ȷ)ε1 + ȷε2 k + (1− ♭) (k−ȷ− 1)ε1 + (ȷ+ 1)ε2 k )∣∣∣∣q d♭) 1 q . As |Φ′|q is cr-h-Godunova-Levin function, we have |Bk(Φ, ε1, ε2)| ⪯cr k−1∑ ȷ=0 ε2 − ε1 2k2 [∫ 1 0 |1− 2♭|d♭ ]1− 1 q [∫ 1 0 |1− 2♭| ( 1 h(♭) ∣∣∣∣Φ′ ( (k−ȷ)ε1 + ȷε2 k )∣∣∣∣q + 1 h(1− ♭) · ∣∣∣∣Φ′ ( (k−ȷ− 1)ε1 + (ȷ+ 1)ε2 k )∣∣∣∣q)d♭ ] 1 q = k−1∑ ȷ=0 ε2 − ε1 2k2 (∫ 1 0 |1− 2♭|d♭ )1− 1 q (∫ 1 0 |1− 2♭| h(♭) ∣∣∣∣Φ′ ( (k−ȷ)ε1 + ȷε2 k )∣∣∣∣q d♭ + ∫ 1 0 |1− 2♭| h(1− ♭) ∣∣∣∣Φ′ ( (k−ȷ− 1)ε1 + (ȷ+ 1)ε2 k )∣∣∣∣q d♭) 1 q = k−1∑ ȷ=0 ε2 − ε1 2k2 [( 1 2 ) q−1 q (∫ 1 0 |1− 2♭| h(♭) · ∣∣∣∣Φ′ ( (k−ȷ)ε1 + ȷε2 k )∣∣∣∣q d♭ + ∫ 1 0 |1− 2♭| h(1− ♭) ∣∣∣∣Φ′ ( (k−ȷ− 1)ε1 + (ȷ+ 1)ε2 k )∣∣∣∣q d♭) 1 q ] . Corollary 1. Setting h(♭) = 1 ♭ and Φ = Φ in Theorem 17, we get |Bk(Φ, ε1, ε2)| = k−1∑ ȷ=0 ε2 − ε1 k2(2) 2+ 1 q (∣∣∣∣Φ′ ( (k−ȷ)ε1 + ȷε2 z )∣∣∣∣q + ∣∣∣∣Φ′ ( (k−ȷ− 1)ε1 + (ȷ+ 1)ε2 k )∣∣∣∣q) 1 q which has been obtained by authors in [29]. J. E. Maćıas-Dı́az et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 4014-4049 4041 Corollary 2. Setting h(♭) = 1 ♭s and Φ = Φ in Theorem 17, we get |Bk(Φ, ε1, ε2)| ⪯cr k−1∑ ε=0 ε2 − ε1 k22 2− 1 q ( 1 2s(s+ 1)(s+ 2) + s (s+ 1)(s+ 2) ) 1 q × [∣∣∣∣Φ′ ( (k−ε)ε1 + εr k )∣∣∣∣q + ∣∣∣∣Φ′ ( (k−ε− 1)ε1 + (ε+ 1)ε2 k )∣∣∣∣q] 1 q , which has been proved by authors in [57]. Theorem 18. Let h : (0, 1) → R+ and h ̸= 0. Let Φ : [ε1, ε2] → R+ I is cr-h-Godunova- Levin mapping, ε1, ε2 ∈ R+, ε1 < ε2 and [h(♭)]q ∈ L1[0, 1],Φ ∈ L1[ε1, ε2]. If |Φ′| is also cr-h-Godunova-Levin mapping on [ε1, ε2], then the following relation |Bk(Φ, ε1, ε2)| ⪯cr k−1∑ ȷ=0 ε2 − ε1 2k2 [( 1 1 + p ) 1 p × (∫ 1 0 ( 1 h(♭) ∣∣∣∣Φ′ ( (k−ȷ)ε1 + ȷε2 k )∣∣∣∣q + 1 h(1− ♭) ∣∣∣∣Φ′ ( (k−ȷ− 1)ε1 + (ȷ+ 1)ε2 k )∣∣∣∣q)d♭ ) 1 q ] holds, where 1 q + 1 p = 1. Proof. Assume that 1 < p. Taking into account Lemma 2.1 and the Hölder inequality, one has |Bk(Φ, ε1, ε2)| ⪯cr k−1∑ ȷ=0 ε2 − ε1 2k2 [(∫ 1 0 ∣∣∣∣(1− 2♭)Φ′ ( ♭ (k−ȷ)ε1 + ȷε2 k + (1− ♭) (k−ȷ− 1)ε1 + (ȷ+ 1)ε2 k )∣∣∣∣ d♭)] ⪯cr k−1∑ ȷ=0 ε2 − ε1 2k2 [(∫ 1 0 |1− 2♭|pd♭ ) 1 p × (∫ 1 0 ∣∣∣∣Φ′ ( ♭ (k−ȷ)ε1 + ȷb k + (1− ♭) (k−ȷ− 1)ε1 + (ȷ+ 1)b k )∣∣∣∣q d♭) 1 q ] . As |Φ′|q is cr-h-Godunova-Levin mapping, one has∫ 1 0 ∣∣∣∣Φ′ ( ♭ (k−ȷ)ε1 + ȷε2 k + (1− ♭) (k−ȷ− 1)ε1 + (ȷ+ 1)ε2 k )∣∣∣∣ d♭ ⪯cr ∫ 1 0 ( 1 h(♭) ∣∣∣∣Φ′ ( (k−ȷ)ε1 + ȷε2 k )∣∣∣∣q + 1 h(1− ♭) ∣∣∣∣Φ′ ( (k−ȷ− 1)ε1 + (ȷ+ 1)ε2 k )∣∣∣∣) d♭. Therefore, we deduce J. E. Maćıas-Dı́az et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 4014-4049 4042 |Bk(Φ, ε1, ε2)| ⪯cr k−1∑ ȷ=0 ε2 − ε1 2k2 [(∫ 1 0 |1− 2♭|pd♭ ) 1 p × (∫ 1 0 ( 1 h(♭) ∣∣∣∣Φ′ ( (k−ȷ)ε1 + ȷε2 k )∣∣∣∣q + 1 h(1− ♭) ∣∣∣∣Φ′ ( (k−ȷ− 1)ε1 + (ȷ+ 1)ε2 k )∣∣∣∣q) d♭ ) 1 q ] ⪯cr k−1∑ ȷ=0 ε2 − ε1 2k2 [( 1 1 + p ) 1 p × (∫ 1 0 ( 1 h(♭) ∣∣∣∣Φ′ ( (k−ȷ)ε1 + ȷε2 k )∣∣∣∣q + 1 h(1− ♭) ∣∣∣∣Φ′ ( (k−ȷ− 1)ε1 + (ȷ+ 1)ε2 k )∣∣∣∣q) d♭ ) 1 q ] . Remark 9. Setting h(♭) = ♭−s and Φ = Φ in Theorem 18, then we get Theorem 6 in [57]. 4. Applications to special means The following section relates some of our main results with special means, and illus- trates some of their applications. Let ε1, ε2 ∈ R, (i) The arithmetic mean: A = A(ε1, ε2) := ε1 + ε2 2 , ε1, ε2 ≥ 0. (ii) The harmonic mean: H = H(ε1, ε2) := 2ε1ε2 ε1 + ε2 , ε1, ε2 > 0. (iii) The logarithmic mean: L = L(ε1, ε2) := { ε1, if ε1 = ε2 ε2−ε1 ln ε2−ln ε1 , if ε1 ̸= ε2, ε1, ε2 > 0. (iv) The p-logarithmic mean: Lp = Lp(ε1, ε2) :=  ε1, if ε1 = ε2[ rp+1−ε1p+1 (p+1)(ε2−ε1) ] 1 p , if ε1 ̸= ε2, p ∈ R\{−1, 0}, ε1, ε2 > 0. Let ε1, ε2 ∈ R, 0 < ε1 < ε2, and m ∈ N, m ≥ 2. Then, the following J. E. Maćıas-Dı́az et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 4014-4049 4043∣∣∣∣∣∣ k−1∑ ȷ=0 1 kȷ A (( (k−ȷ)ε1 + ȷε2 k )m , ( (k−ȷ− 1)ε1 + (ȷ+ 1)ε2 k )m) − Lm m(ε1, ε2) ∣∣∣∣∣∣ ⪯cr k−1∑ ȷ=0 (ε2 − ε1)m 2 2− 1 q k2 [(∫ 1 0 |1− 2♭| h(♭) d♭ )( (k−ȷ)ε1 + ȷε2 k )(m−1)q + (∫ 1 0 |1− 2♭| h(1− ♭) d♭ )( (k−ȷ− 1)ε1 + (ȷ+ 1)ε2 k )(m−1)q )] 1 q holds, for all 1 ⪯cr q. Proof. This proof is proven using Theorem 17 with the following settings Φ(♭) = ♭m, ♭ ∈ [ε1, ε2], m ∈ N, m ≥ 2. Let ε1, ε2 ∈ R, 0 < ε1 < ε2, and m ∈ N, m ≥ 2. Then, the following∣∣∣∣∣∣ k−1∑ ȷ=0 1 k A (( (k−ȷ)ε1 + ȷε2 k )m , ( (k−ȷ− 1)ε1 + (ȷ+ 1)ε2 k )m) − Lm m(ε1, ε2) ∣∣∣∣∣∣ ⪯cr k−1∑ ȷ=0 (ε2 − ε1)m 2 2− 1 q k2 ( 1 p+ 1 ) 1 p × [(∫ 1 0 d♭ h(♭) )( (k−ȷ)ε1 + ȷε2 k )(m−1)q + (∫ 1 0 d♭ h(1− ♭) )( (k−ȷ− 1)ε1 + (ȷ+ 1)ε2 k )(m−1)q )] 1 q holds, for all 1 ⪯cr q. Proof. This proof is proven using Theorem 18 with the following settings Φ(♭) = ♭m, ♭ ∈ [ε1, ε2], m ∈ N, m ≥ 2. 5. Conclusion The primary contribution in this paper to present the different variants of double inequalities by using fractional integral operators involving special functions. Recently, authors in [4, 11, 13] developed several relevant results by using classical integral oper- ators and partial standard order relation. Additionally, we use a number of other well- known inequalities, including Holder’s, Young’s, and Minkowski’s, to extend these upper bounds for Hermite-Hadamard inequality. Furthermore, these types of results involving Godunova-Levin mappings and fractional operators are not initiated, and we believe that our study opens up a whole new path for other relevant classes of Godunova-Levin func- tions, including s-Godunova-Levin, tgs-Godunova-Levin, Harmonic, and various others. In the future, we suggest that readers extend these inequalities to probabilistic and fractional stochastic settings, where randomness is incorporated into the functions them- selves or into the bounds, so they can use them more effectively in risk analysis, finance, or uncertainty quantification. Furthermore, generalizing Hermite-Hadamard inequalities REFERENCES 4044 to multiple variables, especially in convexity spaces like convex domains in Rn, may yield inequalities useful in multivariate optimization, machine learning, and control theory. Acknowledgements The authors acknowledge the financial support from the program PROSNI of the University of Guadalajara, Mexico. References [1] Thabet Abdeljawad. On conformable fractional calculus. Journal of computational and Applied Mathematics, 279:57–66, 2015. [2] Waqar Afzal, Mujahid Abbas, and Omar Mutab Alsalami. Bounds of different integral operators in tensorial hilbert and variable exponent function spaces. Mathematics, 12(16):1–33, 2024. [3] Waqar Afzal, Mujahid Abbas, Waleed Hamali, Ali M Mahnashi, and M De la Sen. Hermite–hadamard-type inequalities via caputo–fabrizio fractional integral for h- godunova–levin and (h 1, h 2)-convex functions. Fractal and Fractional, 7(9):687, 2023. [4] Waqar Afzal, Mujahid Abbas, Jorge E Maćıas-Dı́az, and Savin Treanţă. Some h- godunova–levin function inequalities using center radius (cr) order relation. Fractal and Fractional, 6(9):518, 2022. [5] Waqar Afzal, Najla M Aloraini, Mujahid Abbas, Jong-Suk Ro, and Abdullah A Za- agan. Hermite-hadamard, fejér and trapezoid type inequalities using godunova-levin preinvex functions via bhunia’s order and with applications to quadrature formula and random variable. Mathematical biosciences and engineering: MBE, 21(2):3422–3447, 2024. [6] Waqar Afzal, Thongchai Botmart, W Afzal, and T Botmart. Some novel estimates of jensen and hermite-hadamard inequalities for h-godunova–levin stochastic processes. Aims Math, 8:7277–7291, 2023. [7] Waqar Afzal, Daniel Breaz, Mujahid Abbas, Luminiţa-Ioana Cot̂ırlă, Zareen A Khan, and Eleonora Rapeanu. Hyers–ulam stability of 2 d-convex mappings and some related new hermite–hadamard, pachpatte, and fejér type integral inequalities using novel fractional integral operators via totally interval-order relations with open problem. Mathematics, 12(8):1238, 2024. [8] Waqar Afzal, Khurram Shabbir, Mubashar Arshad, Joshua Kiddy K Asamoah, and Ahmed M Galal. Some novel estimates of integral inequalities for a generalized class of harmonical convex mappings by means of center-radius order relation. Journal of Mathematics, 2023(1):8865992, 2023. REFERENCES 4045 [9] Waqar Afzal, Khurram Shabbir, and Thongchai Botmart. Generalized version of jensen and hermite-hadamard inequalities for interval-valued (h1, h2)-godunova-levin functions. AIMS Math, 7:19372–19387, 2022. [10] Abdullah Ali H Ahmadini, Waqar Afzal, Mujahid Abbas, and Elkhateeb S Aly. Weighted fejér, hermite–hadamard, and trapezium-type inequalities for (h 1, h 2)– godunova–levin preinvex function with applications and two open problems. Mathe- matics, 12(3):382, 2024. [11] Sabila Ali, Rana Safdar Ali, Miguel Vivas-Cortez, Shahid Mubeen, Gauhar Rahman, and Kottakkaran Sooppy Nisar. Some fractional integral inequalities via h-godunova– levin preinvex function. AIMS Math, 7:13832–13844, 2022. [12] Yahya Almalki and Waqar Afzal. Some new estimates of hermite–hadamard inequal- ities for harmonical cr-h-convex functions via generalized fractional integral operator on set-valued mappings. Mathematics, 11(19):4041, 2023. [13] Ohud Almutairi and Adem Kılıçman. Some integral inequalities for h-godunova-levin preinvexity. Symmetry, 11(12):1500, 2019. [14] Xiaohua Bao, Haicen Yuan, Jun Shen, Chunxun Liu, Xiangsheng Chen, and Hongzhi Cui. Numerical analysis of seismic response of a circular tunnel-rectangular underpass system in liquefiable soil. Computers and Geotechnics, 174:106642, 2024. [15] Bandar Bin-Mohsin, Muhammad Zakria Javed, Muhammad Uzair Awan, and Artion Kashuri. On some new ab-fractional inclusion relations. Fractal and Fractional, 7(10):725, 2023. [16] RP Boas Jr and MB Marcus. Generalizations of young’s inequality. J. Math. Anal. Appl, 46:36–40, 1974. [17] Jonathan M Borwein, JP Penot, and M Thera. Conjugate convex operators. Journal of mathematical analysis and applications, 102(2):399–414, 1984. [18] Sever Silvestru Dragomir. Hermite–hadamard’s type inequalities for operator convex functions. Applied Mathematics and Computation, 218(3):766–772, 2011. [19] Asfand Fahad, Youhua Qian, Zammad Ali, and Awais Younus. On generalization of hermite-hadamard-mercer inequalities for interval-valued functions with general- ized geometric-arithmetic convexity. International Journal of Geometric Methods in Modern Physics, 2024. [20] Asfand Fahad, Yuanheng Wang, Zammad Ali, Riaz Hussain, and Shigeru Furuichi. Exploring properties and inequalities for geometrically arithmetically-cr-convex func- tions with cr-order relative entropy. Information Sciences, 662:120219, 2024. REFERENCES 4046 [21] Arran Fernandez and Pshtiwan Mohammed. Hermite-hadamard inequalities in frac- tional calculus defined using mittag-leffler kernels. Mathematical Methods in the Ap- plied Sciences, 44(10):8414–8431, 2021. [22] Guanghe Han, Jiahui Xu, Xin Zhang, and Xin Pan. Efficiency and driving factors of agricultural carbon emissions: A study in chinese state farms. Agriculture, 14(9):1454, 2024. [23] Bao Qing Hu and Song Wang. A novel approach in uncertain programming part i: New arithmetic and order relation for interval numbers. Journal of Industrial and Management Optimization, 2(4):351–371, 2006. [24] Yu Hu, Xu chao Zhang, Guo qiang Wang, Xue Peng Zhang, and Han Zhen Li. Hov- ering efficiency optimization of ducted propeller with large blade tip clearance based on grooved duct configuration. Aerospace Science and Technology, 150:109226, 2024. [25] Yu Hu, Xu chao Zhang, Guo qiang Wang, Xue Peng Zhang, and Han Zhen Li. Hov- ering efficiency optimization of ducted propeller with large blade tip clearance based on grooved duct configuration. Aerospace Science and Technology, 150:109226, 2024. [26] Ben Huang, Fei Kang, Xinyu Li, and Sisi Zhu. Underwater dam crack image genera- tion based on unsupervised image-to-image translation. Automation in Construction, 163:105430, 2024. [27] Zhenhua Huang, Kunhao Li, Yihang Jiang, Zhaohong Jia, Linyuan Lv, and Yunjie Ma. Graph relearn network: Reducing performance variance and improving prediction accuracy of graph neural networks. Knowledge-Based Systems, 301:112311, 2024. [28] Abd-Allah Hyder, Hüseyin Budak, and Areej A Almoneef. Further midpoint in- equalities via generalized fractional operators in riemann–liouville sense. Fractal and Fractional, 6(9):496, 2022. [29] I Iscan, TEKİN Toplu, and FATİH Yetgin. Some new inequalities on generalization of hermite–hadamard and bullen type inequalities, applications to trapezoidal and midpoint formula. J. Math, 45(4):647–657, 2021. [30] Fahd Jarad, Thabet Abdeljawad, and Dumitru Baleanu. Caputo-type modification of the hadamard fractional derivatives. Advances in Difference Equations, 2012:1–8, 2012. [31] Hasan Kara, Hüseyin Budak, Muhammad Aamir Ali, Mehmet Zeki Sarikaya, and Yu- Ming Chu. Weighted hermite–hadamard type inclusions for products of co-ordinated convex interval-valued functions. Advances in Difference Equations, 2021:1–16, 2021. [32] Havva Kavurmacı Önalan, Ahmet Ocak Akdemir, Merve Avcı Ardıç, and Dumitru Baleanu. On new general versions of hermite–hadamard type integral inequalities via fractional integral operators with mittag-leffler kernel. Journal of Inequalities and Applications, 2021:1–16, 2021. REFERENCES 4047 [33] Dawood Khan and Saad Ihsan Butt. Superquadraticity and its fractional perspective via center-radius cr-order relation. Chaos, Solitons & Fractals, 182:114821, 2024. [34] Muhammad Bilal Khan, Jorge E Maćıas-Dı́az, Savin Treant, ǎ, and Mohamed S Soli- man. Some fejér-type inequalities for generalized interval-valued convex functions. Mathematics, 10(20):3851, 2022. [35] Ruonan Liu and Run Xu. Hermite-hadamard type inequalities for harmonical (h1, h2)-convex interval-valued functions. Mathematical Foundations of Computing, 4(2), 2021. [36] Wei Liu, Fangfang Shi, Guoju Ye, and Dafang Zhao. The properties of harmonically cr-h-convex function and its applications. Mathematics, 10(12):2089, 2022. [37] Wei Liu, Fangfang Shi, Guoju Ye, and Dafang Zhao. Some inequalities for cr-log-h- convex functions. Journal of Inequalities and Applications, 2022(1):160, 2022. [38] Yucui Lu, Linyin Qin, Yuanhui Mao, Xianmei Lnong, Qianni Wei, Junwen Su, Shuwen Chen, Zhongshi Wei, Lijing Wang, Xiayun Liao, et al. Antibacterial activity of a polysaccharide isolated from litchi (litchi chinensis sonn.) pericarp against staphylo- coccus aureus and the mechanism investigation. International Journal of Biological Macromolecules, 279:134788, 2024. [39] Jorge E Maćıas-Dı́az, Muhammad Bilal Khan, Muhammad Aslam Noor, A Mousa Abd Allah, and Safar M Alghamdi. Hermite-hadamard inequalities for generalized convex functions in interval-valued calculus. AIMS Math, 7(3):4266–4292, 2022. [40] Sikander Mehmood, Pshtiwan Othman Mohammed, Artion Kashuri, Nejmeddine Chorfi, Sarkhel Akbar Mahmood, and Majeed A Yousif. Some new fractional inequal- ities defined using cr-log-h-convex functions and applications. Symmetry, 16(4):407, 2024. [41] Bandar Bin Mohsin, Muhammad Uzair Awan, Muhammad Zakria Javed, Hüseyin Budak, Awais Gul Khan, and Muhammad Aslam Noor. Inclusions involving interval- valued harmonically co-ordinated convex functions and raina’s fractional double in- tegrals. Journal of Mathematics, 2022(1):5815993, 2022. [42] S Mubeen and GM Habibullah. k-fractional integrals and application. Int. J. Con- temp. Math. Sci, 7(2):89–94, 2012. [43] Uma Devi Patel and Stojan Radenović. An application to nonlinear fractional differ- ential equation via α-γ f-fuzzy contractive mappings in a fuzzy metric space. Mathe- matics, 10(16):2831, 2022. [44] Soubhagya Kumar Sahoo, Eman Al-Sarairah, Pshtiwan Othman Mohammed, Muhammad Tariq, and Kamsing Nonlaopon. Modified inequalities on center-radius order interval-valued functions pertaining to riemann–liouville fractional integrals. Axioms, 11(12):732, 2022. REFERENCES 4048 [45] Soubhagya Kumar Sahoo, Hleil Alrweili, Savin Treanţă, and Zareen A Khan. New fractional integral inequalities pertaining to center-radius (cr)-ordered convex func- tions. Fractal and Fractional, 7(1):81, 2023. [46] Soubhagya Kumar Sahoo, Muhammad Amer Latif, Omar Mutab Alsalami, Savin Treanţă, Weerawat Sudsutad, and Jutarat Kongson. Hermite–hadamard, fejér and pachpatte-type integral inequalities for center-radius order interval-valued preinvex functions. Fractal and Fractional, 6(9):506, 2022. [47] Ahsan Fareed Shah, Serap Özcan, Miguel Vivas-Cortez, Muhammad Shoaib Saleem, and Artion Kashuri. Fractional hermite–hadamard–mercer-type inequalities for interval-valued convex stochastic processes with center-radius order and their related applications in entropy and information theory. Fractal and Fractional, 8(7):408, 2024. [48] Nidhi Sharma, Sanjeev Kumar Singh, Shashi Kant Mishra, and Abdelouahed Hamdi. Hermite–hadamard-type inequalities for interval-valued preinvex functions via riemann–liouville fractional integrals. Journal of Inequalities and Applications, 2021:1–15, 2021. [49] Fangfang Shi, Guoju Ye, Dafang Zhao, and Wei Liu. Some integral inequalities for coordinated log-h-convex interval-valued functions. AIMS Mathematics, 7(1):156– 170, 2022. [50] Hari Mohan Srivastava, Soubhagya Kumar Sahoo, Pshtiwan Othman Mohammed, Dumitru Baleanu, and Bibhakar Kodamasingh. Hermite–hadamard type inequalities for interval-valued preinvex functions via fractional integral operators. International Journal of Computational Intelligence Systems, 15(1):8, 2022. [51] Hari Mohan Srivastava, Soubhagya Kumar Sahoo, Pshtiwan Othman Mohammed, Artion Kashuri, and Nejmeddine Chorfi. Results on minkowski-type inequalities for weighted fractional integral operators. Symmetry, 15(8):1522, 2023. [52] Vuk Stojiljković, Nikola Mirkov, and Stojan Radenović. Variations in the tensorial trapezoid type inequalities for convex functions of self-adjoint operators in hilbert spaces. Symmetry, 16(1):121, 2024. [53] Avanidhar Subrahmanyam, Ke Tang, Jingyuan Wang, and Xuewei Yang. Leverage is a double-edged sword. The Journal of Finance, 79(2):1579–1634, 2024. [54] Chunzheng Wang, Lei Yang, Minghua Hu, Yanjun Wang, and Zheng Zhao. On- demand airport slot management: tree-structured capacity profile and coadapted fire-break setting and slot allocation. Transportmetrica A: Transport Science, pages 1–35, 2024. [55] Li Xinde, Fir Dunkin, and Jean Dezert. Multi-source information fusion: Progress and future. Chinese Journal of Aeronautics, 2023. REFERENCES 4049 [56] Heng Yao, Diego Pugliese, Matthieu Lancry, and Ye Dai. Ultrafast laser direct writ- ing nanogratings and their engineering in transparent materials. Laser & Photonics Reviews, page 2300891. [57] Çetin Yildiz, Büşra Yergöz, and Abdulvahit Yergöz. On new general inequalities for s-convex functions and their applications. Journal of Inequalities and Applications, 2023(1):11, 2023. [58] Hui ZHANG, DAI Songjie, LIU Yang, ZHU Yijun, XU Yangdong, Baotong Li, and DONG Guangneng. Fishbone-like micro-textured surface for unidirectional spreading of droplets and lubricity improvement. Tribology International, 198:109932, 2024. [59] Hui Zhang, Pu Wang, Yang Liu, Songjie Dai, Yijun Zhu, Baotong Li, and Guang- neng Dong. Stretch-controlled branch shape microstructures for switchable unidi- rectional self-driven spreading of oil droplets. ACS Applied Materials & Interfaces, 16(31):41694–41703, 2024. [60] Xiaoju Zhang, Khurram Shabbir, Waqar Afzal, He Xiao, and Dong Lin. Hermite– hadamard and jensen-type inequalities via riemann integral operator for a generalized class of godunova–levin functions. Journal of Mathematics, 2022(1):3830324, 2022. [61] Zhentong Zhang, Xinde Li, Heqing Li, Fir Dunkin, Bing Li, and Zhijun Li. Dual- branch sparse self-learning with instance binding augmentation for adversarial de- tection in remote sensing images. IEEE Transactions on Geoscience and Remote Sensing, 2024. [62] Zhiyue Zhang, Muhammad Aamir Ali, Hüseyin Budak, and Mehmet Zeki Sarıkaya. On hermite-hadamard type inequalities for interval-valued multiplicative integrals. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 69(2):1428–1448, 2020. [63] Dafang Zhao, Guohui Zhao, Guoju Ye, Wei Liu, and Silvestru Sever Dragomir. On hermite–hadamard-type inequalities for coordinated h-convex interval-valued func- tions. Mathematics, 9(19):2352, 2021. [64] Daqiong Zhou, Zaiyun Peng, Zhi Lin, and Jingjing Wang. Continuity of the solution set mappings to parametric unified weak vector equilibrium problems via free-disposal sets. RAIRO-Operations Research, 2024.