EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6413 ISSN 1307-5543 – ejpam.com Published by New York Business Global Hermite-Hadamard Inequalities via Riemann-Liouville Fractional Integrals with Generalized Convex Functions Muhammad Samraiz1, Tahira Atta1, Saima Naheed1, Gauhar Rahman2, Miguel Vivas-Cortez3,∗ 1 Department of Mathematics, University of Sargodha, Sargodha 40100, Pakistan. 2 Department of Mathematics & Statistics, Hazara University, Mansehra 21300, Pakistan. 3 Pontificia Universidad Católica del Ecuador, Faculty of Exact, Natural and Environmental Sciences, FRACTAL Laboratory (Fractional Research in Analysis, Convexity and Their Applications Laboratory), Ecuador. Abstract. In this study, novel fractional integral inequalities for twice-differentiable geometrically arithmetically (α,m)-convex functions are presented. The classical Riemann-Liouville fractional integrals are used to obtain several new identities. By employing the above convexity, Hermite- Hadamard type inequalities are investigated using these identities. The main findings of this work extend the existing literature and are derived as special cases. 2020 Mathematics Subject Classifications: 26A51, 26A33, 26D15 Key Words and Phrases: Geometrically Arithmetically (α,m)-Convex, Hermite-Hadamard Type Inequalities, Hölder’s Inequality, Fractional Integrals 1. Introduction and Preliminaries Fractional calculus explores the integrals and derivatives of arbitrary real or complex orders. It provides a range of tools that can be used to solve differential equations, integral equations, mathematical physics, engineering and machine learning problems. Within the context of Riemann-Liouville fractional calculus, this discussion focuses on the linear operators of fractional integration and differentiation [1, 2]. It also addresses the existence of mild solutions for fractional-order Caputo derivatives in Banach spaces. Moreover, it delves into the existence of mild solutions for nonlocal impulsive differential inclusions [3, 4], considering Neumann boundary conditions in the form where u and v represent ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6413 Email addresses: muhammad.samraiz@uos.edu.pk (M. Samraiz), tahiraatta55@gmail.com (T. Atta), saima.naheed@uos.edu.pk (S. Naheed), drgauhar.rahman@hu.edu.pk, gauhar55uom@gmail.com (G. Rahman), mjvivas@puce.edu.ec (M. Vivas-Cortez) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) M. Samraiz et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6413 2 of 26 typical Caputo fractional derivatives [5]. The investigation further encompasses fractional- order delay differential equations [6], while Volterra-Fredholm integral equations involving the Erdlyi-Kober fractional integral operator are discussed in [7]. The Ulam-Hyers type stability for certain nonlinear differential equations are discussed in [8]. This study also derives several tensorial trapezoid-type inequalities in Hilbert spaces by employing classical analytical identities and exploring the convexity properties of functions involving self- adjoint operators [9]. The applications of non-integer order derivatives can be studied in [10] and numerical solutions for various types of fractional diffusion equations in [11]. Additionally, the applications in the dynamics of particles, fields and media can be explored in the book [12]. The theory of convex functions has experienced rapid development. There are several reasons to study this concept. Firstly, modern analysis involve applications of convex functions directly or indirectly. Secondly, this theory plays a significant role in the con- struction of many existing inequalities, which play a major role in optimization theory. It has been demonstrated that fractional integral inequalities are one of the best tools for the growth of many fields of pure and applied mathematics. Research on Hermite-Hadamard inequality has been ongoing since its introduction in 1893. The Hermite-Hadamard inequality for fractional integrals was initially formu- lated by Sarikaya et al. in [13]. Such inequalities were further studied by Shuang et al. for geometrically arithmetically (GA) s-convex functions in [14]. The Riemann-Liouville fractional Hermite-Hadamard inequalities for twice differentiable geometrically and arith- metically s-convex functions, along with precise error estimates presented in [15, 16] high- lighting the significance of Hermite-Hadamard. The inequality provides bounds on the mean function, assisting in error estimation for trapezoid formulas and the construction of generalized means. Fractional Hermite-Hadamard inequalities involving different types of fractional integrals and various classes of convex functions have attracted significant attention of the scientists. Applications to special means, fractional integral inequalities for differentiable convex maps and midpoint formulas were explored in [17], while the in- tegral inequality of the Ostrowski’s type and Hermite-Hadamard integral inequality were investigated in [18, 19]. References such as [20–23] provide insights into convex functions, s-convex functions, r-convex functions, (s,m)-convex, and (s,m) logarithmical functions, respectively. Two classes of new Hermite-Hadamard type inequalities, requiring Riemann- Liouville fractional integrals, were constructed for once differentiable and twice differen- tiable (s,m) and (α,m)-lgorithmically convex functions in [24, 25]. Hermite-Hadamard inequalities for functions satisfying the s− e-condition and along with a method for solv- ing nonlinear integral equations through the Riemann-Liouville fractional operator were studied in [26]. Researchers have also explored inequalities using fractional continuities and differences. Hermite-Hadamard type integral inequalities involving the k-Riemann- Liouville fractional operator for twice-differentiable h-convex functions are investigated in [27]. Hermite-Hadamard type inequalities for h-convex function, as well as those involving (k − p)-operator with (α,h-m)-p convexity are discussed in [28]. Iqbal et al. investigated Grüss inequalities in [29] by considering the notion of generalized fractional derivative and Samraiz et al. examined Hermite-Hadamard inequalities for differentiable functions M. Samraiz et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6413 3 of 26 in [30]. Convex functions in the second sense were studied in [31] and the fractional Hermite-Hadamard inequalities in the second sense were investigated in [32]. 2. Preliminaries This section contains basic definitions and introductory information need to explore the main results. One of the early mathematicians to investigate the gamma function was Leonhard Euler in 1729, as documented in [33]. The gamma function, as defined in [34], can be expressed using the following definition. Definition 1. The gamma function, for R(λ) > 0, is defined by the relation: Γ(λ) = ∫ +∞ 0 ⊺λ−1e−⊺d ⊺ . The expression below that defines the complete beta function as presented in reference [35]. Definition 2. The complete beta function defined for positive real numbers ā and b̄, where R(ā) > 0 and R(b̄) > 0. B(ā, b̄) = ∫ 1 0 ⊺ā−1(1− ⊺)b̄−1d ⊺ . The incomplete beta function, defined in reference [36], given by the the following definition. Definition 3. Let λ ∈ [0, 1] and ā, b̄ > 0. Then, the incomplete beta function is defined by Bλ(ā, b̄) = ∫ λ 0 ⊺ā−1(1− ⊺)b̄−1d ⊺ . The beta function is also related to the gamma function through the following relationship: Bλ(ā, b̄) = Bλ(b̄, ā) = Γ(ā)Γ(b̄) Γ(ā+ b̄) . The (α,m)-convexity presented in [37] can be considered as a generalization of ordinary convexity and is defined by the following: Definition 4. Let a function 𭟋 : [0, b̄] → R, and (α,m) ∈ (0, 1]2 if, 𭟋(⊺ā+m(1− ⊺)b̄) ≤ ⊺α𭟋(ā) +m(1− ⊺α)𭟋(b̄). (1) is valid for all ā, b̄ ∈ [0, b̄] and ⊺ ∈ [0, 1], then we say 𭟋 is (α,m)-convex on [0, b̄]. This definition generalized the following convexities (i) If we substitute m = 1 in (1), then we get α-convex function. 𭟋(⊺αā+ (1− ⊺α)b̄) ≤ ⊺α𭟋(ā) + (1− ⊺α)𭟋(b̄). M. Samraiz et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6413 4 of 26 (ii) If we substitute α = 1 in (1), then we get m-convex function. 𭟋(⊺ā+m(1− ⊺)b̄) ≤ ⊺𭟋(ā) +m(1− ⊺)𭟋(b̄). (iii) If we substitute α = 1 and m = 1 in (1), then we get convex function. 𭟋(⊺ā+ (1− ⊺)b̄) ≤ ⊺𭟋(ā) + (1− ⊺)𭟋(b̄). The information related to Riemann-Liouville fractional integrals provided in [1] and [2] can be summarized as follows: Definition 5. If 𭟋 ∈ L[ā, b̄], then the Riemann-Liouville fractional integrals of order θ ∈ R+, denoted by χθ ā+𭟋 and χθ ā−𭟋, represent the left and right sided integrals, respectively i.e., (χθ ā+𭟋)(λ) = 1 Γ(θ) λ∫ ā (λ− ⊺)θ−1𭟋(⊺)d⊺, (0 ≤ ā < λ) and (χθ b̄−𭟋)(λ) = 1 Γ(θ) b̄∫ λ (⊺− λ)θ−1𭟋(⊺)d⊺, (0 ≤ λ < b̄). In [24], the following lemma discussed. Lemma 1. For ⊺ ∈ [0, 1], we obtain the following (1− ⊺)ω ≤ 21−ω − ⊺ω, for ω ∈ [0, 1], (1− ⊺)ω ≥ 21−ω − ⊺ω, for ω ∈ [1,+∞). The following lemma presented in [38] stated as follows: Lemma 2. Let 𭟋 : [0, b̄] → R and (α,m) ∈ (0, 1]2. If 𭟋(ā⊺b̄m(1−⊺)) ≤ ⊺α𭟋(ā) +m(1− ⊺α)𭟋(b̄). is valid for all ā, b̄ ∈ [0, b̄] and ⊺ ∈ [0, 1], then we say 𭟋 is GA (α,m)-convex function on [0, b̄]. Mathematicians are exploring fascinating inequalities and extending their reach through various convexities. Generalizations in inequalities through different convexities showcase the versatility and depth of the mathematical approach. Mathematicians not only refine existing inequalities but also explore new insights into the relationships between mathe- matical entities. These generalizations provide a broader understanding of mathematical structures, enriching the field with powerful tools for analysis and applications across var- ious domains. The extensions and generalizations of inequalities via different convexities M. Samraiz et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6413 5 of 26 reflect the dynamic and evolving nature of mathematics, leading to a deeper comprehen- sion of fundamental mathematical principles. The main objective of the present work is to establish more generalized forms of Hermite-Hadamard inequalities by using GA (α,m)-convex functions. It is important to mention here that it is not easy and not al- ways possible to introduce inequalities by only changing the convexity. Sometimes, it is a complex procedure to investigate predicted results by using a convexity involving new parameters, as in our work. We hope this idea motivates researchers to explore more generalized inequalities by using appropriate convexities. 3. Fractional Integral Inequalities for Geometrically-Arithmetically (α,m)-Convex Functions In this section, we derive fundamental identities to explore the Hermite-Hadamard inequalities. The first lemma is presented as follows: Lemma 3. Let 𭟋 : [ā, b̄] → R be a differentiable mapping on (ā, b̄) where ā < mb̄ ≤ b̄, mb̄ = µ and µ ∈ (ā, b̄]. If 𭟋′ ∈ L[ā, b̄], then the following equality for fractional integrals holds. 𭟋(ā) +𭟋(µ) 2 − Γ(θ + 1) 2(µ− ā)θ [ χθ ā+𭟋(µ) + χθ µ−𭟋(ā) ] = µ− ā 2 ∫ 1 0 [(1− ⊺)θ − ⊺θ]𭟋′ (⊺ā+m(1− ⊺)b̄ ) d ⊺ . (2) Proof. Consider I = ∫ 1 0 [(1− ⊺)θ − ⊺θ]𭟋′ (⊺ā+m(1− ⊺)b̄ ) d⊺ = [∫ 1 0 (1− ⊺)θ𭟋′ (⊺ā+m(1− ⊺)b̄ ) d⊺ ] + [ − ∫ 1 0 ⊺θ𭟋′ (⊺ā+m(1− ⊺)b̄ ) d⊺ ] = I1 + I2. (3) Applying integration by parts I1 = ∫ 1 0 (1− ⊺)θ𭟋′ (⊺ā+m(1− ⊺)b̄ ) d⊺ = 𭟋(µ) µ− ā − Γ(θ + 1) (µ− ā)θ+1 χθ µ−𭟋(ā). (4) Similarly I2 = − [∫ 1 0 ⊺θ𭟋′ (⊺ā+m(1− ⊺)b̄ ) d⊺ ] = 𭟋(ā) µ− ā − Γ(θ + 1) (µ− ā)θ+1 χθ ā+𭟋(µ). (5) M. Samraiz et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6413 6 of 26 Using (4) and (5) in (3), it follows that I = 𭟋(ā) +𭟋(µ) µ− ā − Γ(θ + 1) (µ− ā)θ+1 [χθ ā+𭟋(µ) + χθ µ−𭟋(ā)]. (6) Thus, by multiplying both sides of (6) by (µ−ā) 2 , we obtain the required result. Remark 1. By substituting m = 1 in Lemma 3, we arrive at [20, Lemma 1.5] i.e., 𭟋(ā) +𭟋(b̄) 2 − Γ(θ + 1) 2(b̄− ā)θ [ χθ ā+𭟋(b̄) + χθ b̄−𭟋(ā) ] = b̄− ā 2 ∫ 1 0 [(1− ⊺)θ − ⊺θ]𭟋′ (⊺ā+ (1− ⊺)b̄ ) d ⊺ . Lemma 4. Let 𭟋 : [ā, b̄] → R be a twice differentiable mapping on (ā, b̄) with ā < b̄. If 𭟋′′ ∈ L[ā, b̄], then the following fractional integral equality is true. 𭟋(ā) +𭟋(µ) 2 − Γ(θ + 1) 2(µ− ā)θ [ χθ ā+𭟋(µ) + χθ µ−𭟋(ā) ] = (µ− ā)2 2 ∫ 1 0 1− (1− ⊺)θ+1 − ⊺θ+1 θ + 1 𭟋′′ (⊺ā+m(1− ⊺)b̄ ) d ⊺ . (7) Proof. By comparing Lemma 3 and (7), we can write (µ− ā) 2 ∫ 1 0 [(1− ⊺)θ − ⊺θ]𭟋′ (⊺ā+m(1− ⊺)b̄ ) d⊺ = (µ− ā)2 2 ∫ 1 0 1− (1− ⊺)θ+1 − ⊺θ+1 θ + 1 𭟋′′ (⊺ā+m(1− ⊺)b̄ ) d ⊺ . (8) To prove the required result, we are to prove (8). For this purpose, consider (µ− ā) 2 ∫ 1 0 [(1− ⊺)θ − ⊺θ]𭟋′ (⊺ā+m(1− ⊺)b̄ ) d ⊺ . Integrating by parts the following, we have∫ 1 0 [(1− ⊺)θ − ⊺θ]𭟋′ (⊺ā+m(1− ⊺)b̄ ) d⊺ = −𭟋′(ā) +𭟋′(µ) θ + 1 − (µ− ā) ∫ 1 0 (1− ⊺)θ+1 + ⊺θ+1 θ + 1 𭟋′′ (⊺ā+m(1− ⊺)b̄ ) d ⊺ . (9) Note that 𭟋′(µ)−𭟋′(ā) = ∫ µ ā 𭟋′′(λ)dλ. (10) M. Samraiz et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6413 7 of 26 Substituting λ = ⊺ā+m(1− ⊺)b̄, we can write 𭟋′(µ)−𭟋′(ā) = ∫ 1 0 𭟋′′(⊺ā+m(1− ⊺)b̄)(µ− ā)d ⊺ . (11) Submitting (11) in (9), we have∫ 1 0 [(1− ⊺)θ − ⊺θ]𭟋′ (⊺ā+m(1− ⊺)b̄ ) d⊺ = 1 θ + 1 ∫ 1 0 𭟋′′(⊺ā+m(1− ⊺)b̄)(µ− ā)d⊺ − (µ− ā) ∫ 1 0 (1− ⊺)θ+1 + ⊺θ+1 θ + 1 𭟋′′ (⊺ā+m(1− ⊺)b̄ ) d ⊺ . Multiplying by µ−ā 2 , we obtain µ− ā 2 ∫ 1 0 [(1− ⊺)θ − ⊺θ]𭟋′ (⊺ā+m(1− ⊺)b̄ ) d⊺ = (µ− ā)2 2 ∫ 1 0 1− (1− ⊺)θ+1 − ⊺θ+1 θ + 1 𭟋′′ (⊺ā+m(1− ⊺)b̄ ) d ⊺ . Hence, the proof is done. Remark 2. By substituting m = 1 in Lemma 4, we arrive at [? , Lemma 2] i.e., 𭟋(ā) +𭟋(b̄) 2 − Γ(θ + 1) 2(b̄− ā)θ [ χθ ā+𭟋(b̄) + χθ b̄−𭟋(ā) ] = (b̄− ā)2 2 ∫ 1 0 1− (1− ⊺)θ+1 − ⊺θ+1 θ + 1 𭟋′′ (⊺ā+ (1− ⊺)b̄ ) d ⊺ . Lemma 5. Let 𭟋 : [ā, b̄] → R be a twice differentiable mapping on (ā, b̄) with ā < b̄. If 𭟋′′ ∈ L[ā, b̄], then Γ(θ + 1) 2(µ− ā)θ [ χθ ā+𭟋(µ) + χθ µ−𭟋(ā) ] −𭟋 ( ā+ µ 2 ) = (µ− ā)2 2 ∫ 1 0 p0(⊺)𭟋′′ (⊺ā+m(1− ⊺)b̄ ) d⊺, where p0(⊺) = { ⊺− 1−(1−⊺)θ+1−⊺θ+1 θ+1 , ⊺ ∈ [0, 12), 1− ⊺− 1−(1−⊺)θ+1−⊺θ+1 θ+1 , ⊺ ∈ [12 , 1). M. Samraiz et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6413 8 of 26 Proof. Consider∫ 1 0 p0(⊺)𭟋′′ (⊺ā+m(1− ⊺)b̄ ) d⊺ = [∫ 1 2 0 ( ⊺− 1− (1− ⊺)θ+1 − ⊺θ+1 θ + 1 ) 𭟋′′(⊺ā+m(1− ⊺)b̄)d⊺ ∫ 1 1 2 ( 1− ⊺− 1− (1− ⊺)θ+1 − ⊺θ+1 θ + 1 ) 𭟋′′(⊺ā+m(1− ⊺)b̄)d⊺ ] = [(∫ 1 2 0 ⊺𭟋′′(⊺ā+m(1− ⊺)b̄)d ⊺+ ∫ 1 1 2 (1− ⊺)𭟋′′(⊺ā+m(1− ⊺)b̄)d⊺ ) − ∫ 1 0 1− (1− ⊺)θ+1 − ⊺θ+1 θ + 1 𭟋′′(⊺ā+m(1− ⊺)b̄)d⊺ ] . (12) Let I = ∫ 1 2 0 ⊺𭟋′′ (⊺ā+m(1− ⊺)b̄ ) d ⊺+ ∫ 1 1 2 (1− ⊺)𭟋′′ (⊺ā+m(1− ⊺)b̄ ) d⊺ = I1 + I2. (13) Integrating by parts, we have I1 = ∫ 1 2 0 ⊺𭟋′′ (⊺ā+m(1− ⊺)b̄ ) d⊺ = 1 2(ā− µ) 𭟋′ ( ā+ µ 2 ) − 1 (ā− µ) [ 𭟋 ( ā+ µ 2 ) −𭟋(µ) ] , (14) and I2 = ∫ 1 1 2 (1− ⊺)𭟋′′ (⊺ā+m(1− ⊺)b̄ ) d⊺ = − 1 2(ā− µ) 𭟋′ ( ā+ µ 2 ) − 1 (ā− µ) [ 𭟋(ā)−𭟋 ( ā+ µ 2 )] . (15) Substituting (14) and (15) in (13), it follows that I = 𭟋(ā) +𭟋(µ) (µ− ā)2 − 2 (µ− ā)2 𭟋 ( ā+ µ 2 ) . (16) From (12), we obtain∫ 1 0 p0(⊺)𭟋′′ (⊺ā+m(1− ⊺)b̄ ) d⊺ M. Samraiz et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6413 9 of 26 = 𭟋(ā) +𭟋(µ) (µ− ā)2 − 2 (µ− ā)2 𭟋 ( ā+ µ 2 ) − ∫ 1 0 1− (1− ⊺)θ+1 − ⊺θ+1 ⋎(θ + 1) 𭟋′′(⊺ā+m(1− ⊺)b̄)d ⊺ . (17) Thus, by multiplying both sides of (17) by (µ−ā)2 2 , we have (µ− ā)2 2 ∫ 1 0 p0(⊺)𭟋′′ (⊺ā+m(1− ⊺)b̄ ) d⊺ = 𭟋(ā) +𭟋(µ) 2 −𭟋 ( ā+ µ 2 ) − (µ− ā)2 2 ∫ 1 0 1− (1− ⊺)θ+1 − ⊺θ+1 θ + 1 𭟋′′ (⊺ā+m(1− ⊺)b̄ ) d ⊺ . (18) On the other hand, by (7), we obtain 𭟋(ā) +𭟋(µ) 2 − Γ(θ + 1) 2(µ− ā)θ [ χθ ā+𭟋(µ) + χθ µ−𭟋(ā) ] = (µ− ā)2 2 ∫ 1 0 1− (1− ⊺)θ+1 − ⊺θ+1 θ + 1 𭟋′′ (⊺ā+m(1− ⊺)b̄ ) d ⊺ . (19) Combining (18) and (19), we have obtained the conclusion of the proof. Remark 3. By substituting m = 1 in Lemma 5, we arrive at [20, Lemma 2.1] i.e., Γ(θ + 1) 2(b̄− ā)θ [ χθ ā+𭟋(b̄) + χθ b̄−𭟋(ā) ] −𭟋 ( ā+ b̄ 2 ) = (b̄− ā)2 2 ∫ 1 0 p0(⊺)𭟋′′ (⊺ā+ (1− ⊺)b̄ ) d ⊺ . Lemma 6. Let 𭟋 : [ā, b̄] → R be a twice differentiable mapping on (ā, b̄) with ā < b̄. If ⋎ > 0, 𭟋′′ ∈ L[ā, b̄], then 𭟋(ā) +𭟋(µ) ⋎(⋎+ 1) + 2 ⋎+ 1 𭟋 ( ā+ µ 2 ) − Γ(θ + 1) ⋎(µ− ā)θ [ χθ ā+𭟋(µ) + χθ µ−𭟋(ā) ] = (µ− ā)2 ∫ 1 0 q0(⊺)𭟋′′ (⊺ā+m(1− ⊺)b̄ ) d⊺, where q0(⊺) =  1−(1−⊺)θ+1−⊺θ+1 ⋎(θ+1) − ⊺ ⋎+1 , ⊺ ∈ [0, 12), 1−(1−⊺)θ+1−⊺θ+1 ⋎(θ+1) − 1−⊺ ⋎+1 , ⊺ ∈ [12 , 1). M. Samraiz et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6413 10 of 26 Proof. Consider∫ 1 0 q0(⊺)𭟋′′ (⊺ā+m(1− ⊺)b̄ ) d⊺ = [∫ 1 2 0 ( 1− (1− ⊺)θ+1 − ⊺θ+1 ⋎(θ + 1) − ⊺ ⋎+ 1 ) 𭟋′′(⊺ā+m(1− ⊺)b̄)d⊺ ∫ 1 1 2 ( 1− (1− ⊺)θ+1 − ⊺θ+1 ⋎(θ + 1) − 1− ⊺ ⋎+ 1 ) 𭟋′′(⊺ā+m(1− ⊺)b̄)d⊺ ] = [∫ 1 0 1− (1− ⊺)θ+1 − ⊺θ+1 ⋎(θ + 1) 𭟋′′(⊺ā+m(1− ⊺)b̄)d⊺ − 1 ⋎+ 1 (∫ 1 2 0 ⊺𭟋′′(⊺ā+m(1− ⊺)b̄)d ⊺+ ∫ 1 1 2 (1− ⊺)𭟋′′(⊺ā+m(1− ⊺)b̄)d⊺ )] . (20) By using (16), we obtain∫ 1 0 q0(⊺)𭟋′′ (⊺ā+m(1− ⊺)b̄ ) d⊺ = ∫ 1 0 1− (1− ⊺)θ+1 − ⊺θ+1 ⋎(θ + 1) 𭟋′′(⊺ā+m(1− ⊺)b̄)d⊺ − 1 ⋎+ 1 ( 𭟋(ā) +𭟋(µ) (µ− ā)2 − 2 (µ− ā)2 𭟋 ( ā+ µ 2 )) . (21) Multiplying both sides of (21) by (µ− ā)2, we get (µ− ā)2 ∫ 1 0 q0(⊺)𭟋′′ (⊺ā+m(1− ⊺)b̄ ) d⊺ = (µ− ā)2 ∫ 1 0 1− (1− ⊺)θ+1 − ⊺θ+1 ⋎(θ + 1) 𭟋′′(⊺ā+m(1− ⊺)b̄)d⊺ − 𭟋(ā) +𭟋(µ) (⋎+ 1) + 2 ⋎+ 1 𭟋 ( ā+ µ 2 ) . (22) By multiplying 1 ⋎ on both sides of Lemma 4, we can write 𭟋(ā) +𭟋(µ) ⋎ − Γ(θ + 1) ⋎(µ− ā)θ [ χθ ā+𭟋(µ) + χθ µ−𭟋(ā) ] = (µ− ā)2 ∫ 1 0 1− (1− ⊺)θ − ⊺θ ⋎(θ + 1) 𭟋′ (⊺ā+m(1− ⊺)b̄ ) d ⊺ . (23) Combining (22) and (23), we have obtained the conclusion of the proof. M. Samraiz et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6413 11 of 26 Remark 4. By substituting m = 1 in Lemma 6, we arrive at [20, Lemma 2.2] i.e., 𭟋(ā) +𭟋(b̄) ⋎(⋎+ 1) + 2 ⋎+ 1 𭟋 ( ā+ b̄ 2 ) − Γ(θ + 1) ⋎(b̄− ā)θ [ χθ ā+𭟋(ā) + χθ b̄−𭟋(b̄) ] = (b̄− ā)2 ∫ 1 0 q0(⊺)𭟋′′ (⊺ā+ (1− ⊺)b̄ ) d ⊺ . 4. Hermite-Hadamard Inequalities for GA (α,m)-Convex Functions This section is dedicated to explore Hermite-Hadamard type inequalities by using the results proved in the previous section. To establish the inequalities, we need the following lemma. Lemma 7. For ⊺ ∈ [0, 1], ā, b̄ > 0, we have ⊺ā+m(1− ⊺)b̄ ≤ (ā⊺b̄m(1−⊺)). Theorem 1. Let 𭟋 : [ā, b̄] −→ R be a twice differentiable function such that |𭟋′′| is Lebesgue integerable, increasing and GA (α,m)-convex function on [ā, b̄]. Then for given parameters θ ∈ (0,+∞) and (α,m) ∈ (0, 1]2, where 0 ≤ ā < b̄ the following inequality∣∣∣∣𭟋(ā) +𭟋(µ) 2 − Γ(θ + 1) 2(µ− ā)θ [ χθ ā+𭟋(µ) + χθ µ−𭟋(ā) ]∣∣∣∣ ≤ (µ− ā)2 2(θ + 1) [ |𭟋′′(ā)| ( 1 α+ 1 − 1 θ + α+ 2 ) +m|𭟋′′(b̄)| ( α α+ 1 + 1 θ + α+ 2 )] (24) holds true. Proof. By using Lemma 2, Lemma 4 and Lemma 7, we have∣∣∣∣𭟋(ā) +𭟋(µ) 2 − Γ(θ + 1) 2(µ− ā)θ [ χθ ā+𭟋(µ) + χθ µ−𭟋(ā) ]∣∣∣∣ ≤ (µ− ā)2 2 ∫ 1 0 ∣∣∣∣1− (1− ⊺)θ+1 − ⊺θ+1 θ + 1 ∣∣∣∣ ∣∣𭟋′′ (⊺ā+m(1− ⊺)b̄ )∣∣ d⊺ ≤ (µ− ā)2 2 ∫ 1 0 ∣∣∣∣1− (1− ⊺)θ+1 − ⊺θ+1 θ + 1 ∣∣∣∣ ∣∣∣𭟋′′ ( ā⊺b̄m(1−⊺) )∣∣∣ d⊺ ≤ (µ− ā)2 2(θ + 1) ∫ 1 0 ∣∣∣1− (1− ⊺)θ+1 − ⊺θ+1 ∣∣∣ [⊺α ∣∣𭟋′′(ā) ∣∣+m(1− ⊺α) ∣∣𭟋′′(b̄) ∣∣] d⊺ ≤ (µ− ā)2 2(θ + 1) [ |𭟋′′(ā)| ( 1 α+ 1 − 1 θ + α+ 2 ) +m|𭟋′′(b̄)| ( α α+ 1 + 1 θ + α+ 2 )] . Hence, the proof is done. Example 1. Let 𭟋(⊺) = ⊺3 and by taking the values of the parameters M. Samraiz et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6413 12 of 26 • ā = 0, µ = 1 and b̄ = 2 • α = 0.5,m = 0.5 and θ = 1 The LHS of the inequality∣∣∣∣𭟋(ā) +𭟋(µ) 2 − Γ(θ + 1) 2(µ− ā)θ [ χθ ā+𭟋(µ) + χθ µ−𭟋(ā) ]∣∣∣∣∣∣∣∣(0)3 + (1)3 2 − Γ(1 + 1) 2(1− 0)1 [ χ1 0+𭟋(1) + χ1 1−𭟋(0) ]∣∣∣∣∣∣∣∣∣∣(0) 3 + (1)3 2 − Γ(1 + 1) 2(1− 0)1  1 Γ(1) 1∫ 0 (1− ⊺)1−1(⊺)3d ⊺+ 1 Γ(1) 2∫ 1 (⊺− 0)1−1(0)3d⊺ ∣∣∣∣∣∣∣∣∣∣∣12 − Γ(2) 2 [ 1 Γ(1) [ ⊺4 4 ]1 0 + 0 ]∣∣∣∣∣∣∣∣∣12 − 1 2 × 1 4 ∣∣∣∣ |0.5− 0.125| = 0.375. The RHS of the inequality (µ− ā)2 2(θ + 1) [ |𭟋′′(ā)| ( 1 α+ 1 − 1 θ + α+ 2 ) +m|𭟋′′(b̄)| ( α α+ 1 + 1 θ + α+ 2 )] (1− 0)2 2(1 + 1) [ |6(0)| ( 1 0.5 + 1 − 1 1 + 0.5 + 2 ) + 0.5|6(2)| ( 0.5 0.5 + 1 + 1 1 + 0.5 + 2 )] . To simplify, 0.5 1.5 = 1 3 , 1 0.5 = 1 2 and 1 3.5 = 2 7 1 4 [ 0.5(12) ( 0.5 1.5 + 1 3.5 )] ≈ 0.9286 Hence, the chosen values satisfy the inequality 0.375 ≤ 0.9286. This example confirms the validity of the fractional Hermite-Hadamard-type inequality and demonstrates its useful- ness in numerical analysis for estimating error bounds in numerical integration, as well as in inequality theory, fractional differential equations and mathematical modeling. Theorem 2. Let 𭟋 : [ā, b̄] −→ R be a twice differentiable function on [ā, b̄] and 1 < ⋋2 < +∞. If |𭟋′′|⋋2 is Lebesgue integerable, increasing and GA (α,m)-convex on [ā, b̄]. Then for given parameters θ ∈ (0,+∞) and (α,m) ∈ (0, 1]2, where 0 ≤ ā < b̄ the following inequality ∣∣∣∣𭟋(ā) +𭟋(µ) 2 − Γ(θ + 1) 2(µ− ā)θ [ χθ ā+𭟋(µ) + χθ µ−𭟋(ā) ]∣∣∣∣ M. Samraiz et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6413 13 of 26 ≤ (µ− ā)2max(1− 21−θ, 21−θ − 1) 2(θ + 1) ( |𭟋′′(ā)|⋋2 +mα|𭟋′′(b̄)|⋋2 α+ 1 ) 1 ⋋2 (25) holds true. Proof. The proof of this result is split into two situations presented as follows. Case(i) Let θ ∈ (0, 1). By utilizing Lemma 1, 2, 4, 7 and applying Hölder’s inequality, we have ∣∣∣∣𭟋(ā) +𭟋(µ) 2 − Γ(θ + 1) 2(µ− ā)θ [ χθ ā+𭟋(b̄) + χθ b̄−𭟋(ā) ]∣∣∣∣ ≤ (µ− ā)2 2 ∫ 1 0 ∣∣∣∣1− (1− ⊺)θ+1 − ⊺θ+1 θ + 1 ∣∣∣∣ |𭟋′′(⊺ā+ (1− ⊺)b̄)|d⊺ ≤ (µ− ā)2 2 (∫ 1 0 ∣∣∣∣1− (1− ⊺)θ+1 − ⊺θ+1 θ + 1 ∣∣∣∣⋋1 d⊺ ) 1 ⋋1 × (∫ 1 0 |𭟋′′(⊺ā+m(1− ⊺)b̄)|⋋2d⊺ ) 1 ⋋2 ≤ (µ− ā)2 2(θ + 1) (∫ 1 0 ∣∣∣1− (1− ⊺)θ − ⊺θ ∣∣∣⋋1 d⊺ ) 1 ⋋1 (∫ 1 0 |𭟋′′(ā⊺b̄m(1−⊺))|⋋2d⊺ ) 1 ⋋2 ≤ (µ− ā)2 2(θ + 1) (∫ 1 0 |1− (1− ⊺)θ − ⊺θ|⋋1d⊺ ) 1 ⋋1 × (∫ 1 0 (⊺α|𭟋′′(ā)|⋋2 +m(1− ⊺α)|𭟋′′(b̄)|⋋2)d⊺ ) 1 ⋋2 ≤ (µ− ā)2 2(θ + 1) ( |𭟋′′(ā)|⋋2 +mα|𭟋′′(b̄)|⋋2 α+ 1 ) 1 ⋋2 (∫ 1 0 ∣∣∣1− (1− ⊺)θ − ⊺θ ∣∣∣⋋1 ) 1 ⋋1 ≤ (µ− ā)2 2(θ + 1) ( |𭟋′′(ā)|⋋2 +mα|𭟋′′(b̄)|⋋2 α+ 1 ) 1 ⋋2 (∫ 1 0 [(1− ⊺)θ + ⊺θ − 1]⋋1 ) 1 ⋋1 ≤ (µ− ā)2(21−θ − 1) 2(θ + 1) ( |𭟋′′(ā)|⋋2 +mα|𭟋′′(b̄)|⋋2 α+ 1 ) 1 ⋋2 , (26) where 1 ⋋1 + 1 ⋋2 = 1. Case(ii): Let θ ∈ [1,+∞). By utilizing Lemma 1, 2, 3, 7 and applying Hölder’s inequality, we have ∣∣∣∣𭟋(ā) +𭟋(µ) 2 − Γ(θ + 1) 2(µ− ā)θ [ χθ ā+𭟋(µ) + χθ b̄−𭟋(ā) ]∣∣∣∣ ≤ (µ− ā)2 2(θ + 1) ( |𭟋′′(ā)|⋋2 +mα|𭟋′′(b̄)|⋋2 α+ 1 ) 1 ⋋2 (∫ 1 0 [1− (1− ⊺)θ − ⊺θ]⋋1d⊺ ) 1 ⋋1 M. Samraiz et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6413 14 of 26 ≤ (µ− ā)2(1− 21−θ) 2(θ + 1) ( |𭟋′′(ā)|⋋2 +mα|𭟋′′(b̄)|⋋2 α+ 1 ) 1 ⋋2 . (27) From (26) and (27), we have obtained the required result. Remark 5. In accordance with the selection of parameters α = 1 and m = 1 in Theorem 2 and s = 1 in [2, Theorem 3.2], we get∣∣∣∣𭟋(ā) +𭟋(b̄) 2 − Γ(θ + 1) 2(b̄− ā)θ [ χθ ā+𭟋(b̄) + χθ b̄−𭟋(ā) ]∣∣∣∣ ≤ (b̄− ā)2max(1− 21−θ, 21−θ − 1) 2(θ + 1) ( |𭟋′′(ā)|⋋2 + |𭟋′′(b̄)|⋋2 2 ) 1 ⋋2 . The following main result is based on utilization of Lemma 5. Theorem 3. Let 𭟋 : [ā, b̄] −→ R be a twice differentiable function such that |𭟋′′| is Lebesgue integerable, increasing and GA (α,m)-convex function on [ā, b̄]. Then for given parameters θ ∈ (0,+∞) and (α,m) ∈ (0, 1]2, where 0 ≤ ā < b̄ the following inequality∣∣∣∣ Γ(θ + 1) 2(b̄− ā)θ [ χθ ā+𭟋(µ) + χθ µ−𭟋(ā) ] −𭟋 ( ā+ µ 2 )∣∣∣∣ ≤ (µ− ā)2 2(θ + 1) [ |𭟋′′(ā)| ( θ − θ2−α−1 − 2−α−1 α+ 1 − θ + 1 α+ 2 + 1 θ + α+ 2 + 2B0.5(θ + 2, α+ 1) ) +m|𭟋′′(b̄)| ( θ − 3 4 + 1 θ + 2 − θ − θ2−α−1 − 2−α−1 α+ 1 − θ + 1(1− 2−α) α+ 2 − 1 θ + α+ 2 − 2B0.5(α+ 1, θ + 2) )] holds true. Proof. By using Lemma 2, 5 and 7 we have∣∣∣∣ Γ(θ + 1) 2(µ− ā)θ [χθ ā+𭟋(µ) + χθ µ−𭟋(ā)]−𭟋 ( ā+ µ 2 )∣∣∣∣ ≤ (µ− ā)2 2 ∫ 1 0 |p0(⊺)||𭟋′′(⊺ā+m(1− ⊺)b̄)|d⊺ ≤ (µ− ā)2 2 ∫ 1 0 |p0(⊺)||𭟋′′(ā⊺b̄m(1−⊺)|d⊺ ≤ (µ− ā)2 2 ∫ 1 0 |p0(⊺)|(⊺α|𭟋′′(ā)|+m(1− ⊺α)|𭟋′′(b̄)|)d⊺ M. Samraiz et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6413 15 of 26 ≤ (µ− ā)2 2 [∫ 1 2 0 ∣∣∣∣⊺− 1− (1− ⊺)θ+1 − ⊺θ+1 θ + 1 ∣∣∣∣ × ( ⊺α|𭟋′′(ā)|+m(1− ⊺α)|𭟋′′(b̄)| ) d⊺ + ∫ 1 1 2 ∣∣∣∣1− ⊺− 1− (1− ⊺)θ+1 − ⊺θ+1 θ + 1 ∣∣∣∣ (⊺α|𭟋′′(ā)|+m(1− ⊺α)|𭟋′′(b̄)|)d⊺ ] ≤ (µ− ā)2 2(θ + 1) [∫ 1 2 0 ∣∣∣⊺(θ + 1)− 1 + (1− ⊺)θ+1 + ⊺θ+1 ∣∣∣ ×(⊺α|𭟋′′(ā)|+m(1− ⊺α)|𭟋′′(b̄)|)d⊺ + ∫ 1 1 2 ∣∣∣θ − ⊺(θ + 1) + (1− ⊺)θ+1 + ⊺θ+1 ∣∣∣ (⊺α|𭟋′′(ā)|+m(1− ⊺α)|𭟋′′(b̄)|)d⊺ ] . (28) Consider∫ 1 2 0 ∣∣∣⊺(θ + 1)− 1 + (1− ⊺)θ+1 + ⊺θ+1 ∣∣∣ (⊺α|𭟋′′(ā)|+m(1− ⊺α)|𭟋′′(µ)|)d⊺ ≤ |𭟋′′(ā)| ( (θ + 1) (12) α+2 α+ 2 − (12) α+1 α+ 1 +B0.5(α+ 1, θ + 2) + (12) θ+α+2 θ + α+ 2 ) +m|𭟋′′(b̄) ( θ − 3 8 + 2−α−1 α+ 1 − (θ + 1)2−α−2 α+ 2 − 2−θ−α−2 θ + α+ 2 −B0.5(α+ 1, θ + 2) ) . (29) Also, consider∫ 1 1 2 ∣∣∣θ − ⊺(θ + 1) + (1− ⊺)θ+1 + ⊺θ+1 ∣∣∣ (⊺α|𭟋′′(ā)|+m(1− ⊺α)|𭟋′′(b̄)|)d⊺ ≤ |𭟋′′(ā)| ( θ 1− (12) α+1 α+ 1 − (θ + 1) 1− (12) α+2 α+ 2 +B0.5(α+ 1, θ + 2) + 1− (12) θ+α+2 θ + α+ 2 ) +m|𭟋′′(b̄)| ( θ − 3 8 + 1 θ + 2 − θ − θ2−α−1 α+ 1 + (θ + 1)(1− 2−α−2) α+ 2 − 1− 2−θ−α−2 θ + α+ 2 −B0.5(α+ 1, θ + 2) ) . (30) M. Samraiz et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6413 16 of 26 Substituting values of (29) and (30) in (28), we obtain∣∣∣∣ Γ(θ + 1) 2(µ− ā)θ [χθ ā+𭟋(µ) + χθ µ−𭟋(ā)]−𭟋 ( ā+ µ 2 )∣∣∣∣ ≤ (µ− ā)2 2(θ + 1) |𭟋′′(ā)| ( (θ + 1) (12) α+2 α+ 2 − (12) α+1 α+ 1 +B0.5(α+ 1, θ + 2) + (12) θ+α+2 θ + α+ 2 ) + (µ− ā)2 2(θ + 1) m|𭟋′′(b̄) ( θ − 3 8 + (12) α+1 α+ 1 − (θ + 1)(12) α+2 α+ 2 − (12) θ+α+2 θ + α+ 2 −B0.5(α+ 1, θ + 2) ) + (µ− ā)2 2(θ + 1) |𭟋′′(ā)| ( θ 1− (12) α+1 α+ 1 − (θ + 1) 1− (12) α+2 α+ 2 +B(α+ 1, θ + 2) + 1− (12) θ+α+2 θ + α+ 2 ) + (µ− ā)2 2(θ + 1) m|𭟋′′(b̄)| ( θ − 3 8 + 1 θ + 2 − θ − θ2−α−1 α+ 1 + (θ + 1)(1− 2−α−2) α+ 2 − 1− 2−θ−α−2 θ + α+ 2 −B0.5(α+ 1, θ + 2) ) . Hence, the proof is done. Theorem 4. Let 𭟋 : [ā, b̄] −→ R be a twice differentiable function such that |𭟋′′| is Lebesgue integerable, increasing and GA (α,m)-convex function on [ā, b̄]. Then for given parameters θ ∈ (0,+∞) and (α,m) ∈ (0, 1]2, where 0 ≤ ā < b̄. Thus, let 1 < ⋋2 < +∞ the following inequality∣∣∣∣ Γ(θ + 1) 2(µ− ā)θ [ χθ ā+𭟋(µ) + χθ µ−𭟋(ā) ] −𭟋 ( ā+ µ 2 )∣∣∣∣ ≤ (µ− ā)2 2(θ + 1) ( |𭟋′′(ā)|⋋2 +mα|𭟋′′(b̄)|⋋2 α+ 1 ) 1 ⋋2 × ( (θ + 1)2−⋋1−1 + (θ + 0.5)⋋1+1 − θ⋋1+1 ⋋1 + 1 ) 1 ⋋1 holds true, where 1 ⋋1 + 1 ⋋2 = 1. M. Samraiz et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6413 17 of 26 Proof. By utilizing Lemma 5, 7 and applying Hölder’s inequality, we have∣∣∣∣ Γ(θ + 1) 2(µ− ā)θ [χθ ā+𭟋(µ) + χθ µ−𭟋(ā)]−𭟋 ( ā+ µ 2 )∣∣∣∣ ≤ (µ− ā)2 2 ∫ 1 0 |p0(⊺)||𭟋′′(⊺ā+m(1− ⊺)b̄)|d⊺ ≤ (µ− ā)2 2 ∫ 1 0 |p0(⊺)||𭟋′′(ā⊺b̄m(1−⊺)|d⊺ ≤ (µ− ā)2 2 ∫ 1 0 |p0(⊺)|⊺α|𭟋′′(ā)|+m(1− ⊺α)|𭟋′′(b̄)|d⊺ ≤ (µ− ā)2 2 ∫ 1 0 ( |(p0(⊺))⋋1 | ) 1 ⋋1 ( |𭟋′′(ā)|⋋2 ∫ 1 0 ⊺αd ⊺+m|𭟋′′(b̄)|⋋2 ∫ 1 0 (1− ⊺α)d⊺ ) 1 ⋋2 ≤ (µ− ā)2 2 ∫ 1 0 ( |(p0(⊺))⋋1 | ) 1 ⋋1 ( |𭟋′′(ā)|⋋2 1 α+ 1 +m|𭟋′′(b̄)|⋋2 α α+ 1 ) 1 ⋋2 Substituting the value p0(⊺) of Lemma 4. ≤ (µ− ā)2 2 ( |𭟋′′(ā)|⋋2 +mα|𭟋′′(b̄)|⋋2 α+ 1 ) 1 ⋋2 (∫ 1 2 0 ∣∣∣∣⊺− 1− (1− ⊺)θ+1 − ⊺θ+1 θ + 1 ∣∣∣∣ d⊺ + ∫ 1 1 2 ∣∣∣∣1− ⊺− 1− (1− ⊺)θ+1 − ⊺θ+1 θ + 1 ∣∣∣∣ d⊺ ) 1 ⋋1 ≤ (µ− ā)2 2(θ + 1) ( |𭟋′′(ā)|⋋2 +mα|𭟋′′(b̄)|⋋2 α+ 1 ) 1 ⋋2 × (∫ 1 2 0 ∣∣∣θ(⊺) + ⊺− 1 + (1− ⊺)θ+1 + ⊺θ+1 ∣∣∣⋋1 d⊺ + ∫ 1 1 2 ∣∣∣θ − ⊺+ (1− ⊺)θ+1 + ⊺θ+1 ∣∣∣⋋1 d⊺ ) 1 ⋋1 ≤ (µ− ā)2 2(θ + 1) ( |𭟋′′(ā)|⋋2 +mα|𭟋′′(b̄)|⋋2 α+ 1 ) 1 ⋋2 × ( (θ + 1) ∫ 1 2 0 ⊺⋋1d ⊺+ ∫ 1 1 2 (θ − ⊺+ 1)⋋1d⊺ ) 1 ⋋1 ≤ (µ− ā)2 2(θ + 1) ( |𭟋′′(ā)|⋋2 +mα|𭟋′′(b̄)|⋋2 α+ 1 ) 1 ⋋2 ( (θ + 1) (12) ⋋1+1 ⋋1 + 1 − (θ)⋋1+1 ⋋1 + 1 + (θ + 1 2) ⋋1+1 ⋋1 + 1 ) 1 ⋋1 M. Samraiz et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6413 18 of 26 ≤ (µ− ā)2 2(θ + 1) ( |𭟋′′(ā)|⋋2 +mα|𭟋′′(b̄)|⋋2 α+ 1 ) 1 ⋋2 × ( (θ + 1)2−⋋1−1 − θ⋋1+1 + (θ + 0.5)⋋1+1 ⋋1 + 1 ) 1 ⋋1 . Hence, the proof is done. Remark 6. In accordance with the selection of parameters α = 1 and m = 1 in Theorem 4 and s = 1 in [2, Theorem 4.2], we get∣∣∣∣ Γ(θ + 1) 2(b̄− ā)θ [χθ ā+𭟋(b̄) + χθ b̄−𭟋(ā)]−𭟋 ( ā+ b̄ 2 )∣∣∣∣ ≤ (b̄− ā)2 2(θ + 1) ( |𭟋′′(ā)|⋋2 + |𭟋′′(b̄)|⋋2 2 ) 1 ⋋2 × ( (θ + 1)2−⋋1−1 + (θ + 0.5)⋋1+1 − θ⋋1+1 ⋋1 + 1 ) 1 ⋋1 . In the following result, we utilize Lemma 6. Theorem 5. Let 𭟋 : [ā, b̄] −→ R be a twice differentiable function such that |𭟋′′| is Lebesgue integerable, increasing and GA (α,m)-convex function on [ā, b̄]. Then for some given parameters θ ∈ (0,+∞) and (α,m) ∈ (0, 1]2, where 0 ≤ ā < b̄ the following inequality∣∣∣∣𭟋(ā) +𭟋(µ) ⋎(⋎+ 1) + 2 ⋎+ 1 𭟋 ( ā+ µ 2 ) − Γ(θ + 1) ⋎(µ− ā)θ [χθ ā+𭟋(µ) + χθ µ−𭟋(ā)] ∣∣∣∣ ≤ (µ− ā)2 ⋎(θ + 1)(⋎+ 1) max [ [(⋎+ 1)− (⋎+ 1)2−θ] × ( 2−α|𭟋′′(ā) + (α+ 1− 2−α)m|𭟋′′(b̄) 2(α+ 1) ) −⋎ (θ + 1) ( 2−α+1|𭟋′′(ā)|+ (α+ 2− 2−α+1)m|𭟋′′(b̄)| 8(α+ 2) ) ,⋎(θ + 1) ( 2−α+1|𭟋′′(ā)|+ (α+ 2− 2−α+1)m|𭟋′′(b̄)| 8(α+ 2) )] + (µ− ā)2 ⋎(θ + 1)(⋎+ 1) max [ [(⋎+ 1)− (⋎+ 1)2−θ −⋎(θ + 1)] × ( (2− 2−α)|𭟋′′(ā)|+ (α− 1 + 2−α)m|𭟋′′(b̄)| 2(α+ 1) ) +⋎ (θ + 1) ( (8− 2−α+1)|𭟋′′(ā)|+ (3α− 2 + 2−α+1)m|𭟋′′(b̄)| 8(α+ 2) ) ,⋎(θ + 1) ( (2− 2−α)|𭟋′′(ā)|+ (α− 1 + 2−α)m|𭟋′′(b̄)| 2(α+ 1) M. Samraiz et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6413 19 of 26 −(8− 2−α+1)|𭟋′′(ā)|+ (3α− 2 + 2−α+1)m|𭟋′′(b̄) 8(α+ 2) )] holds true. Proof. By using Lemma 2, 6 and 7, we have∣∣∣∣𭟋(ā) +𭟋(µ) ⋎(⋎+ 1) + 2 ⋎+ 1 𭟋 ( ā+ µ 2 ) − Γ(θ + 1) ⋎(µ− ā)θ [χθ ā+𭟋(µ) + χθ µ−𭟋(ā)] ∣∣∣∣ ≤ (µ− ā)2 ∫ 1 0 |q0(⊺)||𭟋′′(⊺ā+m(1− ⊺)b̄)|d⊺ ≤ (µ− ā)2 ∫ 1 0 |q0(⊺)||𭟋′′(ā⊺b̄m(1−⊺)|d⊺ ≤ (µ− ā)2 2 ∫ 1 0 |q0(⊺)|[⊺α|𭟋′′(ā)|+m(1− ⊺α)|𭟋′′(b̄)|]d⊺ ≤ (µ− ā)2 2 (∫ 1 2 0 ∣∣∣∣1− (1− ⊺)θ+1 − ⊺θ+1 ⋎(θ + 1) − ⊺ ⋎+ 1 ∣∣∣∣ ×[⊺α|𭟋′′(ā)|+m(1− ⊺α)|𭟋′′(b̄)|]d⊺ + ∫ 1 1 2 ∣∣∣∣1− (1− ⊺)θ+1 − ⊺θ+1 ⋎(θ + 1) − 1− ⊺ ⋎+ 1 ∣∣∣∣ [⊺α|𭟋′′(ā)|+m(1− ⊺α)|𭟋′′(b̄)|]d⊺ ) ≤ (µ− ā)2 2(θ + 1)(⋎+ 1) × (∫ 1 2 0 [(⋎+ 1)− (⋎+ 1)(1− ⊺)θ+1 − (⋎+ 1) ⊺θ+1 − ⊺ ⋎(θ + 1)] ×(⊺α|𭟋′′(ā)|+m(1− ⊺α)|𭟋′′(b̄)|)d⊺ + ∫ 1 1 2 [(⋎+ 1)− (⋎+ 1)(1− ⊺)θ+1 − (⋎+ 1) ⊺θ+1 −⋎ (θ + 1)(1− ⊺)] ×[⊺α|𭟋′′(ā)|+m(1− ⊺α)|𭟋′′(b̄)|d⊺] ) . (31) Let us consider∫ 1 2 0 [⋎+ 1− (⋎+ 1)[(1− ⊺)θ+1 + ⊺θ+1]− ⊺ ⋎ (θ + 1)] × [⊺α|𭟋′′(ā)|+m(1− ⊺α)|𭟋′′(b̄)|]d⊺ ≤ [(⋎+ 1)− (⋎+ 1)2−θ] ( |𭟋′′(ā)| ∫ 1 2 0 ⊺αd ⊺+m|𭟋′′(b̄)| ∫ 1 2 0 (1− ⊺α)d⊺ ) − ⊺ ⋎ (θ + 1) ( |𭟋′′(ā)| ∫ 1 2 0 ⊺αd ⊺+m|𭟋′′(b̄)| ∫ 1 2 0 (1− ⊺α)d⊺ ) M. Samraiz et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6413 20 of 26 ≤ [(⋎+ 1)− (⋎+ 1)2−θ] ( |𭟋′′(ā)2−α +m|𭟋′′(b̄)(α+ 1− 2−α) 2(α+ 1) ) −⋎(θ + 1) ( |𭟋′′(ā)|2−α+1 +m|𭟋′′(b̄)(α+ 2− 2−α+1) 8(α+ 2) ) , (32) and ∫ 1 2 0 [−⋎−1 + (⋎+ 1)2−θ + ⊺ ⋎ (θ + 1)] [ |𭟋′′(ā)| ⊺α +m|𭟋′′(b̄)|(1− ⊺α) ] d⊺ ≤ [−⋎−1 + (⋎+ 1)2−θ] ∫ 1 2 0 [ |𭟋′′(ā)| ⊺α +m|𭟋′′(b̄)|(1− ⊺α) ] d⊺ +⋎(θ + 1) ∫ 1 2 0 [ |𭟋′′(ā)| ⊺α+1 +m|𭟋′′(b̄)| ⊺ (1− ⊺)α ] d⊺ ≤ ⋎(θ + 1) ( |𭟋′′(ā)|2−α+1 +m|𭟋′′(b̄)(α+ 2− 2−α+1) 8(α+ 2) ) , (33) and∫ 1 1 2 [ (⋎+ 1)− (⋎+ 1)2−θ + ⊺ ⋎ (θ + 1)−⋎(θ + 1) ] × ( |𭟋′′(ā)| ⊺α d ⊺+m|𭟋′′(b̄)|(1− ⊺α)d⊺ ) ≤ [(⋎+ 1)− (⋎+ 1)2−θ −⋎(θ + 1)] ( |𭟋′′(ā)| ∫ 1 1 2 ⊺αd ⊺+m|𭟋′′(b̄)| ∫ 1 1 2 (1− ⊺α)d⊺ ) +⋎(θ + 1) ( |𭟋′′(ā)| ∫ 1 1 2 ⊺α+1d ⊺+m|𭟋′′(b̄)| ∫ 1 1 2 ⊺(1− ⊺α)d⊺ ) ≤ [(⋎+ 1)− (⋎+ 1)2−θ −⋎(θ + 1)] ( (2− 2−α)|𭟋′′(ā)|+ (α− 1 + 2−α)m|𭟋′′(b̄)| 2(α+ 1) ) +⋎(θ + 1) ( (8− 2−α+1)|𭟋′′(ā)|+ (3α− 2 + 2−α+1)m|𭟋′′(b̄)|) 8(α+ 2) ) , (34) and ∫ 1 1 2 [−⋎−1 + (⋎+ 1)2−θ − ⊺ ⋎ (θ + 1) +⋎(θ + 1)]( |𭟋′′(ā)| ⊺α d ⊺+m|𭟋′′(b̄)|(1− ⊺α)d⊺ ) ≤ [−⋎−1 + (⋎+ 1) +⋎(θ + 1)] ( |𭟋′′(ā)| ∫ 1 1 2 ⊺αd ⊺−m|𭟋′′(b̄)| ∫ 1 1 2 (1− ⊺α)d⊺ ) +⋎(θ + 1) ( |𭟋′′(ā)| ∫ 1 1 2 ⊺α+1d ⊺+m|𭟋′′(b̄)| ∫ 1 1 2 ⊺(1− ⊺α)d⊺ ) M. Samraiz et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6413 21 of 26 ≤ ⋎(θ + 1) ( (2− 2−α)|𭟋′′(ā)|(α− 1 + 2−α)m|𭟋′′(b̄)|) 2(α+ 1) −(8− 2−α+1)|𭟋′′(ā)|+ (3α− 2 + 2−α+1)m|𭟋′′(b̄) 8(α+ 2) ) . (35) Substituting the values of (32), (33), (34) and (35) in (31), we obtained the required result. Theorem 6. Let 𭟋 : [ā, b̄] −→ R be a twice differentiable function such that |𭟋′′| is Lebesgue integrable, increasing and GA (α,m)-convex function on [ā, b̄]. Then for given parameters θ ∈ (0,+∞] and (α,m) ∈ (0, 1]2, where 0 ≤ ā < b̄. Thus, let 1 < ⋋2 < +∞ then, the following inequality∣∣∣∣𭟋(ā) +𭟋(µ) ⋎(⋎+ 1) + 2 ⋎+ 1 𭟋 ( ā+ µ 2 ) − Γ(θ + 1) ⋎(µ− ā)θ [χθ ā+𭟋(µ) + χθ µ−𭟋(ā)] ∣∣∣∣ ≤ (µ− ā)2 [⋎(θ + 1)(⋎+ 1)]1+⋋−1 1 ( |𭟋′′(ā)|⋋2 +mα|𭟋′′(b̄)|⋋2 α+ 1 ) 1 ⋋2 × [ max (( ⋎+ 1− (⋎+ 1)2−θ )⋋1+1 − ( 1 + 0.5⋎ (1− θ)− (1 +⋎)2−θ )⋋1+1 , (⋎(θ + 1))⋋1+1 2−⋋1−1 ) +max (( ⋎+ 1− (⋎+ 1)2−θ )⋋1+1 − ( 1 + 0.5⋎ (1− θ)− (1 +⋎)2−θ )⋋1+1 , 0.5⋎ (θ + 1)⋋1+1 )) 1 ⋋1 )] holds true, where 1 ⋋1 + 1 ⋋2 = 1. Proof. By utilizing Definitions 2, 4, Lemma 2, 6, 7 and applying Hölder’s inequality, we have∣∣∣∣𭟋(ā) +𭟋(µ) ⋎(⋎+ 1) + 2 ⋎+ 1 𭟋 ( ā+ µ 2 ) − Γ(θ + 1) ⋎(µ− ā)θ [χθ ā+𭟋(µ) + χθ µ−𭟋(ā)] ∣∣∣∣ ≤ (µ− ā)2 ∫ 1 0 |q0(⊺)||𭟋′′(⊺ā+m(1− ⊺)b̄)|d⊺ ≤ (µ− ā)2 (∫ 1 0 |q0(⊺)|⋋1 ) 1 ⋋1 (∫ 1 0 |𭟋′′(⊺ā+m(1− ⊺)b̄)|⋋2d⊺ ) 1 ⋋2 ≤ (µ− ā)2 (∫ 1 0 |q0(⊺)|⋋1 ) 1 ⋋1 (∫ 1 0 |𭟋′′(ā⊺b̄m(1−⊺)|⋋2d⊺ ) 1 ⋋2 ≤ (µ− ā)2 (∫ 1 0 |q0(⊺)|⋋1 ) 1 ⋋1 (∫ 1 0 ⊺α|𭟋′′(ā)|⋋2 +m(1− ⊺α)|𭟋′′(b̄)|⋋2 ) 1 ⋋2 M. Samraiz et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6413 22 of 26 ≤ (µ− ā)2 ( |𭟋′′(ā)|⋋2 +mα|𭟋′′(b̄)|⋋2 α+ 1 ) 1 ⋋2 (∫ 1 0 |q0(⊺)|⋋1 ) 1 ⋋1 ≤ (µ− ā)2 ( |𭟋′′(ā)|⋋2 +mα|𭟋′′(b̄)|⋋2 α+ 1 ) 1 ⋋2 × (∫ 1 2 0 ∣∣∣∣1− (1− ⊺)θ+1 − ⊺θ+1 ⋎(θ + 1) − ⊺ ⋎+ 1 ∣∣∣∣⋋1 d⊺ + ∫ 1 1 2 ∣∣∣∣1− (1− ⊺)θ+1 − ⊺θ+1 ⋎(θ + 1) − 1− ⊺ ⋎+ 1 ∣∣∣∣⋋1 d⊺ ) 1 ⋋1 ≤ (µ− ā)2 ⋎(⋎+ 1)(θ + 1) ( |𭟋′′(ā)|⋋2 +mα|𭟋′′(b̄)|⋋2 α+ 1 ) 1 ⋋2 × (∫ 1 2 0 ∣∣∣(⋎+ 1)− (⋎+ 1)[(1 + ⊺)θ+1 + ⊺θ+1]− ⊺ ⋎ (θ + 1) ∣∣∣⋋1 d⊺ + ∫ 1 1 2 ∣∣∣(⋎+ 1)− (⋎+ 1)[(1 + ⊺)θ+1 + ⊺θ+1]−⋎(θ + 1)(1− ⊺) ∣∣∣⋋1 d⊺ ) 1 ⋋1 . (36) Let us consider∫ 1 2 0 [ (⋎+ 1)− (⋎+ 1)[(1 + ⊺)θ+1 + ⊺θ+1]− ⊺ ⋎ (θ + 1) ]⋋1 d⊺ ≤ − 1 ⋎(θ + 1) [ [(⋎+ 1)− (⋎+ 1)2−θ − ⊺ ⋎ (θ + 1)]⋋1+1 ⋋1 + 1 ∣∣∣∣ 1 2 0 ] ≤ [(⋎+ 1)− (⋎+ 1)2−θ]⋋1+1 − [1 + 0.5⋎ (1− θ)− (1 +⋎)2−θ]⋋1+1 ⋎(θ + 1)(⋋1 + 1) , (37) and ∫ 1 2 0 [ −(⋎+ 1) + (⋎+ 1)[(1 + ⊺)θ+1 + ⊺θ+1] + ⊺ ⋎ (θ + 1) ]⋋1 d⊺ ≤ 1 ⋎(θ + 1) [ [−⋎−1 + (⋎+ 1)2−θ +⋎(θ + 1)⊺]⋋1+1 ⋋1 + 1 ∣∣∣∣ 1 2 0 ] ≤ [⋎(θ + 1)]⋋1+12−⋋1−1 ⋎(θ + 1)(⋋1 + 1) , (38) and ∫ 1 1 2 [ (⋎+ 1)− (⋎+ 1)[(1 + ⊺)θ+1 + ⊺θ+1]−⋎(θ + 1)(1− ⊺) ]⋋1 d⊺ M. Samraiz et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6413 23 of 26 ≤ − 1 ⋎(θ + 1) [ [(⋎+ 1)− (⋎+ 1)2−θ −⋎(θ + 1)(1− ⊺)]⋋1+1 ⋋1 + 1 ∣∣∣∣1 1 2 ] ≤ [((⋎+ 1)− (⋎+ 1)2−θ)− (1 + 0.5⋎ (1− θ) + (⋎+ 1)2−θ)]⋋1+1 ⋎(θ + 1)(⋋1 + 1) , (39) and ∫ 1 1 2 [ (−⋎−1 + (⋎+ 1)2−θ +⋎(θ + 1)(1− ⊺))d⊺ ]⋋1 ≤ [⋎(θ + 1)]⋋1+12−⋋1−1 ⋎(θ + 1)(⋋1 + 1) . (40) Substituting (37), (38), (39) and (40) in (36), we obtained the required result. Remark 7. In accordance with the selection of parameters α = 1 and m = 1 in Theorem 6 and s = 1 in [2, Theorem 5.2], we obtained the same result i.e.,∣∣∣∣𭟋(ā) +𭟋(b̄) ⋎(⋎+ 1) + 2 ⋎+ 1 𭟋 ( ā+ b̄ 2 ) − Γ(θ + 1) ⋎(b̄− ā)θ [χθ ā+𭟋(b̄) + χθ b̄−𭟋(ā)] ∣∣∣∣ ≤ (b̄− ā)2 [⋎(θ + 1)(⋎+ 1)]1+⋋−1 1 ( |𭟋′′(ā)|⋋2 + |𭟋′′(b̄)|⋋2 2 ) 1 ⋋2 × [ max (( ⋎+ 1− (⋎+ 1)2−θ )⋋1+1 − ( 1 + 0.5⋎ (1− θ)− (1 +⋎)2−θ )⋋1+1 , (⋎(θ + 1))⋋1+1 2−⋋1−1 ) +max (( ⋎+ 1− (⋎+ 1)2−θ )⋋1+1 − ( 1 + 0.5⋎ (1− θ)− (1 +⋎)2−θ )⋋1+1 , 0.5⋎ (θ + 1)⋋1+1 )) 1 ⋋1 )] . 5. Conclusions The importance of convexity and fractional calculus in real-life implications is unavoid- able. Both concepts represent nature well. In the presented work, we utilized both concepts and obtained some good results. This paper introduces new fractional integral inequali- ties that are applicable to twice-differentiable geometrically arithmetically (α,m)-convex functions. The utilization of classical Riemann-Liouville fractional integrals leads to the derivation of new identities. Consequently, we are able to explore the Hermite-Hadamard type inequalities based on the mentioned convexity. Hölder’s inequality is employed to investigate the mean inequalities, which have strong applicability in optimization theory. These findings significantly extend the existing literature, presenting them as special cases M. Samraiz et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6413 24 of 26 of our newly established consequences. This work contributes to a deeper understand- ing of the properties and applications of GA (α,m)-convex functions in the context of fractional integral inequalities. The presented work opens avenues for further exploration and expansion of inequalities applicable in optimization theory. The mathematicians can compare the proposed approach with alternative methods or convexities to evaluate its efficiency and uniqueness in handling similar mathematical problems. Acknowledgements Researchers Supporting Project number (RSPD2024R1060), King Saud University, Riyadh, Saudi Arabia. Declarations Availability of data and material No data were used to support this study. Ethical Approval Not Applicable. Competing interests The authors declare that they have no competing interests. Funding This project is funded by King Saud University, Riyadh, Saudi Arabia. Authors’ contributions All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript. References [1] R. Gorenflo and F. Mainardi. Fractional Calculus: Integral and Differential Equations of Fractional Order, volume 378. Springer, Vienna, 1997. [2] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo. Theory and Applications of Frac- tional Differential Equations, volume 204. Elsevier, 2006. [3] D. Baleanu, A. C. J. Luo, and J. A. T. Machado. Fractional Dynamics and Control. Springer, New York, 2020. [4] K. Diethelm. The Analysis of Fractional Differential Equations, volume 2004 of Lec- ture Notes in Mathematics. Springer, 2010. [5] V. Lakshmikantham, S. Leela, and J. V. Devi. Theory of Fractional Dynamic Systems. Cambridge Science Publishers, 2009. [6] K. S. Miller and B. Ross. An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, 1993. [7] S. K. Paul, L. N. Mishra, V. N. Mishra, and D. Baleanu. Analysis of mixed type nonlinear volterra-fredholm integral equations involving the erdélyi-kober fractional operator. Journal of King Saud University-Science, 35(10):102949, 2023. M. Samraiz et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6413 25 of 26 [8] L. P. Castro and A. M. Simoes. Stabilities of ulam-hyers type for a class of nonlinear fractional differential equations with integral boundary conditions in banach spaces. Filomat, 39(2):617–628, 2025. [9] V. Stojiljkovic, N. Mirkov, and S. Radenovic. Variations in the tensorial trapezoid type inequalities for convex functions of self-adjoint operators in hilbert spaces. Sym- metry, 16(1):121, 2024. [10] M. W. Michalski. Derivatives of Non-Integer Order and Their Application. 1993. [11] I. Podlubny, A. Chechkin, T. Skovranek, Y. Chen, and B. M. V. Jara. Matrix ap- proach to discrete fractional calculus ii: Partial fractional differential equations. Jour- nal of Computational Physics, 228(8):3137–3153, 2009. [12] V. E. Tarasov. Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media. Springer Science and Business Media, 2011. [13] M. Z. Sarikaya, E. Set, H. Yaldiz, and N. Basak. Hermite-hadamard’s inequalities for fractional integrals and related fractional inequalities. Mathematical and Computer Modelling, 57(9-10):2403–2407, 2013. [14] Y. Shuang, H. P. Yin, and F. Qi. Hermite-hadamard type integral inequalities for geometric-arithmetically s-convex functions. Analysis, 33(2):197–208, 2013. [15] Y. Liao, J. Deng, and J. Wang. Riemann-liouville fractional hermite-hadamard in- equalities. part ii: For twice differentiable geometric-arithmetically s-convex func- tions. Journal of Inequalities and Applications, 2013(517):1–13, 2013. [16] H. Kavurmaci, M. Avci, and M. E. Özdemir. New inequalities of hermite-hadamard type for convex functions with applications. Journal of Inequalities and Applications, 2011(86):1–11, 2011. [17] C. Zhu, M. Feckan, and J. Wang. Fractional integral inequalities for differentiable convex mappings and applications to special means and a midpoint formula. Journal of Applied Mathematics, Statistics and Informatics, 8(2):21–28, 2012. [18] M. A. Latif, S. S. Dragomir, and A. E. Matouk. New inequalities of ostrowski type for co-ordinated convex functions via fractional integrals. Journal of Fractional Calculus and its Application, 2(1):1–15, 2012. [19] J. Wang, X. Li, and C. Zhu. Refinements of hermite-hadamard type inequalities involving fractional integrals. Bulletin of the Belgian Mathematical Society-Simon Stevin, 20(4):655–666, 2013. [20] Y. Zhang and J. Wang. On some new hermite-hadamard inequalities involv- ing riemann-liouville fractional integrals. Journal of Inequalities and Applications, 2013(220):1–27, 2013. [21] E. Set. New inequalities of ostrowski type for mappings whose derivatives are s- convex in the second sense via fractional integrals. Computers and Mathematics with Applications, 63(7):1147–1154, 2012. [22] J. Wang, J. Deng, and M. Feckan. Hermite-hadamard-type inequalities for r-convex functions based on the use of riemann-liouville fractional integrals. Ukrains’kyi Matematychnyi Zhurnal, 65(2):175–191, 2013. [23] J. Wang, X. Li, M. Fekan, and Y. Zhou. Hermite-hadamard-type inequalities for riemann-liouville fractional integrals via two kinds of convexity. Applicable Analysis, M. Samraiz et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6413 26 of 26 92(11):2241–2253, 2013. [24] G. Farid, S. Bibi, L. Rathour, L. N. Mishra, and V. N. Mishra. Fractional versions of hadamard inequalities for strongly (s,m)-convex functions via caputo fractional derivatives. Korean Journal of Mathematics, 31(1):75–94, 2023. [25] J. Deng and J. Wang. Fractional hermite-hadamard inequalities for (α,m)- logarithmically convex functions. Journal of Inequalities and Applications, pages 1–11, 2013. [26] J. Wang, J. Deng, and M. Feckan. Exploring s−e-conditions and applications to some ostrowski type inequalities via riemann-liouville fractional integrals. Mathematica Slovaca, 64(6):1381–1396, 2014. [27] S. S. Dragomir. Hermite-hadamard type inequalities for generalized riemann-liouville fractional integrals of h-convex functions. Mathematical Methods in the Applied Sci- ences, 44(3):2364–2380, 2021. [28] V. Stojiljkovic. Hermite-hadamard type inequalities involving kp fractional operator with (a, h −m) − p convexity. European Journal of Pure and Applied Mathematics, 16(1):503–522, 2023. [29] S. Iqbal, M. Samraiz, G. Rahman, K. S. Nisar, and T. Abdeljawad. Some new grüss inequalities associated with generalized fractional derivative. AIMS Mathematics, 8(1):213–227, 2023. Article ID 2023010. [30] M. Samraiz, Z. Perveen, G. Rahman, M. Adil Khan, and K. S. Nisar. Hermite- hadamard fractional inequalities for differentiable functions. Fractal and Fractional, 6(2):60, 2022. [31] S. S. Dragomir and C. E. M. Pearce. Selected topics on hermite-hadamard inequalities and applications. Science Direct Working Paper, S1574-0358(4), 2003. [32] M. Z. Sarikaya, E. Set, H. Yaldiz, and N. Basak. Hermite-hadamard’s inequalities for fractional integrals and related fractional inequalities. Mathematical and Computer Modelling, 57(9–10):2403–2407, 2013. [33] P. J. Davis. Leonhard euler’s integral: A historical profile of the gamma function: In memoriam: Milton abramowitz. The American Mathematical Monthly, 66(10):849– 869, 1959. [34] R. A. Askey and R. Roy. NIST Handbook of Mathematical Functions. Cambridge University Press, New York, 2010. [35] M. A. Chaudhry, A. Qadir, M. Rafique, and S. M. Zubair. Extension of euler’s beta function. Journal of Computational and Applied Mathematics, 78(1):19–32, 1997. [36] A. R. DiDonato and M. P. Jarnagin. The efficient calculation of the incomplete beta- function ratio for half-integer values of the parameters. Mathematics of Computation, 21(100):652–662, 1967. [37] S. Özcan. Hermite-hadamard type inequalities for m-convex and (α,m)-convex func- tions. Journal of Inequalities and Applications, 2020(1), 2020. [38] J.-Y. Wang, H.-P. Yin, W.-L. Sun, and B.-N. Guo. Hermite-hadamard integral in- equalities of (α, s)-ga and (α, s,m)-ga-convex functions. Axioms, 11(11):616, 2022.