EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 1, Article Number 5834 ISSN 1307-5543 – ejpam.com Published by New York Business Global New Inequalities for Differentiable Mappings via Fractional Integral Operators Ohud Bulayhan Almutairi Department of Mathematics, University of Hafr Al Batin, Hafr Al Batin 31991, Saudi Arabia Abstract. In this study, explicit bounds for the midpoint type inequalities for functions whose twice differentiable in absolute value raised to positive real powers are (ρ, s) and (ρ, s,m)- convexities are explored through the integral fractional operator. Several estimate for special functions including Euler gamma, incomplete Beta and hypergeometric functions are presented in the study. 2020 Mathematics Subject Classifications: 26D10, 26A33, 33B15 Key Words and Phrases: Riemann-Liouville integrals, Hölder’s inequality, Integral inequality, (ρ, s,m)-convex functions 1. Introduction Fractional calculus – whose impact on both pure and applied sciences substantially increased in the last two decades – captures the attention of many researchers. In ad- dition, this area of interest invariably remains one of the few disciplines that immensely contribute to not only different areas of mathematics, but also to other natural sciences. One of such examples is associated to the existence of many fractional operators in those disciplines, most of which occur through different formulations differential equa- tions. This area has been developed through the contributions of many researchers; for example, Kilbas et al. [9] gave the background of fractional differential equations, Xing et al. [22] extended Hermite-Hadamard inequalities using fractional integrals, Zhang et al.[7] generalized the inequalities for strongly (s,m)-convexities, and Noor and Awan [11] established the inequalities with two different convexities. These operators have been used to understand many problems, such as the propagation of sound, vibrations of strings and waves in liquids. This leads to the establishment of numerous fractional operators – including Riemann Liouville, Caputo and Atangana Baleanu Caputo – as well as studying many vital concepts for investigating these operators. For example, the existence of unique solution of many fractional differential and integral equations – DOI: https://doi.org/10.29020/nybg.ejpam.v18i1.5834 Email addresses: ohoudbalmutairi@gmail.com, dr.ohudalmutairi@uhb.edu.sa (O. B. Almutairi) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) O. B. Almutairi / Eur. J. Pure Appl. Math, 18 (1) (2025), 5834 2 of 12 for initial and boundary value problems – have been reported through these fractional operators. For further studies, see [10, 13–15, 17]. Fractional calculus, whose broad contents have been rapidly developing in mathe- matical analysis, plays a vital role in approximation theory. One example of this is the frequent use of integral operators in the study of inequalities. Consider an integrable function η : [k1, k2] → R over [k1, k2]. This function is belonging to a space L1 [k1, k2] rep- resenting the set of Lebesgue integrable over the same interval. Using Riemann-Liouville fractional integrals, Sarikaya et al. [19] studied the following new integral inequalities. Theorem 1. [19] Let η : [k1, k2] → R be a positive function with 0 ≤ k1 < k2 and η ∈ L1[k1, k2]. If η is a convex function on [k1, k2], then the following inequalities for fractional integrals hold: η ( k1 + k2 2 ) ≤ Γ(ρ+ 1) 2(k2 − k1)ρ [ Jρ k+1 η(k2) + Jρ k−2 η(k1) ] ≤ η(k1) + η(k2) 2 , ρ > 0. (1) Due to vital roles played by this inequality in both science and engineering [2, 4], many authors improve, extend and generalize inequality (1) through various types of convexities and fractional integrals. For example, Agarwal et al. [1] established new Hermite-Hadamard type inequalities via generalized k-fractional integrals; Almutairi [3] explored fractional inequalities using Euler’s beta function; Budak et al. [5] developed Hermite-Hadamard-type inequalities for interval-valued functions; Du and Peng [6] ex- tended the idea to report Hermite-Hadamard type inequalities involving multiplicative Riemann-Liouville fractional integrals. Noor and Awan [11] established new results related to the left hand side of (1) for twice differentiable s-convex functions using the following lemma. Lemma 1. Let η : [k1, k2] → R be a twice differentiable function on (k1, k2) with k1 < k2. Also, let η′′ ∈ L[k1, k2]. Then the following identity holds true: 2ρ−1Γ(ρ+ 1) (k2 − k1)ρ [ Jρ( k1+k2 2 )−η(k1) + Jρ( k1+k2 2 )+η(k2) ] − η ( k1 + k2 2 ) = (k2 − k1) 2 8(ρ+ 1) ∫ 1 0 (1−ϖ)ρ+1 [ η′′ ( 1 +ϖ 2 k1 + 1−ϖ 2 k2 ) + η′′ ( 1 +ϖ 2 k2 + 1−ϖ 2 k1 )] dϖ (2) Even though a classical convexity - which was later replaced by s-convexity [11] - has been previously used to establish integral inequalities [19], the central idea of our study lies in establishing more generalized integral inequalities through two different classes of convexities. Motivated by these two mentioned independent studies, we opt to obtain new bounds for the midpoint inequalities via fractional integral operator. Two generalized convexities (ρ, s) and (ρ, s,m)-convexity are used to establish the new bounds. Some of our findings - which can be reduced to different inequalities through various convexities - are obtained. The other parts of this paper are organized as follows. Preliminary studies are presented in Section 2. Section 3 presents integral inequalities involving (ρ, s) and (ρ, s,m) convex functions via fractional integral operators. Section 4 concludes the study. O. B. Almutairi / Eur. J. Pure Appl. Math, 18 (1) (2025), 5834 3 of 12 2. Preliminaries Some basic results of different classes of convex functions, Riemann Liouville frac- tional integral operator and special functions are presented in this section. These pre- liminary results and definitions can be later used to establish our main results. Therefore, the definitions of some types of convexities are given as follows. Definition 1. [20] A function η : [0, d] → R0 = [0,∞) is said to be m-convex on [0, d] for some m ∈ (0, 1], if η(ϖk1 +m(1−ϖ)k2) ≤ ϖη(k1) +m(1−ϖ)η(k2), for all k1, k2 ∈ [0, d] and ϖ ∈ [0, 1]. Definition 2. [8] A function η : [k1, k2] ⊂ R → R is said to be s-convex function of the second kind if η((1−ϖ)k1 +ϖk2) ≤ (1−ϖ)sη(k1) +ϖsη(k2), for all k1, k2 ∈ (0,∞], ϖ ∈ [0, 1] and s ∈ (0, 1]. Definition 3. [21] For some s ∈ [−1, 1] and ρ ∈ (0, 1], a function η : I ⊆ R → R is said to be (ρ, s)-convex if η(ϖk1 + (1−ϖ)k2) ≤ ϖρsη(k1) + (1−ϖρ)s η(k2), for all k1, k2 ∈ I and ϖ ∈ (0, 1). Definition 4. [21] The function η : [0, d] → R is said to be (ρ, s,m)-convex, if we have η(ϖk1 +m(1−ϖ)k2) ≤ ϖρsη(k1) +m (1−ϖρ)s η(k2), where k1, k2 ∈ [0, d], ϖ ∈ (0, 1) and for some s ∈ [−1, 1], (ρ,m) ∈ (0, 1]2. Different special cases are considered in the following remark. Remark 1. Definition 4 produces the following: i. If s = 1 in Definition 4, then we get the class of (ρ,m)-convex function. ii. If ρ = 1, then Definition 4 reduces to the definition for (s,m)-convex function. iii. If ρ = m = 1 in Definition 4, then we have the class of extended s-convex function. iv. If ρ = s = m = 1 in Definition 4, then we obtain the classical convex function. We know present the defintions of Riemann-Liouville integrals as follows. Definition 5. [18] Let η ∈ L1[k1, k2]. The Riemann-Liouville fractional integrals Jα k1+ η and Jα k2−η of order ρ > 0 with k1 ≥ 0 are defined by Jρ k+1 η(x) = 1 Γ(ρ) ∫ x k1 (x−ϖ)ρ−1η(ϖ)dϖ, x > k1 O. B. Almutairi / Eur. J. Pure Appl. Math, 18 (1) (2025), 5834 4 of 12 and Jρ k−2 η(x) = 1 Γ(ρ) ∫ k2 x (ϖ − x)ρ−1η(ϖ)dϖ, x < k2 respectively. Here, Γ(ρ) is the Gamma function and J0 k1+ η(x) = J0 k2−η(x) = η(x). We further, present the definitions of some special functions as follows. Definition 6. [9] For any complex numbers and nonpositive integers k1, k2 such that Re(k1) > 0 and Re(k2) > 0. The beta function is defined by B(k1, k2) = ∫ 1 0 ϖk1−1(1−ϖ)k2−1dϖ = Γ(k1)Γ(k2) Γ(k1 + k2) , where Γ is the Gamma function. Definition 7. [12] For any complex numbers k1, k2 with Re (k1) ,Re (k2) > 0, the in- complete beta function is defined as Ba(k1, k2) = ∫ a 0 ϖk1−1(1−ϖ)k2−1dϖ, 0 < a < 1. Definition 8. [9] The integral representation of the hypergeometric function is defined for k1, k2 ∈ C and k3 ∈ C\Z− 0 , Re(k3) > Re(k2) > 0, and |k4| < 1, as follows 2F1(k1, k2, k3; k4) = 1 B(k2, k3 − k2) ∫ 1 0 ϖk2−1(1−ϖ)k3−k2−1(1− k4ϖ)−k1dϖ, where B(.,.) is the beta function. 3. Main Result In this section, we first present some generalized midpoint type inequalities for (ρ, s)- convex functions. Theorem 2. Suppose that η : [k1, k2] → R is a twice differentiable function on (k1, k2) with k1 < k2. If |η′′| is (ρ, s) convex function, where (ρ, s) ∈ (0, 1]2 and η′′ ∈ L[k1, k2], then the following∣∣∣∣∣2ρ−1Γ(ρ+ 1) (k2 − k1)ρ [ Jρ( k1+k2 2 )−η(k1) + Jα( k1+k2 2 )+η(k2) ] − η ( k1 + k2 2 )∣∣∣∣∣ ≤ (k2 − k1) 2 2ρs+3(ρ+ 1) [∣∣η′′(k1)∣∣ 2F1(1,−ρs; 3 + k1;−1) k1 + 2 + ∣∣η′′(k2)∣∣Ω(ρ, s,ϖ) + ∣∣η′′(k1)∣∣ Γ(ρs+ ρ+ 2) Γ(ρs+ ρ+ 3) + ∣∣η′′(k2)∣∣Ω(ρ, s,ϖ) ] . where Ω(ρ, s,ϖ) = ∫ 1 0 (1−ϖ)ρ+1(2ρ − (1 +ϖ)ρ)sdϖ holds. O. B. Almutairi / Eur. J. Pure Appl. Math, 18 (1) (2025), 5834 5 of 12 Proof. We now use identity (2) and (ρ, s)-convexity of |η′′| to obtain the following∣∣∣∣∣2ρ−1Γ(ρ+ 1) (k2 − k1)ρ [ Jρ( k1+k2 2 )−η(k1) + Jρ( k1+k2 2 )+η(k2) ] − η ( k1 + k2 2 )∣∣∣∣∣ = ∣∣∣∣(k2 − k1) 2 8(ρ+ 1) ∫ 1 0 (1−ϖ)ρ+1 [ η′′ ( 1 +ϖ 2 k1 + 1−ϖ 2 k2 ) + η′′ ( 1−ϖ 2 k1 + 1 +ϖ 2 k2 )] dϖ ∣∣∣∣ ≤ ∣∣∣∣(k2 − k1) 2 8(ρ+ 1) ∫ 1 0 (1−ϖ)ρ+1η′′ ( 1 +ϖ 2 k1 + 1−ϖ 2 k2 ) dϖ ∣∣∣∣ + ∣∣∣∣(k2 − k1) 2 8(ρ+ 1) ∫ 1 0 (1−ϖ)ρ+1η′′ ( 1−ϖ 2 k1 + 1 +ϖ 2 k2 ) dϖ ∣∣∣∣ ≤ (k2 − k1) 2 8(ρ+ 1) ∫ 1 0 (1−ϖ)ρ+1 ∣∣∣∣η′′(1 +ϖ 2 k1 + 1−ϖ 2 k2 )∣∣∣∣ dϖ + (k2 − k1) 2 8(ρ+ 1) ∫ 1 0 (1−ϖ)ρ+1 ∣∣∣∣η′′(1−ϖ 2 k1 + 1 +ϖ 2 k2 )∣∣∣∣ dϖ ≤ (k2 − k1) 2 8(ρ+ 1) [∫ 1 0 (1−ϖ)ρ+1 [( 1 +ϖ 2 )ρs ∣∣η′′(k1)∣∣+ ( 1− ( 1−ϖ 2 )ρ)s ∣∣η′′(k2)∣∣] dϖ + ∫ 1 0 (1−ϖ)ρ+1 [( 1−ϖ 2 )ρs ∣∣η′′(k1)∣∣+ ( 1− ( 1 +ϖ 2 )ρ)s ∣∣η′′(k2)∣∣] dϖ] = (k2 − k1) 2 2ρs+3(ρ+ 1) [∣∣η′′(k1)∣∣ ∫ 1 0 (1−ϖ)ρ+1(1 +ϖ)ρsdϖ + ∣∣η′′(k2)∣∣ ∫ 1 0 (1−ϖ)ρ+1(2ρ − (1−ϖ)ρ)sdϖ + ∣∣η′′(k1)∣∣ ∫ 1 0 (1−ϖ)ρ+1(1−ϖ)ρsdϖ + ∣∣η′′(k2)∣∣ ∫ 1 0 (1−ϖ)ρ+1(2ρ − (1 +ϖ)ρ)sdϖ ] = (k2 − k1) 2 2ρs+3(ρ+ 1) [∣∣η′′(k1)∣∣ 2F1(1,−ρs; 3 + k1;−1) k1 + 2 + ∣∣η′′(k2)∣∣ ∫ 1 0 (1−ϖ)ρ+1(2ρ − (1−ϖ)ρ)sdϖ + ∣∣η′′(k1)∣∣ Γ(ρs+ ρ+ 2) Γ(ρs+ ρ+ 3) + ∣∣η′′(k2)∣∣ ∫ 1 0 (1 +ϖ)ρ+1(2ρ − (1 +ϖ)ρ)sdϖ ] ≤ (k2 − k1) 2 2ρs+3(ρ+ 1) [∣∣η′′(k1)∣∣ 2F1(1,−ρs; 3 + k1;−1) k1 + 2 + ∣∣η′′(k2)∣∣Ω(ρ, s,ϖ) + ∣∣η′′(k1)∣∣ Γ(ρs+ ρ+ 2) Γ(ρs+ ρ+ 3) + ∣∣η′′(k2)∣∣Ω(ρ, s,ϖ) ] . Theorem 3. Let η : [k1, k2] → R be twice differentiable function on (k1, k2) with k1 < k2. If η′′ ∈ L[k1, k2] and |η′′|q is (ρ, s)-convex function of second kind, then, we have the following inequality for fractional integrals:∣∣∣∣∣2ρ−1Γ(ρ+ 1) (k2 − k1)ρ [ Jρ( k1+k2 2 )−η(k1) + Jρ( k1+k2 2 )+η(k2) ] − η ( k1 + k2 2 )∣∣∣∣∣ ≤ (k2 − k1) 2 23(ρ+ 1) ( 1 p(ρ+ 1) + 1 ) 1 p (( 2− 2−ρs ρs+ 1 ) + ( 2−ρs ρs+ 1 )) 1 q [ ∣∣η′′(k1)∣∣q + ∣∣η′′(k2)∣∣q] 1q O. B. Almutairi / Eur. J. Pure Appl. Math, 18 (1) (2025), 5834 6 of 12 Proof. Applying Hölder’s inequality, (ρ, s)-convex of |η′′|q and Lemma1, we get∣∣∣∣∣2ρ−1Γ(ρ+ 1) (k2 − k1)α [ Jρ( k1+k2 2 )−η(k1) + Jα( k1+k2 2 )+η(k2) ] − η ( k1 + k2 2 )∣∣∣∣∣ = ∣∣∣∣(k2 − k1) 2 8(α+ 1) ∫ 1 0 (1−ϖ)α+1 [ η′′ ( 1 +ϖ 2 k1 + 1−ϖ 2 k2 ) + η′ ( 1−ϖ 2 k1 + 1 +ϖ 2 k2 )] dϖ ∣∣∣∣ ≤ ∣∣∣∣(k2 − k1) 2 8(ρ+ 1) ∫ 1 0 (1−ϖ)ρ+1η′′ ( 1 +ϖ 2 k1 + 1−ϖ 2 k2 ) dϖ ∣∣∣∣ + ∣∣∣∣k2 − k1 4 ∫ 1 0 (1−ϖ)ρ+1η′′ ( 1−ϖ 2 k1 + 1 +ϖ 2 k2 ) dϖ ∣∣∣∣ ≤ (k2 − k1) 2 8(ρ+ 1) {(∫ 1 0 (1−ϖ)p(ρ+1)dϖ ) 1 p (∫ 1 0 ∣∣∣∣η′′(1 +ϖ 2 k1 + 1−ϖ 2 k2 )∣∣∣∣q dϖ) 1 q + (∫ 1 0 (1−ϖ)p(ρ+1)dϖ ) 1 p (∫ 1 0 ∣∣∣∣η′′(1−ϖ 2 k1 + 1 +ϖ 2 k2 )∣∣∣∣q dϖ) 1 q } ≤ (k2 − k1) 2 8(ρ+ 1) ( 1 p(ρ+ 1) + 1 ) 1 p {(∣∣η′′(k1)∣∣q ∫ 1 0 ( 1 +ϖ 2 )ρs dϖ + ∣∣η′′(k2)∣∣q ∫ 1 0 ( 1− ( 1−ϖ 2 )ρ)s dϖ ) 1 q + (∣∣η′′(k1)∣∣q ∫ 1 0 ( 1−ϖ 2 )ρs dϖ + ∣∣η′′(k2)∣∣q ∫ 1 0 ( 1− ( 1 +ϖ 2 )ρ)s dϖ ) 1 q } ≤ (k2 − k1) 2 8(ρ+ 1) ( 1 p(ρ+ 1) + 1 ) 1 p [{∣∣η′′(k1)∣∣q (2− 2−ρs ρs+ 1 ) + ∣∣η′′(k2)∣∣q (2− 2−ρs ρs+ 1 )} 1 q + {∣∣η′′(k1)∣∣q ( 2−ρs ρs+ 1 ) + ∣∣η′′(k2)∣∣q ( 2−ρs ρs+ 1 )} 1 q ] ≤ (k2 − k1) 2 23(ρ+ 1) ( 1 p(ρ+ 1) + 1 ) 1 p (( 2− 2−ρs ρs+ 1 ) + ( 2−ρs ρs+ 1 )) 1 q [ ∣∣η′′(k1)∣∣q + ∣∣η′′(k2)∣∣q] 1q Using power-mean inequality, we obtain the following integral inequalities. Theorem 4. Let η : [k1, k2] → R be twice differentiable function on (k1, k2) with k1 < k2. If η′′ ∈ L[k1, k2] and |η′′|q is (ρ, s)-convex function of second kind, then, we have the following inequality for fractional integrals:∣∣∣∣∣2ρ−1Γ(ρ+1) (k2−k1)α [ Jρ( k1+k2 2 )−η(k1) + Jρ( k1+k2b 2 )+η(k2) ] − η ( k1+k2 2 )∣∣∣∣∣ ≤ (k2−k1)2 23(ρ+1) ( 1 ρ+2 )1− 1 q {( |η′′(k1)|q 2F1(1,−ρs;3+k1;−1) k1+2 + |η′′(k2)|q Ω(ρ, s,ϖ) ) 1 q + ( |η′′(k1)|q ( Γ(ρs+ρ+2) Γ(ρs+ρ+3) ) + |η′′(k2)|q Ω(ρ, s,ϖ) ) 1 q } O. B. Almutairi / Eur. J. Pure Appl. Math, 18 (1) (2025), 5834 7 of 12 Proof. Applying Lemma 1, power-mean inequality and the fact that |η′′|q is (ρ, s)- convex function, we have∣∣∣∣∣2ρ−1Γ(ρ+ 1) (k2 − k1)ρ [ Jρ( k1+k2 2 )−η(k1) + Jα( k1+k2 2 )+η(k2) ] − η ( k1 + k2 2 )∣∣∣∣∣ = ∣∣∣∣(k2 − k1) 2 8(ρ+ 1) ∫ 1 0 (1−ϖ)ρ+1 [ η′′ ( 1 +ϖ 2 k1 + 1−ϖ 2 k2 ) + η′ ( 1−ϖ 2 k1 + 1 +ϖ 2 k2 )] dϖ ∣∣∣∣ ≤ ∣∣∣∣(k2 − k1) 2 8(ρ+ 1) ∫ 1 0 (1−ϖ)ρ+1η′′ ( 1 +ϖ 2 k1 + 1−ϖ 2 k2 ) dϖ ∣∣∣∣ + ∣∣∣∣k2 − k1 4 ∫ 1 0 (1−ϖ)ρ+1η′′ ( 1−ϖ 2 k1 + 1 +ϖ 2 k2 ) dϖ ∣∣∣∣ ≤ (k2 − k1) 2 8(ρ+ 1) {(∫ 1 0 (1−ϖ)(ρ+1)dϖ )1− 1 q (∫ 1 0 (1−ϖ)(ρ+1) ∣∣∣∣η′′(1 +ϖ 2 k1 + 1−ϖ 2 k2 )∣∣∣∣q dϖ) 1 q + (∫ 1 0 (1−ϖ)(ρ+1)dϖ )1− 1 q (∫ 1 0 (1−ϖ)(ρ+1) ∣∣∣∣η′′(1−ϖ 2 k1 + 1 +ϖ 2 k2 )∣∣∣∣q dϖ) 1 q } ≤ (k2 − k1) 2 23(ρ+ 1) ( 1 ρ+ 2 )1− 1 q {(∣∣η′′(k1)∣∣q 2F1(1,−ρs; 3 + k2;−1) k1 + 2 + ∣∣η′′(k2)∣∣q Ω(ρ, s,ϖ) ) 1 q + (∣∣η′′(k1)∣∣q (Γ(ρs+ ρ+ 2) Γ(ρs+ α+ 3) ) + ∣∣η′′(k2)∣∣q Ω(ρ, s,ϖ) ) 1 q } Now, we present generalized integral inequalities via Riemann-Liouville operators for mappings whose twice differentiable are (ρ, s,m). Theorem 5. Let η : [k1, k2] → R be twice differentiable function on (k1, k2) with k1 < k2. If η′′ ∈ L[k1, k2] and |η′′| is (ρ, s,m) convex function, where (ρ,m) ∈ (0, 1]2, s ∈ (−1, 1] then, we have the following inequality for fractional integrals:∣∣∣∣∣2ρ−1Γ(ρ+ 1) (k2 − k1)ρ [ Jρ( k1+k2 2 )−η(k1) + Jρ( k1+k2 2 )+η(k2) ] − η ( k1 + k2 2 )∣∣∣∣∣ ≤ (k2 − k1) 2 2ρs+3(ρ+ 1) [∣∣η′′(k1)∣∣ 2F1(1,−ρs; 3 + k1;−1) k1 + 2 +m ∣∣∣∣η′′(k2 m )∣∣∣∣Ω(ρ, s,ϖ) + ∣∣η′′(k1)∣∣ Γ(ρs+ ρ+ 2) Γ(ρs+ ρ+ 3) +m ∣∣∣∣η′′(k2 m )∣∣∣∣Ω(ρ, s,ϖ) ] , where Ω(ρ, s,ϖ) = ∫ 1 0 (1−ϖ)ρ+1(2ρ − (1 +ϖ)ρ)sdϖ. Proof. Using Lemma 1 and the fact that |η′′| is (ρ, s,m)-convexity, we get the fol- O. B. Almutairi / Eur. J. Pure Appl. Math, 18 (1) (2025), 5834 8 of 12 lowing∣∣∣∣∣2ρ−1Γ(ρ+ 1) (k2 − k1)ρ [ Jρ( k1+k2 2 )−η(k1) + Jρ( k1+k2 2 )+η(k2) ] − η ( k1 + k2 2 )∣∣∣∣∣ = ∣∣∣∣(k2 − k1) 2 8(ρ+ 1) ∫ 1 0 (1−ϖ)ρ+1 [ η′′ ( 1 +ϖ 2 k1 + 1−ϖ 2 k2 ) + η′′ ( 1−ϖ 2 k1 + 1 +ϖ 2 k2 )] dϖ ∣∣∣∣ ≤ ∣∣∣∣(k2 − k1) 2 8(ρ+ 1) ∫ 1 0 (1−ϖ)ρ+1η′′ ( 1 +ϖ 2 k1 + 1−ϖ 2 k2 ) dϖ ∣∣∣∣ + ∣∣∣∣(k2 − k1) 2 8(ρ+ 1) ∫ 1 0 (1−ϖ)ρ+1η′′ ( 1−ϖ 2 k1 + 1 +ϖ 2 k2 ) dϖ ∣∣∣∣ ≤ (k2 − k1) 2 8(ρ+ 1) ∫ 1 0 (1−ϖ)ρ+1 ∣∣∣∣η′′(1 +ϖ 2 k1 + 1−ϖ 2 k2 )∣∣∣∣ dϖ + (k2 − k1) 2 8(ρ+ 1) ∫ 1 0 (1−ϖ)ρ+1 ∣∣∣∣η′′(1−ϖ 2 k1 + 1 +ϖ 2 k2 )∣∣∣∣ dϖ ≤ (k2 − k1) 2 8(ρ+ 1) [∫ 1 0 (1−ϖ)ρ+1 [( 1 +ϖ 2 )ρs ∣∣η′′(k1)∣∣+ ( 1− ( 1−ϖ 2 )ρ)s m ∣∣∣∣η′′(k2 m )∣∣∣∣] dϖ + ∫ 1 0 (1−ϖ)ρ+1 [( 1−ϖ 2 )ρs ∣∣η′′(k1)∣∣+m ( 1− ( 1 +ϖ 2 )ρ)s ∣∣∣∣η′′(k2 m )∣∣∣∣] dϖ] = (k2 − k1) 2 2ρs+3(ρ+ 1) [∣∣η′′(k1)∣∣ ∫ 1 0 (1−ϖ)ρ+1(1 +ϖ)ρsdϖ +m ∣∣∣∣η′′(k2 m )∣∣∣∣ ∫ 1 0 (1−ϖ)ρ+1(2ρ − (1−ϖ)ρ)sdϖ + ∣∣η′′(k1)∣∣ ∫ 1 0 (1−ϖ)ρ+1(1−ϖ)ρsdϖ +m ∣∣∣∣η′′(k2 m )∣∣∣∣ ∫ 1 0 (1−ϖ)ρ+1(2ρ − (1 +ϖ)ρ)sdϖ ] = (k2 − k1) 2 2ρs+3(ρ+ 1) [∣∣η′′(k1)∣∣ 2F1(1,−ρs; 3 + k1;−1) k1 + 2 +m ∣∣∣∣η′′(k2 m )∣∣∣∣ ∫ 1 0 (1−ϖ)ρ+1(2ρ − (1−ϖ)ρ)sdϖ + ∣∣η′′(k1)∣∣ Γ(ρs+ ρ+ 2) Γ(ρs+ ρ+ 3) +m ∣∣∣∣η′′(k2 m )∣∣∣∣ ∫ 1 0 (1 +ϖ)ρ+1(2ρ − (1 +ϖ)ρ)sdϖ ] ≤ (k2 − k1) 2 2ρs+3(ρ+ 1) [∣∣η′′(k1)∣∣ 2F1(1,−ρs; 3 + k1;−1) k1 + 2 +m ∣∣∣∣η′′(k2 m )∣∣∣∣Ω(ρ, s,ϖ) + ∣∣η′′(k1)∣∣ Γ(ρs+ ρ+ 2) Γ(ρs+ ρ+ 3) +m ∣∣∣∣η′′(k2 m )∣∣∣∣Ω(ρ, s,ϖ) ] . Theorem 6. Let η : [k1, k2] → R be twice differentiable function on (k1, k2) with k1 < k2. If η′′ ∈ L[k1, k2] and |η′′|q is (ρ, s,m) is convex function, where (ρ,m) ∈ (0, 1]2, O. B. Almutairi / Eur. J. Pure Appl. Math, 18 (1) (2025), 5834 9 of 12 s ∈ (−1, 1], then, we have the following inequality for fractional integrals:∣∣∣∣∣2ρ−1Γ(ρ+ 1) (k2 − k1)ρ [ Jρ( k1+k2 2 )−η(k1) + Jρ( k1+k2 2 )+η(k2) ] − η ( k1 + k2 2 )∣∣∣∣∣ ≤ (k2 − k1) 2 8(ρ+ 1) ( 1 p(ρ+ 1) + 1 ) 1 p [{∣∣η′′(k1)∣∣q (2− 2−ρs ρs+ 1 ) +m ∣∣∣∣η′′(k2 m )∣∣∣∣q (2− 2−ρs ρs+ 1 )} 1 q + {∣∣η′′(k1)∣∣q ( 2−ρs ρs+ 1 ) +m ∣∣∣∣η′′(k2 m )∣∣∣∣q ( 2−ρs ρs+ 1 )} 1 q ] Proof. Applying Hölder’s inequality, Lemma 1 and the fact that |η′′|q is (ρ, s,m)- convex function, we have∣∣∣∣∣2ρ−1Γ(ρ+ 1) (k2 − k1)ρ [ Jρ( k1+k2 2 )−η(k1) + Jρ( k1+k2 2 )+η(k2) ] − η ( k1 + k2 2 )∣∣∣∣∣ = ∣∣∣∣(k2 − k1) 2 8(ρ+ 1) ∫ 1 0 (1−ϖ)ρ+1 [ η′′ ( 1 +ϖ 2 k1 + 1−ϖ 2 k2 ) + η′ ( 1−ϖ 2 k1 + 1 +ϖ 2 k2 )] dϖ ∣∣∣∣ ≤ ∣∣∣∣(k2 − k1) 2 8(ρ+ 1) ∫ 1 0 (1−ϖ)ρ+1η′′ ( 1 +ϖ 2 k1 + 1−ϖ 2 k2 ) dϖ ∣∣∣∣ + ∣∣∣∣k2 − k1 4 ∫ 1 0 (1−ϖ)ρ+1η′′ ( 1−ϖ 2 k1 + 1 +ϖ 2 k2 ) dϖ ∣∣∣∣ ≤ (k2 − k1) 2 8(ρ+ 1) {(∫ 1 0 (1−ϖ)p(ρ+1)dϖ ) 1 p (∫ 1 0 ∣∣∣∣η′′(1 +ϖ 2 k1 + 1−ϖ 2 k2 )∣∣∣∣q dϖ) 1 q + (∫ 1 0 (1−ϖ)p(ρ+1)dϖ ) 1 p (∫ 1 0 ∣∣∣∣η′′(1−ϖ 2 k1 + 1 +ϖ 2 k2 )∣∣∣∣q dϖ) 1 q } ≤ (k2 − k1) 2 8(ρ+ 1) ( 1 p(ρ+ 1) + 1 ) 1 p {(∣∣η′′(k1)∣∣q ∫ 1 0 ( 1 +ϖ 2 )ρs dϖ +m ∣∣∣∣η′′(k2 m )∣∣∣∣q ∫ 1 0 ( 1− ( 1−ϖ 2 )ρ)s dϖ ) 1 q + (∣∣η′′(k1)∣∣q ∫ 1 0 ( 1−ϖ 2 )ρs dϖ +m ∣∣∣∣η′′(k2 m )∣∣∣∣q ∫ 1 0 ( 1− ( 1 +ϖ 2 )ρ)s dϖ ) 1 q } ≤ (k2 − k1) 2 8(ρ+ 1) ( 1 p(ρ+ 1) + 1 ) 1 p [{∣∣η′′(k1)∣∣q (2− 2−ρs ρs+ 1 ) +m ∣∣∣∣η′′(k2 m )∣∣∣∣q (2− 2−ρs ρs+ 1 )} 1 q + {∣∣η′′(k1)∣∣q ( 2−ρs ρs+ 1 ) +m ∣∣∣∣η′′(k2 m )∣∣∣∣q ( 2−ρs ρs+ 1 )} 1 q ] . O. B. Almutairi / Eur. J. Pure Appl. Math, 18 (1) (2025), 5834 10 of 12 ∣∣∣∣∣2ρ−1Γ(ρ+ 1) (k2 − k1)ρ [ Jρ( k1+k2 2 )−η(k1) + Jρ( k1+k2 2 )+η(k2) ] − η ( k1 + k2 2 )∣∣∣∣∣ = ∣∣∣∣(k2 − k1) 2 8(ρ+ 1) ∫ 1 0 (1−ϖ)ρ+1 [ η′′ ( 1 +ϖ 2 k1 + 1−ϖ 2 k2 ) + η′ ( 1−ϖ 2 k1 + 1 +ϖ 2 k2 )] dϖ ∣∣∣∣ ≤ (k2 − k1) 2 8(ρ+ 1) [∫ 1 0 (1−ϖ)ρ+1 ∣∣∣∣η′′(1 +ϖ 2 k1 + 1−ϖ 2 k2 )∣∣∣∣ dϖ + ∫ 1 0 (1−ϖ)ρ+1 ∣∣∣∣η′′(1−ϖ 2 k1 + 1 +ϖ 2 k2 )∣∣∣∣ dϖ] ≤ (k2 − k1) 2 8(ρ+ 1) ( 1 p(ρ+ 1) + 1 ) 1 p {[(∫ 1 0 (1−ϖ)p(ρ+1)dϖ ) 1 p (∫ 1 0 ∣∣∣∣η′′(1 +ϖ 2 k1 + 1−ϖ 2 k2 )∣∣∣∣q dϖ) 1 q ] + [(∫ 1 0 (1−ϖ)p(ρ+1)dϖ ) 1 p (∫ 1 0 ∣∣∣∣η′′(1−ϖ 2 k1 + 1 +ϖ 2 k2 )∣∣∣∣q dϖ) 1 q ]} ≤ (k2 − k1) 2 8(ρ+ 1) ( 1 p(ρ+ 1) + 1 ) 1 p [{∣∣η′′(k1)∣∣q (2− 2−ρs ρs+ 1 ) +m ∣∣∣∣η′′(k2 m )∣∣∣∣q (2− 2−ρs ρs+ 1 )} 1 q + {∣∣η′′(k1)∣∣q ( 2−ρs ρs+ 1 ) +m ∣∣∣∣η′′(k2 m )∣∣∣∣q ( 2−ρs ρs+ 1 )} 1 q ] . 4. Conclusion Fractional calculus plays a vital role in understanding problems in pure and applied sciences due to its possession of many interesting integral and differential operators. In this study, therefore, we employed a Riemann-Liouville operator to establish some new fractional integral inequalities for mappings whose second derivatives, raised to positive powers, exhibit (ρ, s)-convexity and (ρ, s,m)-convexities. Our study established new inequalities of midpoint type involving fractional operators through generalized classes of convexities. Several estimates of special functions including incomplete Beta, Euler gamma and hyperglycemic functions are reported in this study. Our findings can enhance the techniques by which the properties of convexity along with their generalizations can be thoroughly studied through Fractional Calculus. The findings of this study can be relevant in different areas of interest where the fractional calculus is extensibility used. Our inequalities can be specifically used to estimate the error bounds for numerical integration by improving the accuracy of quadrature methods. Future studies should include the use of different fractional integral operators, such as the generalized fractional integral operator unifying two existing fractional integral operators [16], together with other classes of convexities to establish different inequalities. O. B. Almutairi / Eur. J. Pure Appl. Math, 18 (1) (2025), 5834 11 of 12 References [1] Praveen Agarwal, Mohamed Jleli, and Muharrem Tomar. Certain hermite- hadamard type inequalities via generalized k-fractional integrals. Journal of in- equalities and applications, 2017:1–10, 2017. [2] Ravi P Agarwal. Inequalities and applications, volume 3. World scientific, 1994. [3] Ohud Bulayhan Almutairi. New fractional integral inequalities via euler’s beta function. Open Mathematics, 21(1):20230163, 2023. [4] Drumi D Bainov and Pavel S Simeonov. 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