EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6159 ISSN 1307-5543 – ejpam.com Published by New York Business Global A Novel Fractional Integral of a Function via Polynomial n-Fractional s-Like Preinvexity Jamshed Nasir1, Hassen Aydi2,3,∗, Saber Mansour4 1 Department of Mathematics and Statistics, Virtual University of Pakistan, Lahore Campus, 54000, Pakistan. 2 Institut Supérieur d’Informatique et des Techniques de Communication, Université de Sousse, H. Sousse 4000, Tunisia. 3 Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Ga-Rankuwa, South Africa. 4 Department of Mathematics, Umm Al-Qura University, Faculty of Sciences, P.O. Box 14035, Holy Makkah 21955, Saudi Arabia. Abstract. In the following numerical novel, we develop a new fractional integral operator that in- corporates polynomials n-fractional with s-like preinvexity, thus expanding the notion of fractional calculus. This new operator provides a more comprehensive framework for examining the behavior of functions exhibiting generalized preinvexity properties which are crucial in many optimization problems. We investigate the existence, uniqueness, and stability of this fractional integral as well as its basic characteristics. In addition, we provide a number of inequalities that show how useful this operator is in the context of applied sciences and mathematical analysis. Our results not only advance the theory of fractional calculus but also pave the way for future investigations into integral inequalities and fractional optimization. 2020 Mathematics Subject Classifications: 26D15, 26D51, 26D07, 26D10 Key Words and Phrases: Preinvex function, Polynomial n-fractional s-like preinvexity, K- fractional operator 1. Introduction and Preliminaries Integral inequalities provide significant bounds for function integrals, making them in- dispensable tools in mathematical analysis (see [1, 2]). When exact evaluation is challeng- ing or impossible, these inequalities can be used to estimate the magnitude or behavior of a function’s integral. Typical instances are the Minkowski inequality, related to Lp spaces and norms, and Hölder’s inequality, which extends the Cauchy-Schwarz inequality ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6159 Email addresses: jamshed@vu.edu.pk (J. Nasir), hassen.aydi@isima.rnu.tn (H. Aydi), samansour@uqu.edu.sa (S. Mansour) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) J. Nasir et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6159 2 of 26 to integrals. In many domains, including partial differential equations, probability theory, and numerical analysis, these inequalities are crucial for enabling the study of function spaces, the convergence of function sequences, and the stability of differential equation so- lutions. Integral inequalities are essential in optimization problems and the demonstration of existence and uniqueness because they set upper and lower bounds. Mathematicians including Leibniz, Liouville, Riemann, and others investigated the idea of extending the nth-order derivative to non-integer values, laying the groundwork for fractional calculus. The fractional derivative, which has multiple definitions (Riemann- Liouville, Caputo, and Grünwald-Letnikov), is one of the fundamental ideas in this disci- pline. Each definition is appropriate for a certain set of features and applications. Convex functions were significantly expanded upon with the introduction of preinvex functions, which broadened the meaning of convexity in optimization theory (see [3–6]). Preinvex functions reduce this requirement by introducing an invex function, whereas convex functions are defined by the fact that any line segment connecting two points on the function’s graph lies above the graph. In particular, an invex function serves as a sort of ”generalized direction” in the domain of a function, and a function is said to be preinvex if it fulfills a specific inequality with regard to it. More applications can be made possible by this generalization, especially when the standard convexity requirements prove to be too restrictive. Preinvex functions are useful in tackling complicated optimization issues, such as those found in game theory, economics, and multi-objective optimization (see [7– 9]). They maintain many of the beneficial aspects of convex functions, such as certain optimality criteria. Thus, the emergence of preinvexity has created new opportunities for study and application in fields where conventional convex analysis would not be enough. Definition 1. [10] The set Xo ⊂ ℜn is said to be invex iwith respect to ς∗(∗, ∗), if for every a1, b1 ∈ Xo and t ∈ [0, 1] a1 + tς∗(b1, a1) ∈ Xo. In definition 1, the set Xo is also known to be a ς∗−connectediset. For every convexiset is invex withirespect to ς∗(b1, a1) = b1 − a1 but there exist invex sets which are noticonvex (see [11]). Definition 2. [12] A mapping F on the invex set Xo is called to be preinvex with respect (w.r.) to ς∗ if F (a1 + tς∗ (b1, a1)) ≤ (1− t)F (a1) + tF (b1) ; ∀a1, b1 ∈ Xo, t ∈ [0, 1]. (1) The function −F is said to be preconcave if and only if F is preinvex. It is necessarily that each convexity becomes preinvexity, but not viceversa [13]. For example, F(t) = −|t| (for all t ∈ ℜ) is not a iconvex mapping but it is apreinvexifunction w.r. to ς∗ (b1, a1) = { b1 − a1 if a1b1 > 0, a1 − b1 if a1b1 < 0. J. Nasir et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6159 3 of 26 The following proposition regarding the mapping on ς∗ is mentioned in [14]. Property−C: Let Xo ⊂ ℜn be an openiinvex subset w.r. to ς∗ : Xo ×Xo ⊂ ℜn. For any a1, b1 ∈ Xo and t ∈ [0, 1], ς∗ (b1, b1 + tς∗ (a1, b1)) = −tς∗ (a1, b1) , ς∗ (a1, b1 + tς∗ (a1, b1)) = (1− t) ς∗ (a1, b1) . (2) For any a1, b1 ∈ Xo and t1, t2 ∈ [0, 1] from property−C, we have ς∗ (b1 + t2ς∗ (a1, b1) , b1 + t1ς∗ (a1, b1)) = (t2 − t1) ς∗ (a1, b1) . (3) If F is a preinvex function on ς∗ (a1, a1 + tς∗ (b1, a1)) and theimapping ς∗ satisfies property–C, then forievery t ∈ [0, 1], from (2), it yields that |F (a1 + tς∗ (b1, a1))| = |F (a1 + ς∗ (b1, a1)) + (1− t) ς∗ (a1, a1 + ς∗ (b1, a1))| ≤ t |F (a1 + ς∗ (b1, a1))|+ (1− t) |F (a1)| and |F (a1 + (1− t) ς∗ (b1, a1))| = |F (a1 + ς∗ (b1, a1)) + tς∗ (a1, a1 + ς∗ (b1, a1))| ≤ (1− t) |F (a1 + ς∗ (b1, a1))|+ t |F (a1)| . In [15], the following inequalities of ’H–H’ have been proved. Theorem 1. [10] Suppose F : X = [a1, a1 + ς∗ (b1, a1)] → (0,∞) is a ipreinvex mapping on the interval of realinumbers Xo with ς∗ (b1, a1) > 0 F ( 2a1 + ς∗ (b1, a1) 2 ) ≤ 1 ς∗ (b1, a1) ∫ a1+ς∗(b1,a1) a1 F(x)dx ≤ F (a1) + F (b1) 2 . (4) Suppose X ⊆ ℜ and F :⊆ ℜ is a mapping on a differentiable at Xo(the interior of X) such that [a1, b1] ∈ Xo with a1 < b1. In this case, the famousiOstrowski inequality [16] is stated as ∣∣∣∣F (x)− 1 b1 − a1 ∫ b1 a1 F(x)dx ∣∣∣∣ ≤ ∣∣∣∣∣14 + ( x− a1+b1 2 )2 (b1 − a1) 2 ∣∣∣∣∣ (b1 − a1)S, (5) for all x ∈ [a1, b1], if |F′| ≤ S. Ostrowski-like inequalities, which give estimations of error for various quadrature crite- ria, are widely used in numericalianalysis. These distinctions have grown and been applied to more fields in recent years (see [17–20]). J. Nasir et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6159 4 of 26 Definition 3. [21, 22] Let s ∈ [0, 1]. A real valued mapping Xo → ℜ is said to be s−like convex on Xo if F (µa1 + (1− µ) b1) ≤ (1− s (1− µ))F (a1) + (1− sµ)F (b1) , (6) for all a1, b1 ∈ Xo and µ ∈ [0, 1]. Definition 4. [23] Let X ⊂ ℜibe a nonempty invexiset with respect to ς∗ : Xo × Xo ⊂ ℜ → ℜ. iThen the mapping Ψ : X → ℜ is saidito be s−like preinvex, if F (b1 + µς∗(a1, b1)) ≤ (1− s (1− µ))F (a1) + (1− sµ)F (b1) . (7) In [24], İşcan gave the definition on n−fractional polynomial convexity as below. Definition 5. [24] Suppose n ∈ N . A non-negative mapping Xo ⊂ ℜ → ℜ is said to be an n−fractional polynomial convex(FPC) mapping if F (µa1 + (1− µ) b1) ≤ 1 n n∑ i=1 µ 1 i F (a1) + 1 n n∑ i=1 (1− µ) 1 i F (b1) , (8) for all a1, b1 ∈ Xo and µ ∈ [0, 1]. Definition 6. [25] Consider F ∈ L[a1, b1]. The left-right-sidediRiemann-Liouville(R– L)ifractional integrals of order ϱ > 0 are defined by Jϱ a1− F (x) = 1 Γ (ϱ) ∫ x a1 (x− t)ϱ−1 F (t) dt; a1 < x (9) and Jϱ b1 + F (x) = 1 Γ (ϱ) ∫ b1 x (t− x)ϱ−1 F (t) dt ; x < b1. (10) Gammaifunction is defined as Γ(ϱ) = ∫∞ 0 e−uuϱ−1du. In [26], Mubeen et al. introduced the following class of fractional integrals. Definition 7. [26] Suppose that F ∈ L[a1, b1]. The K−fractional integrals Jϱ,K a1− F (x) and Jϱ,K b1 + F (x) order ϱ > 0,K > 0 areidefined as Jϱ,K a1− F (x) = 1 KΓK (ϱ) ∫ x a1 (x− t) ϱ K −1 F (t) dt; a1 < x (11) and Jϱ,K b1 + F (x) = 1 KΓK (ϱ) ∫ b1 x (t− x) ϱ K −1 F (t) dt ; x < b1, (12) respectively,iwhere K > 0 and ΓK(ϱ) is the K−gammaifunction is given as ΓK (ϱ) =∫∞ 0 tϱ−1e− tK K dt with the properties ΓK(ϱ + K) = ϱΓK(ϱ) and ΓK(K) = 1. It is noted that J0,K a1− F (x)=J0,K b1 + F (x) = F (x) . J. Nasir et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6159 5 of 26 2. Main Results 2.1. Polynomials on n−fractional s−like preinvex mappings : This section examines the basic algebraic features of a novel fractional integral operator that incorporates polynomials on n−fractional with s−like preinvex mappings. Definition 8. Let us suppose that s ∈ [0, 1], n ∈ N , ai ≥ 0 ( i = 1, n ) , such that ∑n i=1 ai > 0, Xo ⊂ ℜ is an interval. A non-negative mapping Xo × Xo ⊂ ℜ → ℜ is said to be a polynomial on n−fractional s−like preinvex mapping if for every F (a1 + tς∗ (b1, a1)) ≤ ∑n i=1 ai (1− s (1− t)) 1 i∑n i=1 ai F (a1) + ∑n i=1 ai (1− st) 1 i∑n i=1 ai F (b1) , (13) for all t ∈ [0, 1], a1, b1 ∈ Xo. Remark 1. • If we take n = 1 in Definition (8), we attain [21]. • If n = 1 and s = 1 in Definition (8), it will be explored by Weiriand Mond [12]. • With taking n = 1 and ς∗ (b1, a1) = b1 − a1 in Definition (8), then it will attain a published definition named as s−type convexity that was explored by İ. İşcan et al. [21]. We will mention the nature of class with some polynomials on n−fractional s−like preinvex mappings by GFPP−s. Example 1. Consider a mapping F(x) = x2, and with some substitutions as s = 0.4, n = 2, a1 = 1, b1 = 2 and t = 0.4. According to (13), one writes F (a1 + tς∗ (b1, a1)) ≤ ∑n i=1 ai (1− s (1− t)) 1 i∑n i=1 ai F (a1) + ∑n i=1 ai (1− st) 1 i∑n i=1 ai F (b1) . For a1 = 1 and b1 = 3, we have F (1 + 0.4ς∗ (3, 1)) = F(1.8) = 1.82 = 3.24 and the other side will be 1. (1− 0.4 (0.6)) 1 1 + 2. (1− 0.4 (0.6)) 1 2 3 12 + 1. (1− 0.4 (0.6)) 1 1 + 2. (1− 0.4 (0.6)) 1 2 3 32 = 8.35 so that 3.24 ≤ 8.35. . J. Nasir et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6159 6 of 26 2.2. Polynomials on n−fractional s−like preinvex mapping as new exten- sions of H–H like inequalities Now, we will attain a new generalization ofiH–H inequality for the GFPP−s function F. Theorem 2. Let Xo ⊆ ℜ ibe an open invexisubset with respectito ς∗ : Xo×Xo → ℜ and a1, b1 ∈ Xo with b1 + ς∗ (a1, b1) ≤ b1. Suppose that F : [b1 + ς∗ (a1, b1) , b1] and satisfies Property- C with n ∈ N , ai ≥ 0 ( i = 1, n ) , such that ∑n i=1 ai > 0, s ∈ [0, 1], ϱ ∈ [0, 1], K > 0. Then ∑n i=1 ai∑n i=1 ai ( 1− s 2 ) 1 i F ( 2a1 + ς∗ (b1, a1) 2 ) ≤ ΓK (ϱ+ K) ς ϱ K ∗ (b1, a1) { Jϱ,K a1+ F (a1 + ς∗ (b1, a1)) + Jϱ,K (a1+ς∗(b1,a1)) − F (a1) } ≤ [F (a1) + F (a1 + ς∗ (b1, a1))] × ∫ 1 0 t ϱ K −1 {∑n i=1 ai (1− st) 1 i∑n i=1 ai + ∑n i=1 ai (1− s (1− t)) 1 i∑n i=1 ai } dt.. (14) Proof. From the definition of the GFPP−s function F, one obtains F ( x+ ς∗ (y, x) 2 ) ≤ ∑n i=1 ai ( 1− s 2 ) 1 i∑n i=1 ai F (x) + ∑n i=1 ai ( 1− s 2 ) 1 i∑n i=1 ai F (y) F ( x+ ς∗ (y, x) 2 ) ≤ ∑n i=1 ai ( 1− s 2 ) 1 i∑n i=1 ai [F (x) + F (y)]. (15) With substituting the x = a1 + (1− t) ς∗ (b1, a1) and y = a1 + tς∗ (b1, a1) in (15), we get F ( a1 + (1− t) ς∗ (b1, a1) + ς∗ (a1 + tς∗ (b1, a1) , a1 + (1− t) ς∗ (b1, a1)) 2 ) ≤ ∑n i=1 ai ( 1− s 2 ) 1 i∑n i=1 ai [F (a1 + (1− t) ς∗ (b1, a1)) + F (a1 + tς∗ (b1, a1))]. (16) By taking product with the term t ϱ K −1 and antiderivative with respect to t ∈ [0, 1], one gets 1 ϱ K F ( 2a1 + ς∗ (b1, a1) 2 ) ≤ ∑n i=1 ai ( 1− s 2 ) 1 i∑n i=1 ai J. Nasir et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6159 7 of 26 × [ ∫ 1 0 t ϱ K −1F (a1 + (1− t) ς∗ (b1, a1)) dt+ ∫ 1 0 t ϱ K −1F (a1 + tς∗ (b1, a1)) dt ] 1 ϱ K F ( 2a1 + ς∗ (b1, a1) 2 ) ≤ ∑n i=1 ai ( 1− s 2 ) 1 i∑n i=1 ai × [ KΓK (ϱ) ς ϱ K ∗ (b1, a1) { Jϱ,K a1+ F (a1 + ς∗ (b1, a1)) + Jϱ,K (a1+ς∗(b1,a1)) − F (a1) }] , (17) which completes the leftihand side of (14). For theiproof of the secondiinequality in (14), we first note that if F is n−polynomial s−like preinvexity on [a1, a1 + ς∗ (b1, a1)] and the mapping ς∗ satisfies the property-C, then for ievery t ∈ [0, 1], it yields that F (a1 + (1− t) ς∗ (b1, a1)) ≤ ∑n i=1 ai (1− s (1− t)) 1 i∑n i=1 ai F (a1 + ς∗ (b1, a1)) + ∑n i=1 ai (1− st) 1 i∑n i=1 ai F (a1) F (a1 + tς∗ (b1, a1)) ≤ ∑n i=1 ai (1− st) 1 i∑n i=1 ai F (a1 + ς∗ (b1, a1)) + ∑n i=1 ai (1− s (1− t)) 1 i∑n i=1 ai F (a1) . (18) By adding above two inequalities, one gets F (a1 + (1− t) ς∗ (b1, a1)) + F (a1 + tς∗ (b1, a1)) ≤ {∑n i=1 ai (1− s (1− t)) 1 i∑n i=1 ai + ∑n i=1 ai (1− st) 1 i∑n i=1 ai } [F (a1) + F (a1 + ς∗ (b1, a1))]. (19) By taking product with the term t ϱ K −1 and antiderivative with respect to t ∈ [0, 1], one gets F (a1 + (1− t) ς∗ (b1, a1)) + F (a1 + tς∗ (b1, a1)) ≤ {∑n i=1 ai (1− s (1− t)) 1 i∑n i=1 ai + ∑n i=1 ai (1− st) 1 i∑n i=1 ai } [F (a1) + F (a1 + ς∗ (b1, a1))] (20) ∫ 1 0 t ϱ K −1F (a1 + (1− t) ς∗ (b1, a1)) dt+ ∫ 1 0 t ϱ K −1 F (a1 + tς∗ (b1, a1)) dt ≤ [F (a1) + F (a1 + ς∗ (b1, a1))] ∫ 1 0 t ϱ K −1 J. Nasir et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6159 8 of 26 × {∑n i=1 ai (1− s (1− t)) 1 i∑n i=1 ai + ∑n i=1 ai (1− st) 1 i∑n i=1 ai } dt (21) KΓK (ϱ) ς ϱ K ∗ (b1, a1) { Jϱ,K a1+ F (a1 + ς∗ (b1, a1)) + Jϱ,K (a1+ς∗(b1,a1)) − F (a1) } ≤ [F (a1) + F (a1 + ς∗ (b1, a1))] ∫ 1 0 t ϱ K −1 × {∑n i=1 ai (1− s (1− t)) 1 i∑n i=1 ai + ∑n i=1 ai (1− st) 1 i∑n i=1 ai } dt. (22) By combining the inequalities (17) and (22), we can get (14). Corollary 1. If we judge the value s = 1 in Theorem 2, then the following inequalities for GFPP function with K−fractionaliintegral operators:∑n i=1 ai∑n i=1 ai ( 1 2 ) 1 i F ( 2a1 + ς∗ (b1, a1) 2 ) ≤ ΓK (ϱ+ K) ς ϱ k ∗ (b1, a1) { Jϱ,K a1+ F (a1 + ς∗ (b1, a1)) + Jϱ,K (a1+ς∗(b1,a1)) − F (a1) } ≤ [F (a1) + F (a1 + ς∗ (b1, a1))]∑n i=1 ai × ∫ 1 0 n∑ i=1 ait ϱ K −1{(1− t) 1 i + (t) 1 i }dt. (23) Corollary 2. If we judge the value K = 1 in Corollary 1, then we get the following inequalities for GFPP function with RL−fractionaliintegral operators:∑n i=1 ai∑n i=1 ai ( 1 2 ) 1 i F ( 2a1 + ς∗ (b1, a1) 2 ) ≤ Γ (ϱ+ 1) ςϱ∗ (b1, a1) { Jϱ a1+ F (a1 + ς∗ (b1, a1)) + Jϱ (a1+ς∗(b1,a1)) − F (a1) } ≤ [F (a1) + F (a1 + ς∗ (b1, a1))]∑n i=1 ai × ∫ 1 0 n∑ i=1 ait ϱ−1{(1− t) 1 i + (t) 1 i }dt. (24) Remark 2. If we take ϱ = 1 andin = 1, in Corollary 2, then one can get the inequalities (4). Through out the article, we take U∗ = Let us suppose that n ∈ N , ai ≥ 0 ( i = 1, n ) , such that ∑n i=1 ai > 0, s ∈ [0, 1], a1 < a1 + ς∗ (b1, a1) and ϱ,K > 0. J. Nasir et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6159 9 of 26 2.3. New generalizations of Ostrowski type inequalities using polynomial n−fractional s−like preinvex functions This portion explores the new inequalities for the derivatives of first and second with the GFPP−s function. In the mean while, we will develop a following new lemma. Lemma 1. Suppose F : [a1, a1 + ς∗ (b1, a1)] → ℜ is a differentiable mapping on (a1, a1 + ς∗ (b1, a1)) with a1 < a1 + ς∗ (b1, a1) . If F ′ ∈ L[a1, a1 + ς∗ (b1, a1)], ϱ > 0,K > 0, then the followingiequality for K−fractional integral operator is as follows ς ϱ K ∗ (x, a1) + ς ϱ K ∗ (x, b1) ς∗ (b1, a1) F (x)− ΓK (ϱ+ K) ς ϱ K ∗ (b1, a1) × { Jϱ,K (a1+ς∗(x,a1)) − F (a1) + Jϱ,K (b1+ς∗(x,b1)) + F (a1 + ς∗ (b1, a1)) } = ς ϱ K +1 ∗ (x, a1) ς∗ (b1, a1) ∫ 1 0 t ϱ KF (a1 + tς∗ (x, a1)) dt+ ς ϱ K +1 ∗ (x, b1) ς∗ (b1, a1) ∫ 1 0 t ϱ KF (b1 + tς∗ (x, b1)) dt. (25) Proof. Let us assume that ς ϱ K +1 ∗ (x, a1) ς∗ (b1, a1) ∫ 1 0 t ϱ KF (a1 + tς∗ (x, a1)) dt+ ς ϱ K +1 ∗ (x, b1) ς∗ (b1, a1) ∫ 1 0 t ϱ KF (b1 + tς∗ (x, b1)) dt. (26) By using integrationiby parts and suitable substitution, we get I1 = ∫ 1 0 t ϱ KF (a1 + tς∗ (x, a1)) dt = F (a1 + ς∗ (x, a1)) ς∗ (x, a1) − ΓK (ϱ+ K) ς ϱ K +1 ∗ (x, a1) .Jϱ,K (a1+ς∗(x,a1)) − F (a1) . (27) Similarly, we can find I2 = ∫ 1 0 t ϱ KF (b1 + tς∗ (x, b1)) dt = F (b1 + ς∗ (x, b1)) ς∗ (x, b1) − ΓK (ϱ+ K) ς ϱ K +1 ∗ (x, b1) .Jϱ,K (b1+ς∗(x,b1)) + F (a1 + ς∗ (b1, a1)) . (28) Substituting the values of I1 and I2 in (26), we can get (25). J. Nasir et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6159 10 of 26 Theorem 3. Suppose F : X = [a1, a1 + ς∗ (b1, a1)] → ℜ is a differentiable function on Xo such that F′ ∈ L[a1, a1 + ς∗ (b1, a1)] and consideration with U∗. Let |F′| be a GFPP−s function on X with |F′| ≤ S, for all x ∈ [a1, a1 + ς∗ (b1, a1)]. Then ∣∣∣∣ ς ϱ K ∗ (x, a1) + ς ϱ K ∗ (x, b1) ς∗ (b1, a1) F (x)− ΓK (ϱ+ K) ς ϱ K ∗ (b1, a1) × { Jϱ,K (a1+ς∗(x,a1)) − F (a1) + Jϱ,K (b1+ς∗(x,b1)) + F (a1 + ς∗ (b1, a1)) }∣∣∣∣ ≤ ( ς ϱ K +1 ∗ (x, a1) + ς ϱ K +1 ∗ (x, b1) ς∗ (b1, a1) ) × S∑n i=1 ai . n∑ i=1 ai [ ∫ 1 0 t ϱ k ( (1− s (1− t)) 1 i ) dt+ ∫ 1 0 t ϱ k (1− st) 1 i dt ] . (29) Proof. From Lemma 1 and a modulus property of the GFPP−s function |F′|, one has ∣∣∣∣ ς ϱ K ∗ (x, a1) + ς ϱ K ∗ (x, b1) ς∗ (b1, a1) F (x)− ΓK (ϱ+ K) ς ϱ K ∗ (b1, a1) × { Jϱ,K (a1+ς∗(x,a1)) − F (a1) + Jϱ,K (b1+ς∗(x,b1)) + F (a1 + ς∗ (b1, a1)) }∣∣∣∣ ≤ ς ϱ K +1 ∗ (x, a1) ς∗ (b1, a1) ∫ 1 0 t ϱ K ∣∣F′ (a1 + tς∗ (x, a1)) ∣∣dt+ ς ϱ K +1 ∗ (x, b1) ς∗ (b1, a1) ∫ 1 0 t ϱ K ∣∣F′ (b1 + tς∗ (x, b1)) ∣∣dt ≤ ς ϱ K +1 ∗ (x, a1) ς∗ (b1, a1) ∫ 1 0 t ϱ K [∑n i=1 ai (1− s (1− t)) 1 i∑n i=1 ai ∣∣F′ (x) ∣∣+ ∑n i=1 ai (1− st) 1 i∑n i=1 ai ∣∣F′ (a1) ∣∣ ]dt + ς ϱ K +1 ∗ (x, b1) ς∗ (b1, a1) ∫ 1 0 t ϱ K [∑n i=1 ai (1− s (1− t)) 1 i∑n i=1 ai ∣∣F′ (x) ∣∣+ ∑n i=1 ai (1− st) 1 i∑n i=1 ai ∣∣F′ (b1) ∣∣ ]dt ≤ ( ς ϱ k +1 ∗ (x, a1) + ς ϱ k +1 ∗ (x, b1) ς∗ (b1, a1) ) × S∑n i=1 ai . n∑ i=1 ai [ ∫ 1 0 t ϱ K ( (1− s (1− t)) 1 i ) dt+ ∫ 1 0 t ϱ K (1− st) 1 i dt ] . (30) Corollary 3. If we judge the value s = 1 iniTheorem 3, then we have the follow- ingiinequalities for GFPP function with K−fractionaliintegral operators:∣∣∣∣ ς ϱ K ∗ (x, a1) + ς ϱ K ∗ (x, b1) ς∗ (b1, a1) F (x)− ΓK (ϱ+ K) ς ϱ K ∗ (b1, a1) J. Nasir et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6159 11 of 26 × { Jϱ,K (a1+ς∗(x,a1)) − F (a1) + Jϱ,K (b1+ς∗(x,b1)) + F (a1 + ς∗ (b1, a1)) }∣∣∣∣ ≤ ( ς ϱ k +1 ∗ (x, a1) + ς ϱ k +1 ∗ (x, b1) ς∗ (b1, a1) ) S∑n i=1 ai . n∑ i=1 ai [ ∫ 1 0 t ϱ k t 1 i dt+ ∫ 1 0 t ϱ k (1− t) 1 i dt ] . Corollary 4. If we judge the value K = 1 in Corollary 3, then we have the following inequalities for GFPP function with RL−fractionaliintegral operators:∣∣∣∣ ςϱ∗ (x, a1) + ςϱ∗ (x, b1) ς∗ (b1, a1) F (x)− Γ (ϱ+ 1) ςϱ∗ (b1, a1) × { Jϱ (a1+ς∗(x,a1)) − F (a1) + Jϱ (b1+ς∗(x,b1)) + F (a1 + ς∗ (b1, a1)) }∣∣∣∣ ≤ ( ςϱ+1 ∗ (x, a1) + ςϱ+1 ∗ (x, b1) ς∗ (b1, a1) ) S∑n i=1 ai . n∑ i=1 ai [ ∫ 1 0 tϱt 1 i dt+ ∫ 1 0 tϱ (1− t) 1 i dt ] . Remark 3. If we take ϱ = 1 and n = 1 and ς∗(b1, a1) = b1 − a1, in Corollary 4, then one can get the inequalities (5). Theorem 4. Suppose F : X =: [a1, a1+ς∗ (b1, a1)] → ℜ is a differentiable function function on Xo such that F′ ∈ L[a1, a1 + ς∗ (b1, a1)] and consideration with U∗. Let for some q > 1, |F′|q be a GFPP−s function on X with |F′| ≤ S, for all x ∈ [a1, a1 + ς∗ (b1, a1)]. Then ∣∣∣∣ ς ϱ K ∗ (x, a1) + ς ϱ K ∗ (x, b1) ς∗ (b1, a1) F (x)− ΓK (ϱ+ K) ς ϱ K ∗ (b1, a1) × { Jϱ,K (a1+ς∗(x,a1)) − F (a1) + Jϱ,K (b1+ς∗(x,b1)) + F (a1 + ς∗ (b1, a1)) }∣∣∣∣ ≤ ( K K+ ϱ )1− 1 q ( ς ϱ K +1 ∗ (x, a1) + ς ϱ K +1 ∗ (x, b1) ς∗ (b1, a1) ) × Sq∑n i=1 ai . n∑ i=1 ai [ ∫ 1 0 t ϱ K ( (1− s (1− t)) 1 i ) dt+ ∫ 1 0 t ϱ K (1− st) 1 i dt ] 1 q . (31) Proof. From Lemma 1 and a propertyiof the GFPP−s function |F′|q, and the power meaniinequality, one has ∣∣∣∣ ς ϱ K ∗ (x, a1) + ς ϱ K ∗ (x, b1) ς∗ (b1, a1) F (x)− ΓK (ϱ+ K) ς ϱ K ∗ (b1, a1) × { Jϱ,K (a1+ς∗(x,a1)) − F (a1) + Jϱ,K (b1+ς∗(x,b1)) + F (a1 + ς∗ (b1, a1)) }∣∣∣∣ J. Nasir et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6159 12 of 26 ≤ ς ϱ K +1 ∗ (x, a1) ς∗ (b1, a1) ∫ 1 0 t ϱ K ∣∣F′ (a1 + tς∗ (x, a1)) ∣∣dt + ς ϱ K +1 ∗ (x, b1) ς∗ (b1, a1) ∫ 1 0 t ϱ K ∣∣F′ (b1 + tς∗ (x, b1)) ∣∣dt (32) ≤ ς ϱ K +1 ∗ (x, a1) ς∗ (b1, a1) (∫ 1 0 t ϱ Kdt )1− 1 q (∫ 1 0 t ϱ K ∣∣F′ (a1 + tς∗ (x, a1)) ∣∣qdt) 1 q + ς ϱ K +1 ∗ (x, b1) ς∗ (b1, a1) (∫ 1 0 t ϱ Kdt )1− 1 q (∫ 1 0 t ϱ K ∣∣F′ (b1 + tς∗ (x, b1)) ∣∣qdt) 1 q ≤ ( K K+ ϱ )1− 1 q [ ς ϱ K +1 ∗ (x, a1) ς∗ (b1, a1) {∑n i=1 ai ∫ 1 0 t ϱ K ( (1− s (1− t)) 1 i ) ∑n i=1 ai ∣∣F′ (x) ∣∣q dt + ∑n i=1 ai ∫ 1 0 t ϱ K ( (1− st) 1 i ) ∑n i=1 ai ∣∣F′ (a1) ∣∣q dt} 1 q + ς ϱ K +1 ∗ (x, b1) ς∗ (b1, a1) {∑n i=1 ai ∫ 1 0 t ϱ K ( (1− s (1− t)) 1 i ) ∑n i=1 ai ∣∣F′ (x) ∣∣q dt + ∑n i=1 ai ∫ 1 0 t ϱ K ( (1− st) 1 i ) ∑n i=1 ai ∣∣F′ (b1) ∣∣q dt} 1 q ] ≤ ( K K+ ϱ )1− 1 q ( ς ϱ K +1 ∗ (x, a1) + ς ϱ K +1 ∗ (x, b1) ς∗ (b1, a1) ) × Sq∑n i=1 ai . n∑ i=1 ai [ ∫ 1 0 t ϱ K ( (1− s (1− t)) 1 i ) dt+ ∫ 1 0 t ϱ K (1− st) 1 i dt ] 1 q . (33) Corollary 5. If one can take s = 1 iniTheorem 4, then we have the followingiinequalities for GFPP function with K−fractionaliintegral operators:∣∣∣∣ ς ϱ K ∗ (x, a1) + ς ϱ K ∗ (x, b1) ς∗ (b1, a1) F (x)− ΓK (ϱ+ K) ς ϱ K ∗ (b1, a1) × { Jϱ,K (a1+ς∗(x,a1)) − F (a1) + Jϱ,K (b1+ς∗(x,b1)) + F (a1 + ς∗ (b, a1)) }∣∣∣∣ ≤ ( K K+ ϱ )1− 1 q ( ς ϱ K +1 ∗ (x, a1) + ς ϱ K +1 ∗ (x, b1) ς∗ (b1, a1) ) × Sq∑n i=1 ai . n∑ i=1 ai [ ∫ 1 0 t ϱ K t 1 i dt+ ∫ 1 0 t ϱ K (1− t) 1 i dt ] 1 q . J. Nasir et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6159 13 of 26 Corollary 6. If one can takes K = 1 in Corollary 5, then we have the following inequalities for GFPP function with RL−fractional integral operators:∣∣∣∣ ςϱ∗ (x, a1) + ςϱ∗ (x, b1) ς∗ (b1, a1) F (x)− Γ (ϱ+ 1) ςϱ∗ (b1, a1) × { Jϱ (a1+ς∗(x,a1)) − F (a1) + Jϱ (b1+ς∗(x,b1)) + F (a1 + ς∗ (b1, a1)) }∣∣∣∣ ≤ ( 1 1+ ϱ )1− 1 q ( ςϱ+1 ∗ (x, a1) + ςϱ+1 ∗ (x, b1) ς∗ (b1, a1) ) × Sq∑n i=1 ai . n∑ i=1 ai [ ∫ 1 0 tϱt 1 i dt+ ∫ 1 0 tϱ (1− t) 1 i dt ] 1 q . Theorem 5. Suppose F : X = [a1, a1+ς∗ (b1, a1)] → ℜ is a differentiableifunction function on Xo such that F′ ∈ L[a1, a1 + ς∗ (b1, a1)] and consideration with U∗. Let for some p, q > 1, with 1 p + 1 q = 1, |F′|q be a GFPP−s function on X with |F′| ≤ S, for all x ∈ [a1, a1 + ς∗ (b1, a1)]. Then ∣∣∣∣ ς ϱ K ∗ (x, a1) + ς ϱ K ∗ (x, b1) ς∗ (b1, a1) F (x)− ΓK (ϱ+ K) ς ϱ K ∗ (b1, a1) × { Jϱ,K (a1+ς∗(x,a1)) − F (a1) + Jϱ,K (b1+ς∗(x,b1)) + F (a1 + ς∗ (b1, a1)) }∣∣∣∣ ≤ ( K K+ pϱ ) 1 p ( ς ϱ K +1 ∗ (x, a1) + ς ϱ K +1 ∗ (x, b1) ς∗ (b1, a1) ) × Sq∑n i=1 ai . n∑ i=1 ai [ ∫ 1 0 ( (1− s (1− t)) 1 i ) dt+ ∫ 1 0 (1− st) 1 i dt ] 1 q . (34) Proof. From Lemma 1 and a propertyiof the GFPP−s function |F′|q, and the Hölder inequality, one has ∣∣∣∣ ς ϱ K ∗ (x, a1) + ς ϱ K ∗ (x, b1) ς∗ (b1, a1) F (x)− ΓK (ϱ+ K) ς ϱ K ∗ (b1, a1) × { Jϱ,K (a1+ς∗(x,a1)) − F (a1) + Jϱ,K (b1+ς∗(x,b1)) + F (a1 + ς∗ (b1, a1)) }∣∣∣∣ ≤ ς ϱ K +1 ∗ (x, a1) ς∗ (b1, a1) ∫ 1 0 t ϱ K ∣∣F′ (a1 + tς∗ (x, a1)) ∣∣dt + ς ϱ K +1 ∗ (x, b1) ς∗ (b1, a1) ∫ 1 0 t ϱ K ∣∣F′ (b1 + tς∗ (x, b1)) ∣∣dt J. Nasir et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6159 14 of 26 ≤ ς ϱ K +1 ∗ (x, a1) ς∗ (b1, a1) (∫ 1 0 t ϱ Kdt ) 1 p (∫ 1 0 ∣∣F′ (a1 + tς∗ (x, a1)) ∣∣qdt) 1 q + ς ϱ K +1 ∗ (x, b1) ς∗ (b1, a1) (∫ 1 0 t ϱ Kdt ) 1 p (∫ 1 0 ∣∣F′ (b1 + tς∗ (x, b1)) ∣∣qdt) 1 q ≤ ( K K+ pϱ ) 1 p [ ς ϱ K +1 ∗ (x, a1) ς∗ (b1, a1) {∑n i=1 ai ∫ 1 0 ( (1− s (1− t)) 1 i ) ∑n i=1 ai ∣∣F′ (x) ∣∣q dt + ∑n i=1 ai ∫ 1 0 ( (1− st) 1 i ) ∑n i=1 ai ∣∣F′ (a1) ∣∣q dt} 1 q + ς ϱ K +1 ∗ (x, b1) ς∗ (b1, a1) {∑n i=1 ai ∫ 1 0 ( (1− s (1− t)) 1 i ) ∑n i=1 ai ∣∣F′ (x) ∣∣q dt + ∑n i=1 ai ∫ 1 0 ( (1− st) 1 i ) ∑n i=1 ai ∣∣F′ (b1) ∣∣q dt} 1 q ] ≤ ( K K+ pϱ ) 1 p ( ς ϱ K +1 ∗ (x, a1) + ς ϱ K +1 ∗ (x, b1) ς∗ (b1, a1) ) × Sq∑n i=1 ai . n∑ i=1 ai [ ∫ 1 0 ( (1− s (1− t)) 1 i ) dt+ ∫ 1 0 (1− st) 1 i dt ] 1 q . (35) Corollary 7. If one can take s = 1 iniTheorem 5, then we have the following inequalities for a GFPP function with K−fractional integral operators:∣∣∣∣ ς ϱ K ∗ (x, a1) + ς ϱ K ∗ (x, b1) ς∗ (b1, a1) F (x)− ΓK (ϱ+ K) ς ϱ K ∗ (b1, a1) × { Jϱ,K (a1+ς∗(x,a1)) − F (a1) + Jϱ,K (b1+ς∗(x,b1)) + F (a1 + ς∗ (b1, a1)) }∣∣∣∣ ≤ ( K K+ pϱ ) 1 p ( ς ϱ K +1 ∗ (x, a1) + ς ϱ K +1 ∗ (x, b1) ς∗ (b1, a1) )[ Sq∑n i=1 ai . n∑ i=1 ai ( 2i i+ 1 )] 1 q . Corollary 8. If one can take K = 1 in Corollary 7, then we have the following inequalities for a GFPP function with RL−fractional integral operators:∣∣∣∣ ςϱ∗ (x, a1) + ςϱ∗ (x, b1) ς∗ (b1, a1) F (x)− Γ (ϱ+ 1) ςϱ∗ (b1, a1) × { Jϱ (a1+ς∗(x,a1)) − F (a1) + Jϱ (b1+ς∗(x,b1)) + F (a1 + ς∗ (b1, a1)) }∣∣∣∣ ≤ ( 1 1+ pϱ ) 1 p ( ςϱ+1 ∗ (x, a1) + ςϱ+1 ∗ (x, b1) ς∗ (b1, a1) )[ Sq∑n i=1 ai . n∑ i=1 ai ( 2i i+ 1 )] 1 q . J. Nasir et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6159 15 of 26 Now, we develop some new Ostrowskiitype inequalities for twiceidifferentiable func- tions. First, we will give the new following lemma: Lemma 2. Suppose F : [a1, a1+ς∗ (b1, a1)] → ℜ is twice differentiable mapping on (a1, a1+ ς∗ (b1, a1)) with a1 < a1 + ς∗ (b1, a1) . If F ′′ ∈ L[a1, a1 + ς∗ (b1, a1)], ϱ > 0,K > 0, then we have the followingiequality for K−fractional integral operator (1− λ) [ ς ϱ K ∗ (b1, x)− ς ϱ K ∗ (x, a1) ς∗ (b1, a1) ] F′ (x) + ( 1 + ϱ K − λ )[ ς ϱ K ∗ (b1, x) + ς ϱ K ∗ (x, a1) ς∗ (b1, a1) ] F (x) + λ [ ς ϱ K ∗ (b1, x)F (b1) + ς ϱ K ∗ (x, a1)F (a1) ς∗ (b1, a1) ] − ΓK (ϱ+ 2K) ς∗ (b1, a1) { Jϱ,K (a1+ς∗(x,a1)) − F (a1) + Jϱ,K (b1+ς∗(x,b1)) + F (b1) } = ς ϱ K +2 ∗ (x, a1) ς∗ (b1, a1) ∫ 1 0 t ( λ− t ϱ K ) F′′ (a1 + tς∗ (x, a1)) dt + ς ϱ K +2 ∗ (b1, x) ς∗ (b1, a1) ∫ 1 0 t ( λ− t ϱ K ) F′′ (b1 + tς∗ (x, b1)) dt, holds forall x ∈ [a1, + ς∗ (b1, a1)], λ ∈ [0, 1]. (36) Proof. It can easily be proved as similar Lemma 1. Theorem 6. Suppose F : X = [a1, a1 + ς∗ (b1, a1)] → ℜ is a twice differentiable function on Xo such that F′′ ∈ L[a1, a1 + ς∗ (b1, a1)] and consideration with U∗. Let |F′′(x)| be a GFPP−s function on X. Then ∣∣∣∣ (1− λ) [ ς ϱ K ∗ (b1, x)− ς ϱ K ∗ (x, a1) ς∗ (b1, a1) ] F′ (x) + ( 1 + ϱ K − λ )[ ς ϱ K ∗ (b1, x) + ς ϱ K ∗ (x, a1) ς∗ (b1, a1) ] F (x) + λ [ ς ϱ K ∗ (b1, x)F (b1) + ς ϱ K ∗ (x, a1)F (a1) ς∗ (b1, a1) ] − ΓK (ϱ+ 2K) ς∗ (b1, a1) { Jϱ,K (a1+ς∗(x,a1)) − F (a1) + Jϱ,K (b1+ς∗(x,b1)) + F (b1) }∣∣∣∣ ≤ [ ς ϱ K +2 ∗ (x, a1) |F′′ (a1)|+ ς ϱ K +2 ∗ (b1, x) |F′′ (b1)| ς∗ (b1, a1) ] 1∑n i=1 ai n∑ i=1 ai ∫ 1 0 t ( λ− t ϱ K ) (1− st) 1 i dt (37) + ς ϱ K +2 ∗ (x, a1) + ς ϱ K +2 ∗ (b1, x)∑n i=1 ai ς∗ (b1, a1) ∣∣F′′ (x) ∣∣ n∑ i=1 ai ∫ 1 0 t ( λ− t ϱ K ) (1− s (1− t)) 1 i dt holds ∀ ∈ [a1, b1] and λ ∈ [0, 1]. J. Nasir et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6159 16 of 26 Proof. From Lemma 2 and a modulus property of the GFPP−s function |F′′|, one has∣∣∣∣ (1− λ) [ ς ϱ K ∗ (b1, x)− ς ϱ K ∗ (x, a1) ς∗ (b1, a1) ] F′ (x) + ( 1 + ϱ K − λ )[ ς ϱ K ∗ (b1, x) + ς ϱ K ∗ (x, a1) ς∗ (b1, a1) ] F (x) + λ [ ς ϱ K ∗ (b1, x)F (b1) + ς ϱ K ∗ (x, a1)F (a1) ς∗ (b1, a1) ] − ΓK (ϱ+ 2K) ς∗ (b1, a1) { Jϱ,K (a1+ς∗(x,a1)) − F (a1) + Jϱ,K (b1+ς∗(x,b1)) + F (b1) }∣∣∣∣ ≤ ς ϱ K +2 ∗ (x, a1) ς∗ (b1, a1) ∫ 1 0 t ( λ− t ϱ K ) |F′′ (a1 + tς∗ (x, a1)) |dt + ς ϱ K +2 ∗ (b1, x) ς∗ (b1, a1) ∫ 1 0 t ( λ− t ϱ K ) |F′′ (b1 + tς∗ (x, b1)) |dt ≤ ς ϱ K +2 ∗ (x, a1) ς∗ (b1, a1) ∫ 1 0 t ( λ− t ϱ K ) ×[∑n i=1 ai (1− s (1− t)) 1 i∑n i=1 ai ∣∣F′′ (x) ∣∣+ ∑n i=1 ai (1− st) 1 i∑n i=1 ai ∣∣F′′ (a1) ∣∣ ]dt + ς ϱ K +2 ∗ (b1, x) ς∗ (b1, a1) ∫ 1 0 t ( λ− t ϱ K ) ×[∑n i=1 ai (1− s (1− t)) 1 i∑n i=1 ai ∣∣F′′ (x) ∣∣+ ∑n i=1 ai (1− st) 1 i∑n i=1 ai ∣∣F′′ (b1) ∣∣ ]dt ≤ [ ς ϱ K +2 ∗ (x, a1) |F′′ (a1)|+ ς ϱ K +2 ∗ (b1, x) |F′′ (b1)| ς∗ (b1, a1) ] 1∑n i=1 ai n∑ i=1 ai ∫ 1 0 t ( λ− t ϱ k ) (1− st) 1 i dt + ς ϱ K +2 ∗ (x, a1) + ς ϱ K +2 ∗ (b1, x)∑n i=1 ai ς∗ (b1, a1) ∣∣F′′ (x) ∣∣ n∑ i=1 ai ∫ 1 0 t ( λ− t ϱ K ) (1− s (1− t)) 1 i dt. (38) Corollary 9. If one can take s = 1 in Theorem 6, then we have the followingiinequalities for GFPP function with K−fractional integral operators:∣∣∣∣ (1− λ) [ ς ϱ K ∗ (b1, x)− ς ϱ K ∗ (x, a1) ς∗ (b1, a1) ] F′ (x) + ( 1 + ϱ K − λ )[ ς ϱ K ∗ (b1, x) + ς ϱ K ∗ (x, a1) ς∗ (b1, a1) ] F (x) + λ [ ς ϱ K ∗ (b1, x)F (b1) + ς ϱ K ∗ (x, a1)F (a1) ς∗ (b1, a1) ] − ΓK (ϱ+ 2K) ς∗ (b1, a1) { Jϱ,K (a1+ς∗(x,a1)) − F (a1) + Jϱ,K (b1+ς∗(x,b1)) + F (b1) }∣∣∣∣ ≤ [ ς ϱ K +2 ∗ (x, a1) |F′′ (a1)|+ ς ϱ K +2 ∗ (b1, x) |F′′ (b1)| ς∗ (b1, a1) ] 1∑n i=1 ai n∑ i=1 ai ∫ 1 0 t ( λ− t ϱ K ) (1− t) 1 i dt J. Nasir et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6159 17 of 26 + ς ϱ K +2 ∗ (x, a1) + ς ϱ K +2 ∗ (b1, x)∑n i=1 ai ς∗ (b1, a1) ∣∣F′′ (x) ∣∣ n∑ i=1 ai ∫ 1 0 ( λ− t ϱ K ) (t) 1 i +1 dt. Corollary 10. If one can take K = 1 in Corollary 9, then we have the following inequalities for a GFPP function with RL−fractional integral operators:∣∣∣∣ (1− λ) [ ςϱ∗ (b1, x)− ςϱ∗ (x, a1) ς∗ (b1, a1) ] F′ (x) + (1 + ϱ− λ) [ ςϱ∗ (b1, x) + ςϱ∗ (x, a1) ς∗ (b1, a1) ] F (x) + λ [ ςϱ∗ (b1, x)F (b1) + ςϱ∗ (x, a1)F (a1) ς∗ (b1, a1) ] − Γ (ϱ+ 2) ς∗ (b1, a1) { Jϱ (a1+ς∗(x,a1)) − F (a1) + Jϱ (b1+ς∗(x,b1)) + F (b1) }∣∣∣∣ ≤ [ ςϱ+2 ∗ (x, a1) |F′′ (a1)|+ ςϱ+2 ∗ (b1, x) |F′′ (b1)| ς∗ (b1, a1) ] 1∑n i=1 ai n∑ i=1 ai ∫ 1 0 t (λ− tϱ) (1− t) 1 i dt + ςϱ+2 ∗ (x, a1) + ςϱ+2 ∗ (b1, x)∑n i=1 ai ς∗ (b1, a1) ∣∣F′′ (x) ∣∣ n∑ i=1 ai ∫ 1 0 (λ− tϱ) (t) 1 i +1 dt. Theorem 7. Suppose F : X =: [a1, a1 + ς∗ (b1, a1)] → ℜ is twice differentiable function function on Xo such that F′′ ∈ L[a1, a1 + ς∗ (b1, a1)] and consideration with U∗. Let for some q > 1, |F′′(x)|q be a GFPP−s function on X, for all x ∈ [a1, a1 + ς∗ (b1, a1)]. Then∣∣∣∣ (1− λ) [ ς ϱ K ∗ (b1, x)− ς ϱ K ∗ (x, a1) ς∗ (b1, a1) ] F′ (x) + ( 1 + ϱ K − λ )[ ς ϱ K ∗ (b1, x) + ς ϱ K ∗ (x, a1) ς∗ (b1, a1) ] F (x) + λ [ ς ϱ K ∗ (b1, x)F (b1) + ς ϱ K ∗ (x, a1)F (a1) ς∗ (b1, a1) ] − ΓK (ϱ+ 2K) ς∗ (b1, a1) { Jϱ,K (a1+ς∗(x,a1)) − F (a1) + Jϱ,K (b1+ς∗(x,b1)) + F (b1) }∣∣∣∣ ≤ M 1− 1 q (ϱ,K, λ) × [ ς ϱ K +2 ∗ (x, a1) ς∗ (b1, a1) {∫ 1 0 t ( λ− t ϱ K ){∑n i=1 ai (1− st) 1 i∑n i=1 ai ∣∣F′′ (a1) ∣∣q + ∑n i=1 ai (1− s (1− t)) 1 i∑n i=1 ai ∣∣F′′ (x) ∣∣q }dt} 1 q + ς ϱ K +2 ∗ (b1, x) ς∗ (b1, a1) {∫ 1 0 t ( λ− t ϱ K ) {∑n i=1 ai (1− st) 1 i∑n i=1 ai ∣∣F′′ (b1) ∣∣q + ∑n i=1 ai (1− s (1− t)) 1 i∑n i=1 ai ∣∣F′′ (x) ∣∣q }dt} 1 q ] , where M (ϱ,K, λ) = ∫ 1 0 [t ( λ− t ϱ K ) ]qdt J. Nasir et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6159 18 of 26 = K λ K(1+q)+ϱq q ϱ [ Γ (1 + q) Γ ( K (1 + q) + ϱ ϱ ) 2F1 ( 1, 1 + q, 2 + q + K (1 + q) ϱ , 1 ) + β ( 1 + q,−K (1 + iq) + ϱq qϱ ) − β ( λ, 1 + iq,−K (1 + iq) + ϱq qϱ )] . (39) Proof. From Lemma 2 and a property of the GFPP−s function |F′′|q, and the power meaniinequality, one has ∣∣∣∣ (1− λ) [ ς ϱ K ∗ (b1, x)− ς ϱ K ∗ (x, a1) ς∗ (b1, a1) ] F′ (x) + ( 1 + ϱ K − λ )[ ς ϱ K ∗ (b1, x) + ς ϱ K ∗ (x, a1) ς∗ (b1, a1) ] F (x) + λ [ ς ϱ K ∗ (b1, x)F (b1) + ς ϱ K ∗ (x, a1)F (a1) ς∗ (b1, a1) ] − ΓK (ϱ+ 2K) ς∗ (b1, a1) { Jϱ,K (a1+ς∗(x,a1)) − F (a1) + Jϱ,K (b1+ς∗(x,b1)) + F (b1) }∣∣∣∣ ≤ ς ϱ K +2 ∗ (x, a1) ς∗ (b1, a1) ∫ 1 0 |t ( λ− t ϱ K ) ||F′′ (a1 + tς∗ (x, a1)) |dt + ς ϱ K +2 ∗ (b1, x) ς∗ (b1, a1) ∫ 1 0 |t ( λ− t ϱ K ) ||F′′ (b1 + tς∗ (x, b1)) |dt ≤ (∫ 1 0 |tq ( λ− t ϱ K )q dt ) 1 q × [ ς ϱ K +2 ∗ (x, a1) ς∗ (b1, a1) ∫ 1 0 t ( λ− t ϱ K ) × [∑n i=1 ai (1− st) 1 i∑n i=1 ai ∣∣F′′ (a1) ∣∣q + ∑n i=1 ai (1− s (1− t)) 1 i∑n i=1 ai ∣∣F′′ (x) ∣∣q dt] 1 q + ς ϱ K +2 ∗ (b1, x) ς∗ (b1, a1) ∫ 1 0 t ( λ− t ϱ K ) × [∑n i=1 ai (1− st) 1 i∑n i=1 ai ∣∣F′′ (b1) ∣∣q + ∑n i=1 ai (1− s (1− t)) 1 i∑n i=1 ai ∣∣F′′ (x) ∣∣q dt] 1 q ] ≤ M 1− 1 q (ϱ,K, λ) × [ ς ϱ K +2 ∗ (x, a1) ς∗ (b1, a1) {∫ 1 0 t ( λ− t ϱ K ){∑n i=1 ai (1− st) 1 i∑n i=1 ai ∣∣F′′ (a1) ∣∣q + ∑n i=1 ai (1− s (1− t)) 1 i∑n i=1 ai ∣∣F′′ (x) ∣∣q }dt} 1 q + ς ϱ K +2 ∗ (b1, x) ς∗ (b1, a1) {∫ 1 0 t ( λ− t ϱ K ) {∑n i=1 ai (1− st) 1 i∑n i=1 ai ∣∣F′′ (b1) ∣∣q J. Nasir et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6159 19 of 26 + ∑n i=1 ai (1− s (1− t)) 1 i∑n i=1 ai ∣∣F′′ (x) ∣∣q }dt} 1 q ] . Corollary 11. If one can take s = 1 in Theorem 7, then we have the following inequalities for a GFPP function with K−fractional integral operators:∣∣∣∣ (1− λ) [ ς ϱ K ∗ (b1, x)− ς ϱ K ∗ (x, a1) ς∗ (b1, a1) ] F′ (x) + ( 1 + ϱ K − λ )[ ς ϱ K ∗ (b1, x) + ς ϱ K ∗ (x, a1) ς∗ (b1, a1) ] F (x) + λ [ ς ϱ K ∗ (b1, x)F (b1) + ς ϱ K ∗ (x, a1)F (a1) ς∗ (b1, a1) ] − ΓK (ϱ+ 2K) ς∗ (b1, a1) { Jϱ,K (a1+ς∗(x,a1)) − F (a1) + Jϱ,K (b1+ς∗(x,b1)) + F (b1) }∣∣∣∣ ≤ M 1− 1 q (ϱ,K, λ) × [ ς ϱ K +2 ∗ (x, a1) ς∗ (b1, a1) {∫ 1 0 t ( λ− t ϱ K ){∑n i=1 ai (1− t) 1 i∑n i=1 ai ∣∣F′′ (a1) ∣∣q + ∑n i=1 ai (t) 1 i∑n i=1 ai ∣∣F′′ (x) ∣∣q }dt} 1 q + ς ϱ K +2 ∗ (b1, x) ς∗ (b1, a1) {∫ 1 0 t ( λ− t ϱ K ) {∑n i=1 ai (1− t) 1 i∑n i=1 ai ∣∣F′′ (b1) ∣∣q + ∑n i=1 ait 1 i∑n i=1 ai ∣∣F′′ (x) ∣∣q }dt} 1 q ] . Corollary 12. If one can take K = 1 in Corollary 11, then we have the following inequal- ities for GFPP function with RL−fractional integral operators:∣∣∣∣ (1− λ) [ ςϱ∗ (b1, x)− ςϱ∗ (x, a1) ς∗ (b1, a1) ] F′ (x) + (1 + ϱ− λ) [ ςϱ∗ (b1, x) + ςϱ∗ (x, a1) ς∗ (b1, a1) ] F (x) + λ [ ςϱ∗ (b1, x)F (b1) + ςϱ∗ (x, a1)F (a1) ς∗ (b1, a1) ] − Γ (ϱ+ 2) ς∗ (b1, a1) { Jϱ (a1+ς∗(x,a1)) − F (a1) + Jϱ (b1+ς∗(x,b1)) + F (b1) }∣∣∣∣ ≤ M 1− 1 q (ϱ, λ) × [ ςϱ+2 ∗ (x, a1) ς∗ (b1, a1) {∫ 1 0 t (λ− tϱ) {∑n i=1 ai (1− t) 1 i∑n i=1 ai ∣∣F′′ (a1) ∣∣q + ∑n i=1 ait 1 i∑n i=1 ai ∣∣F′′ (x) ∣∣q }dt} 1 q + ςϱ+2 ∗ (b1, x) ς∗ (b1, a1) {∫ 1 0 t (λ− tϱ) {∑n i=1 ai (1− t) 1 i∑n i=1 ai ∣∣F′′ (b1) ∣∣q J. Nasir et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6159 20 of 26 + ∑n i=1 ait 1 i∑n i=1 ai ∣∣F′′ (x) ∣∣q }dt} 1 q ] where M (ϱ, λ) = ∫ 1 0 [t (λ− tϱ)]qdt = λ (1+iq)+ϱq q ϱ [ Γ (1 + iq) Γ ( (1 + iq) + ϱ ϱ ) 2F1 ( 1, 1 + iq, 2 + q + (1 + q) ϱ , 1 ) + β ( 1 + iq,−(1 + iq) + ϱq qϱ ) − β ( λ, 1 + iq,−(1 + iq) + ϱq qϱ )] . (40) Theorem 8. Suppose F : X = [a1, a1 + ς∗ (b1, a1)] → ℜ is a twice differentiable function function on Xo such that F′′ ∈ L[a1, a1+ς∗ (b1, a1)] and consideration with U∗. Let for some q > 1, with 1 p + 1 q = 1, |F′′|q be a GFPP−s function on X, for all x ∈ [a1, a1 + ς∗ (b1, a1)]. Then ∣∣∣∣ (1− λ) [ ς ϱ K ∗ (b1, x)− ς ϱ K ∗ (x, a1) ς∗ (b1, a1) ] F′ (x) + ( 1 + ϱ K − λ )[ ς ϱ K ∗ (b1, x) + ς ϱ K ∗ (x, a1) ς∗ (b1, a1) ] F (x) + λ [ ς ϱ K ∗ (b1, x)F (b1) + ς ϱ K ∗ (x, a1)F (a1) ς∗ (b1, a1) ] − ΓK (ϱ+ 2K) ς∗ (b1, a1) { Jϱ,K (a1+ς∗(x,a1)) − F (a1) + Jϱ,K (b1+ς∗(x,b1)) + F (b1) }∣∣∣∣ ≤ M 1 p (ϱ,K, λ) × [ ς ϱ K +2 ∗ (x, a1) ς∗ (b1, a1) { 1∑n i=1 ai n∑ i=1 ai ∫ 1 0 { (1− st) 1 i ∣∣F′′ (a1) ∣∣q + (1− s (1− t)) 1 i ∣∣F′′ (x) ∣∣q }dt} 1 q + ς ϱ K +2 ∗ (x, b1) ς∗ (b1, a1) × { 1∑n i=1 ai n∑ i=1 ai ∫ 1 0 { (1− st) 1 i ∣∣F′′ (b1) ∣∣q + (1− s (1− t)) 1 i ∣∣F′′ (x) ∣∣q }dt} 1 q ] . (41) Proof. From Lemma 2 and a propertyiof the GFPP−s function |F′′|q, and the Hölderiinequality, one has ∣∣∣∣ (1− λ) [ ς ϱ K ∗ (b1, x)− ς ϱ K ∗ (x, a1) ς∗ (b1, a1) ] F′ (x) + ( 1 + ϱ K − λ )[ ς ϱ K ∗ (b1, x) + ς ϱ K ∗ (x, a1) ς∗ (b1, a1) ] F (x) + λ [ ς ϱ K ∗ (b1, x)F (b1) + ς ϱ K ∗ (x, a1)F (a1) ς∗ (b1, a1) ] J. Nasir et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6159 21 of 26 − ΓK (ϱ+ 2K) ς∗ (b1, a1) { Jϱ,K (a1+ς∗(x,a1)) − F (a1) + Jϱ,K (b1+ς∗(x,b1)) + F (b1) }∣∣∣∣ ≤ ς ϱ K +2 ∗ (x, a1) ς∗ (b1, a1) ∫ 1 0 |t ( λ− t ϱ K ) ||F′′ (a1 + tς∗ (x, a1)) |dt + ς ϱ K +2 ∗ (b1, x) ς∗ (b1, a1) ∫ 1 0 |t ( λ− t ϱ K ) ||F′′ (b1 + tς∗ (x, b1)) |dt (42) ≤ (∫ 1 0 ∣∣∣t(λ− t ϱ k )∣∣∣p dt) 1 p × [ ς ϱ k +2 ∗ (x, a1) ς∗ (b1, a1) {∫ 1 0 {∑n i=1 ai (1− st) 1 i∑n i=1 ai ∣∣F′′ (a1) ∣∣q + ∑n i=1 ai (1− s (1− t)) 1 i∑n i=1 ai ∣∣F′′ (x) ∣∣q }dt} 1 q + ς ϱ k +1 ∗ (x, b1) ς∗ (b1, a1) {∫ 1 0 { ∑n i=1 ai (1− st) 1 i∑n i=1 ai ∣∣F′′ (b1) ∣∣q + ∑n i=1 ai (1− s (1− t)) 1 i∑n i=1 ai ∣∣F′′ (x) ∣∣q}dt} 1 q ] (43) ≤ M 1 p (ϱ,K, λ) × [ ς ϱ K +2 ∗ (x, a1) ς∗ (b1, a1) { 1∑n i=1 ai n∑ i=1 ai ∫ 1 0 { (1− st) 1 i ∣∣F′′ (a1) ∣∣q + (1− s (1− t)) 1 i ∣∣F′′ (x) ∣∣q }dt} 1 q + ς ϱ K +2 ∗ (x, b1) ς∗ (b1, a1) × { 1∑n i=1 ai n∑ i=1 ai ∫ 1 0 { (1− st) 1 i ∣∣F′′ (b1) ∣∣q + (1− s (1− t)) 1 i ∣∣F′′ (x) ∣∣q }dt} 1 q ] . Corollary 13. If one can take s = 1 iniTheorem 8, then we have the following inequalities for GFPP function with K−fractional integral operators:∣∣∣∣ (1− λ) [ ς ϱ K ∗ (b1, x)− ς ϱ K ∗ (x, a1) ς∗ (b1, a1) ] F′ (x) + ( 1 + ϱ K − λ )[ ς ϱ K ∗ (b1, x) + ς ϱ K ∗ (x, a1) ς∗ (b1, a1) ] F (x) + λ [ ς ϱ K ∗ (b1, x)F (b1) + ς ϱ K ∗ (x, a1)F (a1) ς∗ (b1, a1) ] − ΓK (ϱ+ 2K) ς∗ (b1, a1) { Jϱ,K (a1+ς∗(x,a1)) − F (a1) + Jϱ,K (b1+ς∗(x,b1)) + F (b1) }∣∣∣∣ J. Nasir et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6159 22 of 26 ≤ M 1 p (ϱ,K, λ) × [ ς ϱ K +2 ∗ (x, a1) ς∗ (b1, a1) { 1∑n i=1 ai n∑ i=1 ai ∫ 1 0 { (1− t) 1 i ∣∣F′′ (a1) ∣∣q + t 1 i ∣∣F′′ (x) ∣∣q }dt} 1 q + ς ϱ K +2 ∗ (x, b1) ς∗ (b1, a1) { 1∑n i=1 ai n∑ i=1 ai ∫ 1 0 { (1− t) 1 i ∣∣F′′ (b1) ∣∣q + t 1 i ∣∣F′′ (x) ∣∣q }dt} 1 q ] . Corollary 14. If one can take K = 1 in Corollary 13, then we have the following inequal- ities for GFPP function with RL−fractional integral operators:∣∣∣∣ (1− λ) [ ςϱ∗ (b1, x)− ςϱ∗ (x, a1) ς∗ (b1, a1) ] F′ (x) + (1 + ϱ− λ) [ ςϱ∗ (b1, x) + ςϱ∗ (x, a1) ς∗ (b1, a1) ] F (x) + λ [ ςϱ∗ (b1, x)F (b1) + ςϱ∗ (x, a1)F (a1) ς∗ (b1, a1) ] − Γ (ϱ+ 2) ς∗ (b1, a1) { Jϱ (a1+ς∗(x,a1)) − F (a1) + Jϱ (b1+ς∗(x,b1)) + F (b1) }∣∣∣∣ ≤ M 1 p (ϱ, λ) × [ ςϱ+2 ∗ (x, a1) ς∗ (b1, a1) { 1∑n i=1 ai n∑ i=1 ai ∫ 1 0 { (1− t) 1 i ∣∣F′′ (a1) ∣∣q + t 1 i ∣∣F′′ (x) ∣∣q }dt} 1 q + ςϱ+2 ∗ (x, b1) ς∗ (b1, a1) { 1∑n i=1 ai n∑ i=1 ai ∫ 1 0 { (1− t) 1 i ∣∣F′′ (b1) ∣∣q + t 1 i ∣∣F′′ (x) ∣∣q }dt} 1 q ] . 3. Application to matrices Example: Denote by Pn the set of n× n compleximatrices, by Nn the algebra of n× n complex matrices, and by N+ n the strictly positiveimatrices in Nn. That is, D ∈ N+ n if ⟨Dx, x⟩ > 0 for all nonzero x ∈ iPn. In [27], Sababheh proved that the following mapping F(u) = ∥∥DuXB1−u +D1−uXBu ∥∥ , D, iB ∈ M+ n , iX ∈ Nn isiconvex for all u ∈ [0, 1]. Then by using Theorem 2, we have ∑n i=1 ai∑n i=1 ai ( 1− s 2 ) 1 i × ∥∥∥∥D( 2a1+ς∗(b1,a1) 2 ) XB 1− ( 2a1+ς∗(b1,a1) 2 ) +D 1− ( 2a1+ς∗(b1,a1) 2 ) XB ( 2a1+ς∗(b1,a1) 2 )∥∥∥∥ ≤ ΓK (ϱ+ K) ς ϱ K ∗ (b1, a1) J. Nasir et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6159 23 of 26 × { Jϱ,K a1+ ∥∥∥D(a1+ς∗(b1,a1))XB1−(a1+ς∗(b1,a1)) +D1−(a1+ς∗(b1,a1))XB(a1+ς∗(b1,a1)) ∥∥∥ + Jϱ,K (a1+ς∗(b1,a1)) − ∥∥∥D(a1)XB1−(a1) +D1−(a1)XB(a1) ∥∥∥} ≤ [ ∥∥∥D(a1)XB1−(a1) +D1−(a1)XB(a1) ∥∥∥ + ∥∥∥D(a1+ς∗(b1,a1))XB1−(a1+ς∗(b1,a1)) +D1−(a1+ς∗(b1,a1))XB(a1+ς∗(b1,a1)) ∥∥∥ ] × ∫ 1 0 t ϱ K −1 {∑n i=1 ai (1− st) 1 i∑n i=1 ai + ∑n i=1 ai (1− s (1− t)) 1 i∑n i=1 ai } dt.. 4. Applications with bivariates In this section, we recall the following special means of two positive numbers a1, b1 with a1 < b1 (see [28]): (i) The arithmetic mean A = A(a1, b1) = a1 + b1 2 . (ii) The harmonic mean H = H(a1, b1) = 2a1b1 a1 + b1 . The next relationship is well-knowniin the literature: H(a1, b1) ≤ G(a1, b1) ≤ A(a1, b1). Proposition 1. Suppose that 0 < a1 < b1 and s ∈ [0, 1], then∑n i=1 ai∑n i=1 ai ( 1− s 2 ) 1 i A(2a1, ς∗ (b1, a1)) ≤ ΓK (ϱ+ K) ς ϱ K ∗ (b1, a1) { Jϱ,K a1+ 2A (a1, ς∗ (b1, a1)) + Jϱ,K (a1+ς∗(b1,a1)) − (a1) } ≤ 2A (a1, a1 + ς∗ (b1, a1)) × ∫ 1 0 t ϱ K −1 {∑n i=1 ai (1− st) 1 i∑n i=1 ai + ∑n i=1 ai (1− s (1− t)) 1 i∑n i=1 ai } dt. (44) Proof. We attain the above inequality from Proposition 1 if we put F(u) = u for u > 0 in Theorem 2. J. Nasir et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6159 24 of 26 Proposition 2. Suppose that 0 < a1 < b1 and s ∈ [0, 1], then∑n i=1 ai∑n i=1 ai ( 1− s 2 ) 1 i A−1(2a1, ς∗ (b1, a1)) ≤ ΓK (ϱ+ K) ς ϱ K ∗ (b1, a1) { Jϱ,K a1+ 2A−1 (a1, ς∗ (b1, a1)) + Jϱ,K (a1+ς∗(b1,a1)) − ( a1 −1 )} ≤ 2H−1 (a1, a1 + ς∗ (b1, a1)) × ∫ 1 0 t ϱ K −1 {∑n i=1 ai (1− st) 1 i∑n i=1 ai + ∑n i=1 ai (1− s (1− t)) 1 i∑n i=1 ai } dt. (45) Proof. We attain the above inequality from Proposition 2 if we put F(u) = 1 u for u > 0 in Theorem 2. 5. Conclusion In this article, we introduced a novel fractional integral operator for functions, lever- aging the concept of polynomial n-fractional s-like preinvexity. The proposed operator extends the classical fractional calculus framework by incorporating polynomial and prein- vexity properties, offering a more versatile tool for analyzing functions with specific con- vexity and fractional characteristics. We established key properties of the new operator, including its convergence, boundedness, and applicability to various classes of functions. Furthermore, we demonstrated its utility in solving fractional differential equations and optimizing problems involving preinvex functions. The results presented herein not only generalize existing fractional integral operators but also open new avenues for research in fractional calculus and its applications in optimization, mathematical modeling, and applied sciences. Future work could explore the extension of this operator to higher di- mensions, its application in real-world problems, and its relationship with other fractional operators. Authors’ Contributions All authors contribute equally in this paper. Conflict of interest The authors declare that they have no conflict of interest. Acknowledgments The authors extend their appreciation to Umm Al-Qura University, Saudi Arabia for funding this research work through grant number: 25UQU4331214GSSR06 J. Nasir et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6159 25 of 26 Funding This Research work was funded by Umm Al-Qura University, Saudi Arabia under grant number: 25UQU4331214GSSR06. References [1] P. Agarwal, S. S. Dragomir, M. Jleli, and B. Samet. 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