Double Fractional Integral Inequalities for Co-ordinated Convex Functions
Abstract
2nd Kocaeli Science Congress (KOSC-2025), 19-21 November 2025, Kocaeli, TÜRKİYE https://fefkongre.kocaeli.edu.tr/en
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M31-1 2nd KOCAELI SCIENCE CONGRESS (KOSC-2025) Kocaeli University, Faculty of Arts and Sciences November 19-21, 2025, İzmit, Kocaeli, Türkiye Double Fractional Integral Inequalities for Co-ordinated Convex Functions Samet ERDEN1, Burçin Gökkurt Özdemir2 1Department of Mathematics,Faculty of Science, Bartın University, Bartın, Turkey 2Department of Mathematics and Science Education, Faculty of Education, Bartn University, Bartn, Turkey Corresponding author: [email protected] ORCID IDs: First Author: 0000-0001-8430-7533 Second Author: 0000-0002-1551-0113 DOI : 10.5281/zenodo.18032820 Abstract The primary objective of this study is to establish novel Ostrowski-type fractional integral inequalities for functions whose powers of the absolute values of partial derivatives are convex on the coordinates. By utilizing a fractional integral identity previously proposed in the literature, several generalized versions of these inequalities are derived. In addition, various special cases and related corollaries are provided to emphasize the importance and applicability of the obtained results. Keywords:Fractional integrals, Co-ordinated convex functions, Ostrowski type inequalities, Double integrals. 1. Introduction The study one of the most fundamental and influential integral inequalities in mathematical analysis is the celebrated Ostrowski inequality, established by Alexander Markovich Ostrowski [16] in 1938. In its classical form, this inequality provides an upper bound for the error between the value of a function at any point 𝑥∈[𝑎,𝑏] and its integral average over that interval. This bound is expressed in terms of the maximum norm of the function's first derivative, 𝑓′. The significance of this inequality stems from its utility as a powerful tool in various fields, particularly in error analysis for numerical integration and probability theory. The practical significance of the Ostrowski inequality is particularly evident in numerical analysis and the theory of special means. It is extensively applied to estimate the error bounds for various quadrature formulae, including the well-known midpoint, trapezoid, and Simpson rules. Firstly, we give the definitions of Riemann-Liouville fractional integrals:
M31-2 2nd Kocaeli Science Congress, November 19-21, 2025 Definition 1. [11] Let 𝑓∈𝐿[𝑎,𝑏]. The Riemann-Liouville integrals 𝐽 𝑓 and 𝐽 𝑓 of order 𝛼>0 with 𝑎≥0 are defined by 𝐽 𝑓(𝑥)= 1 𝛤(𝛼) (𝑥−𝑡)𝑓(𝑡)𝑑𝑡, 𝑥>𝑎 and 𝐽 𝑓(𝑥)= 1 𝛤(𝛼) (𝑡−𝑥)𝑓(𝑡)𝑑𝑡, 𝑥<𝑏 respectively. Here, 𝛤(𝛼) is the Gamma function and 𝐽 𝑓(𝑥)=𝐽 𝑓(𝑥)=𝑓(𝑥). An essential component of our investigation involves the Riemann-Liouville fractional integrals for functions of two variables, defined as follows. Definition 2. [19] Let 𝑓∈𝐿([𝑎,𝑏]×[𝑐,𝑑]). The Riemann-Liouville fractional integrals 𝐽, , ,𝐽, , ,𝐽, , and 𝐽, , are defined by 𝐽, , 𝑓(𝑥,𝑦)= 1 𝛤(𝛼)𝛤(𝛽) (𝑥−𝑡)(𝑦−𝑠)𝑓(𝑡,𝑠)𝑑𝑠𝑑𝑡,𝑥>𝑎,𝑦>𝑐, 𝐽, , 𝑓(𝑥,𝑦)= 1 𝛤(𝛼)𝛤(𝛽) (𝑥−𝑡)(𝑠−𝑦)𝑓(𝑡,𝑠)𝑑𝑠𝑑𝑡,𝑥>𝑎,𝑦<𝑑, 𝐽, , 𝑓(𝑥,𝑦)= 1 𝛤(𝛼)𝛤(𝛽) (𝑡−𝑥)(𝑦−𝑠)𝑓(𝑡,𝑠)𝑑𝑠𝑑𝑡,𝑥<𝑏,𝑦>𝑐, and 𝐽, , 𝑓(𝑥,𝑦)= 1 𝛤(𝛼)𝛤(𝛽) (𝑡−𝑥)(𝑠−𝑦)𝑓(𝑡,𝑠)𝑑𝑠𝑑𝑡,𝑥<𝑏,𝑦<𝑑. Finally, the concept of co-ordinates convex, which will be used in this article, will be mentioned. A formal definition for co-ordinated convex function may be stated as follows: Definition 3. A function 𝑓:𝛥→𝑅 will be called co-ordinated canvex on 𝛥 , for all 𝑡,𝑠∈[0,1] and (𝑥,𝑦),(𝑢,𝑣)∈𝛥 , if the following inequality holds: 𝑓(𝑡𝑥+ (1 −𝑡)𝑦,𝑠𝑢+ (1 −𝑠)𝑣) ≤𝑡𝑠𝑓(𝑥,𝑢)+𝑠(1−𝑡)𝑓(𝑦,𝑢)+𝑡(1−𝑠)𝑓(𝑥,𝑣)+(1−𝑡)(1−𝑠)𝑓(𝑦,𝑣). Clearly, every convex function is co-ordinated convex. Furthermore, there exist co-ordinated convex function which is not convex, (see, [2]). Following the presentation of the basic definitions and notions utilized throughout this work, the next step is to discuss the main studies in the literature concerning these topics. In the referenced works [3]- [5], Dragomir employed identities involving the sum of the rightand left-sided Riemann--Liouville
M31-3 2nd Kocaeli Science Congress, November 19-21, 2025 fractional integrals to establish a number of Ostrowski-type inequalities for functions belonging to different Lebesgue norm spaces, including those of bounded variation, Hölder continuous, Lipschitz, and absolutely continuous functions. Hermite-Hadamard inequality and Ostrowski inequality for fractional integrals of two variable functions are obtained in [13] and [19], respectively. Recently, Erden et al. [8] provided some Ostrowski type inequalities including Riemann-Liouville fractional integrals for functions in class of functions 𝐿,𝐿 and 𝐿, respectively. Also, Erden et al. [9] established new double fractional inequalities of Ostowski type for functions of bounded variaton with two variables. In addition to these foundational works, a wide range of research has focused on inequalities concerning functions that are convex on the coordinates or that involve Riemann--Liouville fractional integrals. For several recent results concerning Hermite-Hadamard's inequality for some convex function on the co-ordinates on a rectangle from the plane 𝑅 , we refer the reader to ([7], [10], [14], [17], [18], [21]-[25]). There are also several papers on fractional Ostrowski type inequalities for one or two variable functions, you can find some of them in the references ([1], [6], [9], [12], [15], [20], [26]). In the introduction, we review the fundamental definitions of the fractional integral operators under consideration, along with the notion of coordinated convexity, which form the basis of our analysis. The structure of the paper is as follows: Section 2 presents our main results. By applying a key fractional integral identity established in a previous work [8], we derive several new Ostrowski type inequalities including the definitions of Riemann-Liouville fractional integral for co-ordinated convex functions. In addition, a midpoint-type formulation of the main result is presented as a corollary. 2. Double Integral Inequalities Involving Riemann-Liouville Fractional Integrals The identity to be used in constructing this section is given below. This identity, which was previously employed by Erdem et al. in [8], will be utilized here to derive integral inequalities for co-ordinated convex functions. Lemma 1.[8] Let 𝑓 : 𝛥=:[𝑎,𝑏]×[𝑐,𝑑]→𝑅 be an absolutely continuous function such that the partial derivative of order 2 exists and is continuous on 𝛥 in 𝑅. Then, for any (𝑥,𝑦)∈𝛥, we have 1 𝛤(𝛼)𝛤(𝛽) 𝛺(𝑡,𝑠) 𝜕𝑓(𝜍,𝜏) 𝜕𝜍𝜕𝜏 𝑑𝜏𝑑𝜍𝑑𝑠𝑑𝑡 (1) =𝐽, , 𝑓(𝑏,𝑑)+𝐽, , 𝑓(𝑏,𝑐)+𝐽, , 𝑓(𝑎,𝑑)+𝐽, , 𝑓(𝑎,𝑐) −(𝑦−𝑐)+(𝑑−𝑦) 𝛤(𝛽+1)[𝐽 𝑓(𝑏,𝑦)+𝐽 𝑓(𝑎,𝑦)] −(𝑥−𝑎)+(𝑏−𝑥) 𝛤(𝛼+1)𝐽, 𝑓(𝑥,𝑑)+𝐽, 𝑓(𝑥,𝑐) +(𝑥−𝑎)+(𝑏−𝑥) 𝛤(𝛼+1)(𝑦−𝑐)+(𝑑−𝑦) 𝛤(𝛽+1)𝑓(𝑥,𝑦) =: 𝐺(𝑥,𝑦;𝑎,𝑏,𝑐,𝑑) where 𝛺(𝑡,𝑠) is defined by
M31-4 2nd Kocaeli Science Congress, November 19-21, 2025 𝛺(𝑡,𝑠) := ⎩ ⎪ ⎨ ⎪ ⎧ (𝑡−𝑎)(𝑠−𝑐), 𝑎≤𝑡<𝑥 and 𝑐≤𝑠<𝑦 (𝑡−𝑎)(𝑑−𝑠), 𝑎≤𝑡<𝑥 and 𝑦≤𝑠≤𝑑 (𝑏−𝑡)(𝑠−𝑐), 𝑥≤𝑡≤𝑏 and 𝑐≤𝑠<𝑦 (𝑏−𝑡)(𝑑−𝑠), 𝑥≤𝑡≤𝑏 and 𝑦≤𝑠≤𝑑. By employing the above identity, we derive several double integral inequalities in the framework of Riemann-Liouville fractional operators for functions whose powers are convex on the coordinates. Theorem 1. Let 𝑓:𝛥→𝑅 be an absolutely continuous function such that the partial derivative of order 2 exists and is continuous for all (𝑡,𝑠)∈𝛥 in 𝑅. If (,) is a co-ordinated convex function on 𝛥 for 𝑝,𝑞>1 with + =1, then we have the Riemann-Liouville fractional inequality |𝐺(𝑥,𝑦;𝑎,𝑏,𝑐,𝑑)| (2) ≤𝛤1+ 𝛤1+ 4 𝛤𝛼+1+ 𝛤𝛽+1+ ×(𝑥−𝑎)(𝑦−𝑐)𝑓(𝑎,𝑐)+𝑓(𝑎,𝑦)+𝑓(𝑥,𝑐)+𝑓(𝑥,𝑦) +(𝑥−𝑎)(𝑑−𝑦)𝑓(𝑎,𝑦)+𝑓(𝑎,𝑑)+𝑓(𝑥,𝑦)+𝑓(𝑥,𝑑) +(𝑏−𝑥)(𝑦−𝑐)𝑓(𝑥,𝑐)+𝑓(𝑥,𝑦)+𝑓(𝑏,𝑐)+𝑓(𝑏,𝑦) +(𝑏−𝑥)(𝑑−𝑦)𝑓(𝑥,𝑦)+𝑓(𝑥,𝑑)+𝑓(𝑏,𝑦)+𝑓(𝑏,𝑑) for all (𝑥,𝑦)∈𝛥 and 𝛼,𝛽>0. Proof. Taking absolute value of both sides of the equality (1), because of the definition of 𝛺(𝑡,𝑠), it follows that |𝐺(𝑥,𝑦;𝑎,𝑏,𝑐,𝑑)| (3) ≤1 𝛤(𝛼)𝛤(𝛽) (𝑡−𝑎)(𝑠−𝑐) 𝜕𝑓(𝜍,𝜏) 𝜕𝜍𝜕𝜏 𝑑𝜏𝑑𝜍𝑑𝑠𝑑𝑡 +1 𝛤(𝛼)𝛤(𝛽) (𝑡−𝑎)(𝑑−𝑠) 𝜕𝑓(𝜍,𝜏) 𝜕𝜍𝜕𝜏 𝑑𝜏𝑑𝜍𝑑𝑠𝑑𝑡 +1 𝛤(𝛼)𝛤(𝛽) (𝑏−𝑡)(𝑠−𝑐) 𝜕𝑓(𝜍,𝜏) 𝜕𝜍𝜕𝜏 𝑑𝜏𝑑𝜍𝑑𝑠𝑑𝑡 +1 𝛤(𝛼)𝛤(𝛽) (𝑏−𝑡)(𝑑−𝑠) 𝜕𝑓(𝜍,𝜏) 𝜕𝜍𝜕𝜏 𝑑𝜏𝑑𝜍𝑑𝑠𝑑𝑡. Since |𝑓(𝑡,𝑠)| is a convex function on the co-ordinates on (𝜍,𝜏)∈[𝑎,𝑥]×[𝑐,𝑦], we have
M31-5 2nd Kocaeli Science Congress, November 19-21, 2025 𝑓𝑥−𝜍 𝑥−𝑎𝑎+𝜍−𝑎 𝑥−𝑎𝑥,𝑦−𝜏 𝑦−𝑐𝑐+𝜏−𝑐 𝑦−𝑐𝑦 (4) ≤(𝑥−𝜍)(𝑦−𝜏) (𝑥−𝑎)(𝑦−𝑐)𝑓(𝑎,𝑐)+(𝑥−𝜍)(𝜏−𝑐) (𝑥−𝑎)(𝑦−𝑐)𝑓(𝑎,𝑦) +(𝜍−𝑎)(𝑦−𝜏) (𝑥−𝑎)(𝑦−𝑐)𝑓(𝑥,𝑐)+(𝜍−𝑎)(𝜏−𝑐) (𝑥−𝑎)(𝑦−𝑐)𝑓(𝑥,𝑦). By employing the above inequality (4), which was obtained for a function whose absolute value of the partial derivatives is convex on the coordinates, together with the use of Hölder's inequality, it is clear that 𝜕𝑓(𝜍,𝜏) 𝜕𝜍𝜕𝜏 𝑑𝜏𝑑𝜍≤ 𝜕𝑓(𝜍,𝜏) 𝜕𝜍𝜕𝜏 𝑑𝜏𝑑𝜍 ≤ 𝑑𝜏𝑑𝜍 𝜕𝑓(𝜍,𝜏) 𝜕𝜍𝜕𝜏 𝑑𝜏𝑑𝜍 ≤(𝑥−𝑡) (𝑦−𝑠) 𝜕𝑓(𝜍,𝜏) 𝜕𝜍𝜕𝜏 𝑑𝜏𝑑𝜍 ≤(𝑥−𝑡) (𝑦−𝑠) (𝑥−𝑎)(𝑦−𝑐) 4 ×𝑓(𝑎,𝑐)+𝑓(𝑎,𝑦)+𝑓(𝑥,𝑐)+𝑓(𝑥,𝑦) . Then, it is clear that 1 𝛤(𝛼)𝛤(𝛽) (𝑡−𝑎)(𝑠−𝑐) 𝜕𝑓(𝜍,𝜏) 𝜕𝜍𝜕𝜏 𝑑𝜏𝑑𝜍𝑑𝑠𝑑𝑡 (5) ≤𝑓(𝑎,𝑐)+𝑓(𝑎,𝑦)+𝑓(𝑥,𝑐)+𝑓(𝑥,𝑦) 𝛤(𝛼)𝛤(𝛽) ×(𝑥−𝑎)(𝑦−𝑐) 4 (𝑡−𝑎)(𝑠−𝑐)(𝑥−𝑡) (𝑦−𝑠) 𝑑𝑠𝑑𝑡 =𝑓(𝑎,𝑐)+𝑓(𝑎,𝑦)+𝑓(𝑥,𝑐)+𝑓(𝑥,𝑦) 𝛤(𝛼)𝛤(𝛽) (𝑥−𝑎)(𝑦−𝑐) 4 × (𝑡−𝑎)(𝑥−𝑡) 𝑑𝑡 (𝑠−𝑐)(𝑦−𝑠) 𝑑𝑠 To complete the proof, we must calculate two integrals in the right side of the result (5). Applying the change of the variable =𝑢 and 𝑑𝑡=(𝑥−𝑎)𝑑𝑢 for the first integral, it is found that (𝑡−𝑎)(𝑥−𝑡) 𝑑𝑡 = (𝑡−𝑎)(𝑥−𝑎−(𝑡−𝑎)) 𝑑𝑡 =(𝑥−𝑎) 𝑢(1−𝑢) 𝑑𝑢
M31-6 2nd Kocaeli Science Congress, November 19-21, 2025 =(𝑥−𝑎) 𝐵𝛼,1+1 𝑝. And similarly, we have (𝑠−𝑐)(𝑦−𝑠) 𝑑𝑠 =(𝑦−𝑐) 𝐵𝛽,1+1 𝑝 Then, we possess 1 𝛤(𝛼)𝛤(𝛽) (𝑡−𝑎)(𝑠−𝑐) 𝜕𝑓(𝜍,𝜏) 𝜕𝜍𝜕𝜏 𝑑𝜏𝑑𝜍𝑑𝑠𝑑𝑡 ≤(𝑥−𝑎) (𝑦−𝑐) 𝐵𝛼,1+ 𝐵𝛽,1+ 𝛤(𝛼)𝛤(𝛽) (𝑥−𝑎)(𝑦−𝑐) 4 ×𝑓(𝑎,𝑐)+𝑓(𝑎,𝑦)+𝑓(𝑥,𝑐)+𝑓(𝑥,𝑦) =𝛤1+ 𝛤1+ 4 𝛤𝛼+1+ 𝛤𝛽+1+ (𝑥−𝑎)(𝑦−𝑐) ×𝑓(𝑎,𝑐)+𝑓(𝑎,𝑦)+𝑓(𝑥,𝑐)+𝑓(𝑥,𝑦) Similarly, applying the change of the variable =𝑣 and 𝑑𝑡=(𝑥−𝑏)𝑑𝑣 for calculating the other integrals in the right hand side of the inequality (3), it is found that (𝑏−𝑡)(𝑡−𝑥) 𝑑𝑡 = (𝑏−𝑡)(𝑏−𝑥−(𝑏−𝑡)) 𝑑𝑡 =(𝑏−𝑥) 𝐵𝛼,1+1 𝑝. ,and one has (𝑑−𝑠)(𝑠−𝑦) 𝑑𝑠 =(𝑑−𝑦) 𝐵𝛽,1+1 𝑝 Then, since |𝑓(𝑡,𝑠)| is a convex function on the co-ordinates on (𝜍,𝜏)∈[𝑎,𝑥]×[𝑦,𝑑], one has 1 𝛤(𝛼)𝛤(𝛽) (𝑥−𝑡)(𝑠−𝑦) 𝜕𝑓(𝜍,𝜏) 𝜕𝜍𝜕𝜏 𝑑𝜏𝑑𝜍𝑑𝑠𝑑𝑡
M31-7 2nd Kocaeli Science Congress, November 19-21, 2025 ≤𝛤1+ 𝛤1+ 4 𝛤𝛼+1+ 𝛤𝛽+1+ (𝑥−𝑎)(𝑑−𝑦) ×𝑓(𝑎,𝑦)+𝑓(𝑎,𝑑)+𝑓(𝑥,𝑦)+𝑓(𝑥,𝑑) Since |𝑓(𝑡,𝑠)| is a convex function on the co-ordinates on (𝜍,𝜏)∈[𝑥,𝑏]×[𝑐,𝑦], we have 1 𝛤(𝛼)𝛤(𝛽) (𝑡−𝑥)(𝑦−𝑠) 𝜕𝑓(𝜍,𝜏) 𝜕𝜍𝜕𝜏 𝑑𝜏𝑑𝜍𝑑𝑠𝑑𝑡 ≤𝛤1+ 𝛤1+ 4 𝛤𝛼+1+ 𝛤𝛽+1+ (𝑏−𝑥)(𝑦−𝑐) ×𝑓(𝑥,𝑐)+𝑓(𝑥,𝑦)+𝑓(𝑏,𝑐)+𝑓(𝑏,𝑦) , and since |𝑓(𝑡,𝑠)| is a convex function on the co-ordinates on (𝜍,𝜏)∈[𝑥,𝑏]×[𝑦,𝑑], we have 1 𝛤(𝛼)𝛤(𝛽) (𝑡−𝑥)(𝑠−𝑦) 𝜕𝑓(𝜍,𝜏) 𝜕𝜍𝜕𝜏 𝑑𝜏𝑑𝜍𝑑𝑠𝑑𝑡 ≤𝛤1+ 𝛤1+ 4 𝛤𝛼+1+ 𝛤𝛽+1+ (𝑏−𝑥)(𝑑−𝑦) ×𝑓(𝑥,𝑦)+𝑓(𝑥,𝑑)+𝑓(𝑏,𝑦)+𝑓(𝑏,𝑑) . If the above four results are substituted in right hand side of (3) inequality, the desired inequality (2) can be attained. Corollary 1. If we choose 𝑥= and 𝑦= in (3), then we have the Midpoint type inequality 𝐽 , , 𝑓(𝑏,𝑑)+𝐽 , , 𝑓(𝑏,𝑐)+𝐽 , , 𝑓(𝑎,𝑑)+𝐽 , , 𝑓(𝑎,𝑐) −(𝑑−𝑐) 2𝛤(𝛽+1)𝐽 𝑓𝑏,𝑐+𝑑 2+𝐽 𝑓𝑎,𝑐+𝑑 2 −(𝑏−𝑎) 2𝛤(𝛼+1)𝐽 , 𝑓𝑎+𝑏 2,𝑑+𝐽 , 𝑓𝑎+𝑏 2,𝑐 +(𝑏−𝑎)(𝑑−𝑐) 2𝛤(𝛼+1)𝛤(𝛽+1)𝑓𝑎+𝑏 2,𝑐+𝑑 2 ≤(𝑏−𝑎)(𝑑−𝑐)𝛤1+ 𝛤1+ 2 𝛤𝛼+1+ 𝛤𝛽+1+ ×𝑓(𝑎,𝑐)+𝑓(𝑎,𝑦)+𝑓(𝑥,𝑐)+𝑓(𝑥,𝑦) +𝑓(𝑎,𝑦)+𝑓(𝑎,𝑑)+𝑓(𝑥,𝑦)+𝑓(𝑥,𝑑)
M31-8 2nd Kocaeli Science Congress, November 19-21, 2025 +𝑓(𝑥,𝑐)+𝑓(𝑥,𝑦)+𝑓(𝑏,𝑐)+𝑓(𝑏,𝑦) +𝑓(𝑥,𝑦)+𝑓(𝑥,𝑑)+𝑓(𝑏,𝑦)+𝑓(𝑏,𝑑) References [1] Anastassiou, G.,Hooshmandasl, M. R., Ghasemi A. & Moftakharzadeh, F. (2009). “Montgomery identities for fractional integrals and related fractional inequalities.” J. Inequal. in Pure and Appl. Math, 10(4), 2009, Art. 97, 6 pp. [2] Dragomir, S. S. (2001). “On Hadamard's inequality for convex functions on the co-ordinates in a rectangle from the plane.” Taiwanese Journal of Mathematics, 4, 775-788. [3] Dragomir, S.S. (2017). “Ostrowski Type inequalities for riemann-Liouville fractional integrals of absolutely continuous functions in terms of ∞ − norms.” RGMIA Research Report Collection, 20 (2017), Article 49. [4] Dragomir, S.S. (2017). “Ostrowski Type inequalities for riemann-Liouville fractional integrals of absolutely continuous functions in terms of p− norms.” RGMIA Research Report Collection, 20 (2017), Article 50. [5] Dragomir, S.S.,Barmett N.S. & Cerone, P. (2003). “An Ostrowski type inequality for double integrals in terms of Lp-norms and applications in numerical integration.” Anal. Num. Theor. Approx., 32 (2), 161-169. [6] Dragomir, S. S. (2017). “On some Ostrowski type inequalities for generalized Riemann--Liouville fractional integrals.” RGMIA Res. Rep. Coll., 20, Art 67, pp. 13. [7] Erden,S. & Sarikaya, M. Z. (2017). “On the Hermite-Hadamard type and Ostrowski type inequalities for the co-ordinated convex functions, Palestine Journal of Mathematics.” 6(1), 257-270. [8] Erden, S., Budak, H., Sarikaya, M. Z., Iftikhar, S. & Kumam, P. (2020). “Fractional Ostrowski type inequalities for bounded functions.” Journal of Inequalities and Applications, 123, 1-11. [9] Erden, S., Budak, H., & Sarikaya, M. Z. (2020). “Fractional Ostrowski type inequalities for functions of bounded variaton with two variables.” Miskolc Mathematical Notes, 21(1), 171-188. [10] Erden, S., & Sarıkaya, M. Z. (2024). New weighted inequalities for functions whose higher-order partial derivatives are co-ordinated convex. Fundamental Journal of Mathematics and Applications, 7(2), 77-86. [11] Kilbas, A. A.,Srivastava H. M. & Trujillo, J. J. (2006).“Theory and Applications of Fractional Differential Equations.” North-Holland Mathematics Studies, 204, Elsevier Sci. B.V., Amsterdam, 2006. [12] Lakoud, A. G.& Aissaoui, F. (2013). “New fractional inequalities of Ostrowski type.” Transylv. J. Math. Mech., 5 (2), 103-106. [13] Latif, M. A.,Dragomir S. S. & Matouk, A. E. (2012). “New inequalities of Ostrowski type for coordinated convex functions via fractional integrals.” J. Fract. Calc. Appl, 2(1), 2012.
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