A modified class of Ostrowski-type inequalities and error bounds of Hermite–Hadamard inequalities

This paper aims to extend the application of the Ostrowski inequality, a crucial tool for figuring out the error bounds of various numerical quadrature rules, including Simpson’s, trapezoidal, and midpoint rules. Specifically, we develop a more comprehensive class of Ostrowski-type inequalities by utilizing the weighted version of Riemann–Liouville (RL) fractional integrals on an increasing function. We apply our findings to estimate the error bounds of Hadamard-type inequalities. Our results are more comprehensive, since we obtain the results of the existing literatures as particular cases for certain parameter values. This research motivates researchers to apply this concept to other fractional operators.


Introduction
Fractional calculus has a rich academic history and represents a natural extension of traditional calculus.Recent advances in fractional calculus can be found in mathematical physics, biology, chemistry, engineering, signal processing, fluid mechanics, viscoelasticity, mathematical biology, and electrochemistry [1][2][3].The well-known RL fractional operators include single kernels and are used to examine and evaluate memory effect phenomena in mathematical physics [4].Fractional calculus operators with various types of kernels are important for generalizing classical mathematical inequalities.The kernels involved in other mathematical conceptions serve a critical role in their existence and applications [5,6].Our findings can encompass a wide range of fractional operators as the kernel includes a strictly increasing function.This allows us to extend and unify numerous previously published results in the literature.A variety of fractional integral operators that reduce to the traditional RL fractional integral operator have been developed [7][8][9][10][11].Weighted fractional integral operators are revealed to be bound in the Lebesgue space, and various classical fractional integral and differential operators are found as special instances [12].Fractional operators are used to generalize integrals, derivatives, and in particular integrals involving inequalities [13].We generalize the inequalities for a family of n positive functions by employing the (k, ς)-fractional integral operator [14,15].
Inequalities are a modern mathematical analysis model that depicts the rate of advancement in the mathematical analysis competition [16].The study of fractional partial and ordinary differential equations can greatly benefit from Hermite-Hadamard inequalities that involve fractional integral operators [17,18].Fractional integral inequalities have shown to be one of the most powerful and broad-reaching tools for advancing many fields of pure and applied mathematics [19,20].Due to their distinctive applications in numerical quadrature, transform theory, probability, and statistical issues, these inequalities have gained tremendous prominence and relevance during the last few decades [21,22].The Ostrowski inequality holds significant value and utility in the fields of mathematical analysis, numerical analysis, and engineering [23].It provides an estimate of the integral mean of a function [24][25][26].Such inequalities are used to derive explicit limits for the perturbed trapezoidal, midpoint, Simpson, Newton-Cotes, and left and right rectangle rules.These are also applied to various composite quadrature rules, and the analysis allows for the calculation of the partition necessary for the accuracy of the result to be within a certain error tolerance [27].It also specifies the error boundaries of specific mean relations and numerous numerical quadrature rules of integration [28].
The Ostrowski inequality was first proposed by Ostrowski [29] in 1938 and can be expressed as follows.
The Hadamard inequality provides an interesting viewpoint on convex functions in the Cartesian plane as in the following statement.Theorem 1.2 Suppose that γ is positive convex function defined on a real interval J. Then we have the following inequality for λ 1 , λ 2 ∈ J such that λ 1 < λ 2 : To begin, let us review some definitions and notions.The gamma and k-gamma functions can be represented as integrals, as outlined in [10].Specifically, these functions are defined as follows.
Definition 1.3 If (μ) > 0, then the k-gamma function is defined as with the property Setting k = 1 yields the classical gamma function.
Let us revisit the definitions of RL fractional integrals and their general forms [30].The left-and right-sided RL k-fractional integrals are defined in [10].The idea of RL fractional integrals is extended by the (k, ς)-RL fractional integrals described in [9].Definition 1. 4 The right and left RL fractional integrals of order μ > 0 for a continuous function γ 1 on the finite real interval [λ 1 , λ 2 ] is given by Definition 1.6 Let γ 1 be a continuous function on [λ 1 , λ 2 ].Then the right and left (k, ς)-RL fractional integrals of order μ, k > 0 are defined as and where ς ∈ R \ {-1}.
An extension of RL fractional integrals for an increasing function is presented in [31], expressed in the following definition.Definition 1.8 Let γ 1 be a continuous function on [λ 1 , λ 2 ], let γ 2 be a strictly increasing function, and let σ be a nonzero increasing weight function.Then the right and left generalized weighted RL fractional integrals of order μ > 0 are defined as and where σ (θ ) = 0.
Reference [32] presents the Hadamard inequality in fractional form by utilizing RL fractional integrals.The following theorem presents a variant of the fractional Hadamard inequality for RL fractional integrals, as defined in Definition 1.4.

Theorem 1.9 Let γ be a convex function on
. Then the RL fractional integrals obey the following inequality: By utilizing generalized RL fractional integrals Farid et al. [33] achieved a generalization of the Hadamard inequality.The following theorem presents the generalized version of the Hadamard inequality for generalized RL fractional integrals defined in Definition 1.5.

Theorem 1.10 Let γ be a convex function on
. Then the RL k-fractional integrals satisfy the following inequality: Reference [34] illustrates the fractional version of the aforementioned fractional Hadamard inequality as follows.

Theorem 1.11 Let γ be a convex function on
. Then the RL fractional integrals satisfy the following inequality: Reference [33] illustrates the k-fractional version of the aforementioned fractional Hadamard inequality as follows.

Theorem 1.12 Let γ be a convex function on
. Then the RL k-fractional integrals satisfy the following inequality: The Ostrowski-type inequalities for RL fractional integrals are reported in [35].Inspired by the research discussed earlier, we intend to propose a new class of Ostrowski-type inequalities.Our approach involves the utilization of generalized weighted (k, ς)-RL fractional operators.

Weighted fractional integral inequalities via generalized fractional operator
In this section, we proof the Ostrowski-type inequalities by using weighted (k, ς)-RL fractional integral operator with respect to an increasing function.
Next, we present a more extended fractional Ostrowski-type inequality for the weighted (k, ς)-RL fractional integrals operators.

General forms of weighted fractional integrals inequalities
This section is devoted to presenting general forms of the outcomes from the previous section.In this section, Theorem 2.3 takes the following specific form.
Theorem 3.1 Under the assumptions of Theorem 2.3, we have The notation id[•, •] represents the identity function on the interval Proof The result is proved in the same way as Theorem 2.3.
In particular, Theorem 2.10 is transformed into the following form.
Theorem 3.2 Assuming that the conditions of Theorem 2.10 are satisfied, we have

The notation id[•, •] represents the identity function on the interval [•, •].
Proof The result is proved in the same way as Theorem 2.10.
In particular, Theorem 2.17 takes the following form.
Theorem 3.3 Under the assumptions of Theorem 2.17, we have Proof The result is proved in the same way as Theorem 2.17.
In particular, Theorem 2.23 takes the following form.
Theorem 3.4 Under the assumptions of Theorem 2.23, we have Proof The result is proved in the same way as Theorem 2.23.
The results presented in this section also provide the fractional integral inequalities for weighted RL fractional integrals, specifically, when the function γ 2 behaves like the identity function.

Applications to main results
This section pertains to the applications of our main results.The first application is presented for Theorem 2.3.
Theorem 4.1 Under the conditions of Theorem 2.3, we have

Corollary 4.2
Under the conditions of Theorem 2.3, for μ = ν in (4.1), we obtain Next, we derive an approximation for the Hadamard inequality for RL k-fractional integrals given in [37, Theorem 2.1].

Conclusions
Inequalities are a crucial concept in mathematics that is used extensively in various fields of study.It allows us to compare and contrast the relative values of different mathematical expressions, leading to a deeper understanding of the relationships between them.Inequalities are not only essential for theoretical purposes, but also for practical applications, such as optimization problems and statistical data analysis.Understanding inequalities is a critical component of mathematical literacy, enabling individuals to evaluate and interpret quantitative information and make informed decisions in various aspects of their lives.The presented work includes generalized fractional integral inequalities for the family of generalized weighted RL fractional integrals.These operators have been extensively studied and utilized by researchers across different fields.We specifically investigate the weighted RL k-fractional integral operators and extend the established in the direction of weighted version.Our study provides a simple method for proving Östrowski-type inequalities using the weighted fractional integral operators.We explored a more comprehensive form of the Östrowski-type inequalities that is more inclusive than the current ones in the literature.These inequalities have various applications in numerical analysis, specifically in numerical integration.Furthermore, we determine the best possible error bounds for Hadamard-type inequalities.Our results, which are related to the existing literature, are obtained through the application of Theorems 2.3, 2.10, 2.17, and 2.23.These findings have significant implications in establishing error bounds for Hadamard inequalities in fractional calculus.