Some New Fractional Boole’s Inequalities with Machine Learning Validations

Abstract

The present work proposes a new class of weighted Boole-type fractional inequalities formulated by integrating the Riemann–Liouville $\left(\mathcal{RL}\right)$ fractional integral operators with symmetric weight functions. Unlike the classical Boole error estimate which relies on the boundedness of the sixth derivative, the present approach is based on a newly derived weighted fractional identity, which is the key analytical tool of the study. Based on this identity, a family of inequalities is established for several particular classes of functions: convex functions, s-convex functions, bounded functions, Lipschitz continuous functions and bounded-variation functions, which provides meaningful error bounds under substantially weaker smoothness hypotheses. The flexibility of the proposed framework is illustrated by providing some examples of the proposed inequality limit cases, showing that the derived inequalities unify and extend the existing ones in the literature. Also, a few interesting applications to special means and a composite quadrature rule are presented. To test the inequality, we built a computational framework and generated a structured dataset of 5400 samples using six convex functions with polynomial, exponential, and hyperbolic growth. For all samples generated, a value between the left-hand-side ($\mathrm{LHS}$) and right-hand-side ($\mathrm{RHS}$) is numerically verified and the slack variable is numerically kept non-negative in the whole computational domain. The good consistency obtained between data-driven predictions and the analytical results confirms the applicability of the proposed framework and provides a logical bridge between the fractional inequality theory, the numerical approximation, and the explainable artificial intelligence.

1. Introduction

The theory of inequalities plays an important role in studying a diverse range of problems in nonlinear analysis, including differential equations, optimization, operational research and numerical error analysis. Based on their significance, researchers have extensively studied and extended them for higher dimensions and function classes to eliminate the shortcomings of classical counterparts. In the exponential growth of inequalities, the role of convexity is unparalleled. A central theme of this theory is the Hermite–Hadamard inequality, which is closely related to convex analysis and numerical integration. Many refinements and extensions of it have been achieved under various convexity and differentiability assumptions; see [1,2,3].

Integral inequalities provide fundamental tools for deriving analytical estimates and approximation errors. In the following perspective, Dragomir et al. [4] introduced Simpson-type inequalities under the bounded-variation and Lipschitz conditions. Later on, Shuang and his coauthors obtained three-point inequalities for $(s,m)$ convex functions in [5]. Cheng obtained a class of Ostrowski–Grüss-type estimates in [6]. Liu derived the error bounds for higher order differentiable functions for three-point closed formulas in [7]. Acu et al. [8] and Yang and Tseng [9] further contributed to the analysis of the numerical integration methods and approximation inequalities, respectively, while Delavar et al. [10] delivered the weighted Simpson-type inequalities under convexity assumptions. Alomari [11] studied approximation inequalities for the Milne quadrature formula. Javed et al. [12,13] studied classical approximation inequalities along with their applications based on a generic identity and upper bounds for the remainder term of Boole’s quadrature rule.

Fractional calculus is one of the most active fields of mathematical analysis, and involves the generalization of the classical notions of differentiation and integration to real (and complex) orders, and a natural language for describing memory-dependent and hereditary phenomena occurring in all fields of physics, engineering and applied sciences [14]. One of the important directions where this theory has proven fruitful is the study of integral inequalities which are used in the estimation of integrals, in the control of approximation errors and in the analysis of the solution of differential and integral equations.

The fractional Chebyshev inequalities were studied in [15] by Belarbi and Dahmani and the fractional framework was extended to Minkowski-type inequalities in [16]. The Hadamard inequality and its right estimates using $\left(\mathcal{RL}\right)$ fractional and generalized operators were discussed in [17,18]. Later on, Mohsen et al. [19] extended the left Hadamard inequality for $\sigma$-fractional operators. Mateen et al. [20] applied the fractional concepts to explore Weddle-type inequalities. The incorporation of weight functions into fractional inequalities enables the approximation of weighted fractional integrals. Almoneef et al. [21] obtained weighted Newton-type inequalities for various classes of functions via $\mathcal{RL}$ integrals, while the weighted fractional Euler–Maclaurin inequalities for convex and bounded-variation functions were examined in [22,23].

In parallel, powerful computational tools have emerged for the validation of analysis-based theoretical estimates: neural networks approximate highly nonlinear relationships present in fractional operators [24], deep learning frameworks solve forward and inverse problems for differential and integral equations [25], and fractional physics-informed neural networks embed fractional operators directly into their learning architecture [26]. Recently, an artificial neural network was used to predict the left, middle and right sides of fractional Bullen-type inequalities with strong agreement against analytical results [27]. Symbolic Regression [28] and extreme gradient boosting (XGBoost) [29] have proven highly effective for learning strong nonlinearities in tabular regression, together creating a fertile link between analytical mathematics and computational intelligence.

Motivated by recent contributions in weighted inequalities, the present work introduces a new class of weighted fractional Boole-type inequalities involving the $\mathcal{RL}$ fractional integral operator and symmetric weight functions. To derive the main inequalities, a weighted fractional identity that establishes a natural correspondence between classical Boole-type quadrature structures and fractional integral operators will be developed. Under assumptions of s-convexity, boundedness, Lipschitz continuity and bounded variation, several inequalities are derived that provide meaningful error bounds under weaker hypotheses than those required by classical Boole estimates, where a sixth derivative regularity is assumed. The fractional parameter $\mu \in (0,1]$ and symmetric weight function endow important flexibility, recover several known inequalities as special cases, and unify existing results. Also, we will discuss some interesting special cases for different choices of weights relating to renowned classes of integrals. In parallel with these analytical results, a theorem-consistent computational framework is developed over 5400 samples using Symbolic Regression, XGBoost and SHAP (SHapley Additive exPlanations) analysis to investigate the contribution of input features that predicts exactly that the $\mathrm{LHS}$ is less than the $\mathrm{RHS}$ of the fractional inequality.

Our proposed results are novel due to their generic nature. They unify the fractional and classical inequalities for a diverse range of function classes using symmetric weight functions. Moreover, for $s=1$, we get weighted Boole-type inequalities under convexity assumptions, which are also new in the literature. The weighted fractional inequalities enable us to discuss the error analysis of Boole’s rule for special kinds of integrals like logarithm-, Jacobi-, and Chebyshev-type integrals. The computational analysis via machine learning techniques is another fascinating part. It focuses on the validation and analysis of all contributors utilizing the technique of SHAP. This is the first study where the comparison of different machine learning techniques for inequalities has been presented.

The current study is structured as follows: Section 2 contains essential notions, classical inequalities, and fractional integral operators required for the derivation of the main results; Section 3 offers the primary outcomes with numerical examples and applications; Section 5 provides the computational analysis via two machine learning techniques; while Section 6 concludes the study and indicates directions for future research.

2. Preliminaries

In the following sections the basic definitions, classical inequalities and fractional integral operators will be gathered as a basis for the subsequent analysis.

Definition 1 

([1]). A function $\mathsf{\Psi }:[c_1,z_2]\to \mathbb{R}$ is said to be convex if

$$\mathsf{\Psi }(\xi c_1+(1-\xi )z_2)\le \xi \mathsf{\Psi }\left(c_1\right)+(1-\xi )\mathsf{\Psi }\left(z_2\right),forany\xi \in [0,1].$$

The theory of integral inequalities is closely related to the concept of convexity which offers the geometric framework of numerous classical estimates. One of the most important results in this direction is the Hermite–Hadamard inequality, which gives two-sided bounds for the integral mean of a convex function in terms of its values at the midpoint and the endpoints and it is stated as shown below.

Theorem 1 

([1]). For a convex function $\mathsf{\Psi }:[c_1,z_2]\to \mathbb{R}$, the following inequality holds

$$\mathsf{\Psi }\left({\displaystyle \frac{c_1+z_2}{2}}\right)\le {\displaystyle \frac{1}{z_2-c_1}}{\int }_{c_1}^{z_2}\mathsf{\Psi }\left(x\right)dx\le {\displaystyle \frac{\mathsf{\Psi }\left(c_1\right)+\mathsf{\Psi }\left(z_2\right)}{2}}.$$

Next, we provide the definition of s-convex functions.

Definition 2 

([1]). A function $\mathsf{\Psi }:[c_1,z_2]\to \mathbb{R}$ is said to be s-convex if

$$\mathsf{\Psi }(\xi c_1+(1-\xi )z_2)\le {\xi }^s\mathsf{\Psi }\left(c_1\right)+{(1-\xi )}^s\mathsf{\Psi }\left(z_2\right),\forall \xi \in [0,1],ands\in (0,1].$$

The Euler Gamma function is defined as follows.

Definition 3. 

For $\mu >0$ the Euler Gamma function is defined as

$$\mathsf{\Gamma }\left(\mu \right)={\int }_0^{\infty }{\xi }^{\mu -1}{e}^{-\xi }d\xi .$$

The left-sided and right-sided $\mathcal{RL}$ fractional integrals are defined in [14] as follows.

Definition 4. 

Let $\mathsf{\Psi }\in L^1[c_1,z_2]$ and let $\mu >0$. The left-sided and right-sided $\mathcal{RL}$ fractional integrals of Ψ are defined by

$$J_{c_1+}^{\mu }\mathsf{\Psi }\left(z_2\right)={\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mu \right)}}{\int }_{c_1}^{z_2}{(z_2-\xi )}^{\mu -1}\mathsf{\Psi }\left(\xi \right)d\xi ,$$

and

$$J_{z_2-}^{\mu }\mathsf{\Psi }\left(c_1\right)={\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mu \right)}}{\int }_{c_1}^{z_2}{(\xi -c_1)}^{\mu -1}\mathsf{\Psi }\left(\xi \right)d\xi .$$

Boole’s rule is a particular five-point Newton–Cotes quadrature formula which is exact for all polynomials of a degree of at most 4. It is given by

$${\int }_{c_1}^{z_2}\mathsf{\Psi }\left(\xi \right)d\xi \approx {\displaystyle \frac{z_2-c_1}{90}}\left[7\mathsf{\Psi }\left(c_1\right)+32\mathsf{\Psi }\left({\displaystyle \frac{3c_1+z_2}{4}}\right)+12\mathsf{\Psi }\left({\displaystyle \frac{c_1+z_2}{2}}\right)+32\mathsf{\Psi }\left({\displaystyle \frac{c_1+3z_2}{4}}\right)+7\mathsf{\Psi }\left(z_2\right)\right].$$

The classical error estimate was introduced by Philip J. Davis and Philip Rabinowitz in Methods of Numerical Integration [30] and is stated as follows.

Theorem 2. 

Let $\mathsf{\Psi }:[c_1,z_2]\to \mathbb{R}$ be a six-times continuously differentiable function on $(c_1,z_2)$ such that

$${\parallel {\mathsf{\Psi }}^{\left(6\right)}\parallel }_{\infty }=\underset{x\in (c_1,z_2)}{\sup }\left|{\mathsf{\Psi }}^{\left(6\right)}\left(x\right)\right|<\infty .$$

Then

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{90}}\left[7\mathsf{\Psi }\left(c_1\right)+32\mathsf{\Psi }\left({\displaystyle \frac{3c_1+z_2}{4}}\right)+12\mathsf{\Psi }\left({\displaystyle \frac{c_1+z_2}{2}}\right)+32\mathsf{\Psi }\left({\displaystyle \frac{c_1+3z_2}{4}}\right)+7\mathsf{\Psi }\left(z_2\right)\right]-{\displaystyle \frac{1}{z_2-c_1}}{\int }_{c_1}^{z_2}\mathsf{\Psi }\left(x\right)dx\right|\hfill \\ \hfill & \le {\displaystyle \frac{{(z_2-c_1)}^6}{\mathrm{1,935,360}}}{\parallel {\mathsf{\Psi }}^{\left(6\right)}\parallel }_{\infty }.\hfill \end{array}$$

This estimate indicates that the classical form of Boole’s rule is highly accurate, and shows that the classical form is significantly linked to strong smoothness assumptions. As a consequence, numerous subsequent investigations have tried to obtain Boole-type inequalities under weaker smoothness assumptions. These extensions can be very useful when the function is not six-times differentiable.

The second result is a Boole-type inequality where convexity conditions have been imposed on the absolute value of the first derivative.

Theorem 3 

([13]). Let $\mathsf{\Psi }:[c_1,z_2]\to \mathbb{R}$ be a continuously differentiable function. If $|{\mathsf{\Psi }}^{\prime }|$ is convex on $[c_1,z_2]$, then

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{90}}\left[7\mathsf{\Psi }\left(c_1\right)+32\mathsf{\Psi }\left({\displaystyle \frac{3c_1+z_2}{4}}\right)+12\mathsf{\Psi }\left({\displaystyle \frac{c_1+z_2}{2}}\right)+32\mathsf{\Psi }\left({\displaystyle \frac{c_1+3z_2}{4}}\right)+7\mathsf{\Psi }\left(z_2\right)\right]-{\displaystyle \frac{1}{z_2-c_1}}{\int }_{c_1}^{z_2}\mathsf{\Psi }\left(x\right)dx\right|\hfill \\ \hfill & \le {\displaystyle \frac{239(z_2-c_1)}{6480}}\left[|{\mathsf{\Psi }}^{\prime }\left(c_1\right)|+|{\mathsf{\Psi }}^{\prime }\left(z_2\right)|\right].\hfill \end{array}$$

3. Results and Discussion

The subsequent section is devoted to introducing a new fractional identity which will be the basic analytical tool on which the proposed Boole-type inequalities will be developed.

3.1. Fractional Auxiliary Result

Let us assume that $\omega :[c_1,z_2]\to \mathbb{R}$ is a positive integrable function and symmetric with respect to ${\displaystyle \frac{c_1+z_2}{2}}$, i.e., $\omega \left(c\right)=\omega (c_1+z_2-c)$ for all $c\in [c_1,z_2]$. Let us now give a definition of the map $\mathcal{W}$:

$$\begin{array}{c}\mathcal{W}\left(\mu \right)={\int }_0^1{\xi }^{\mu -1}\omega \left(\xi z_2+(1-\xi )c_1\right)d\xi .\hfill \end{array}$$

Since the function $\omega$ is symmetric with respect to ${\displaystyle \frac{c_1+z_2}{2}}$, we can write

$$\begin{array}{c}\mathcal{W}\left(\mu \right)={\displaystyle \frac{\mathsf{\Gamma }\left(\mu \right)}{{(z_2-c_1)}^{\mu }}}{J}_{{z_2}^{-}}^{\mu }\omega \left(c_1\right)={\displaystyle \frac{\mathsf{\Gamma }\left(\mu \right)}{{(z_2-c_1)}^{\mu }}}{J}_{{c_1}^{+}}^{\mu }\omega \left(z_2\right)={\displaystyle \frac{\mathsf{\Gamma }\left(\mu \right)}{2{(z_2-c_1)}^{\mu }}}\left[J_{{c_1}^{+}}^{\mu }\omega \left(z_2\right)+J_{{z_2}^{-}}^{\mu }\omega \left(c_1\right)\right].\hfill \end{array}$$

Particularly, we have

$$\begin{array}{c}\mathcal{W}\left(1\right)={\displaystyle \frac{1}{z_2-c_1}}{\int }_{c_1}^{z_2}\omega \left(\xi \right)d\xi .\hfill \end{array}$$

We now prove a new weighted fractional equation, which will serve as the central result for developing the weighted Boole-type inequalities.

Lemma 1. 

Consider $\mathsf{\Psi }:[c_1,z_2]\to \mathbb{R}$ to be a differentiable function on $(c_1,z_2)$, where $c_1<z_2$ and ${\mathsf{\Psi }}^{\prime }\in L[c_1,z_2]$. Then,

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{90}}\left[7\mathsf{\Psi }\left(c_1\right)+32\mathsf{\Psi }\left({\displaystyle \frac{3c_1+z_2}{4}}\right)+12\mathsf{\Psi }\left({\displaystyle \frac{c_1+z_2}{2}}\right)+32\mathsf{\Psi }\left({\displaystyle \frac{c_1+3z_2}{4}}\right)+7\mathsf{\Psi }\left(z_2\right)\right]\mathcal{W}\left(\mu \right)\hfill \\ & -{\displaystyle \frac{2^{\mu -1}\mathsf{\Gamma }\left(\mu \right)}{{(z_2-c_1)}^{\mu }}}\left[J_{{c_1}^{+}}^{\mu }\mathsf{\Psi }\omega \left({\displaystyle \frac{c_1+z_2}{2}}\right)+J_{{z_2}^{-}}^{\mu }\mathsf{\Psi }\omega \left({\displaystyle \frac{c_1+z_2}{2}}\right)\right]\hfill \\ \hfill & ={\displaystyle \frac{z_2-c_1}{4}}{\displaystyle \sum _{k=1}^2}{\chi }_k,\hfill \end{array}$$

(1)

where,

$$\begin{array}{c}\left\{\begin{array}{cc}\hfill & {\displaystyle {\chi }_1={\int }_0^{\frac{1}{2}}{r}_1(\mu ,\xi )\left[{\mathsf{\Psi }}^{\prime }\left(\frac{1-\xi }{2}{c}_1+\frac{1+\xi }{2}{z}_2\right)-{\mathsf{\Psi }}^{\prime }\left(\frac{1+\xi }{2}{c}_1+\frac{1-\xi }{2}{z}_2\right)\right]\mathrm{d}\xi ,}\hfill \\ & {\displaystyle {\chi }_2={\int }_{\frac{1}{2}}^1{r}_2(\mu ,\xi )\left[{\mathsf{\Psi }}^{\prime }\left(\frac{1-\xi }{2}{c}_1+\frac{1+\xi }{2}{z}_2\right)-{\mathsf{\Psi }}^{\prime }\left(\frac{1+\xi }{2}{c}_1+\frac{1-\xi }{2}{z}_2\right)\right]\mathrm{d}\xi ,}\hfill \end{array}\right.\hfill \end{array}$$

with

$$\begin{array}{cc}\hfill & r_1(\mu ,\xi )={\int }_0^{\xi }{\xi }^{\mu -1}\omega \left({\displaystyle \frac{1-\xi }{2}}{c}_1+{\displaystyle \frac{1+\xi }{2}}{z}_2\right)d\xi -{\displaystyle \frac{6}{45}}\mathcal{W}\left(\mu \right),\hfill \\ & r_2(\mu ,\xi )={\int }_0^{\xi }{\xi }^{\mu -1}\omega \left({\displaystyle \frac{1-\xi }{2}}{c}_1+{\displaystyle \frac{1+\xi }{2}}{z}_2\right)d\xi -{\displaystyle \frac{38}{45}}\mathcal{W}\left(\mu \right).\hfill \end{array}$$

Proof. 

Utilizing the principles of integration, we have

$$\begin{array}{cc}\hfill {\chi }_1& ={\int }_0^{\frac{1}{2}}{r}_1(\mu ,\xi )\left[{\mathsf{\Psi }}^{\prime }\left({\displaystyle \frac{1-\xi }{2}}{c}_1+{\displaystyle \frac{1+\xi }{2}}{z}_2\right)-{\mathsf{\Psi }}^{\prime }\left({\displaystyle \frac{1+\xi }{2}}{c}_1+{\displaystyle \frac{1-\xi }{2}}{z}_2\right)\right]\mathrm{d}\xi \hfill \\ \hfill & ={\displaystyle \frac{2}{z_2-c_1}}\left[{\int }_0^{\frac{1}{2}}{\xi }^{\mu -1}\omega \left({\displaystyle \frac{1-\xi }{2}}{c}_1+{\displaystyle \frac{1+\xi }{2}}{z}_2\right)d\xi -{\displaystyle \frac{6}{45}}\mathcal{W}\left(\mu \right)\right]\left[\mathsf{\Psi }\left({\displaystyle \frac{3c_1+z_2}{4}}\right)+\mathsf{\Psi }\left({\displaystyle \frac{c_1+3z_2}{4}}\right)\right]\hfill \\ \hfill & +{\displaystyle \frac{24}{45(z_2-c_1)}}\mathsf{\Psi }\left({\displaystyle \frac{c_1+z_2}{2}}\right)\hfill \\ \hfill & -{\displaystyle \frac{2}{z_2-c_1}}{\int }_0^{\frac{1}{2}}{\xi }^{\mu -1}\omega \left({\displaystyle \frac{1-\xi }{2}}{c}_1+{\displaystyle \frac{1+\xi }{2}}{z}_2\right)\left[\mathsf{\Psi }\left({\displaystyle \frac{1-\xi }{2}}{c}_1+{\displaystyle \frac{1+\xi }{2}}{z}_2\right)+\mathsf{\Psi }\left({\displaystyle \frac{1+\xi }{2}}{c}_1+{\displaystyle \frac{1-\xi }{2}}{z}_2\right)\right]d\xi ,\hfill \end{array}$$

(2)

and similarly, we obtain

$$\begin{array}{cc}\hfill {\chi }_2& ={\int }_{\frac{1}{2}}^1{r}_2(\mu ,\xi )\left[{\mathsf{\Psi }}^{\prime }\left({\displaystyle \frac{1-\xi }{2}}{c}_1+{\displaystyle \frac{1+\xi }{2}}{z}_2\right)-{\mathsf{\Psi }}^{\prime }\left({\displaystyle \frac{1+\xi }{2}}{c}_1+{\displaystyle \frac{1-\xi }{2}}{z}_2\right)\right]\mathrm{d}\xi \hfill \\ \hfill & ={\displaystyle \frac{14}{45(z_2-c_1)}}\left[\mathsf{\Psi }\left(c_1\right)+\mathsf{\Psi }\left(z_2\right)\right]\hfill \\ \hfill & -{\displaystyle \frac{2}{z_2-c_1}}\left[{\int }_0^{\frac{1}{2}}{\xi }^{\mu -1}\omega \left({\displaystyle \frac{1-\xi }{2}}{c}_1+{\displaystyle \frac{1+\xi }{2}}{z}_2\right)d\xi -{\displaystyle \frac{38}{45}}\mathcal{W}\left(\mu \right)\right]\left[\mathsf{\Psi }\left({\displaystyle \frac{3c_1+z_2}{4}}\right)+\mathsf{\Psi }\left({\displaystyle \frac{c_1+3z_2}{4}}\right)\right]\hfill \\ \hfill & -{\displaystyle \frac{2}{z_2-c_1}}{\int }_{\frac{1}{2}}^1{\xi }^{\mu -1}\omega \left({\displaystyle \frac{1-\xi }{2}}{c}_1+{\displaystyle \frac{1+\xi }{2}}{z}_2\right)\left[\mathsf{\Psi }\left({\displaystyle \frac{1-\xi }{2}}{c}_1+{\displaystyle \frac{1+\xi }{2}}{z}_2\right)+\mathsf{\Psi }\left({\displaystyle \frac{1+\xi }{2}}{c}_1+{\displaystyle \frac{1-\xi }{2}}{z}_2\right)\right]d\xi .\hfill \end{array}$$

(3)

Summing (2) and (3), we obtain

$$\begin{array}{cc}\hfill {\displaystyle \sum _{k=1}^2}{\chi }_k& ={\displaystyle \frac{14}{45(z_2-c_1)}}\left[\mathsf{\Psi }\left(c_1\right)+\mathsf{\Psi }\left(z_2\right)\right]\mathcal{W}\left(\mu \right)+{\displaystyle \frac{12}{45(z_2-c_1)}}\left({\displaystyle \frac{c_1+z_2}{2}}\right)\mathcal{W}\left(\mu \right)\hfill \\ & +{\displaystyle \frac{64}{45(z_2-c_1)}}\left[\mathsf{\Psi }\left({\displaystyle \frac{3c_1+z_2}{4}}\right)+\left({\displaystyle \frac{c_1+3z_2}{4}}\right)\right]\mathcal{W}\left(\mu \right)\hfill \\ \hfill & -{\displaystyle \frac{2}{z_2-c_1}}{\int }_0^{\frac{1}{2}}{\xi }^{\mu -1}\omega \left({\displaystyle \frac{1-\xi }{2}}{c}_1+{\displaystyle \frac{1+\xi }{2}}{z}_2\right)\left[\mathsf{\Psi }\left({\displaystyle \frac{1-\xi }{2}}{c}_1+{\displaystyle \frac{1+\xi }{2}}{z}_2\right)+\mathsf{\Psi }\left({\displaystyle \frac{1+\xi }{2}}{c}_1+{\displaystyle \frac{1-\xi }{2}}{z}_2\right)\right]d\xi \hfill \\ \hfill & -{\displaystyle \frac{2}{z_2-c_1}}{\int }_{\frac{1}{2}}^1{\xi }^{\mu -1}\omega \left({\displaystyle \frac{1-\xi }{2}}{c}_1+{\displaystyle \frac{1+\xi }{2}}{z}_2\right)\left[\mathsf{\Psi }\left({\displaystyle \frac{1-\xi }{2}}{c}_1+{\displaystyle \frac{1+\xi }{2}}{z}_2\right)+\mathsf{\Psi }\left({\displaystyle \frac{1+\xi }{2}}{c}_1+{\displaystyle \frac{1-\xi }{2}}{z}_2\right)\right]d\xi \hfill \\ & ={\displaystyle \frac{2}{45(z_2-c_1)}}\left[7\mathsf{\Psi }\left(c_1\right)+32\mathsf{\Psi }\left({\displaystyle \frac{3c_1+z_2}{4}}\right)+12\mathsf{\Psi }\left({\displaystyle \frac{c_1+z_2}{2}}\right)+32\mathsf{\Psi }\left({\displaystyle \frac{c_1+3z_2}{4}}\right)+7\mathsf{\Psi }\left(z_2\right)\right]\mathcal{W}\left(\mu \right)\hfill \\ \hfill & -{\displaystyle \frac{2}{z_2-c_1}}{\int }_0^1{\xi }^{\mu -1}\omega \left({\displaystyle \frac{1-\xi }{2}}{c}_1+{\displaystyle \frac{1+\xi }{2}}{z}_2\right)\left[\mathsf{\Psi }\left({\displaystyle \frac{1-\xi }{2}}{c}_1+{\displaystyle \frac{1+\xi }{2}}{z}_2\right)+\mathsf{\Psi }\left({\displaystyle \frac{1+\xi }{2}}{c}_1+{\displaystyle \frac{1-\xi }{2}}{z}_2\right)\right]d\xi \hfill \\ & ={\displaystyle \frac{2}{45(z_2-c_1)}}\left[7\mathsf{\Psi }\left(c_1\right)+32\mathsf{\Psi }\left({\displaystyle \frac{3c_1+z_2}{4}}\right)+12\mathsf{\Psi }\left({\displaystyle \frac{c_1+z_2}{2}}\right)+32\mathsf{\Psi }\left({\displaystyle \frac{c_1+3z_2}{4}}\right)+7\mathsf{\Psi }\left(z_2\right)\right]\mathcal{W}\left(\mu \right)\hfill \\ \hfill & -{\displaystyle \frac{2}{z_2-c_1}}{\int }_0^1{\xi }^{\mu -1}\omega \left({\displaystyle \frac{1+\xi }{2}}{c}_1+{\displaystyle \frac{1-\xi }{2}}{z}_2\right)\mathsf{\Psi }\left({\displaystyle \frac{1+\xi }{2}}{c}_1+{\displaystyle \frac{1-\xi }{2}}{z}_2\right)\hfill \\ \hfill & -{\displaystyle \frac{2}{z_2-c_1}}{\int }_0^1{\xi }^{\mu -1}\omega \left({\displaystyle \frac{1-\xi }{2}}{c}_1+{\displaystyle \frac{1+\xi }{2}}{z}_2\right)\mathsf{\Psi }\left({\displaystyle \frac{1-\xi }{2}}{c}_1+{\displaystyle \frac{1+\xi }{2}}{z}_2\right).\hfill \end{array}$$

(4)

By substituting $\gamma =\left({\displaystyle \frac{1+\xi }{2}}{c}_1+{\displaystyle \frac{1-\xi }{2}}{z}_2\right)$ and $\beta =\left({\displaystyle \frac{1-\xi }{2}}{c}_1+{\displaystyle \frac{1+\xi }{2}}{z}_2\right)$ for $\xi \in [0,1]$ and using the fact that $\omega$ is symmetrical with respect to ${\displaystyle \frac{c_1+z_2}{2}}$, (4) can be expressed as

$$\begin{array}{cc}\hfill {\displaystyle \sum _{k=1}^2}{\chi }_k& ={\displaystyle \frac{2}{45(z_2-c_1)}}\left[7\mathsf{\Psi }\left(c_1\right)+32\mathsf{\Psi }\left({\displaystyle \frac{3c_1+z_2}{4}}\right)+12\mathsf{\Psi }\left({\displaystyle \frac{c_1+z_2}{2}}\right)+32\mathsf{\Psi }\left({\displaystyle \frac{c_1+3z_2}{4}}\right)+7\mathsf{\Psi }\left(z_2\right)\right]\mathcal{W}\left(\mu \right)\hfill \\ & -{\displaystyle \frac{1}{z_2-c_1}}\left[{\displaystyle \frac{2^{\mu }}{{(z_2-c_1)}^{\mu }}}{\int }_{c_1}^{{\displaystyle \frac{c_1+z_2}{2}}}{(\gamma -c_1)}^{\mu -1}\mathsf{\Psi }\omega \left(\gamma \right)d\gamma -{\displaystyle \frac{2^{\mu }}{{(z_2-c_1)}^{\mu }}}{\int }_{\frac{c_1+z_2}{2}}^{z_2}{(z_2-\beta )}^{\mu -1}\mathsf{\Psi }\omega \left(\beta \right)d\beta \right]\hfill \\ & ={\displaystyle \frac{2}{45(z_2-c_1)}}\left[7\mathsf{\Psi }\left(c_1\right)+32\mathsf{\Psi }\left({\displaystyle \frac{3c_1+z_2}{4}}\right)+12\mathsf{\Psi }\left({\displaystyle \frac{c_1+z_2}{2}}\right)+32\mathsf{\Psi }\left({\displaystyle \frac{c_1+3z_2}{4}}\right)+7\mathsf{\Psi }\left(z_2\right)\right]\mathcal{W}\left(\mu \right)\hfill \\ & -{\displaystyle \frac{2^{\mu }}{{(z_2-c_1)}^{\mu +1}}}{\int }_{c_1}^{\frac{c_1+z_2}{2}}{(\gamma -c_1)}^{\mu -1}\mathsf{\Psi }\omega \left(\gamma \right)d\gamma -{\displaystyle \frac{2^{\mu }}{{(z_2-c_1)}^{\mu +1}}}{\int }_{\frac{c_1+z_2}{2}}^{z_2}{(z_2-\beta )}^{\mu -1}\mathsf{\Psi }\omega \left(\beta \right)d\beta \hfill \\ & ={\displaystyle \frac{2}{45(z_2-c_1)}}\left[7\mathsf{\Psi }\left(c_1\right)+32\mathsf{\Psi }\left({\displaystyle \frac{3c_1+z_2}{4}}\right)+12\mathsf{\Psi }\left({\displaystyle \frac{c_1+z_2}{2}}\right)+32\mathsf{\Psi }\left({\displaystyle \frac{c_1+3z_2}{4}}\right)+7\mathsf{\Psi }\left(z_2\right)\right]\mathcal{W}\left(\mu \right)\hfill \\ & -{\displaystyle \frac{2^{\mu }\mathsf{\Gamma }\left(\mu \right)}{{(z_2-c_1)}^{\mu +1}}}\left[J_{{c_1}^{+}}^{\mu }\mathsf{\Psi }\omega \left({\displaystyle \frac{c_1+z_2}{2}}\right)+J_{{z_2}^{-}}^{\mu }\mathsf{\Psi }\omega \left({\displaystyle \frac{c_1+z_2}{2}}\right)\right].\hfill \end{array}$$

(5)

Thus, by multiplying (5) by ${\displaystyle \frac{z_2-c_1}{2}}$, we deduce the intended identity (1). □

Corollary 1. 

For $\omega \left(\xi \right)=1$, Lemma 1 reduces to the following identity

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{90}}\left[7\mathsf{\Psi }\left(c_1\right)+32\mathsf{\Psi }\left({\displaystyle \frac{3c_1+z_2}{4}}\right)+12\mathsf{\Psi }\left({\displaystyle \frac{c_1+z_2}{2}}\right)+32\mathsf{\Psi }\left({\displaystyle \frac{c_1+3z_2}{4}}\right)+7\mathsf{\Psi }\left(z_2\right)\right]\hfill \\ & -{\displaystyle \frac{2^{\mu -1}\mathsf{\Gamma }\left(\mu \right)}{{(z_2-c_1)}^{\mu }}}\left[J_{{c_1}^{+}}^{\mu }\mathsf{\Psi }\left({\displaystyle \frac{c_1+z_2}{2}}\right)+J_{{z_2}^{-}}^{\mu }\mathsf{\Psi }\left({\displaystyle \frac{c_1+z_2}{2}}\right)\right]\hfill \\ \hfill & ={\displaystyle \frac{z_2-c_1}{4}}{\displaystyle \sum _{k=3}^4}{\chi }_k,\hfill \end{array}$$

(6)

here,

$$\begin{array}{cc}\hfill & \left\{\begin{array}{c}{\displaystyle {\chi }_3={\int }_0^{\frac{1}{2}}{\xi }^{\mu -1}-\frac{6}{45}\left[{\mathsf{\Psi }}^{\prime }\left(\frac{1-\xi }{2}{c}_1+\frac{1+\xi }{2}{z}_2\right)-{\mathsf{\Psi }}^{\prime }\left(\frac{1+\xi }{2}{c}_1+\frac{1-\xi }{2}{z}_2\right)\right]\mathrm{d}\xi ,}\hfill \\ {\displaystyle {\chi }_4={\int }_{\frac{1}{2}}^1{\xi }^{\mu -1}-\frac{38}{45}\left[{\mathsf{\Psi }}^{\prime }\left(\frac{1-\xi }{2}{c}_1+\frac{1+\xi }{2}{z}_2\right)-{\mathsf{\Psi }}^{\prime }\left(\frac{1+\xi }{2}{c}_1+\frac{1-\xi }{2}{z}_2\right)\right]\mathrm{d}\xi .}\hfill \end{array}\right.\hfill \end{array}$$

Corollary 2. 

For $\mu =1$, Corollary 1 reduces to the following identity, which was proved in [13] (Lemma 1).

3.2. Estimates of Weighted Boole’s Inequality Using Convex Functions

In this subsection, we present the primary fractional estimates under Breckner’s convexity assumptions.

Theorem 4. 

Assume that $|{\mathsf{\Psi }}^{\prime }|$ possesses the s-convexity and meets all the credentials of Lemma 1. Then

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{90}}\left[7\mathsf{\Psi }\left(c_1\right)+32\mathsf{\Psi }\left({\displaystyle \frac{3c_1+z_2}{4}}\right)+12\mathsf{\Psi }\left({\displaystyle \frac{c_1+z_2}{2}}\right)+32\mathsf{\Psi }\left({\displaystyle \frac{c_1+3z_2}{4}}\right)+7\mathsf{\Psi }\left(z_2\right)\right]\mathcal{W}\left(\mu \right)\right.\hfill \\ & \left.-{\displaystyle \frac{2^{\mu -1}\mathsf{\Gamma }\left(\mu \right)}{{(z_2-c_1)}^{\mu }}}\left[J_{{c_1}^{+}}^{\mu }\mathsf{\Psi }\omega \left({\displaystyle \frac{c_1+z_2}{2}}\right)+J_{{z_2}^{-}}^{\mu }\mathsf{\Psi }\omega \left({\displaystyle \frac{c_1+z_2}{2}}\right)\right]\right|\hfill \\ \hfill & \le {\displaystyle \frac{z_2-c_1}{4}}\left[|{\mathsf{\Psi }}^{\prime }\left(c_1\right)|+|{\mathsf{\Psi }}^{\prime }\left(z_2\right)|\right]{\displaystyle \sum _{k=1}^4}{\chi }_k^1\left(\mu \right),\hfill \end{array}$$

(7)

where

$$\left\{\begin{array}{cc}\hfill & {\displaystyle {\chi }_1^1\left(\mu \right)={\int }_0^{\frac{1}{2}}\left|r_1(\mu ,\xi )\right|{\left(\frac{1-\xi }{2}\right)}^{s}d\xi ,{\chi }_2^1\left(\mu \right)={\int }_0^{\frac{1}{2}}\left|r_1(\mu ,\xi )\right|{\left(\frac{1+\xi }{2}\right)}^{s}d\xi ,}\hfill \\ & {\displaystyle {\chi }_3^1\left(\mu \right)={\int }_{\frac{1}{2}}^1\left|r_2(\mu ,\xi )\right|{\left(\frac{1-\xi }{2}\right)}^{s}d\xi ,{\chi }_4^1\left(\mu \right)={\int }_{\frac{1}{2}}^1\left|r_2(\mu ,\xi )\right|{\left(\frac{1+\xi }{2}\right)}^{s}d\xi .}\hfill \end{array}\right.$$

Proof. 

By employing the s-convexity of $|{\mathsf{\Psi }}^{\prime }|$ on Lemma 1, we get

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{90}}\left[7\mathsf{\Psi }\left(c_1\right)+32\mathsf{\Psi }\left({\displaystyle \frac{3c_1+z_2}{4}}\right)+12\mathsf{\Psi }\left({\displaystyle \frac{c_1+z_2}{2}}\right)+32\mathsf{\Psi }\left({\displaystyle \frac{c_1+3z_2}{4}}\right)+7\mathsf{\Psi }\left(z_2\right)\right]\mathcal{W}\left(\mu \right)\right.\hfill \\ & \left.-{\displaystyle \frac{2^{\mu -1}\mathsf{\Gamma }\left(\mu \right)}{{(z_2-c_1)}^{\mu }}}\left[J_{{c_1}^{+}}^{\mu }\mathsf{\Psi }\omega \left({\displaystyle \frac{c_1+z_2}{2}}\right)+J_{{z_2}^{-}}^{\mu }\mathsf{\Psi }\omega \left({\displaystyle \frac{c_1+z_2}{2}}\right)\right]\right|\hfill \\ & \le {\displaystyle \frac{z_2-c_1}{4}}\left[{\int }_0^{\frac{1}{2}}\left|r_1(\mu ,\xi )\right|\left[\left|{\mathsf{\Psi }}^{\prime }\left({\displaystyle \frac{1-\xi }{2}}{c}_1+{\displaystyle \frac{1+\xi }{2}}{z}_2\right)\right|+\left|{\mathsf{\Psi }}^{\prime }\left({\displaystyle \frac{1+\xi }{2}}{c}_1+{\displaystyle \frac{1-\xi }{2}}{z}_2\right)\right|\right]\mathrm{d}\xi \right.\hfill \\ \hfill & \left.+{\int }_{\frac{1}{2}}^1\left|r_2(\mu ,\xi )\right|\left[\left|{\mathsf{\Psi }}^{\prime }\left({\displaystyle \frac{1-\xi }{2}}{c}_1+{\displaystyle \frac{1+\xi }{2}}{z}_2\right)\right|+\left|{\mathsf{\Psi }}^{\prime }\left({\displaystyle \frac{1+\xi }{2}}{c}_1+{\displaystyle \frac{1-\xi }{2}}{z}_2\right)\right|\right]\mathrm{d}\xi \right]\hfill \\ \hfill & \le {\displaystyle \frac{z_2-c_1}{4}}\left[|{\mathsf{\Psi }}^{\prime }\left(c_1\right)|+|{\mathsf{\Psi }}^{\prime }\left(z_2\right)|\right]\left[{\int }_0^{\frac{1}{2}}\left|r_1(\mu ,\xi )\right|\left[{\left({\displaystyle \frac{1-\xi }{2}}\right)}^s+{\left({\displaystyle \frac{1+\xi }{2}}\right)}^s\right]\mathrm{d}\xi \right.\left[4\right]\hfill \\ & \left.+{\int }_{\frac{1}{2}}^1\left|r_2(\mu ,\xi )\right|\left[{\left({\displaystyle \frac{1-\xi }{2}}\right)}^s+{\left({\displaystyle \frac{1+\xi }{2}}\right)}^s\right]\mathrm{d}\xi \right]\hfill \\ \hfill & \le {\displaystyle \frac{z_2-c_1}{4}}\left[|{\mathsf{\Psi }}^{\prime }\left(c_1\right)|+|{\mathsf{\Psi }}^{\prime }\left(z_2\right)|\right]\left[{\int }_0^{\frac{1}{2}}\left|r_1(\mu ,\xi )\right|{\left({\displaystyle \frac{1-\xi }{2}}\right)}^s\mathrm{d}\xi +{\int }_0^{\frac{1}{2}}\left|r_1\left(\mu \xi \right)\right|{\left({\displaystyle \frac{1+\xi }{2}}\right)}^s\mathrm{d}\xi \right.\hfill \\ & \left.+{\int }_{\frac{1}{2}}^1\left|r_2(\mu ,\xi )\right|{\left({\displaystyle \frac{1-\xi }{2}}\right)}^s\mathrm{d}\xi +{\int }_{\frac{1}{2}}^1\left|r_2(\mu ,\xi )\right|{\left({\displaystyle \frac{1+\xi }{2}}\right)}^s\mathrm{d}\xi \right]\hfill \\ & \le {\displaystyle \frac{z_2-c_1}{4}}\left[|{\mathsf{\Psi }}^{\prime }\left(c_1\right)|+|{\mathsf{\Psi }}^{\prime }\left(z_2\right)|\right]{\displaystyle \sum _{k=1}^4}{\chi }_k^1\left(\mu \right).\hfill \end{array}$$

This completes the proof. □

Remark 1. 

If $\omega \left(\xi \right)=1$ in (7), then the following fractional Boole’s inequality holds

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{90}}\left[7\mathsf{\Psi }\left(c_1\right)+32\mathsf{\Psi }\left({\displaystyle \frac{3c_1+z_2}{4}}\right)+12\mathsf{\Psi }\left({\displaystyle \frac{c_1+z_2}{2}}\right)+32\mathsf{\Psi }\left({\displaystyle \frac{c_1+3z_2}{4}}\right)+7\mathsf{\Psi }\left(z_2\right)\right]\right.\hfill \\ & \left.-{\displaystyle \frac{2^{\mu -1}\mathsf{\Gamma }(\mu +1)}{{(z_2-c_1)}^{\mu }}}\left[J_{{c_1}^{+}}^{\mu }\mathsf{\Psi }\left({\displaystyle \frac{c_1+z_2}{2}}\right)+J_{{z_2}^{-}}^{\mu }\mathsf{\Psi }\left({\displaystyle \frac{c_1+z_2}{2}}\right)\right]\right|\hfill \\ & \le {\displaystyle \frac{z_2-c_1}{4}}\left[|{\mathsf{\Psi }}^{\prime }\left(c_1\right)|+|{\mathsf{\Psi }}^{\prime }\left(z_2\right)|\right]{\displaystyle \sum _{k=1}^2}{\widehat{\chi }}_k\left(\mu \right).\hfill \end{array}$$

where

$$\left\{\begin{array}{cc}\hfill & {\displaystyle {\widehat{\chi }}_1\left(\mu \right)={\int }_0^{\frac{1}{2}}\left|{\xi }^{\mu }-\frac{6}{45}\right|\left[{\left(\frac{1-\xi }{2}\right)}^s+{\left(\frac{1+\xi }{2}\right)}^s\right]d\xi ,}\hfill \\ & {\displaystyle {\widehat{\chi }}_2\left(\mu \right)={\int }_{\frac{1}{2}}^1\left|{\xi }^{\mu }-\frac{38}{45}\right|\left[{\left(\frac{1-\xi }{2}\right)}^s+{\left(\frac{1+\xi }{2}\right)}^s\right]d\xi .}\hfill \end{array}\right.$$

For $s=1$, we obtain Boole’s inequality, which is given in [31].

Corollary 3. 

If $\mu =1$ in Theorem 4, then the following weighted Boole’s inequality holds

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{90}}\left[7\mathsf{\Psi }\left(c_1\right)+32\mathsf{\Psi }\left({\displaystyle \frac{3c_1+z_2}{4}}\right)+12\mathsf{\Psi }\left({\displaystyle \frac{c_1+z_2}{2}}\right)+32\mathsf{\Psi }\left({\displaystyle \frac{c_1+3z_2}{4}}\right)+7\mathsf{\Psi }\left(z_2\right)\right]\right.\hfill \\ & \left.\times {\displaystyle \frac{2}{z_2-c_1}}{\int }_{{\displaystyle \frac{c_1+z_2}{2}}}^{z_2}\omega \left(\xi \right)d\xi -{\displaystyle \frac{2}{z_2-c_1}}{\int }_{c_1}^{z_2}\mathsf{\Psi }\left(\xi \right)\omega \left(\xi \right)d\xi \right|\hfill \\ & \le {\displaystyle \frac{z_2-c_1}{4}}\left[|{\mathsf{\Psi }}^{\prime }\left(c_1\right)|+|{\mathsf{\Psi }}^{\prime }\left(z_2\right)|\right]{\displaystyle \sum _{k=1}^2}{\breve{\chi }}_k.\hfill \end{array}$$

where,

$$\begin{array}{c}\left\{\begin{array}{cc}\hfill & {\displaystyle {\breve{\chi }}_1={\int }_0^{\frac{1}{2}}\left|r_1(1,\xi )\right|\left[{\left(\frac{1-\xi }{2}\right)}^s+{\left(\frac{1+\xi }{2}\right)}^s\right]d\xi ,}\hfill \\ & {\displaystyle {\breve{\chi }}_2={\int }_{\frac{1}{2}}^1\left|r_2(1,\xi )\right|\left[{\left(\frac{1-\xi }{2}\right)}^s+{\left(\frac{1+\xi }{2}\right)}^s\right]d\xi ,}\hfill \end{array}\right.\hfill \end{array}$$

with

$$\begin{array}{c}\left\{\begin{array}{c}{\displaystyle r_1(1,\xi )={\int }_0^{\xi }\omega \left(\frac{1-\xi }{2}{c}_1+\frac{1+\xi }{2}{z}_2\right)d\xi -{\displaystyle \frac{12}{45(z_2-c_1)}}{\int }_{\frac{c_1+z_2}{2}}^{z_2}\omega \left(\xi \right)d\xi ,}\hfill \\ {\displaystyle r_2(1,\xi )={\int }_0^{\xi }\omega \left(\frac{1-\xi }{2}{c}_1+\frac{1+\xi }{2}{z}_2\right)d\xi -{\displaystyle \frac{76}{45(z_2-c_1)}}{\int }_{\frac{c_1+z_2}{2}}^{z_2}\omega \left(\xi \right)d\xi .}\hfill \end{array}\right.\hfill \end{array}$$

Corollary 4. 

If $\omega \left(\xi \right)=1$ and $s=1$ in Corollary 3, then the following Boole’s inequality holds, which was proved in [13] (Theorem 3).

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{90}}\left[7\mathsf{\Psi }\left(c_1\right)+32\mathsf{\Psi }\left({\displaystyle \frac{3c_1+z_2}{4}}\right)+12\mathsf{\Psi }\left({\displaystyle \frac{c_1+z_2}{2}}\right)+32\mathsf{\Psi }\left({\displaystyle \frac{c_1+3z_2}{4}}\right)+7\mathsf{\Psi }\left(z_2\right)\right]\right.\hfill \\ & \left.-{\displaystyle \frac{1}{z_2-c_1}}{\int }_{c_1}^{z_2}\mathsf{\Psi }\left(\xi \right)d\xi \right|\hfill \\ & \le {\displaystyle \frac{239(z_2-c_1)}{6480}}\left[|{\mathsf{\Psi }}^{\prime }\left(c_1\right)|+|{\mathsf{\Psi }}^{\prime }\left(z_2\right)|\right].\hfill \end{array}$$

Remark 2. 

The inequality (7) in Theorem 4 is sharp for constant functions. It provides a unified way to discuss the error analysis of several classes of integrals with symmetric weights.

Theorem 5. 

Assume that $|{\mathsf{\Psi }}^{\prime }{|}^{q_2}$ is s-convex on $[c_1,z_2]$ with $p_2,q_2>1$ and meets all the conditions of Lemma 1. In the case ${\displaystyle \frac{1}{p_2}+\frac{1}{q_2}=1}$, we obtain:

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{90}}\left[7\mathsf{\Psi }\left(c_1\right)+32\mathsf{\Psi }\left({\displaystyle \frac{3c_1+z_2}{4}}\right)+12\mathsf{\Psi }\left({\displaystyle \frac{c_1+z_2}{2}}\right)+32\mathsf{\Psi }\left({\displaystyle \frac{c_1+3z_2}{4}}\right)+7\mathsf{\Psi }\left(z_2\right)\right]\mathcal{W}\left(\mu \right)\right.\hfill \\ & \left.-{\displaystyle \frac{2^{\mu -1}\mathsf{\Gamma }\left(\mu \right)}{{(z_2-c_1)}^{\mu }}}\left[J_{{c_1}^{+}}^{\mu }\mathsf{\Psi }\omega \left({\displaystyle \frac{c_1+z_2}{2}}\right)+J_{{z_2}^{-}}^{\mu }\mathsf{\Psi }\omega \left({\displaystyle \frac{c_1+z_2}{2}}\right)\right]\right|\hfill \\ \hfill & \le {\displaystyle \frac{z_2-c_1}{4}}\left\{{\left({\int }_0^{\frac{1}{2}}{\left|r_1(\mu ,\xi )\right|}^{p_2}d\xi \right)}^{1/p_2}\right.\hfill \\ & \times \left[{\left({\displaystyle \frac{2^{-1-2s}\left(-1+2^{1+s}\right)|{\mathsf{\Psi }}^{\prime }\left(c_1\right){|}^{q_2}+2(-1+{\left(\frac{3}{2}\right)}^{1+s}){\left|{\mathsf{\Psi }}^{\prime }\left(z_2\right)\right|}^{q_2}}{(1+s)}}\right)}^{\frac{1}{q_2}}\right.\hfill \\ \hfill & \left.+{\left({\displaystyle \frac{2(-1+{\left(\frac{3}{2}\right)}^{1+s})|{\mathsf{\Psi }}^{\prime }\left(c_1\right){|}^{q_2}+2^{-1-2s}\left(-1+2^{1+s}\right){\left|{\mathsf{\Psi }}^{\prime }\left(z_2\right)\right|}^{q_2}}{(1+s)}}\right)}^{\frac{1}{q_2}}\right]\hfill \\ \hfill & +{\left({\int }_{\frac{1}{2}}^1{\left|r_2(\mu ,\xi )\right|}^{p_2}d\xi \right)}^{1/p_2}\hfill \\ \hfill & \times \left[{\left({\displaystyle \frac{2^{-1-2s}|{\mathsf{\Psi }}^{\prime }\left(c_1\right){|}^{q_2}+2-2^{-1-2s}{3}^{1+s}{\left|{\mathsf{\Psi }}^{\prime }\left(z_2\right)\right|}^{q_2}}{1+s}}\right)}^{\frac{1}{q_2}}\right.\hfill \\ & \left.\left.+{\left({\displaystyle \frac{2^{-1-2s}|{\mathsf{\Psi }}^{\prime }\left(z_2\right){|}^{q_2}+2-2^{-1-2s}{3}^{1+s}{\left|{\mathsf{\Psi }}^{\prime }\left(c_1\right)\right|}^{q_2}}{1+s}}\right)}^{\frac{1}{q_2}}\right]\right\}.\hfill \end{array}$$

(8)

Proof. 

Using Hölder’s inequality, we get

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{90}}\left[7\mathsf{\Psi }\left(c_1\right)+32\mathsf{\Psi }\left({\displaystyle \frac{3c_1+z_2}{4}}\right)+12\mathsf{\Psi }\left({\displaystyle \frac{c_1+z_2}{2}}\right)+32\mathsf{\Psi }\left({\displaystyle \frac{c_1+3z_2}{4}}\right)+7\mathsf{\Psi }\left(z_2\right)\right]\mathcal{W}\left(\mu \right)\right.\hfill \\ & \left.-{\displaystyle \frac{2^{\mu -1}\mathsf{\Gamma }\left(\mu \right)}{{(z_2-c_1)}^{\mu }}}\left[J_{{c_1}^{+}}^{\mu }\mathsf{\Psi }\omega \left({\displaystyle \frac{c_1+z_2}{2}}\right)+J_{{z_2}^{-}}^{\mu }\mathsf{\Psi }\omega \left({\displaystyle \frac{c_1+z_2}{2}}\right)\right]\right|\hfill \\ \hfill & \le {\displaystyle \frac{z_2-c_1}{4}}\left\{{\left({\int }_0^{\frac{1}{2}}{\left|r_1(\mu ,\xi )\right|}^{p_2}\mathrm{d}\xi \right)}^{{\displaystyle \frac{1}{p_2}}}\left[{\left({\int }_0^{\frac{1}{2}}{\left|{\mathsf{\Psi }}^{\prime }\left({\displaystyle \frac{1-\xi }{2}}{c}_1+{\displaystyle \frac{1+\xi }{2}}{z}_2\right)\right|}^{q_2}\mathrm{d}\xi \right)}^{\frac{1}{q_2}}\right.\right.\hfill \\ \hfill & \left.+{\left({\int }_0^{\frac{1}{2}}{\left|{\mathsf{\Psi }}^{\prime }\left({\displaystyle \frac{1+\xi }{2}}{c}_1+{\displaystyle \frac{1-\xi }{2}}{z}_2\right)\right|}^{q_2}\mathrm{d}\xi \right)}^{\frac{1}{q_2}}\right]\hfill \\ \hfill & +{\left({\int }_{\frac{1}{2}}^1{\left|r_2(\mu ,\xi )\right|}^{p_2}\mathrm{d}\xi \right)}^{{\displaystyle \frac{1}{p_2}}}\left[{\left({\int }_{\frac{1}{2}}^1{\left|{\mathsf{\Psi }}^{\prime }\left({\displaystyle \frac{1-\xi }{2}}{c}_1+{\displaystyle \frac{1+\xi }{2}}{z}_2\right)\right|}^{q_2}\mathrm{d}\xi \right)}^{\frac{1}{q_2}}\right.\hfill \\ \hfill & \left.\left.+{\left({\int }_{\frac{1}{2}}^1{\left|{\mathsf{\Psi }}^{\prime }\left({\displaystyle \frac{1+\xi }{2}}{c}_1+{\displaystyle \frac{1-\xi }{2}}{z}_2\right)\right|}^{q_2}\mathrm{d}\xi \right)}^{\frac{1}{q_2}}\right]\right\}.\hfill \end{array}$$

Since $|{\mathsf{\Psi }}^{\prime }{|}^{q_2}$ is s-convex, then we have

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{90}}\left[7\mathsf{\Psi }\left(c_1\right)+32\mathsf{\Psi }\left({\displaystyle \frac{3c_1+z_2}{4}}\right)+12\mathsf{\Psi }\left({\displaystyle \frac{c_1+z_2}{2}}\right)+32\mathsf{\Psi }\left({\displaystyle \frac{c_1+3z_2}{4}}\right)+7\mathsf{\Psi }\left(z_2\right)\right]\mathcal{W}\left(\mu \right)\right.\hfill \\ & \left.-{\displaystyle \frac{2^{\mu -1}\mathsf{\Gamma }\left(\mu \right)}{{(z_2-c_1)}^{\mu }}}\left[J_{{c_1}^{+}}^{\mu }\mathsf{\Psi }\omega \left({\displaystyle \frac{c_1+z_2}{2}}\right)+J_{{z_2}^{-}}^{\mu }\mathsf{\Psi }\omega \left({\displaystyle \frac{c_1+z_2}{2}}\right)\right]\right|\hfill \\ \hfill & \le {\displaystyle \frac{z_2-c_1}{4}}\left\{{\left({\int }_0^{\frac{1}{2}}{\left|r_1(\mu ,\xi )\right|}^{p_2}\mathrm{d}\xi \right)}^{{\displaystyle \frac{1}{p_2}}}\left[{\left({\int }_0^{\frac{1}{2}}\left[{\left({\displaystyle \frac{1-\xi }{2}}\right)}^s{\left|{\mathsf{\Psi }}^{\prime }\left(c_1\right)\right|}^{q_2}+{\left({\displaystyle \frac{1+\xi }{2}}\right)}^s{\left|{\mathsf{\Psi }}^{\prime }\left(z_2\right)\right|}^{q_2}\right]\mathrm{d}\xi \right)}^{\frac{1}{q_2}}\right.\right.\hfill \\ \hfill & \left.+{\left({\int }_0^{\frac{1}{2}}\left[{\left({\displaystyle \frac{1+\xi }{2}}\right)}^s{\left|{\mathsf{\Psi }}^{\prime }\left(c_1\right)\right|}^{q_2}+{\left({\displaystyle \frac{1-\xi }{2}}\right)}^s{\left|{\mathsf{\Psi }}^{\prime }\left(z_2\right)\right|}^{q_2}\right]\mathrm{d}\xi \right)}^{\frac{1}{q_2}}\right]\hfill \\ \hfill & +\left({\int }_{\frac{1}{2}}^1{\left|r_2{(\mu ,\xi )|}^{p_2}\mathrm{d}\xi \right)}^{{\displaystyle \frac{1}{p_2}}}\right)\left[{\left({\int }_{\frac{1}{2}}^1\left[{\left({\displaystyle \frac{1-\xi }{2}}\right)}^s{\left|{\mathsf{\Psi }}^{\prime }\left(c_1\right)\right|}^{q_2}+{\left({\displaystyle \frac{1+\xi }{2}}\right)}^s{\left|{\mathsf{\Psi }}^{\prime }\left(z_2\right)\right|}^{q_2}\right]\mathrm{d}\xi \right)}^{\frac{1}{q_2}}\right.\hfill \\ \hfill & \left.\left.+{\left({\int }_{\frac{1}{2}}^1\left[{\left({\displaystyle \frac{1+\xi }{2}}\right)}^s{\left|{\mathsf{\Psi }}^{\prime }\left(c_1\right)\right|}^{q_2}+{\left({\displaystyle \frac{1-\xi }{2}}\right)}^s{\left|{\mathsf{\Psi }}^{\prime }\left(z_2\right)\right|}^{q_2}\right]\mathrm{d}\xi \right)}^{\frac{1}{q_2}}\right]\right\}\hfill \\ \hfill & \le {\displaystyle \frac{z_2-c_1}{4}}\left\{{\left({\int }_0^{\frac{1}{2}}{\left|r_1(\mu ,\xi )\right|}^{p_2}\mathrm{d}\xi \right)}^{{\displaystyle \frac{1}{p_2}}}{\left({\int }_0^{\frac{1}{2}}\left[{\left({\displaystyle \frac{1-\xi }{2}}\right)}^s{\left|{\mathsf{\Psi }}^{\prime }\left(c_1\right)\right|}^{q_2}+{\left({\displaystyle \frac{1+\xi }{2}}\right)}^s{\left|{\mathsf{\Psi }}^{\prime }\left(z_2\right)\right|}^{q_2}\right]\mathrm{d}\xi \right)}^{\frac{1}{q_2}}\right.\hfill \\ \hfill & +{\left({\int }_0^{\frac{1}{2}}{\left|r_1(\mu ,\xi )\right|}^{p_2}\mathrm{d}\xi \right)}^{{\displaystyle \frac{1}{p_2}}}{\left({\int }_0^{\frac{1}{2}}\left[{\left({\displaystyle \frac{1+\xi }{2}}\right)}^s{\left|{\mathsf{\Psi }}^{\prime }\left(c_1\right)\right|}^{q_2}+{\left({\displaystyle \frac{1-\xi }{2}}\right)}^s{\left|{\mathsf{\Psi }}^{\prime }\left(z_2\right)\right|}^{q_2}\right]\mathrm{d}\xi \right)}^{\frac{1}{q_2}}\hfill \\ \hfill & +{\left({\int }_{\frac{1}{2}}^1{\left|r_2(\mu ,\xi )\right|}^{p_2}\mathrm{d}\xi \right)}^{{\displaystyle \frac{1}{p_2}}}{\left({\int }_{\frac{1}{2}}^1\left[{\left({\displaystyle \frac{1-\xi }{2}}\right)}^s{\left|{\mathsf{\Psi }}^{\prime }\left(c_1\right)\right|}^{q_2}+{\left({\displaystyle \frac{1+\xi }{2}}\right)}^s{\left|{\mathsf{\Psi }}^{\prime }\left(z_2\right)\right|}^{q_2}\right]\mathrm{d}\xi \right)}^{\frac{1}{q_2}}\hfill \\ \hfill & \left.+{\left({\int }_{\frac{1}{2}}^1{\left|r_2(\mu ,\xi )\right|}^{p_2}\mathrm{d}\xi \right)}^{{\displaystyle \frac{1}{p_2}}}{\left({\int }_{\frac{1}{2}}^1\left[{\left({\displaystyle \frac{1+\xi }{2}}\right)}^s{\left|{\mathsf{\Psi }}^{\prime }\left(c_1\right)\right|}^{q_2}+{\left({\displaystyle \frac{1-\xi }{2}}\right)}^s{\left|{\mathsf{\Psi }}^{\prime }\left(z_2\right)\right|}^{q_2}\right]\mathrm{d}\xi \right)}^{\frac{1}{q_2}}\right\}\hfill \\ \hfill & \le {\displaystyle \frac{z_2-c_1}{4}}\left\{{\left({\int }_0^{\frac{1}{2}}{\left|r_1(\mu ,\xi )\right|}^{p_2}d\xi \right)}^{1/p_2}\right.\hfill \\ & \times \left[{\left({\displaystyle \frac{2^{-1-2s}\left(-1+2^{1+s}\right)|{\mathsf{\Psi }}^{\prime }\left(c_1\right){|}^{q_2}+2(-1+{\left(\frac{3}{2}\right)}^{1+s}){\left|{\mathsf{\Psi }}^{\prime }\left(z_2\right)\right|}^{q_2}}{(1+s)}}\right)}^{\frac{1}{q_2}}\right.\hfill \\ \hfill & \left.+{\left({\displaystyle \frac{2(-1+{\left(\frac{3}{2}\right)}^{1+s})|{\mathsf{\Psi }}^{\prime }\left(c_1\right){|}^{q_2}+2^{-1-2s}\left(-1+2^{1+s}\right){\left|{\mathsf{\Psi }}^{\prime }\left(z_2\right)\right|}^{q_2}}{(1+s)}}\right)}^{\frac{1}{q_2}}\right]\hfill \\ \hfill & +{\left({\int }_{\frac{1}{2}}^1{\left|r_2(\mu ,\xi )\right|}^{p_2}d\xi \right)}^{1/p_2}\hfill \\ \hfill & \times \left[{\left({\displaystyle \frac{2^{-1-2s}|{\mathsf{\Psi }}^{\prime }\left(c_1\right){|}^{q_2}+2-2^{-1-2s}{3}^{1+s}{\left|{\mathsf{\Psi }}^{\prime }\left(z_2\right)\right|}^{q_2}}{1+s}}\right)}^{\frac{1}{q_2}}\right.\hfill \\ & \left.\left.+{\left({\displaystyle \frac{2^{-1-2s}|{\mathsf{\Psi }}^{\prime }\left(z_2\right){|}^{q_2}+2-2^{-1-2s}{3}^{1+s}{\left|{\mathsf{\Psi }}^{\prime }\left(c_1\right)\right|}^{q_2}}{1+s}}\right)}^{\frac{1}{q_2}}\right]\right\}.\hfill \end{array}$$

The desired inequality is now achieved. □

Example 1. 

A function $\omega :[0,2]\to \mathbb{R}$ is defined as: $\omega \left(x\right)={\left(x-1\right)}^2$, and so ω is symmetric about 1. Let $\mathsf{\Psi }:[0,2]\to \mathbb{R}$ be an s-convex function, first defined by $\mathsf{\Psi }\left(x\right)=x^3$ and then defined by $\mathsf{\Psi }\left(x\right)=e^x$. Then, by the variation in $\mu >0$, we obtain, respectively:

Theorem 5 is illustrated through the graphical and numerical approaches in Figure 1 and Figure 2 and

Table 1. Numerical illustration of Theorem 5.

μ	Ψ(x) = x3		Ψ(x) = ex
	Left-Term	Right-Term	Left-Term	Right-Term
0.1	2.178	5.2863	16.843	18.653
0.3	0.58356	1.32003	8.706	9.512
0.5	0.304586	0.644687	4.190	4.689
0.7	0.20496	0.405814	1.980	2.223
0.9	0.161638	0.301005	0.657	0.881

.

Theorem 6. 

Assume that $|{\mathsf{\Psi }}^{\prime }{|}^{q_2}$ possesses the s-convexity and meets all the assumptions of Lemma 1. Then

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{90}}\left[7\mathsf{\Psi }\left(c_1\right)+32\mathsf{\Psi }\left({\displaystyle \frac{3c_1+z_2}{4}}\right)+12\mathsf{\Psi }\left({\displaystyle \frac{c_1+z_2}{2}}\right)+32\mathsf{\Psi }\left({\displaystyle \frac{c_1+3z_2}{4}}\right)+7\mathsf{\Psi }\left(z_2\right)\right]\mathcal{W}\left(\mu \right)\right.\hfill \\ & \left.-{\displaystyle \frac{2^{\mu -1}\mathsf{\Gamma }\left(\mu \right)}{{(z_2-c_1)}^{\mu }}}\left[J_{{c_1}^{+}}^{\mu }\mathsf{\Psi }\omega \left({\displaystyle \frac{c_1+z_2}{2}}\right)+J_{{z_2}^{-}}^{\mu }\mathsf{\Psi }\omega \left({\displaystyle \frac{c_1+z_2}{2}}\right)\right]\right|\hfill \\ \hfill & \le {\displaystyle \frac{(z_2-c_1)}{4}}\left\{{\stackrel{`}{\chi }}_1^{1-\frac{1}{q_2}}\left(\mu \right)\left[{\left({\chi }_1^1\left(\mu \right)|{\mathsf{\Psi }}^{\prime }\left(c_1\right){|}^{q_2}+{\chi }_2^1\left(\mu \right){\left|{\mathsf{\Psi }}^{\prime }\left(z_2\right)\right|}^{q_2}\right)}^{\frac{1}{q_2}}+{\left({\chi }_2^1\left(\mu \right)|{\mathsf{\Psi }}^{\prime }\left(c_1\right){|}^{q_2}+{\chi }_1^1\left(\mu \right){\left|{\mathsf{\Psi }}^{\prime }\left(z_2\right)\right|}^{q_2}\right)}^{\frac{1}{q_2}}\right]\right.\hfill \\ \hfill & \left.+{\stackrel{`}{\chi }}_2^{1-\frac{1}{q_2}}\left(\mu \right)\left[{\left({\chi }_3^1\left(\mu \right)|{\mathsf{\Psi }}^{\prime }\left(c_1\right){|}^{q_2}+{\chi }_4^1\left(\mu \right){\left|{\mathsf{\Psi }}^{\prime }\left(z_2\right)\right|}^{q_2}\right)}^{\frac{1}{q_2}}+{\left({\chi }_4^1\left(\mu \right)|{\mathsf{\Psi }}^{\prime }\left(c_1\right){|}^{q_2}+{\chi }_3^1\left(\mu \right){\left|{\mathsf{\Psi }}^{\prime }\left(z_2\right)\right|}^{q_2}\right)}^{\frac{1}{q_2}}\right]\right\},\hfill \end{array}$$

(9)

where $q_2\ge 1$, ${\chi }_k^1\left(\mu \right)$ for $k=:1,2,3,4$ is already given in Theorem 4, and

$$\begin{array}{c}\left\{\begin{array}{cc}\hfill & {\displaystyle {\stackrel{`}{\chi }}_1\left(\mu \right)={\int }_0^{\frac{1}{2}}\left|r_1(\mu ,\xi )\right|d\xi ,}\hfill \\ & {\displaystyle {\stackrel{`}{\chi }}_2\left(\mu \right)={\int }_{\frac{1}{2}}^1\left|r_2(\mu ,\xi )\right|d\xi .}\hfill \end{array}\right.\hfill \end{array}$$

Proof. 

Through the application of power mean inequality, we get

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{90}}\left[7\mathsf{\Psi }\left(c_1\right)+32\mathsf{\Psi }\left({\displaystyle \frac{3c_1+z_2}{4}}\right)+12\mathsf{\Psi }\left({\displaystyle \frac{c_1+z_2}{2}}\right)+32\mathsf{\Psi }\left({\displaystyle \frac{c_1+3z_2}{4}}\right)+7\mathsf{\Psi }\left(z_2\right)\right]\mathcal{W}\left(\mu \right)\right.\hfill \\ & \left.-{\displaystyle \frac{2^{\mu -1}\mathsf{\Gamma }\left(\mu \right)}{{(z_2-c_1)}^{\mu }}}\left[J_{{c_1}^{+}}^{\mu }\mathsf{\Psi }\omega \left({\displaystyle \frac{c_1+z_2}{2}}\right)+J_{{z_2}^{-}}^{\mu }\mathsf{\Psi }\omega \left({\displaystyle \frac{c_1+z_2}{2}}\right)\right]\right|\hfill \\ & \le {\displaystyle \frac{z_2-c_1}{4}}\{{\left({\int }_0^{\frac{1}{2}}\left|r_1(\mu ,\xi )\right|d\xi \right)}^{1-\frac{1}{q_2}}\hfill \\ & \times \left[{\left({\int }_0^{\frac{1}{2}}\left|r_1(\mu ,\xi )\right|{\left|{\mathsf{\Psi }}^{\prime }\left({\displaystyle \frac{1-\xi }{2}}{c}_1+{\displaystyle \frac{1+\xi }{2}}{z}_2\right)\right|}^{q_2}d\xi \right)}^{\frac{1}{q_2}}\right.\hfill \\ & \left.+{\left({\int }_0^{\frac{1}{2}}\left|r_1(\mu ,\xi )\right|{\left|{\mathsf{\Psi }}^{\prime }\left({\displaystyle \frac{1+\xi }{2}}{c}_1+{\displaystyle \frac{1-\xi }{2}}{z}_2\right)\right|}^{q_2}d\xi \right)}^{\frac{1}{q_2}}\right]\hfill \\ & +{\left({\int }_{\frac{1}{2}}^1\left|r_2(\mu ,\xi )\right|d\xi \right)}^{1-\frac{1}{q_2}}\hfill \\ & \times \left[{\left({\int }_{\frac{1}{2}}^1\left|r_2(\mu ,\xi )\right|{\left|{\mathsf{\Psi }}^{\prime }\left({\displaystyle \frac{1-\xi }{2}}{c}_1+{\displaystyle \frac{1+\xi }{2}}{z}_2\right)\right|}^{q_2}d\xi \right)}^{\frac{1}{q_2}}\right.\hfill \\ & \left.+{\left({\int }_{\frac{1}{2}}^1\left|r_2(\mu ,\xi )\right|{\left|{\mathsf{\Psi }}^{\prime }\left({\displaystyle \frac{1+\xi }{2}}{c}_1+{\displaystyle \frac{1-\xi }{2}}{z}_2\right)\right|}^{q_2}d\xi \right)}^{\frac{1}{q_2}}\right]\}.\hfill \end{array}$$

Since $|{\mathsf{\Psi }}^{\prime }{|}^{q_2}$ is s-convex, then

$$\begin{array}{c}{\left|{\mathsf{\Psi }}^{\prime }\left({\displaystyle \frac{1-\xi }{2}}{c}_1+{\displaystyle \frac{1+\xi }{2}}{z}_2\right)\right|}^{q_2}\le {\left({\displaystyle \frac{1+\xi }{2}}\right)}^s|{\mathsf{\Psi }}^{\prime }\left(z_2\right){|}^{q_2}+{\left({\displaystyle \frac{1-\xi }{2}}\right)}^s{\left|{\mathsf{\Psi }}^{\prime }\left(c_1\right)\right|}^{q_2},\hfill \end{array}$$

and finally we get:

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{90}}\left[7\mathsf{\Psi }\left(c_1\right)+32\mathsf{\Psi }\left({\displaystyle \frac{3c_1+z_2}{4}}\right)+12\mathsf{\Psi }\left({\displaystyle \frac{c_1+z_2}{2}}\right)+32\mathsf{\Psi }\left({\displaystyle \frac{c_1+3z_2}{4}}\right)+7\mathsf{\Psi }\left(z_2\right)\right]\mathcal{W}\left(\mu \right)\right.\hfill \\ & \left.-{\displaystyle \frac{2^{\mu -1}\mathsf{\Gamma }\left(\mu \right)}{{(z_2-c_1)}^{\mu }}}\left[J_{{c_1}^{+}}^{\mu }\mathsf{\Psi }\omega \left({\displaystyle \frac{c_1+z_2}{2}}\right)+J_{{z_2}^{-}}^{\mu }\mathsf{\Psi }\omega \left({\displaystyle \frac{c_1+z_2}{2}}\right)\right]\right|\hfill \\ & \le {\displaystyle \frac{z_2-c_1}{4}}\{{\left({\int }_0^{\frac{1}{2}}\left|r_1(\mu ,\xi )\right|d\xi \right)}^{1-\frac{1}{q_2}}\left[{\left({\int }_0^{\frac{1}{2}}\left|r_1(\mu ,\xi )\right|\left({\left({\displaystyle \frac{1-\xi }{2}}\right)}^s|\mathsf{\Psi }{\left(z_2\right)}^{\prime }{|}^{q_2}+{\left({\displaystyle \frac{1+\xi }{2}}\right)}^s\left|\mathsf{\Psi }{\left(c_1\right)}^{\prime }\right|\right)d\xi \right)}^{\frac{1}{q_2}}\right.\hfill \\ & \left.+{\left({\int }_0^{\frac{1}{2}}\left|r_1(\mu ,\xi )\right|\left({\left({\displaystyle \frac{1-\xi }{2}}\right)}^s|\mathsf{\Psi }{\left(c_1\right)}^{\prime }{|}^{q_2}+{\left({\displaystyle \frac{1+\xi }{2}}\right)}^s\left|\mathsf{\Psi }{\left(z_2\right)}^{\prime }\right|\right)d\xi \right)}^{\frac{1}{q_2}}\right]\hfill \\ & +{\left({\int }_{\frac{1}{2}}^1\left|r_2(\mu ,\xi )\right|d\xi \right)}^{1-\frac{1}{q_2}}\left[{\left({\int }_{\frac{1}{2}}^1\left|r_2(\mu ,\xi )\right|\left({\left({\displaystyle \frac{1-\xi }{2}}\right)}^s|\mathsf{\Psi }{\left(z_2\right)}^{\prime }{|}^{q_2}+{\left({\displaystyle \frac{1+\xi }{2}}\right)}^s\left|\mathsf{\Psi }{\left(c_1\right)}^{\prime }\right|\right)d\xi \right)}^{\frac{1}{q_2}}\right.\hfill \\ & \left.+{\left({\int }_{\frac{1}{2}}^1\left|r_2(\mu ,\xi )\right|\left({\left({\displaystyle \frac{1-\xi }{2}}\right)}^s|\mathsf{\Psi }{\left(c_1\right)}^{\prime }{|}^{q_2}+{\left({\displaystyle \frac{1+\xi }{2}}\right)}^s\left|\mathsf{\Psi }{\left(z_2\right)}^{\prime }\right|\right)d\xi \right)}^{\frac{1}{q_2}}\right]\}\hfill \\ & \le {\displaystyle \frac{z_2-c_1}{4}}\left\{{\left({\int }_0^{\frac{1}{2}}\left|r_1(\mu ,\xi )\right|d\xi \right)}^{1-\frac{1}{q_2}}{\left({\int }_0^{\frac{1}{2}}\left|r_1(\mu ,\xi )\right|\left[{\left({\displaystyle \frac{1-\xi }{2}}\right)}^s|{\mathsf{\Psi }}^{\prime }\left(z_2\right){|}^{q_2}+{\left({\displaystyle \frac{1+\xi }{2}}\right)}^s{\left|{\mathsf{\Psi }}^{\prime }\left(c_1\right)\right|}^{q_2}\right]d\xi \right)}^{\frac{1}{q_2}}\right.\hfill \\ \hfill & +{\left({\int }_0^{\frac{1}{2}}\left|r_1(\mu ,\xi )\right|d\xi \right)}^{1-\frac{1}{q_2}}{\left({\int }_0^{\frac{1}{2}}\left|r_1(\mu ,\xi )\right|\left[{\left({\displaystyle \frac{1-\xi }{2}}\right)}^s|{\mathsf{\Psi }}^{\prime }\left(c_1\right){|}^{q_2}+{\left({\displaystyle \frac{1+\xi }{2}}\right)}^s{\left|{\mathsf{\Psi }}^{\prime }\left(z_2\right)\right|}^{q_2}\right]d\xi \right)}^{\frac{1}{q_2}}\hfill \\ \hfill & +{\left({\int }_{\frac{1}{2}}^1\left|r_2(\mu ,\xi )\right|d\xi \right)}^{1-\frac{1}{q_2}}{\left({\int }_{\frac{1}{2}}^1\left|r_2(\mu ,\xi )\right|\left[{\left({\displaystyle \frac{1-\xi }{2}}\right)}^s|{\mathsf{\Psi }}^{\prime }\left(z_2\right){|}^{q_2}+{\left({\displaystyle \frac{1+\xi }{2}}\right)}^s{\left|{\mathsf{\Psi }}^{\prime }\left(c_1\right)\right|}^{q_2}\right]d\xi \right)}^{\frac{1}{q_2}}\hfill \\ \hfill & \left.+{\left({\int }_{\frac{1}{2}}^1\left|r_2(\mu ,\xi )\right|d\xi \right)}^{1-\frac{1}{q_2}}{\left({\int }_{\frac{1}{2}}^1\left|r_2(\mu ,\xi )\right|\left[{\left({\displaystyle \frac{1-\xi }{2}}\right)}^s|{\mathsf{\Psi }}^{\prime }\left(c_1\right){|}^{q_2}+{\left({\displaystyle \frac{1+\xi }{2}}\right)}^s{\left|{\mathsf{\Psi }}^{\prime }\left(z_2\right)\right|}^{q_2}\right]d\xi \right)}^{\frac{1}{q_2}}\right\}\hfill \\ \hfill & \le {\displaystyle \frac{(z_2-c_1)}{4}}\left\{{\stackrel{`}{\chi }}_1^{1-\frac{1}{q_2}}\left(\mu \right)\left[{\left({\chi }_1^1\left(\mu \right)|{\mathsf{\Psi }}^{\prime }\left(c_1\right){|}^{q_2}+{\chi }_2^1\left(\mu \right){\left|{\mathsf{\Psi }}^{\prime }\left(z_2\right)\right|}^{q_2}\right)}^{\frac{1}{q_2}}+{\left({\chi }_2^1\left(\mu \right)|{\mathsf{\Psi }}^{\prime }\left(c_1\right){|}^{q_2}+{\chi }_1^1\left(\mu \right){\left|{\mathsf{\Psi }}^{\prime }\left(z_2\right)\right|}^{q_2}\right)}^{\frac{1}{q_2}}\right]\right.\hfill \\ \hfill & \left.+{\stackrel{`}{\chi }}_2^{1-\frac{1}{q_2}}\left(\mu \right)\left[{\left({\chi }_3^1\left(\mu \right)|{\mathsf{\Psi }}^{\prime }\left(c_1\right){|}^{q_2}+{\chi }_4^1\left(\mu \right){\left|{\mathsf{\Psi }}^{\prime }\left(z_2\right)\right|}^{q_2}\right)}^{\frac{1}{q_2}}+{\left({\chi }_4^1\left(\mu \right)|{\mathsf{\Psi }}^{\prime }\left(c_1\right){|}^{q_2}+{\chi }_3^1\left(\mu \right){\left|{\mathsf{\Psi }}^{\prime }\left(z_2\right)\right|}^{q_2}\right)}^{\frac{1}{q_2}}\right]\right\}.\hfill \end{array}$$

The intended inequality is now obtained. □

3.3. Fractional Weighted Boole’s Inequalities for Bounded Functions

In this subsection, using the proposed fractional identity and appropriate bounded-ness conditions, several weighted fractional Boole-type inequalities are derived.

Theorem 7. 

Assume that all the conditions of Lemma 1 hold. If there exist $m,M\in \mathbb{R}$ such that $m\le {\mathsf{\Psi }}^{\prime }\left(\xi \right)\le M,\forall \xi \in [c_1,z_2]$, then

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{90}}\left[7\mathsf{\Psi }\left(c_1\right)+32\mathsf{\Psi }\left({\displaystyle \frac{3c_1+z_2}{4}}\right)+12\mathsf{\Psi }\left({\displaystyle \frac{c_1+z_2}{2}}\right)+32\mathsf{\Psi }\left({\displaystyle \frac{c_1+3z_2}{4}}\right)+7\mathsf{\Psi }\left(z_2\right)\right]\mathcal{W}\left(\mu \right)\right.\hfill \\ & \left.-{\displaystyle \frac{2^{\mu -1}\mathsf{\Gamma }\left(\mu \right)}{{(z_2-c_1)}^{\mu }}}\left[J_{{c_1}^{+}}^{\mu }\mathsf{\Psi }\omega \left({\displaystyle \frac{c_1+z_2}{2}}\right)+J_{{z_2}^{-}}^{\mu }\mathsf{\Psi }\omega \left({\displaystyle \frac{c_1+z_2}{2}}\right)\right]\right|\hfill \\ & \le {\displaystyle \frac{z_2-c_1}{4}}(M-m){\displaystyle \sum _{k=1}^2}{\stackrel{`}{\chi }}_k\left(\mu \right),\hfill \end{array}$$

(10)

where $q_2\ge 1$, and ${\stackrel{`}{\chi }}_1\left(\mu \right)$ and ${\stackrel{`}{\chi }}_2\left(\mu \right)$ are already given in Theorem 6.

Proof. 

Starting from the integral identity in Lemma 1, we have

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{90}}\left[7\mathsf{\Psi }\left(c_1\right)+32\mathsf{\Psi }\left({\displaystyle \frac{3c_1+z_2}{4}}\right)+12\mathsf{\Psi }\left({\displaystyle \frac{c_1+z_2}{2}}\right)+32\mathsf{\Psi }\left({\displaystyle \frac{c_1+3z_2}{4}}\right)+7\mathsf{\Psi }\left(z_2\right)\right]\mathcal{W}\left(\mu \right)\hfill \\ & -{\displaystyle \frac{2^{\mu -1}\mathsf{\Gamma }\left(\mu \right)}{{(z_2-c_1)}^{\mu }}}\left[J_{{c_1}^{+}}^{\mu }\mathsf{\Psi }\omega \left({\displaystyle \frac{c_1+z_2}{2}}\right)+J_{{z_2}^{-}}^{\mu }\mathsf{\Psi }\omega \left({\displaystyle \frac{c_1+z_2}{2}}\right)\right]\hfill \\ \hfill & ={\displaystyle \frac{z_2-c_1}{4}}\left\{{\int }_0^{\frac{1}{2}}{r}_1(\mu ,\xi )\left[\mathsf{\Psi }\left({\displaystyle \frac{1-\xi }{2}}{c}_1+{\displaystyle \frac{1+\xi }{2}}{z}_2\right)-{\displaystyle \frac{m+M}{2}}\right]d\xi \right.\hfill \\ & +{\int }_0^{\frac{1}{2}}{r}_1(\mu ,\xi )\left[{\displaystyle \frac{m+M}{2}}-\mathsf{\Psi }\left({\displaystyle \frac{1+\xi }{2}}{c}_1+{\displaystyle \frac{1-\xi }{2}}{z}_2\right)\right]d\xi \hfill \\ \hfill & +{\int }_{\frac{1}{2}}^1{r}_2(\mu ,\xi )\left[\mathsf{\Psi }\left({\displaystyle \frac{1-\xi }{2}}{c}_1+{\displaystyle \frac{1+\xi }{2}}{z}_2\right)-{\displaystyle \frac{m+M}{2}}\right]d\xi \hfill \\ & \left.+{\int }_{\frac{1}{2}}^1{r}_2(\mu ,\xi )\left[{\displaystyle \frac{m+M}{2}}-\mathsf{\Psi }\left({\displaystyle \frac{1+\xi }{2}}{c}_1+{\displaystyle \frac{1-\xi }{2}}{z}_2\right)\right]d\xi \right\}.\hfill \end{array}$$

Triangular inequality yields

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{90}}\left[7\mathsf{\Psi }\left(c_1\right)+32\mathsf{\Psi }\left({\displaystyle \frac{3c_1+z_2}{4}}\right)+12\mathsf{\Psi }\left({\displaystyle \frac{c_1+z_2}{2}}\right)+32\mathsf{\Psi }\left({\displaystyle \frac{c_1+3z_2}{4}}\right)+7\mathsf{\Psi }\left(z_2\right)\right]\mathcal{W}\left(\mu \right)\right.\hfill \\ & \left.-{\displaystyle \frac{2^{\mu -1}\mathsf{\Gamma }\left(\mu \right)}{{(z_2-c_1)}^{\mu }}}\left[J_{{c_1}^{+}}^{\mu }\mathsf{\Psi }\omega \left({\displaystyle \frac{c_1+z_2}{2}}\right)+J_{{z_2}^{-}}^{\mu }\mathsf{\Psi }\omega \left({\displaystyle \frac{c_1+z_2}{2}}\right)\right]\right|\hfill \\ \hfill & \le {\displaystyle \frac{z_2-c_1}{4}}\left\{{\int }_0^{\frac{1}{2}}\left|r_1(\mu ,\xi )\right||\mathsf{\Psi }\left({\displaystyle \frac{1-\xi }{2}}{c}_1+{\displaystyle \frac{1+\xi }{2}}{z}_2\right)-{\displaystyle \frac{m+M}{2}}|d\xi \right.\hfill \\ & +{\int }_0^{\frac{1}{2}}\left|r_1(\mu ,\xi )\right||{\displaystyle \frac{m+M}{2}}-\mathsf{\Psi }\left({\displaystyle \frac{1+\xi }{2}}{c}_1+{\displaystyle \frac{1-\xi }{2}}{z}_2\right)|d\xi \hfill \\ \hfill & +{\int }_{\frac{1}{2}}^1\left|r_2(\mu ,\xi )\right||\mathsf{\Psi }\left({\displaystyle \frac{1-\xi }{2}}{c}_1+{\displaystyle \frac{1+\xi }{2}}{z}_2\right)-{\displaystyle \frac{m+M}{2}}|d\xi \hfill \\ & \left.+{\int }_{\frac{1}{2}}^1\left|r_2(\mu ,\xi )\right||{\displaystyle \frac{m+M}{2}}-\mathsf{\Psi }\left({\displaystyle \frac{1+\xi }{2}}{c}_1+{\displaystyle \frac{1-\xi }{2}}{z}_2\right)|d\xi \right\}.\hfill \end{array}$$

(11)

Since $m\le {\mathsf{\Psi }}^{\prime }\left(\xi \right)\le M$ on $[c_1,z_2]$, for all $\xi \in [c_1,z_2]$, we have

$$\begin{array}{c}\left|\mathsf{\Psi }\left({\displaystyle \frac{1-\xi }{2}}{c}_1+{\displaystyle \frac{1+\xi }{2}}{z}_2\right)-{\displaystyle \frac{m+M}{2}}\right|\le {\displaystyle \frac{M-m}{2}},\hfill \end{array}$$

and

$$\begin{array}{c}\left|{\displaystyle \frac{m+M}{2}}-\mathsf{\Psi }\left({\displaystyle \frac{1-\xi }{2}}{c}_1+{\displaystyle \frac{1+\xi }{2}}{z}_2\right)\right|\le {\displaystyle \frac{M-m}{2}}.\hfill \end{array}$$

Applying upper bounds to (11), we obtain

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{90}}\left[7\mathsf{\Psi }\left(c_1\right)+32\mathsf{\Psi }\left({\displaystyle \frac{3c_1+z_2}{4}}\right)+12\mathsf{\Psi }\left({\displaystyle \frac{c_1+z_2}{2}}\right)+32\mathsf{\Psi }\left({\displaystyle \frac{c_1+3z_2}{4}}\right)+7\mathsf{\Psi }\left(z_2\right)\right]\mathcal{W}\left(\mu \right)\right.\hfill \\ & \left.-{\displaystyle \frac{2^{\mu -1}\mathsf{\Gamma }\left(\mu \right)}{{(z_2-c_1)}^{\mu }}}\left[J_{{c_1}^{+}}^{\mu }{\mathsf{\Psi }}_{\omega }\left({\displaystyle \frac{c_1+z_2}{2}}\right)+J_{{z_2}^{-}}^{\mu }{\mathsf{\Psi }}_{\omega }\left({\displaystyle \frac{c_1+z_2}{2}}\right)\right]\right|\hfill \\ \hfill & \le {\displaystyle \frac{z_2-c_1}{4}}\left\{{\int }_0^{\frac{1}{2}}\left|r_1(\mu ,\xi )\right|\left({\displaystyle \frac{M-m}{2}}\right)d\xi +{\int }_0^{\frac{1}{2}}\left|r_1(\mu ,\xi )\right|\left({\displaystyle \frac{M-m}{2}}\right)d\xi \right.\hfill \\ \hfill & \left.+{\int }_{\frac{1}{2}}^1\left|r_2(\mu ,\xi )\right|\left({\displaystyle \frac{M-m}{2}}\right)d\xi +{\int }_{\frac{1}{2}}^1\left|r_2(\mu ,\xi )\right|\left({\displaystyle \frac{M-m}{2}}\right)d\xi \right\}\hfill \\ \hfill & \le {\displaystyle \frac{z_2-c_1}{4}}\left({\int }_0^{\frac{1}{2}}|r_1(\mu ,\xi )|d\xi +{\int }_{\frac{1}{2}}^1\left|r_2(\mu ,\xi )\right|d\xi \right)(M-m)\hfill \\ & \le {\displaystyle \frac{z_2-c_1}{4}}(M-m){\displaystyle \sum _{k=1}^2}{\stackrel{`}{\chi }}_k\left(\mu \right).\hfill \end{array}$$

This completes the proof. □

3.4. $\left(\mathcal{RL}\right)$-Weighted Boole’s Inequality for L-Lipschitzian Function

This subsection provides a Boole-type inequality for the Lipschitzian function for fractional integrals.

Theorem 8. 

Assume that $|{\mathsf{\Psi }}^{\prime }|$ possesses the L-Lipschitz property and meets all the conditions of Lemma 1. Then

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{90}}\left[7\mathsf{\Psi }\left(c_1\right)+32\mathsf{\Psi }\left({\displaystyle \frac{3c_1+z_2}{4}}\right)+12\mathsf{\Psi }\left({\displaystyle \frac{c_1+z_2}{2}}\right)+32\mathsf{\Psi }\left({\displaystyle \frac{c_1+3z_2}{4}}\right)+7\mathsf{\Psi }\left(z_2\right)\right]\mathcal{W}\left(\mu \right)\right.\hfill \\ & \left.-{\displaystyle \frac{2^{\mu -1}\mathsf{\Gamma }\left(\mu \right)}{{(z_2-c_1)}^{\mu }}}\left[J_{{c_1}^{+}}^{\mu }\mathsf{\Psi }\omega \left({\displaystyle \frac{c_1+z_2}{2}}\right)+J_{{z_2}^{-}}^{\mu }\mathsf{\Psi }\omega \left({\displaystyle \frac{c_1+z_2}{2}}\right)\right]\right|\hfill \\ \hfill & \le {\displaystyle \frac{L{(z_2-c_1)}^2}{4}}\left[{\acute{\chi }}_5\left(\mu \right)+{\acute{\chi }}_6\left(\mu \right)\right],\hfill \end{array}$$

(12)

where

$$\begin{array}{c}\left\{\begin{array}{cc}\hfill & {\displaystyle {\acute{\chi }}_5\left(\mu \right)={\int }_0^{\frac{1}{2}}\left|r_1(\mu ,\xi )\right|\left|\xi \right|d\xi ,}\hfill \\ & {\displaystyle {\acute{\chi }}_6\left(\mu \right)={\int }_{\frac{1}{2}}^1\left|r_2(\mu ,\xi )\right|\left|\xi \right|d\xi .}\hfill \end{array}\right.\hfill \end{array}$$

Proof. 

Incorporating this with Lemma 1, we have

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{90}}\left[7\mathsf{\Psi }\left(c_1\right)+32\mathsf{\Psi }\left({\displaystyle \frac{3c_1+z_2}{4}}\right)+12\mathsf{\Psi }\left({\displaystyle \frac{c_1+z_2}{2}}\right)+32\mathsf{\Psi }\left({\displaystyle \frac{c_1+3z_2}{4}}\right)+7\mathsf{\Psi }\left(z_2\right)\right]\mathcal{W}\left(\mu \right)\right.\hfill \\ & \left.-{\displaystyle \frac{2^{\mu -1}\mathsf{\Gamma }\left(\mu \right)}{{(z_2-c_1)}^{\mu }}}\left[J_{{c_1}^{+}}^{\mu }\mathsf{\Psi }\omega \left({\displaystyle \frac{c_1+z_2}{2}}\right)+J_{{z_2}^{-}}^{\mu }\mathsf{\Psi }\omega \left({\displaystyle \frac{c_1+z_2}{2}}\right)\right]\right|\hfill \\ \hfill & \le {\displaystyle \frac{z_2-c_1}{4}}\left\{{\int }_0^{\frac{1}{2}}\left|r_1(\mu ,\xi )\right|\left|{\mathsf{\Psi }}^{\prime }\left({\displaystyle \frac{1-\xi }{2}}{c}_1+{\displaystyle \frac{1+\xi }{2}}{z}_2\right)-{\mathsf{\Psi }}^{\prime }\left({\displaystyle \frac{1+\xi }{2}}{c}_1+{\displaystyle \frac{1-\xi }{2}}{z}_2\right)\right|\mathrm{d}\xi \right.\hfill \\ \hfill & \left.+{\int }_{\frac{1}{2}}^1\left|r_2(\mu ,\xi )\right|\left|{\mathsf{\Psi }}^{\prime }\left({\displaystyle \frac{1-\xi }{2}}{c}_1+{\displaystyle \frac{1+\xi }{2}}{z}_2\right)-{\mathsf{\Psi }}^{\prime }\left({\displaystyle \frac{1+\xi }{2}}{c}_1+{\displaystyle \frac{1-\xi }{2}}{z}_2\right)\right|\mathrm{d}\xi \right\}.\hfill \end{array}$$

Using the Lipschitz condition of $\mathsf{\Psi }$, we have

$$\begin{array}{c}\left|\mathsf{\Psi }\left({\displaystyle \frac{1+\xi }{2}}{c}_1+{\displaystyle \frac{1-\xi }{2}}{z}_2\right)-\mathsf{\Psi }\left({\displaystyle \frac{1-\xi }{2}}{z}_2+{\displaystyle \frac{1+\xi }{2}}{c}_1\right)\right|\le L\left|\xi \right|(z_2-c_1).\hfill \end{array}$$

Thus,

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{90}}\left[7\mathsf{\Psi }\left(c_1\right)+32\mathsf{\Psi }\left({\displaystyle \frac{3c_1+z_2}{4}}\right)+12\mathsf{\Psi }\left({\displaystyle \frac{c_1+z_2}{2}}\right)+32\mathsf{\Psi }\left({\displaystyle \frac{c_1+3z_2}{4}}\right)+7\mathsf{\Psi }\left(z_2\right)\right]\mathcal{W}\left(\mu \right)\right.\hfill \\ & \left.-{\displaystyle \frac{2^{\mu -1}\mathsf{\Gamma }\left(\mu \right)}{{(z_2-c_1)}^{\mu }}}\left[J_{{c_1}^{+}}^{\mu }\mathsf{\Psi }\omega \left({\displaystyle \frac{c_1+z_2}{2}}\right)+J_{{z_2}^{-}}^{\mu }\mathsf{\Psi }\omega \left({\displaystyle \frac{c_1+z_2}{2}}\right)\right]\right|\hfill \\ \hfill & \le {\displaystyle \frac{z_2-c_1}{4}}\left\{{\int }_0^{\frac{1}{2}}\left|r_1(\mu ,\xi )\right|L\left|\xi \right|(z_2-c_1)\mathrm{d}\xi +{\int }_{\frac{1}{2}}^1\left|r_2(\mu ,\xi )\right|L\left|\xi \right|(z_2-c_1)\mathrm{d}\xi \right\}\hfill \\ \hfill & \le {\displaystyle \frac{L{(z_2-c_1)}^2}{4}}\left[{\int }_0^{\frac{1}{2}}\left|r_1(\mu ,\xi )\right|\left|\xi \right|\mathrm{d}\xi +{\int }_{\frac{1}{2}}^1\left|r_2(\mu ,\xi )\right|\left|\xi \right|\mathrm{d}\xi \right]\hfill \\ \hfill & \le {\displaystyle \frac{L{(z_2-c_1)}^2}{4}}\left[{\acute{\chi }}_5\left(\mu \right)+{\acute{\chi }}_6\left(\mu \right)\right].\hfill \end{array}$$

This completes the proof. □

Remark 3. 

For $\omega \left(\xi \right)=1$ and $\mu =1$ in Theorem 12, we obtain

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{90}}\left[7\mathsf{\Psi }\left(c_1\right)+32\mathsf{\Psi }\left({\displaystyle \frac{3c_1+z_2}{4}}\right)+12\mathsf{\Psi }\left({\displaystyle \frac{c_1+z_2}{2}}\right)+32\mathsf{\Psi }\left({\displaystyle \frac{c_1+3z_2}{4}}\right)+7\mathsf{\Psi }\left(z_2\right)\right]-{\displaystyle \frac{1}{z_2-c_1}}{\int }_{c_1}^{z_2}\mathsf{\Psi }\left(\xi \right)d\xi \right|\hfill \\ & \le {\displaystyle \frac{\mathrm{80,627}\parallel {\mathsf{\Psi }}^{\prime }{\parallel }_{\infty }{(z_2-c_1)}^2}{\mathrm{1,093,500}}}.\hfill \end{array}$$

3.5. $\left(\mathcal{RL}\right)$-Fractional Weighted Boole’s Estimate via Functions of Bounded Variation

By using the concepts of the bounded-variation function, we develop an interesting inequality via an appropriate kernel.

Theorem 9. 

Assume that function $\mathsf{\Psi }:[c_1,z_2]\to \mathbb{R}$ meets the condition of bounded variations. Then

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{90}}\left[7\mathsf{\Psi }\left(c_1\right)+32\mathsf{\Psi }\left({\displaystyle \frac{3c_1+z_2}{4}}\right)+12\mathsf{\Psi }\left({\displaystyle \frac{c_1+z_2}{2}}\right)+32\mathsf{\Psi }\left({\displaystyle \frac{c_1+3z_2}{4}}\right)+7\mathsf{\Psi }\left(z_2\right)\right]\mathcal{W}\left(\mu \right)\right.\hfill \\ & \left.-{\displaystyle \frac{2^{\mu -1}\mathsf{\Gamma }\left(\mu \right)}{{(z_2-c_1)}^{\mu }}}\left[J_{{c_1}^{+}}^{\mu }\mathsf{\Psi }\omega \left({\displaystyle \frac{c_1+z_2}{2}}\right)+J_{{z_2}^{-}}^{\mu }\mathsf{\Psi }\omega \left({\displaystyle \frac{c_1+z_2}{2}}\right)\right]\right|\hfill \\ \hfill & \le {\displaystyle \frac{1}{{(z_2-c_1)}^{\mu }}}\max \{{\displaystyle \frac{6}{45}}{(z_2-c_1)}^{\mu }\mathcal{W}\left(\mu \right),\hfill \\ \hfill & \left|{\int }_{c_1}^{\frac{c_1+z_2}{2}}{(\xi -c_1)}^{\mu -1}\omega \left(\xi \right)d\xi -{\displaystyle \frac{6}{45}}{(z_2-c_1)}^{\mu }\mathcal{W}\left(\mu \right)\right|,\hfill \\ \hfill & \left|{\int }_{c_1}^{\frac{c_1+3z_2}{4}}{(\xi -c_1)}^{\mu -1}\omega \left(\xi \right)d\xi -{\displaystyle \frac{38}{45}}{(z_2-c_1)}^{\mu }\mathcal{W}\left(\mu \right)\right|\}{\displaystyle \underset{c_1}{\overset{z_2}{\bigvee }}}\left(\mathsf{\Psi }\right).\hfill \end{array}$$

(13)

Here ${\bigvee }_{c_1}^{z_2}\left(\mathsf{\Psi }\right)$ is the total variation of Ψ on $[c_1,z_2]$, as defined in [32].

Proof. 

Define ${\mathcal{F}}_{\mu }\left(v\right)$ as the piece-wise function.

$${\mathcal{F}}_{\mu }\left(v\right)=\left\{\begin{array}{cc}{\int }_v^{z_2}{(z_2-\xi )}^{\mu -1}\omega \left(\xi \right)d\xi -{\displaystyle \frac{38}{45}}{(z_2-c_1)}^{\mu }\mathcal{W}\left(\mu \right),\hfill & c_1\le v<{\displaystyle \frac{3c_1+z_2}{4}},\hfill \\ {\int }_v^{z_2}{(z_2-\xi )}^{\mu -1}\omega \left(\xi \right)d\xi -{\displaystyle \frac{6}{45}}{(z_2-c_1)}^{\mu }\mathcal{W}\left(\mu \right),\hfill & {\displaystyle \frac{3c_1+z_2}{4}}\le v<{\displaystyle \frac{c_1+z_2}{2}},\hfill \\ {\int }_{c_1}^v{(\xi -c_1)}^{\mu -1}\omega \left(\xi \right)d\xi -{\displaystyle \frac{6}{45}}{(z_2-c_1)}^{\mu }\mathcal{W}\left(\mu \right),\hfill & {\displaystyle \frac{c_1+z_2}{2}}\le v<{\displaystyle \frac{c_1+3z_2}{4}},\hfill \\ {\int }_{c_1}^v{(\xi -c_1)}^{\mu -1}\omega \left(\xi \right)d\xi -{\displaystyle \frac{38}{45}}{(z_2-c_1)}^{\mu }\mathcal{W}\left(\mu \right),\hfill & {\displaystyle \frac{c_1+3z_2}{4}}\le v<z_2.\hfill \end{array}\right.$$

Using the Definition of ${\mathcal{F}}_{\mu }\left(v\right)$, we obtain the following identity

$$\begin{array}{cc}{\int }_{c_1}^{z_2}{\mathcal{F}}_{\mu }\left(v\right)d\mathsf{\Psi }\left(v\right)\hfill & ={\int }_{c_1}^{\frac{3c_1+z_2}{4}}\left({\int }_v^{z_2}{(z_2-\xi )}^{\mu -1}\omega \left(\xi \right)d\xi -{\displaystyle \frac{38}{45}}{(z_2-c_1)}^{\mu }\mathcal{W}\left(\mu \right)\right)d\mathsf{\Psi }\left(v\right)\hfill \\ \hfill & +{\int }_{\frac{3c_1+z_2}{4}}^{\frac{c_1+z_2}{2}}\left({\int }_v^{z_2}{(z_2-\xi )}^{\mu -1}\omega \left(\xi \right)d\xi -{\displaystyle \frac{6}{45}}{(z_2-c_1)}^{\mu }\mathcal{W}\left(\mu \right)\right)d\mathsf{\Psi }\left(v\right)\hfill \\ & +{\int }_{\frac{c_1+z_2}{2}}^{\frac{c_1+3z_2}{4}}\left({\int }_{c_1}^v{(\xi -c_1)}^{\mu -1}\omega \left(\xi \right)d\xi -{\displaystyle \frac{6}{45}}{(z_2-c_1)}^{\mu }\mathcal{W}\left(\mu \right)\right)d\mathsf{\Psi }\left(v\right)\hfill \\ \hfill & +{\int }_{\frac{c_1+3z_2}{4}}^{z_2}\left({\int }_{c_1}^v{(\xi -c_1)}^{\mu -1}\omega \left(\xi \right)d\xi -{\displaystyle \frac{38}{45}}{(z_2-c_1)}^{\mu }\mathcal{W}\left(\mu \right)\right)d\mathsf{\Psi }\left(v\right).\hfill \end{array}$$

Integration by parts yields the following expression

$$\begin{array}{cc}{\int }_{c_1}^{z_2}{\mathcal{F}}_{\mu }\left(v\right)d\mathsf{\Psi }\left(v\right)\hfill & =\left({\int }_v^{z_2}{(z_2-\xi )}^{\mu -1}\omega \left(\xi \right)d\xi -{\displaystyle \frac{38}{45}}{(z_2-c_1)}^{\mu }\mathcal{W}\left(\mu \right)\right)\mathsf{\Psi }\left(v\right){|}_{c_1}^{\frac{3c_1+z_2}{4}}\hfill \\ \hfill & -{\int }_{c_1}^{\frac{3c_1+z_2}{4}}{(z_2-v)}^{\mu -1}\omega \left(v\right)\mathsf{\Psi }\left(v\right)dv\hfill \\ \hfill & +\left({\int }_v^{z_2}{(z_2-\xi )}^{\mu -1}\omega \left(\xi \right)d\xi -{\displaystyle \frac{6}{45}}{(z_2-c_1)}^{\mu }\mathcal{W}\left(\mu \right)\right)\mathsf{\Psi }\left(v\right){|}_{\frac{3c_1+z_2}{4}}^{\frac{c_1+z_2}{2}}\hfill \\ \hfill & -{\int }_{\frac{3c_1+z_2}{4}}^{\frac{c_1+z_2}{2}}{(z_2-v)}^{\mu -1}\omega \left(v\right)\mathsf{\Psi }\left(v\right)dv\hfill \\ & +\left({\int }_{c_1}^v{(\xi -c_1)}^{\mu -1}\omega \left(\xi \right)d\xi -{\displaystyle \frac{6}{45}}{(z_2-c_1)}^{\mu }\mathcal{W}\left(\mu \right)\right)\mathsf{\Psi }\left(v\right){|}_{\frac{c_1+z_2}{2}}^{\frac{c_1+3z_2}{4}}\hfill \\ \hfill & -{\int }_{\frac{c_1+z_2}{2}}^{\frac{c_1+3z_2}{4}}{(v-c_1)}^{\mu -1}\omega \left(v\right)\mathsf{\Psi }\left(v\right)dv\hfill \\ \hfill & +\left({\int }_{c_1}^v{(\xi -c_1)}^{\mu -1}\omega \left(\xi \right)d\xi -{\displaystyle \frac{38}{45}}{(z_2-c_1)}^{\mu }\mathcal{W}\left(\mu \right)\right)\mathsf{\Psi }\left(v\right){|}_{\frac{c_1+3z_2}{4}}^{z_2}\hfill \\ \hfill & -{\int }_{\frac{c_1+3z_2}{4}}^{z_2}{(v-c_1)}^{\mu -1}\omega \left(v\right)\mathsf{\Psi }\left(v\right)dv.\hfill \end{array}$$

Simplifying the aforementioned expression, we obtain

$$\begin{array}{cc}\hfill & {\int }_{c_1}^{z_2}{\mathcal{F}}_{\mu }\left(v\right)d\mathsf{\Psi }\left(v\right)\hfill \\ & ={\displaystyle \frac{1}{90}}\left[7\mathsf{\Psi }\left(c_1\right)+32\mathsf{\Psi }\left({\displaystyle \frac{3c_1+z_2}{4}}\right)+12\mathsf{\Psi }\left({\displaystyle \frac{c_1+z_2}{2}}\right)+32\mathsf{\Psi }\left({\displaystyle \frac{c_1+3z_2}{4}}\right)+7\mathsf{\Psi }\left(z_2\right)\right]\mathcal{W}\left(\mu \right)\hfill \\ & -{\displaystyle \frac{2^{\mu -1}\mathsf{\Gamma }\left(\mu \right)}{{(z_2-c_1)}^{\mu }}}\left[J_{{c_1}^{+}}^{\mu }\mathsf{\Psi }\omega \left({\displaystyle \frac{c_1+z_2}{2}}\right)+J_{{z_2}^{-}}^{\mu }\mathsf{\Psi }\omega \left({\displaystyle \frac{c_1+z_2}{2}}\right)\right].\hfill \end{array}$$

(14)

It is known that functions $\phi ,\mathsf{\Psi }:[c_1,z_2]\to \mathbb{R}$ are such that $\phi$ is continuous on $[c_1,z_2]$ and $\mathsf{\Psi }$ is of bounded variation on $[c_1,z_2]$, so we have

$$\begin{array}{c}\left|{\int }_{c_1}^{z_2}\phi \left(\xi \right)d\mathsf{\Psi }\left(\xi \right)\right|\le \underset{\xi \in [c_1,z_2]}{\sup }\left|\phi \left(\xi \right)\right|{\displaystyle \underset{c_1}{\overset{z_2}{\bigvee }}}\left(\mathsf{\Psi }\right).\hfill \end{array}$$

(15)

Using (15) in (14), we obtain

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{90}}\left[7\mathsf{\Psi }\left(c_1\right)+32\mathsf{\Psi }\left({\displaystyle \frac{3c_1+z_2}{4}}\right)+12\mathsf{\Psi }\left({\displaystyle \frac{c_1+z_2}{2}}\right)+32\mathsf{\Psi }\left({\displaystyle \frac{c_1+3z_2}{4}}\right)+7\mathsf{\Psi }\left(z_2\right)\right]\mathcal{W}\left(\mu \right)\right.\hfill \\ & \left.-{\displaystyle \frac{2^{\mu -1}\mathsf{\Gamma }\left(\mu \right)}{{(z_2-c_1)}^{\mu }}}\left[J_{{c_1}^{+}}^{\mu }\mathsf{\Psi }\omega \left({\displaystyle \frac{c_1+z_2}{2}}\right)+J_{{z_2}^{-}}^{\mu }\mathsf{\Psi }\omega \left({\displaystyle \frac{c_1+z_2}{2}}\right)\right]\right|\hfill \\ \hfill & \le \left|{\int }_{c_1}^{z_2}\mathcal{F}\left(v\right)d\mathsf{\Psi }\left(v\right)\right|\hfill \\ & =\left|{\int }_{c_1}^{\frac{3c_1+z_2}{4}}\left[{\int }_v^{z_2}{(z_2-\xi )}^{\mu -1}\omega \left(\xi \right)d\xi -{\displaystyle \frac{38}{45}}{(z_2-c_1)}^{\mu }\mathcal{W}\left(\mu \right)\right]d\mathsf{\Psi }\left(v\right)\right|\hfill \\ \hfill & +\left|{\int }_{\frac{3c_1+z_2}{4}}^{\frac{c_1+z_2}{2}}\left[{\int }_v^{z_2}{(z_2-\xi )}^{\mu -1}\omega \left(\xi \right)d\xi -{\displaystyle \frac{6}{45}}{(z_2-c_1)}^{\mu }\mathcal{W}\left(\mu \right)\right]d\mathsf{\Psi }\left(v\right)\right|\hfill \\ & +\left|{\int }_{\frac{c_1+z_2}{2}}^{\frac{c_1+3z_2}{4}}\left[{\int }_{c_1}^v{(\xi -c_1)}^{\mu -1}\omega \left(\xi \right)d\xi -{\displaystyle \frac{6}{45}}{(z_2-c_1)}^{\mu }\mathcal{W}\left(\mu \right)\right]d\mathsf{\Psi }\left(v\right)\right|\hfill \\ \hfill & +\left|{\int }_{\frac{c_1+3z_2}{4}}^{z_2}\left[{\int }_{c_1}^v{(\xi -c_1)}^{\mu -1}\omega \left(\xi \right)d\xi -{\displaystyle \frac{38}{45}}{(z_2-c_1)}^{\mu }\mathcal{W}\left(\mu \right)\right]d\mathsf{\Psi }\left(v\right)\right|\hfill \\ & \le \underset{\xi \in [c_1,\frac{3c_1+z_2}{4}]}{\sup }\left|{\int }_v^{z_2}{(z_2-\xi )}^{\mu -1}\omega \left(\xi \right)d\xi -{\displaystyle \frac{38}{45}}{(z_2-c_1)}^{\mu }\mathcal{W}\left(\mu \right)\right|{\displaystyle \underset{c_1}{\overset{\frac{3c_1+z_2}{4}}{\bigvee }}}\left(\mathsf{\Psi }\right)\hfill \\ & +\underset{\xi \in [\frac{3c_1+z_2}{4},{\displaystyle \frac{c_1+z_2}{2}}]}{\sup }\left|{\int }_v^{z_2}{(z_2-\xi )}^{\mu -1}\omega \left(\xi \right)d\xi -{\displaystyle \frac{6}{45}}{(z_2-c_1)}^{\mu }\mathcal{W}\left(\mu \right)\right|{\displaystyle \underset{\frac{3c_1+z_2}{4}}{\overset{\frac{c_1+z_2}{2}}{\bigvee }}}\left(\mathsf{\Psi }\right)\hfill \\ & +\underset{\xi \in [\frac{c_1+z_2}{2},\frac{c_1+3z_2}{4}]}{\sup }\left|{\int }_{c_1}^v{(\xi -c_1)}^{\mu -1}\omega \left(\xi \right)d\xi -{\displaystyle \frac{6}{45}}{(z_2-c_1)}^{\mu }\mathcal{W}\left(\mu \right)\right|{\displaystyle \underset{\frac{c_1+z_2}{2}}{\overset{\frac{c_1+3z_2}{4}}{\bigvee }}}\left(\mathsf{\Psi }\right)\hfill \\ & +\underset{\xi \in [\frac{c_1+3z_2}{4},z_2]}{\sup }\left|{\int }_{c_1}^v{(\xi -c_1)}^{\mu -1}\omega \left(\xi \right)d\xi -{\displaystyle \frac{38}{45}}{(z_2-c_1)}^{\mu }\mathcal{W}\left(\mu \right)\right|{\displaystyle \underset{\frac{c_1+3z_2}{4}}{\overset{z_2}{\bigvee }}}\left(\mathsf{\Psi }\right)\hfill \\ & =max\left|{\int }_{c_1}^{\frac{3c_1+z_2}{4}}{(z_2-\xi )}^{\mu -1}\omega \left(\xi \right)d\xi -{\displaystyle \frac{38}{45}}{(z_2-c_1)}^{\mu }\mathcal{W}\left(\mu \right)\right|{\displaystyle \underset{c_1}{\overset{\frac{3c_1+z_2}{4}}{\bigvee }}}\left(\mathsf{\Psi }\right)\hfill \\ & +max\left|{\int }_{\frac{3c_1+z_2}{4}}^{\frac{c_1+z_2}{2}}{(z_2-\xi )}^{\mu -1}\omega \left(\xi \right)d\xi -{\displaystyle \frac{6}{45}}{(z_2-c_1)}^{\mu }\mathcal{W}\left(\mu \right)\right|{\displaystyle \underset{\frac{3c_1+z_2}{4}}{\overset{\frac{c_1+z_2}{2}}{\bigvee }}}\left(\mathsf{\Psi }\right)\hfill \\ & +max\left|{\int }_{\frac{c_1+z_2}{2}}^{\frac{c_1+3z_2}{4}}{(\xi -c_1)}^{\mu -1}\omega \left(\xi \right)d\xi -{\displaystyle \frac{6}{45}}{(z_2-c_1)}^{\mu }\mathcal{W}\left(\mu \right)\right|{\displaystyle \underset{\frac{c_1+z_2}{2}}{\overset{\frac{c_1+3z_2}{4}}{\bigvee }}}\left(\mathsf{\Psi }\right)\hfill \\ & +max\left|{\int }_{\frac{c_1+3z_2}{4}}^{z_2}{(\xi -c_1)}^{\mu -1}\omega \left(\xi \right)d\xi -{\displaystyle \frac{38}{45}}{(z_2-c_1)}^{\mu }\mathcal{W}\left(\mu \right)\right|{\displaystyle \underset{\frac{c_1+3z_2}{4}}{\overset{z_2}{\bigvee }}}\left(\mathsf{\Psi }\right)\hfill \\ \hfill & \le {\displaystyle \frac{1}{{(z_2-c_1)}^{\mu }}}\max \left\{{\displaystyle \frac{6}{45}}{(z_2-c_1)}^{\mu }\mathcal{W}\left(\mu \right),\right.\hfill \\ \hfill & \left|{\int }_{c_1}^{\frac{c_1+z_2}{2}}{(\xi -c_1)}^{\mu -1}\omega \left(\xi \right)d\xi -{\displaystyle \frac{6}{45}}{(z_2-c_1)}^{\mu }\mathcal{W}\left(\mu \right)\right|,\hfill \\ \hfill & \left.\left|{\int }_{c_1}^{\frac{c_1+3z_2}{4}}{(\xi -c_1)}^{\mu -1}\omega \left(\xi \right)d\xi -{\displaystyle \frac{38}{45}}{(z_2-c_1)}^{\mu }\mathcal{W}\left(\mu \right)\right|\right\}{\displaystyle \underset{c_1}{\overset{z_2}{\bigvee }}}\left(\mathsf{\Psi }\right).\hfill \end{array}$$

(16)

The required result is now achieved. □

Corollary 5. 

If $\mu =1$ in Theorem 9, then the following Boole-type inequality holds

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{90}}\left[7\mathsf{\Psi }\left(c_1\right)+32\mathsf{\Psi }\left({\displaystyle \frac{3c_1+z_2}{4}}\right)+12\mathsf{\Psi }\left({\displaystyle \frac{c_1+z_2}{2}}\right)+32\mathsf{\Psi }\left({\displaystyle \frac{c_1+3z_2}{4}}\right)+7\mathsf{\Psi }\left(z_2\right)\right]\right.\hfill \\ & \left.\times {\displaystyle \frac{2}{z_2-c_1}}{\int }_{\frac{c_1+z_2}{2}}^{z_2}\omega \left(\xi \right)d\xi -{\displaystyle \frac{2}{z_2-c_1}}{\int }_{c_1}^{z_2}\mathsf{\Psi }\left(\xi \right)\omega \left(\xi \right)d\xi \right|\hfill \\ & \le \max \left\{{\displaystyle \frac{6}{45}}{\int }_{c_1}^{z_2}\omega \left(u\right)du,\left|{\int }_{c_1}^{\frac{c_1+z_2}{2}}\omega \left(\xi \right)d\xi -{\displaystyle \frac{6}{45}}{\int }_{c_1}^{z_2}\omega \left(u\right)du\right|,\left|{\int }_{c_1}^{\frac{c_1+3z_2}{4}}\omega \left(\xi \right)d\xi -{\displaystyle \frac{38}{45}}{\int }_{c_1}^{z_2}\omega \left(u\right)du\right|\right\}{\displaystyle \underset{c_1}{\overset{z_2}{\bigvee }}}\left(\mathsf{\Psi }\right).\hfill \end{array}$$

Remark 4 

([33]). If $\mathsf{\Psi }:[c_1,z_2]\to \mathbb{R}$ is a convex function then f satisfies the Lipschitzian, absolute continuity and bounded conditions.

4. Error Approximations of Some Special Kinds of Integrals

For different choices of weight functions, we present the error approximations of various renowned classes of integrals.

4.1. Logarithmic Weight

Let $\mathsf{\Psi }:(0,1)\to \mathbb{R}$ be a derivable function whose derivative is bounded with the finite integral ${\displaystyle {\int }_0^1\mathsf{\Psi }\left(\xi \right)ln\left(\frac{1}{\xi }\right)d\xi }$, then

$$\begin{array}{cc}\hfill & |{\displaystyle \frac{1}{90}}\left[\mathsf{\Psi }\left(0\right)+32\mathsf{\Psi }\left({\displaystyle \frac{1}{4}}\right)+12\mathsf{\Psi }\left(\frac{1}{2}\right)+32\mathsf{\Psi }\left({\displaystyle \frac{3}{4}}\right)+\mathsf{\Psi }\left(1\right)\right]-{\int }_0^1\mathsf{\Psi }\left(\xi \right)ln\left({\displaystyle \frac{1}{\xi }}\right)d\xi |\hfill \\ & \le \left|\right|{\mathsf{\Psi }}^{\prime }{\left|\right|}_{\infty }\left(0.044\right),\hfill \end{array}$$

(17)

for all $x\in (0,1)$. Particularly for $\mathsf{\Psi }\left(x\right)=x$, we get the following relation from (17)

$${\displaystyle \frac{0.055}{2}}<0.044.$$

4.2. Jacobi Weight

Let $\mathsf{\Psi }:(0,1)\to \mathbb{R}$ be a derivable function whose derivative is bounded with the finite integral ${\displaystyle {\int }_0^1\frac{\mathsf{\Psi }\left(\xi \right)}{\sqrt{\xi }}d\xi }$, then

$$\begin{array}{cc}\hfill & |{\displaystyle \frac{1}{90}}\left[\mathsf{\Psi }\left(0\right)+32\mathsf{\Psi }\left({\displaystyle \frac{1}{4}}\right)+12\mathsf{\Psi }\left(\frac{1}{2}\right)+32\mathsf{\Psi }\left({\displaystyle \frac{3}{4}}\right)+\mathsf{\Psi }\left(1\right)\right]-{\displaystyle \frac{1}{2}}{\int }_0^1{\displaystyle \frac{\mathsf{\Psi }\left(\xi \right)}{\sqrt{\xi }}}d\xi |\hfill \\ & \le \left|\right|{\mathsf{\Psi }}^{\prime }{\left|\right|}_{\infty }\left(0.1344\right),\hfill \end{array}$$

(18)

for all $x\in (0,1)$. Particularly for $\mathsf{\Psi }\left(x\right)=x$, we get the following relation from (18)

$$0.006<0.268.$$

4.3. Chebyshev Weight

Let $\mathsf{\Psi }:(-1,1)\to \mathbb{R}$ be a derivable function whose derivative is bounded with finite integral ${\displaystyle {\int }_{-1}^1\frac{\mathsf{\Psi }\left(\xi \right)}{\sqrt{1-{\xi }^2}}d\xi }$, then

$$\begin{array}{cc}\hfill & |{\displaystyle \frac{1}{90}}\left[\mathsf{\Psi }(-1)+32\mathsf{\Psi }\left({\displaystyle \frac{-1}{2}}\right)+12\mathsf{\Psi }\left(0\right)+32\mathsf{\Psi }\left(\frac{1}{2}\right)+\mathsf{\Psi }\left(1\right)\right]-{\displaystyle \frac{1}{\pi }}{\int }_{-1}^1{\displaystyle \frac{\mathsf{\Psi }\left(\xi \right)}{\sqrt{1-{\xi }^2}}}d\xi |\hfill \\ & \le {\displaystyle \frac{\left|\right|{\mathsf{\Psi }}^{\prime }{\left|\right|}_{\infty }}{\pi }}\left(0.0628\right),\hfill \end{array}$$

(19)

for all $x\in (-1,1)$. Particularly for $\mathsf{\Psi }\left(x\right)=x$, we get the following relation from (19)

$$0<{\displaystyle \frac{0.0628}{\pi }}.$$

4.4. Application to Means

To derive several inequalities associated with special means, we recall the following classical means of two positive real numbers:

• ${\displaystyle A(c_1,z_2)=\frac{c_1+z_2}{2};}$
• ${\displaystyle L_n(c_1,z_2)={\left[\frac{{z_2}^{n+1}-{c_1}^{n+1}}{(z_2-c_1)(n+1)}\right]}^{\frac{1}{n}},n\in \mathbb{Z}\setminus \{0,-1\};}$
• ${\displaystyle A_{\omega }(c_1,z_2;{\omega }_1,{\omega }_2)=\frac{{\omega }_1{c}_1+{\omega }_2{z}_2}{{\omega }_1+{\omega }_2}.}$

The quantities are called the arithmetic mean, the generalized logarithmic mean, and the weighted arithmetic mean, respectively.

Proposition 1. 

Let $c_1,z_2>$ 0 and $q_2\ge$ 1. Then we have the following inequality

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{90}}\left[14A({c_1}^4,{z_2}^4)+32A_{\omega }^4(c_1,z_2;3,1)+32A_{\omega }^4(c_1,z_2;1,3)+12A^4(c_1,z_2)\right]-L_4^4(c_1,z_2)\right|\hfill \\ & \le {\displaystyle \frac{717(z_2-c_1)}{3240}}\left[A\left(\right|{c_1}^3|,|{z_2}^3\left|\right)\right].\hfill \end{array}$$

Proof. 

The assertion follows from Theorem 4 for $\mathsf{\Psi }\left(x\right)=x^4$, $\omega \left(\xi \right)=1$, and $\mu =1$. □

4.5. Composite Weighted Fractional Inequality

Theorem 10. 

Consider a partition $\Pi :c_1=h_0<h_1<h_2<\dots <h_{\iota }<h_{\iota +1}<\dots h_n=z_2$, where $[h_{\iota },h_{\iota +1}]$ is a subinterval in $[c_1,z_2]$ for $\iota =0,1,2\dots n-1$. Then, we have

$$\begin{array}{c}{\displaystyle \frac{2^{\mu -1}\mathsf{\Gamma }(\mu +1)}{{(z_2-c_1)}^{\mu }}}\left[J_{{c_1}^{+}}^{\mu }\mathsf{\Psi }\left({\displaystyle \frac{c_1+z_2}{2}}\right)+J_{{z_2}^{-}}^{\mu }\mathsf{\Psi }\left({\displaystyle \frac{c_1+z_2}{2}}\right)\right]=\mathcal{K}(\mathsf{\Psi },h)+\mathcal{N}(\mathsf{\Psi },h),\hfill \end{array}$$

where

$$\begin{array}{c}\mathcal{K}(\mathsf{\Psi },h)={\displaystyle \frac{1}{90}}{\displaystyle \sum _{\iota =0}^{n-1}}\left[7{\gamma }_{\iota }\mathsf{\Psi }\left(h_{\iota }\right)+32{\gamma }_{\iota }\mathsf{\Psi }\left({\displaystyle \frac{3h_{\iota }+h_{\iota +1}}{4}}\right)+12{\gamma }_{\iota }\mathsf{\Psi }\left({\displaystyle \frac{h_{\iota }+h_{\iota +1}}{2}}\right)+32{\gamma }_{\iota }\mathsf{\Psi }\left({\displaystyle \frac{h_{\iota }+3h_{\iota +1}}{4}}\right)+7{\gamma }_{\iota }\mathsf{\Psi }\left(h_{\iota +1}\right)\right],\hfill \end{array}$$

and $\mathcal{N}(\mathsf{\Psi },h)$ is the remainder value.

By Theorem 7, we get the following bound

$$\begin{array}{cc}\hfill & \left|\mathcal{N}\right(\mathsf{\Psi },h\left)\right|\hfill \\ \hfill & \le {\displaystyle \frac{(M-m)}{4}}{\displaystyle \sum _{\iota =0}^{n-1}}{(h_{\iota +1}-h_{\iota })}^2\left[{\int }_0^{\frac{1}{2}}\left|{\int }_0^{\xi }\omega \left(\xi \right)d\xi -{\displaystyle \frac{6{\gamma }_{\iota }}{45(h_{\iota +1}-h_{\iota })}}\right|d\xi \right.\hfill \\ & \left.+{\int }_{\frac{1}{2}}^1\left|{\int }_0^{\xi }\omega \left(\xi \right)d\xi -{\displaystyle \frac{38{\gamma }_{\iota }}{45(h_{\iota +1}-h_{\iota })}}\right|d\xi \right].\hfill \end{array}$$

Proof. 

The result follows directly from Theorem 7. □

Remark 5. 

It is already revealed that under the same regularity assumptions, Boole’s quadrature can give better higher-order approximate values for definite integrals than Simpson-type quadratures. Furthermore, the composite version of a quadrature rule is often used to decrease the approximation error. The $\left(\mathcal{RL}\right)$-fractional integral operators occur in systems with nonlocal and memory-dependent behaviour such as anomalous diffusion, fractional viscoelasticity and fractional-order control systems. It is not always easy to numerically evaluate their solutions due to the weakly singular power-law kernel and the often missing closed-form solutions. In the present context the composite fractional Boole rule is a useful tool for the numerical approximation of such a fractional integral with better accuracy. The novelty of our result gives upper bounds on the approximation error, provided that the absolute value of the first derivative is s-convex. Our bounds are obtained under much weaker first-order differentiability assumptions as compared to the classical Boole error inequality.

5. Numerical Experiments and Machine Learning Validation

Solving the fractional integral inequalities analytically for different values of fractional order is very difficult, except for some simple polynomial functions. To resolve this problem and to reduce the computational cost, we have used machine learning techniques. This section provides a computational study that complements the analytical development of Theorem 4 and illustrates the practical application of the proposed framework. The study has two main goals: first, to numerically verify the validity of the weighted fractional Boole-type inequality for a large class of parameters and function classes, and second, to develop data-driven models for predicting the exact $\mathrm{LHS}$ of the inequality.

The computational framework consists of three steps: mathematical formulation, numerical evaluation, and data-driven modeling, as shown in Figure 3. During the first stage the analytical quantities corresponding to the inequality are checked over a family of strictly convex test functions, and the inequality is tested sample-wise. In the second stage, we use two complementary learning paradigms to approximate the nonlinear structure of the exact fractional Boole-type expression: Symbolic Regression, which is known for its interpretability and ability to generate explicit closed-form approximation solutions, and XGBoost, which is known for its high accuracy when applied to nonlinear tabular data.

The computational study uses a synthetic dataset of 5400 valid samples described directly from the proposed weighted fractional Boole-type inequality framework. Each sample is determined by random sampling of the interval endpoints $c_1$ and $z_2$ ($z_2>c_1$), random sampling of the fractional order $\mu \in (0,1]$ and selection of one of six convex benchmark functions by a function identifier. The benchmark functions

$$\begin{array}{c}\mathsf{\Psi }\left(x\right)\in \{(1+x^4)e^{2x},e^x\cosh \left(3x\right),e^{x^2+x^3},e^{x+x^4},e^{3x+x^2}\cosh \left(x\right),(1+x^2)e^{x+x^2}\}\hfill \end{array}$$

are carefully selected to cover exponential and hyperbolic growth and the inequality is studied across a variety of nonlinear growth. The numerical values of the exact $\mathrm{LHS}$ and the upper bound $\mathrm{RHS}$ given in Theorem 4 are evaluated for each of the generated samples. The sampling parameters, the evaluation of the endpoint and midpoint function values, the evaluation of the endpoint derivatives, the transformed interval quantities, and the fractional integral components are recorded in the dataset. The exact $\mathrm{LHS}$ is kept, and the $\mathrm{RHS}$ and the slack variable

$$\begin{array}{c}\mathrm{Slack}=\mathrm{RHS}-\mathrm{LHS}\hfill \end{array}$$

are stored as auxiliary quantities used for theorem validation. There are 17 variables out of a total of 20 variables that are registered and used as predictors for the ML models.

Table 2. Summary of the synthetic dataset generated from the proposed weighted fractional Boole-type inequality framework.

Item	Description
Number of samples	5400 valid synthetic samples
Dataset size	20 variables
Function families	Six convex test functions: $(1+x^4)e^{2x}$, $e^x\cosh \left(3x\right)$, $e^{x^2+x^3}$, $e^{x+x^4}$, $e^{3x+x^2}\cosh \left(x\right)$, $(1+x^2)e^{x+x^2}$
Fractional order range	$0<\mu \le 1$
Interval construction	Randomly sampled endpoints $(c_1,z_2)$ with $z_2>c_1$
Predictor variables	17 features (sampling, function, derivative, and fractional quantities)
Target variable	$\mathrm{LHS}$ and $\mathrm{RHS}$ of Theorem 4
Auxiliary outputs	$\mathrm{LHS}$, $\mathrm{RHS}$ and slack $=\mathrm{RHS}-\mathrm{LHS}$

and

Table 3. Feature representation employed for predictive modeling of the weighted fractional Boole-type expression.

Category	Variables	Description
Sampling variables	$c_1,z_2,\mu ,\mathrm{function}\text{\_}\mathrm{id}$	Interval endpoints, fractional order, and function identifier
Function evaluations	$\mathsf{\Psi }\left(c_1\right),\mathsf{\Psi }\left(z_2\right),{\mathsf{\Psi }}^{\prime }\left(c_1\right),{\mathsf{\Psi }}^{\prime }\left(z_2\right),\mathsf{\Psi }\left(m\right)$	Endpoint and midpoint function evaluations
Derived interval terms	$h,m,h^{\mu },\log \left(h\right)$	Interval transformations and fractional quantities
Fractional components	$X_1,X_2,X_1+X_2$	Theorem-based fractional integral components
Derivative structure	${\mathsf{\Psi }}_{sum}^{\prime }$	Sum of endpoint derivative magnitudes
Target variable	$\mathrm{LHS}$, $\mathrm{RHS}$	Exact fractional Boole-type expression

provide a summary of the structure of the dataset and the representation of features.

The resulting dataset is first numerically validated against Theorem 4. The condition $\mathrm{LHS}\le \mathrm{RHS}$, called admissibility, is checked for each pair that is computed. The results show that the inequality holds for all 5400 generated samples. The global validation is shown in Figure 4. All sampled points lie on or above the diagonal line $\mathrm{LHS}=\mathrm{RHS}$, which shows that the numerical consistency and robustness of the proposed bound is maintained throughout the computational domain.

The behaviour of the exact $\mathrm{LHS}$ and analytical $\mathrm{RHS}$ is investigated for different values of $\mu \in (0,1]$ to understand the sensitivity of the inequality to the fractional integration order for each of the six convex test functions, as shown in Figure 5. In all cases, the inequality holds for all admissible values of $\mu$, while the variation and size of both sides depend on the nonlinear growth of the function. The higher-order derivative behaviour gives the bound a greater tightness, with rapidly growing functions like $e^{x+x^4}$ and $e^{3x+x^2}\cosh \left(x\right)$ having significantly larger ranges and dispersions than the more moderate functions like $(1+x^2)e^{x+x^2}$ and $(1+x^4)e^{2x}$.

The tightness and stability of the proposed bound are quantified using the slack variable. A non-negative slack over all the samples is equivalent to the global satisfaction of the inequality, whereas the distribution of the slack gives an indication of how conservative the bound is for each function class. The empirical distributions of the slack are given in Figure 6.

After the numerical correctness of the inequality has been established, the focus shifts to the predictive modeling of the exact $\mathrm{LHS}$ and $\mathrm{RHS}$, which are regarded as the regression targets noted as y. The data is divided into a training set and a test set in $80:20$ ratios, with 17 predictor variables listed in Table 3. They are used as inputs to the model. Symbolic Regression performs a search over the space of analytical expressions in order to discover a closed-form approximation to the $\mathrm{LHS}$, $\widehat{y}=\mathsf{\Psi }\left(\mathbf{x}\right)$, that optimizes a dynamic trade-off between accuracy and structural complexity, thereby revealing explicitly the dependence of the $\mathrm{LHS}$ on the underlying variables. In contrast, XGBoost builds an additive ensemble of regression trees,

$$\begin{array}{c}{\widehat{y}}_i={\displaystyle \sum _{k=1}^K}{\mathsf{\Psi }}_k\left({\mathbf{x}}_i\right),{\mathsf{\Psi }}_k\in \mathcal{F},\hfill \end{array}$$

where $\mathcal{F}$ is the space of regression trees and K is the number of boosting rounds, with each successive tree being fit to the residual of the previous ones. The additive structure in this model makes it stage-wise, which makes Symbolic Regression particularly suited to fit the strong nonlinearity in the fractional Boole-type expression.

The performance of the models is evaluated using the Root Mean Squared Error (RMSE), the Mean Absolute Error (MAE) and coefficient of determination $R^2$, as given below:

$$\begin{array}{cc}\mathrm{RMSE}\hfill & =\sqrt{{\displaystyle \frac{1}{n}}\sum _{i=1}^n{\left(y_i-{\widehat{y}}_i\right)}^2},\mathrm{MAE}={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n}\left|y_i-{\widehat{y}}_i\right|,\hfill \\ \hfill R^2& =1-{\displaystyle \frac{{\sum }_{i=1}^n{\left(y_i-{\widehat{y}}_i\right)}^2}{{\sum }_{i=1}^n{\left(y_i-\overline{y}\right)}^2}},\hfill \end{array}$$

where $y_i$ is the desired target and ${\widehat{y}}_i$ is the predicted target, $\overline{y}$ is the mean of the observed targets, and n is the number of samples. Lower RMSE and MAE values indicate better accuracy and an $R^2$ value closer to one indicates stronger correspondence between the prediction and observation. Five-fold cross-validation is also performed to check the generalization ability of the boosting model.

Table 4. Comparative predictive performance of the Symbolic Regression and XGBoost models on the held-out test set for $\mathrm{LHS}$.

Model	RMSE	MAE	R2
Symbolic Regression	0.0465	0.0123	0.99924
XGBoost	0.1282	0.0752	0.99419

and

Table 5. Comparative predictive performance of the Symbolic Regression and XGBoost models on the held-out test set for $\mathrm{RHS}$.

Model	RMSE	MAE	R2
Symbolic Regression	0.0097	0.0031	0.99998
XGBoost	0.1211	0.0772	0.99714

report the quantitative performance of both models on the held-out test set. While each model successfully captures the nonlinear behaviour of the weighted fractional Boole-type expression, Symbolic Regression outperforms XGBoost across all evaluation metrics. For $\mathrm{LHS}$, Symbolic Regression attains $R^2=0.99924$ with RMSE and MAE values of $0.0465$ and $0.0123$, compared with $R^2=0.99419$, RMSE $=0.1282$, and MAE $=0.0752$ for XGBoost. Also for $\mathrm{RHS}$, Symbolic Regression attains $R^2=0.99998$ with RMSE and MAE values of $0.0097$ and $0.0031$, compared with $R^2=0.99714$, RMSE $=0.1211$, and MAE $=0.0772$ for XGBoost. The slack distribution for Symbolic Regression attains $R^2=0.99997$ with RMSE and MAE values of $0.0127$ and $0.0050$, compared with $R^2=0.99708$, RMSE $=0.1206$, and MAE $=0.0745$ for XGBoost, which confirms the strong generalization ability of the Symbolic Regression.

These error metrics are compared in Figure 7 and Figure 8. The RMSE–MAE comparison shows a significant improvement in prediction error when using Symbolic Regression and the $R^2$ comparison shows that the Symbolic Regression model closely agrees with the exact value.

The comparisons between the approximated and exact values of the $\mathrm{LHS}$ and $\mathrm{RHS}$ of Theorem 4 for all six test functions are presented in Figure 9. Clearly, the result is validated through line graphs using exact and approximated values through Symbolic Regression.

Figure 10 and Figure 11 further illustrate the validity of the $\mathrm{LHS}$ and $\mathrm{RHS}$ of Theorem 4 for all admissible functions using exact and XGBoost approximations.

The error histograms of both sides across all test functions using XGboost approximations with log transformation of residues and counts on the horizontal and vertical axis, respectively, are given in Figure 12. Most of the residues are centered around zero for all functions.

SHAP is a widely employed technique to assess the impact of each contributor on the prediction process following the game theory. Mainly, it splits the credit of each prediction over all features. Moreover it provides detailed information about the local and global behaviour of input variables. The mathematical formulation is given in [34] as:

$$\begin{array}{c}{\widehat{y}}_i={\overline{y}}_{base}+{\overline{y}}_{i1}+{\overline{y}}_{i2}+\dots +{\overline{y}}_{in},\hfill \end{array}$$

where ${\overline{y}}_i$, ${\overline{y}}_{base}$ and ${\overline{y}}_{ij}$ are the predicted value of ith sample, average SHAP value, and SHAP value jth contributor in the ith sample, respectively.

The SHAP waterfall analyses of both the $\mathrm{LHS}$ and $\mathrm{RHS}$ are presented in Figure 13 and Figure 14. One can observe that the most effective contributor is the right endpoint $z_2$. But the behaviour is slightly different for all under-consideration functions.

Now we discuss the global analysis of major contributors for all functions for both the $\mathrm{LHS}$ and $\mathrm{RHS}$ in Figure 15 and Figure 16, respectively. The role of the right endpoint is very crucial to predict any value, while the fractional order produces minimal changes as compared with $z_2$.

The SHAP feature importance in a controlled environment across all test functions for the $\mathrm{LHS}$ and $\mathrm{RHS}$ is discussed in Figure 17 and Figure 18, respectively. These visuals are the straightforward verification of global SHAP analysis.

6. Conclusions

In this study, a new class of weighted fractional Boole-type inequalities based on the $\mathcal{RL}$ fractional integral operator and symmetric weight functions has been introduced. By using the newly derived weighted fractional identity instead of the classical sixth-derivative error estimate, several inequalities were derived under convexity, s-convexity, boundedness, Lipschitz continuity and bounded-variation assumptions, which is a relaxation of the strong smoothness requirements on the classical formulation. The flexibility of the proposed method can be seen from the fact that for suitable choices of the fractional parameter $0<\mu \le 1$ and weight function, the inequalities generalize several known inequalities as special cases. The applications to special means and to a composite weighted quadrature rule, along with the examples and graphical illustrations, further highlight the analytical and practical relevance of the results.

This theoretical study was conducted using 5400 samples obtained from the proposed inequality to verify the theoretical framework. For each sample the bound was satisfied, and the slack variable was non-negative throughout the entire computation domain, thus verifying that the proposed estimate is consistent and stable for various choices of parameters and functions. In terms of modeling, Symbolic Regression provided the interpretable closed-form approximation to the theorem while being able to make predictions close to 1. All these findings are brought together within a single scalable framework, linking fractional calculus, numerical approximation and explainable artificial intelligence. Several directions exist for future research: an extension of these types of inequalities to other generalized fractional operators including Atangana–Baleanu fractional operators, a wider class of weight functions, and higher order quadrature structures, and applying other techniques of computational intelligence to the analysis and prediction of related fractional integral inequalities. In future, the proposed inequalities can be explored for a wide range of other fractional operators and generalized classes of convexity. Also, a good problem to investigate is the weighted Boole-type inequalities of higher order differentiable functions. One can transform the proposed framework into a discrete structure using quantum calculus. One can discuss the coordinated weighted Boole-type inequalities under different assumptions. Our next target is to develop an ML-based calculator for fractional inequalities to approximate the bounds for any admissible functions to reduce the computational cost. Hopefully, the machine learning techniques involved in this study will play a crucial role in the computational analysis of inequalities.