Numerical Approximations and Fractional Calculus: Extending Boole’s Rule with Riemann–Liouville Fractional Integral Inequalities

Abstract

This paper develops integral inequalities for first-order differentiable convex functions within the framework of fractional calculus, extending Boole-type inequalities to this domain. An integral equality involving Riemann–Liouville fractional integrals is established, forming the foundation for deriving novel fractional Boole-type inequalities tailored to differentiable convex functions. The proposed framework encompasses a wide range of functional classes, including Lipschitzian functions, bounded functions, convex functions, and functions of bounded variation, thereby broadening the applicability of these inequalities to diverse mathematical settings. The research emphasizes the importance of the Riemann–Liouville fractional operator in solving problems related to non-integer-order differentiation, highlighting its pivotal role in enhancing classical inequalities. These newly established inequalities offer sharper error bounds for various numerical quadrature formulas in classical calculus, marking a significant advancement in computational mathematics. Numerical examples, computational analysis, applications to quadrature formulas and graphical illustrations substantiate the efficacy of the proposed inequalities in improving the accuracy of integral approximations, particularly within the context of fractional calculus. Future directions for this research include extending the framework to incorporate q-calculus, symmetrized q-calculus, alternative fractional operators, multiplicative calculus, and multidimensional spaces. These extensions would enable a comprehensive exploration of Boole’s formula and its associated error bounds, providing deeper insights into its performance across a broader range of mathematical and computational settings.

1. Introduction

Modeling complicated systems with memory and hereditary features requires the use of fractional calculus, which is the expansion of derivatives and integrals to non-integer orders. While its origins trace back to 1695 [1], with foundational insights from Leibniz and Euler, fractional calculus saw formal advancements beginning in the 18th century. Joseph Liouville’s work in the 19th century established fractional derivatives as a rigorous concept, and the field gained further traction in the 1920s with S. Saks’s definition of fractional integrals, which offered a flexible tool for addressing differential equations. Unlike classical integer-order derivatives, fractional derivatives provide a more generalized framework capable of capturing nonlocal dynamics an invaluable feature in disciplines such as physics, engineering, and finance, where local models are often inadequate. By facilitating differentiation and integration to arbitrary orders, fractional calculus enables the accurate modeling of phenomena with long-range dependencies, frequently observed in viscoelastic materials, anomalous diffusion processes, and biological systems. The development of fractional differential equations has further underscored the power of fractional calculus as a modeling tool. Combined with rigorous mathematical tools like Boole’s type inequalities, fractional calculus offers a refined analytical framework for examining solution behavior, including stability, uniqueness, and boundedness. These Boole-type inequalities adapted to fractional differential and integral equations yield precise bounds that enhance theoretical and computational approaches, making fractional calculus an increasingly valuable component of modern mathematical and applied research.

Definition 1

([2,3]). The Riemann–Liouville fractional integrals $J_{{\sigma }^{+}}^{\alpha }\mathrm{\Theta }$ and $J_{{\rho }^{-}}^{\alpha }\mathrm{\Theta }$ of order $\alpha >0$, defined for $\sigma \ge 0$ and $\mathrm{\Theta }\in L_1\left([\sigma ,\rho ]\right)$, are given as follows:

$$J_{{\sigma }^{+}}^{\alpha }\mathrm{\Theta }\left(\varpi \right)={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}}{\int }_{\sigma }^{\varpi }{\left(\varpi -\vartheta \right)}^{\alpha -1}\mathrm{\Theta }\left(\vartheta \right)d\vartheta ,\varpi >\sigma ,$$

and

$$J_{{\rho }^{-}}^{\alpha }\mathrm{\Theta }\left(\varpi \right)={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}}{\int }_{\varpi }^{\rho }{\left(\vartheta -\varpi \right)}^{\alpha -1}\mathrm{\Theta }\left(\vartheta \right)d\vartheta ,\varpi <\rho .$$

The mathematical function known as the Gamma function, represented by the symbol $\mathrm{\Gamma }\left(\alpha \right)$, has the following definition

$$\mathrm{\Gamma }\left(\alpha \right)={\int }_0^{\infty }{e}^{-\varpi }{\varpi }^{\alpha -1}d\varpi ,$$

and $J_{{\sigma }^{+}}^0\mathrm{\Theta }\left(\varpi \right)=J_{{\rho }^{-}}^0\mathrm{\Theta }\left(\varpi \right)=\mathrm{\Theta }\left(\varpi \right).$

The reduction from fractional calculus to classical calculus occurs when $\alpha =1$. More information on fractional calculus may be found in [4,5,6,7,8].

The concept of convexity has its roots in ancient geometry with Euclid (circa 300 BC) studying convex shapes. However, its formal analysis emerged much later, with significant contributions from Leonhard Euler (1707–1783) on convex curves. Further advancements were made by Johann Radon, Hermann Minkowski, and Constantin Carathodory, who developed the field of convex analysis and optimization. Today, convex optimization plays a vital role in areas such as engineering and machine learning, particularly through methods like interior-point algorithms. For a deeper exploration of the history of convexity, refer to [9]. Below are the definitions of convex functions.

Definition 2

([9]). Consider ψ to be a convex subset of a real vector space. A function $\mathrm{\Theta }:\psi \subset \mathfrak{R}\to \mathfrak{R}$ is said to be convex if

$$\mathrm{\Theta }(\vartheta \sigma +(1-\vartheta \left)\rho \right)\le \vartheta \mathrm{\Theta }\left(\sigma \right)+(1-\vartheta )\mathrm{\Theta }\left(\rho \right),$$

(1)

for each value of $\vartheta \in \left[0,1\right]$ and $\sigma ,\rho \in \psi$.

Some important mathematical inequalities involving convex mappings are Hermite–Hadamard type inequalities, which are stated as follows:

Let $\mathrm{\Theta }:[\sigma ,\rho ]\subseteq \mathfrak{R}\to {\mathfrak{R}}^{+}$ be a convex function defined on the interval $[\sigma ,\rho ]$ of real numbers. Hermite–Hadamard inequalities are the double inequalities listed below:

$$\mathrm{\Theta }\left({\displaystyle \frac{\sigma +\rho }{2}}\right)\le {\displaystyle \frac{1}{\rho -\sigma }}{\int }_{\sigma }^{\rho }\mathrm{\Theta }\left(\varpi \right)d\varpi \le {\displaystyle \frac{\mathrm{\Theta }\left(\sigma \right)+\mathrm{\Theta }\left(\rho \right)}{2}}.$$

(2)

For more information on these inequalities, see [10,11].

Definition 3

([12]). A function $\mathrm{\Theta }:I\subset \mathfrak{R}\to \mathfrak{R}$ is the Lipschitzian function if it satisfies the inequality given below:

$$\left|\mathrm{\Theta }\left(\rho \right)-\mathrm{\Theta }\left(\sigma \right)\right|\le \mathcal{L}\left|\rho -\sigma \right|,$$

(3)

for all $\sigma ,\rho \in I$ and $\mathcal{L}\ge 0,$ and the smallest such constant $\mathcal{L}$ is known as the Lipschitz constant.

Definition 4

([13]). A real-valued function $\mathrm{\Theta }:I\subset \mathfrak{R}\to \mathfrak{R}$ is a bounded variation on the given interval if there exists a constant M such that for any partition $I_n$ of the interval $\left[\sigma ,\rho \right]$,

$$I_n:\sigma ={\varpi }_0<{\varpi }_1<{\varpi }_2<\dots <{\varpi }_{n-1}<{\varpi }_n=\rho ,$$

the following sum is known as the variation of Θ on $I_n$, satisfying

$$\underset{\sigma }{\overset{\rho }{\bigvee }}\left(\mathrm{\Theta },I_n\right)=\underset{I_n}{sup}\sum _{i=1}^n\left|\mathrm{\Theta }\left({\varpi }_i\right)-\mathrm{\Theta }\left({\varpi }_{i-1}\right)\right|\le M,$$

where ${\displaystyle \underset{\sigma }{\overset{\rho }{\bigvee }}}\left(\mathrm{\Theta }\right)$ must be finite over the given partition.

In 1915, the term “numerical integration” was first introduced in the publication by David Gibb [14]. In the context of numerical integration with lower error bounds, the third Simpson’s rule, also known as Simpson’s $2/45$ rule or Boole’s rule, is named after George Boole

$$\begin{array}{cc}\hfill {\int }_{\sigma }^{\rho }\mathrm{\Theta }\left(\varpi \right)d\varpi & ={\displaystyle \frac{2h}{45}}\left[7\mathrm{\Theta }\left(\sigma \right)+32\mathrm{\Theta }\left({\displaystyle \frac{3\sigma +\rho }{4}}\right)+12\mathrm{\Theta }\left({\displaystyle \frac{\sigma +\rho }{2}}\right)+32\mathrm{\Theta }\left({\displaystyle \frac{\sigma +3\rho }{4}}\right)\right.\hfill \\ \hfill & \left.+7\mathrm{\Theta }\left(\rho \right)\right]+E\left(\mathrm{\Theta }\right),\hfill \end{array}$$

(4)

where $E\left(\mathrm{\Theta }\right)$ is an error term. To know more about numerical integration and its applications, one can visit [15,16,17]. The error bound for Boole’s rule approximation is described as

$$\begin{array}{c}\hfill E\left(\mathrm{\Theta }\right)=-{\displaystyle \frac{8{\left(\rho -\sigma \right)}^7}{945}}M,\left|{\mathrm{\Theta }}^{\left(vi\right)}\left(\varpi \right)\right|\le M.\end{array}$$

Budak et al. [18] established Milne’s inequality for convex functions using Riemann–Liouville fractional integrals. Mateen et al. [19] developed fractional versions of Simpson’s and Newton’s inequalities within the framework of multiplicative calculus. Dragomir [20,21] investigated the remainder term in Simpson’s formula, while [22] introduced fractional Simpson’s inequalities for functions exhibiting absolute value convexity in their second derivatives. Mateen et al. [23] derived error bounds for Weddle-type inequalities via Riemann–Liouville fractional integrals. Gao et al. [24], and Krukowski [25] generalized these inequalities for convex functions and functions of bounded variation. For further studies on Newton-type inequalities, see [26,27,28,29,30].

Theorem 1

([31]). Suppose $\mathrm{\Theta }:[\sigma ,\rho ]\to \mathfrak{R}$ is a differentiable mapping whose derivative is continuous on $(\sigma ,\rho )$ and ${\mathrm{\Theta }}^{\prime }\in L_1[\sigma ,\rho ]$. If $|{\mathrm{\Theta }}^{\prime }|$ is a convex function, then the subsequent inequality is valid:

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{6}}\left[\mathrm{\Theta }\left(\sigma \right)+4\mathrm{\Theta }\left({\displaystyle \frac{\sigma +\rho }{2}}\right)+\mathrm{\Theta }\left(\rho \right)\right]-{\displaystyle \frac{\mathrm{\Gamma }(\alpha +1)}{2{(\rho -\sigma )}^{\alpha }}}\left[J_{\rho -}^{\alpha }\mathrm{\Theta }\left(\sigma \right)+J_{\sigma +}^{\alpha }\mathrm{\Theta }\left(\rho \right)\right]\right|\hfill \\ \hfill & \le {\displaystyle \frac{\rho -\sigma }{2}}\left[{\displaystyle \frac{{\left(\frac{1}{6}\right)}^{\frac{1}{\alpha }}+5{\left(\frac{5}{6}\right)}^{\frac{1}{\alpha }}-4}{3}}+{\displaystyle \frac{1+2^{\alpha }(1-2{\left(\frac{1}{6}\right)}^{1+\alpha }-2{\left(\frac{5}{6}\right)}^{1+\alpha })}{2^{\alpha }(\alpha +1)}}\right]\left[|{\mathrm{\Theta }}^{\prime }\left(\sigma \right)|+|{\mathrm{\Theta }}^{\prime }\left(\rho \right)|\right].\hfill \end{array}$$

(5)

Theorem 2

([32]). Suppose $\mathrm{\Theta }:[\sigma ,\rho ]\to \mathfrak{R}$ is a differentiable mapping whose derivative is continuous on $(\sigma ,\rho )$ and ${\mathrm{\Theta }}^{\prime }\in L_1[\sigma ,\rho ]$. If $|{\mathrm{\Theta }}^{\prime }|$ is a convex function, then the subsequent inequality is valid:

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{2}}\left[\mathrm{\Theta }\left(\sigma \right)+\mathrm{\Theta }\left(\rho \right)\right]-{\displaystyle \frac{\mathrm{\Gamma }(\alpha +1)}{2{(\rho -\sigma )}^{\alpha }}}\left[J_{\rho -}^{\alpha }\mathrm{\Theta }\left(\sigma \right)+J_{\sigma +}^{\alpha }\mathrm{\Theta }\left(\rho \right)\right]\right|\hfill \\ \hfill & \le {\displaystyle \frac{\rho -\sigma }{2}}{\int }_0^1\left[{(1-\vartheta )}^{\alpha }-{\vartheta }^{\alpha }\right]\left[|{\mathrm{\Theta }}^{\prime }\left(\sigma \right)|+|{\mathrm{\Theta }}^{\prime }\left(\rho \right)|\right].\hfill \end{array}$$

(6)

In this study, fractional-type inequalities for Boole’s formula applied to functions whose derivatives exhibit convexity are derived utilizing Riemann–Liouville fractional integrals. The proposed inequalities offer a method for approximating error bounds in Boole’s formula without requiring higher derivatives. Fractional calculus broadens the applicability of traditional calculus by extending the concept of derivatives from integer to non-integer orders, thus encompassing a more comprehensive class of functions not captured by classical integer-order methods. These contributions advance the theoretical framework of numerical integration while enhancing its practical utility, improving both the solidity and efficiency of integration techniques.

This paper is organized as follows: Section 2 outlines the main findings, emphasizing fractional Boole’s formula-type inequalities. Section 3 provides numerical examples and graphical analysis to demonstrate the validity of the results. Section 3 delves into applications of the proposed inequalities to quadrature formulas, showcasing their practical significance. Lastly, Section 5 summarizes the conclusions and offers recommendations for future research.

2. Main Results

In this section, we substantiate integral equality to elucidate the primary conclusions outlined in this study.

Lemma 1.

Suppose $\mathrm{\Theta }:[\sigma ,\rho ]\to \mathfrak{R}$ is a differentiable function whose derivative is continuous on $(\sigma ,\rho )$ and ${\mathrm{\Theta }}^{\prime }\in L_1[\sigma ,\rho ]$. Then, the following equality holds:

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{90}}\left[7\mathrm{\Theta }\left(\sigma \right)+32\mathrm{\Theta }\left({\displaystyle \frac{3\sigma +\rho }{4}}\right)+12\mathrm{\Theta }\left({\displaystyle \frac{\sigma +\rho }{2}}\right)+32\mathrm{\Theta }\left({\displaystyle \frac{\sigma +3\rho }{4}}\right)+7\mathrm{\Theta }\left(\rho \right)\right]\hfill \\ \hfill & -{\displaystyle \frac{\mathrm{\Gamma }(\alpha +1)}{2{(\rho -\sigma )}^{\alpha }}}\left[J_{\rho -}^{\alpha }\mathrm{\Theta }\left(\sigma \right)+J_{\sigma +}^{\alpha }\mathrm{\Theta }\left(\rho \right)\right]\hfill \\ \hfill & ={\displaystyle \frac{\rho -\sigma }{2}}\left[{\int }_0^{{\displaystyle \frac{1}{4}}}\left({\vartheta }^{\alpha }-{\displaystyle \frac{7}{90}}\right)\left[{\mathrm{\Theta }}^{\prime }(\vartheta \rho +(1-\vartheta )\sigma )-{\mathrm{\Theta }}^{\prime }(\vartheta \sigma +(1-\vartheta )\rho )\right]d\vartheta \right.\hfill \\ \hfill & +{\int }_{{\displaystyle \frac{1}{4}}}^{{\displaystyle \frac{1}{2}}}\left({\vartheta }^{\alpha }-{\displaystyle \frac{39}{90}}\right)\left[{\mathrm{\Theta }}^{\prime }(\vartheta \rho +(1-\vartheta )\sigma )-{\mathrm{\Theta }}^{\prime }(\vartheta \sigma +(1-\vartheta )\rho )\right]d\vartheta \hfill \\ \hfill & +{\int }_{{\displaystyle \frac{1}{2}}}^{{\displaystyle \frac{3}{4}}}\left({\vartheta }^{\alpha }-{\displaystyle \frac{51}{90}}\right)\left[{\mathrm{\Theta }}^{\prime }(\vartheta \rho +(1-\vartheta )\sigma )-{\mathrm{\Theta }}^{\prime }(\vartheta \sigma +(1-\vartheta )\rho )\right]d\vartheta \hfill \\ \hfill & \left.+{\int }_{{\displaystyle \frac{3}{4}}}^1\left({\vartheta }^{\alpha }-{\displaystyle \frac{83}{90}}\right)\left[{\mathrm{\Theta }}^{\prime }(\vartheta \rho +(1-\vartheta )\sigma )-{\mathrm{\Theta }}^{\prime }(\vartheta \sigma +(1-\vartheta )\rho )\right]d\vartheta \right].\hfill \end{array}$$

Proof. 

Using the basic principles of integration, we obtain the following result:

$$\begin{array}{cc}\hfill {\mathcal{I}}_1& ={\int }_0^{{\displaystyle \frac{1}{4}}}\left({\vartheta }^{\alpha }-{\displaystyle \frac{7}{90}}\right){\mathrm{\Theta }}^{\prime }(\vartheta \rho +(1-\vartheta )\sigma )d\vartheta \hfill \\ \hfill & ={\left.{\displaystyle \frac{1}{\rho -\sigma }}\left({\vartheta }^{\alpha }-{\displaystyle \frac{7}{90}}\right)\mathrm{\Theta }(\vartheta \rho +(1-\vartheta )\sigma )\right|}_0^{{\displaystyle \frac{1}{4}}}-{\displaystyle \frac{\alpha }{\rho -\sigma }}{\int }_0^{{\displaystyle \frac{1}{4}}}{\vartheta }^{\alpha -1}\mathrm{\Theta }(\vartheta \rho +(1-\vartheta )\sigma )d\vartheta \hfill \\ \hfill & ={\displaystyle \frac{1}{\rho -\sigma }}\left[\left({\displaystyle \frac{1}{4^{\alpha }}}-{\displaystyle \frac{7}{90}}\right)\mathrm{\Theta }\left({\displaystyle \frac{3\sigma +\rho }{4}}\right)+{\displaystyle \frac{7}{90}}\mathrm{\Theta }\left(\sigma \right)\right]-{\displaystyle \frac{\alpha }{\rho -\sigma }}{\int }_0^{{\displaystyle \frac{1}{4}}}{\vartheta }^{\alpha -1}\mathrm{\Theta }(\vartheta \rho +(1-\vartheta )\sigma )d\vartheta ,\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill {\mathcal{I}}_2& ={\int }_{{\displaystyle \frac{1}{4}}}^{{\displaystyle \frac{1}{2}}}\left({\vartheta }^{\alpha }-{\displaystyle \frac{39}{90}}\right){\mathrm{\Theta }}^{\prime }(\vartheta \rho +(1-\vartheta )\sigma )d\vartheta \hfill \\ \hfill & ={\displaystyle \frac{1}{\rho -\sigma }}\left[\left({\displaystyle \frac{1}{2^{\alpha }}}-{\displaystyle \frac{39}{90}}\right)\mathrm{\Theta }\left({\displaystyle \frac{\sigma +\rho }{2}}\right)-\left({\displaystyle \frac{1}{4^{\alpha }}}-{\displaystyle \frac{39}{90}}\right)\mathrm{\Theta }\left({\displaystyle \frac{3\sigma +\rho }{4}}\right)\right]\hfill \\ \hfill & -{\displaystyle \frac{\alpha }{\rho -\sigma }}{\int }_{{\displaystyle \frac{1}{4}}}^{{\displaystyle \frac{1}{2}}}{\vartheta }^{\alpha -1}\mathrm{\Theta }(\vartheta \rho +(1-\vartheta )\sigma )d\vartheta ,\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill {\mathcal{I}}_3& ={\int }_{{\displaystyle \frac{1}{2}}}^{{\displaystyle \frac{3}{4}}}\left({\vartheta }^{\alpha }-{\displaystyle \frac{51}{90}}\right){\mathrm{\Theta }}^{\prime }(\vartheta \rho +(1-\vartheta )\sigma )d\vartheta \hfill \\ \hfill & ={\displaystyle \frac{1}{\rho -\sigma }}\left[\left({\displaystyle \frac{3^{\alpha }}{4^{\alpha }}}-{\displaystyle \frac{51}{90}}\right)\mathrm{\Theta }\left({\displaystyle \frac{\sigma +3\rho }{4}}\right)-\left({\displaystyle \frac{1}{2^{\alpha }}}-{\displaystyle \frac{51}{90}}\right)\mathrm{\Theta }\left({\displaystyle \frac{\sigma +\rho }{2}}\right)\right]\hfill \\ \hfill & -{\displaystyle \frac{\alpha }{\rho -\sigma }}{\int }_{{\displaystyle \frac{1}{2}}}^{{\displaystyle \frac{3}{4}}}{\vartheta }^{\alpha -1}\mathrm{\Theta }(\vartheta \rho +(1-\vartheta )\sigma )d\vartheta ,\hfill \end{array}$$

(9)

and

$$\begin{array}{cc}\hfill {\mathcal{I}}_4& ={\int }_{{\displaystyle \frac{3}{4}}}^1\left({\vartheta }^{\alpha }-{\displaystyle \frac{83}{90}}\right){\mathrm{\Theta }}^{\prime }(\vartheta \rho +(1-\vartheta )\sigma )d\vartheta \hfill \\ \hfill & ={\displaystyle \frac{1}{\rho -\sigma }}\left[{\displaystyle \frac{7}{90}}\mathrm{\Theta }\left(\rho \right)-\left({\displaystyle \frac{3^{\alpha }}{4^{\alpha }}}-{\displaystyle \frac{83}{90}}\right)\mathrm{\Theta }\left({\displaystyle \frac{\sigma +3\rho }{4}}\right)\right]-{\displaystyle \frac{\alpha }{\rho -\sigma }}{\int }_{{\displaystyle \frac{3}{4}}}^1{\vartheta }^{\alpha -1}\mathrm{\Theta }(\vartheta \rho +(1-\vartheta )\sigma )d\vartheta .\hfill \end{array}$$

(10)

By the Equations (7)–(10), we observe

$$\begin{array}{cc}\hfill & (\rho -\sigma )\left[{\mathcal{I}}_1+{\mathcal{I}}_2+{\mathcal{I}}_3+{\mathcal{I}}_4\right]\hfill \\ \hfill & ={\displaystyle \frac{1}{90}}\left[7\mathrm{\Theta }\left(\sigma \right)+32\mathrm{\Theta }\left({\displaystyle \frac{3\sigma +\rho }{4}}\right)+12\mathrm{\Theta }\left({\displaystyle \frac{\sigma +\rho }{2}}\right)+32\mathrm{\Theta }\left({\displaystyle \frac{\sigma +3\rho }{4}}\right)+7\mathrm{\Theta }\left(\rho \right)\right]\hfill \\ \hfill & -{\displaystyle \frac{\alpha }{\rho -\sigma }}{\int }_0^1{\vartheta }^{\alpha -1}\mathrm{\Theta }(\vartheta \rho +(1-\vartheta )\sigma )d\vartheta .\hfill \end{array}$$

(11)

By the change of variable $\varpi =\vartheta \rho +(1-\vartheta )\sigma$, then the equality (11) can be recast as follows:

$$\begin{array}{cc}\hfill & (\rho -\sigma )\left[{\mathcal{I}}_1+{\mathcal{I}}_2+{\mathcal{I}}_3+{\mathcal{I}}_4\right]\hfill \\ \hfill & ={\displaystyle \frac{1}{90}}\left[7\mathrm{\Theta }\left(\sigma \right)+32\mathrm{\Theta }\left({\displaystyle \frac{3\sigma +\rho }{4}}\right)+12\mathrm{\Theta }\left({\displaystyle \frac{\sigma +\rho }{2}}\right)+32\mathrm{\Theta }\left({\displaystyle \frac{\sigma +3\rho }{4}}\right)+7\mathrm{\Theta }\left(\rho \right)\right]\hfill \\ \hfill & -{\displaystyle \frac{\mathrm{\Gamma }(\alpha +1)}{{(\rho -\sigma )}^{\alpha }}}{J}_{\rho -}^{\alpha }\mathrm{\Theta }\left(\sigma \right).\hfill \end{array}$$

(12)

In the same manner, we have

$$\begin{array}{cc}\hfill & (\rho -\sigma )\left[{\mathcal{I}}_5+{\mathcal{I}}_6+{\mathcal{I}}_7+{\mathcal{I}}_8\right]\hfill \\ \hfill & =-{\displaystyle \frac{1}{90}}\left[7\mathrm{\Theta }\left(\sigma \right)+32\mathrm{\Theta }\left({\displaystyle \frac{3\sigma +\rho }{4}}\right)+12\mathrm{\Theta }\left({\displaystyle \frac{\sigma +\rho }{2}}\right)+32\mathrm{\Theta }\left({\displaystyle \frac{\sigma +3\rho }{4}}\right)+7\mathrm{\Theta }\left(\rho \right)\right]\hfill \\ \hfill & +{\displaystyle \frac{\mathrm{\Gamma }(\alpha +1)}{{(\rho -\sigma )}^{\alpha }}}{J}_{\sigma +}^{\alpha }\mathrm{\Theta }\left(\rho \right).\hfill \end{array}$$

(13)

Consequently, the desired equality is obtained by substituting Equations (12) and (13). □

We begin with the following estimate integrals, which will be used to derive our main results:

$$\begin{array}{cc}\hfill & {\mathsf{A}}_1\left(\alpha \right)={\int }_0^{{\displaystyle \frac{1}{4}}}\left|{\vartheta }^{\alpha }-{\displaystyle \frac{7}{90}}\right|d\vartheta \hfill \\ \hfill & =\left\{\begin{array}{cc}{\displaystyle \frac{2\alpha }{\alpha +1}}{\left({\displaystyle \frac{7}{90}}\right)}^{1+{\displaystyle \frac{1}{\alpha }}}+{\displaystyle \frac{1}{\alpha +1}}{\left({\displaystyle \frac{1}{4}}\right)}^{\alpha +1}-{\displaystyle \frac{7}{360}},\hfill & 0<\alpha \le {\displaystyle \frac{ln(7/90)}{ln(1/4)}},\hfill \\ \hfill & \\ {\displaystyle \frac{7}{360}}-{\displaystyle \frac{1}{\alpha +1}}{\left({\displaystyle \frac{1}{4}}\right)}^{\alpha +1},\hfill & {\displaystyle \frac{ln(7/90)}{ln(1/4)}}<\alpha ,\hfill \end{array}\right.\hfill \\ \hfill & {\mathsf{A}}_2\left(\alpha \right)={\int }_{{\displaystyle \frac{1}{4}}}^{{\displaystyle \frac{1}{2}}}\left|{\vartheta }^{\alpha }-{\displaystyle \frac{39}{90}}\right|d\vartheta \hfill \\ \hfill & =\left\{\begin{array}{cc}{\displaystyle \frac{1}{\alpha +1}}{\left({\displaystyle \frac{1}{2}}\right)}^{\alpha +1}-{\displaystyle \frac{1}{\alpha +1}}{\left({\displaystyle \frac{1}{4}}\right)}^{\alpha +1}-{\displaystyle \frac{13}{120}},\hfill & 0<\alpha \le {\displaystyle \frac{ln(39/90)}{ln(1/4)}},\hfill \\ \hfill & \\ {\displaystyle \frac{2\alpha }{\alpha +1}}{\left({\displaystyle \frac{39}{90}}\right)}^{1+{\displaystyle \frac{1}{\alpha }}}+{\displaystyle \frac{1}{\alpha +1}}{\left({\displaystyle \frac{1}{4}}\right)}^{\alpha +1}+{\displaystyle \frac{1}{\alpha +1}}{\left({\displaystyle \frac{1}{2}}\right)}^{\alpha +1}-{\displaystyle \frac{39}{120}},\hfill & {\displaystyle \frac{ln\left(39/90\right)}{ln(1/4)}}<\alpha \le {\displaystyle \frac{ln(39/90)}{ln(1/2)}},\hfill \\ \hfill & \\ {\displaystyle \frac{13}{120}}+{\displaystyle \frac{1}{\alpha +1}}{\left({\displaystyle \frac{1}{4}}\right)}^{\alpha +1}-{\displaystyle \frac{1}{\alpha +1}}{\left({\displaystyle \frac{1}{2}}\right)}^{\alpha +1},\hfill & \alpha >{\displaystyle \frac{ln(39/90)}{ln(1/2)}},\hfill \end{array}\right.\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill & {\mathsf{A}}_3\left(\alpha \right)={\int }_{{\displaystyle \frac{1}{2}}}^{{\displaystyle \frac{3}{4}}}\left|{\vartheta }^{\alpha }-{\displaystyle \frac{51}{90}}\right|d\vartheta \hfill \\ \hfill & =\left\{\begin{array}{cc}{\displaystyle \frac{1}{\alpha +1}}{\left({\displaystyle \frac{3}{4}}\right)}^{\alpha +1}-{\displaystyle \frac{1}{\alpha +1}}{\left({\displaystyle \frac{1}{2}}\right)}^{\alpha +1}-{\displaystyle \frac{17}{120}},\hfill & 0<\alpha \le {\displaystyle \frac{ln(51/90)}{ln(1/2)}},\hfill \\ \hfill & \\ {\displaystyle \frac{2\alpha }{\alpha +1}}{\left({\displaystyle \frac{51}{90}}\right)}^{1+{\displaystyle \frac{1}{\alpha }}}+{\displaystyle \frac{1}{\alpha +1}}{\left({\displaystyle \frac{1}{2}}\right)}^{\alpha +1}+{\displaystyle \frac{1}{\alpha +1}}{\left({\displaystyle \frac{3}{4}}\right)}^{\alpha +1}-{\displaystyle \frac{17}{24}},\hfill & {\displaystyle \frac{ln\left(51/90\right)}{ln(1/2)}}<\alpha \le {\displaystyle \frac{ln(51/90)}{ln(3/4)}},\hfill \\ \hfill & \\ {\displaystyle \frac{13}{120}}+{\displaystyle \frac{1}{\alpha +1}}{\left({\displaystyle \frac{1}{4}}\right)}^{\alpha +1}-{\displaystyle \frac{1}{\alpha +1}}{\left({\displaystyle \frac{1}{2}}\right)}^{\alpha +1},\hfill & \alpha >{\displaystyle \frac{ln(51/90)}{ln(3/4)}},\hfill \end{array}\right.\hfill \\ \hfill & {\mathsf{A}}_4\left(\alpha \right)={\int }_{{\displaystyle \frac{3}{4}}}^1\left|{\vartheta }^{\alpha }-{\displaystyle \frac{83}{90}}\right|d\vartheta \hfill \\ \hfill & =\left\{\begin{array}{cc}{\displaystyle \frac{1}{\alpha +1}}-{\displaystyle \frac{1}{\alpha +1}}{\left({\displaystyle \frac{3}{4}}\right)}^{\alpha +1}-{\displaystyle \frac{83}{360}},\hfill & 0<\alpha \le {\displaystyle \frac{ln(83/90)}{ln(3/4)}},\hfill \\ \hfill & \\ {\displaystyle \frac{2\alpha }{\alpha +1}}{\left({\displaystyle \frac{83}{90}}\right)}^{1+{\displaystyle \frac{1}{\alpha }}}+{\displaystyle \frac{1}{\alpha +1}}{\left({\displaystyle \frac{3}{4}}\right)}^{\alpha +1}+{\displaystyle \frac{1}{\alpha +1}}-{\displaystyle \frac{581}{360}},\hfill & \alpha >{\displaystyle \frac{ln(83/90)}{ln(3/4)}},\hfill \end{array}\right.\hfill \end{array}$$

$$\begin{array}{cc}\hfill & {\mathsf{A}}_5\left(\alpha \right)={\int }_0^{{\displaystyle \frac{1}{4}}}\vartheta \left|{\vartheta }^{\alpha }-{\displaystyle \frac{7}{90}}\right|d\vartheta \hfill \\ \hfill & =\left\{\begin{array}{cc}{\displaystyle \frac{\alpha }{\alpha +2}}{\left({\displaystyle \frac{7}{90}}\right)}^{1+{\displaystyle \frac{2}{\alpha }}}+{\displaystyle \frac{1}{\alpha +2}}{\left({\displaystyle \frac{1}{4}}\right)}^{\alpha +2}-{\displaystyle \frac{7}{2880}},\hfill & 0<\alpha \le {\displaystyle \frac{ln(7/90)}{ln(1/4)}},\hfill \\ \hfill & \\ {\displaystyle \frac{7}{2880}}-{\displaystyle \frac{1}{\alpha +2}}{\left({\displaystyle \frac{1}{4}}\right)}^{\alpha +2},\hfill & {\displaystyle \frac{ln(7/90)}{ln(1/4)}}<\alpha ,\hfill \end{array}\right.\hfill \end{array}$$

$$\begin{array}{cc}\hfill & {\mathsf{A}}_6\left(\alpha \right)={\int }_{{\displaystyle \frac{1}{4}}}^{{\displaystyle \frac{1}{2}}}\vartheta \left|{\vartheta }^{\alpha }-{\displaystyle \frac{39}{90}}\right|d\vartheta \hfill \\ \hfill & =\left\{\begin{array}{cc}{\displaystyle \frac{1}{\alpha +2}}{\left({\displaystyle \frac{1}{2}}\right)}^{\alpha +2}-{\displaystyle \frac{1}{\alpha +2}}{\left({\displaystyle \frac{1}{4}}\right)}^{\alpha +2}-{\displaystyle \frac{39}{960}},\hfill & 0<\alpha \le {\displaystyle \frac{ln(39/90)}{ln(1/4)}},\hfill \\ \hfill & \\ {\displaystyle \frac{\alpha }{\alpha +2}}{\left({\displaystyle \frac{39}{90}}\right)}^{1+{\displaystyle \frac{2}{\alpha }}}+{\displaystyle \frac{1}{\alpha +2}}{\left({\displaystyle \frac{1}{4}}\right)}^{\alpha +2}+{\displaystyle \frac{1}{\alpha +2}}{\left({\displaystyle \frac{1}{2}}\right)}^{\alpha +2}-{\displaystyle \frac{13}{192}},\hfill & {\displaystyle \frac{ln\left(39/90\right)}{ln(1/4)}}<\alpha \le {\displaystyle \frac{ln(39/90)}{ln(1/2)}},\hfill \\ \hfill & \\ {\displaystyle \frac{39}{960}}+{\displaystyle \frac{1}{\alpha +2}}{\left({\displaystyle \frac{1}{4}}\right)}^{\alpha +2}-{\displaystyle \frac{1}{\alpha +2}}{\left({\displaystyle \frac{1}{2}}\right)}^{\alpha +2},\hfill & \alpha >{\displaystyle \frac{ln(39/90)}{ln(1/2)}},\hfill \end{array}\right.\hfill \end{array}$$

$$\begin{array}{cc}\hfill & {\mathsf{A}}_7\left(\alpha \right)={\int }_{{\displaystyle \frac{1}{2}}}^{{\displaystyle \frac{3}{4}}}\vartheta \left|{\vartheta }^{\alpha }-{\displaystyle \frac{51}{90}}\right|d\vartheta \hfill \\ \hfill & =\left\{\begin{array}{cc}{\displaystyle \frac{1}{\alpha +2}}{\left({\displaystyle \frac{3}{4}}\right)}^{\alpha +2}-{\displaystyle \frac{1}{\alpha +2}}{\left({\displaystyle \frac{1}{2}}\right)}^{\alpha +2}-{\displaystyle \frac{17}{192}},\hfill & 0<\alpha \le {\displaystyle \frac{ln(51/90)}{ln(1/2)}},\hfill \\ \hfill & \\ {\displaystyle \frac{\alpha }{\alpha +2}}{\left({\displaystyle \frac{51}{90}}\right)}^{1+{\displaystyle \frac{2}{\alpha }}}+{\displaystyle \frac{1}{\alpha +2}}{\left({\displaystyle \frac{1}{2}}\right)}^{\alpha +2}+{\displaystyle \frac{1}{\alpha +2}}{\left({\displaystyle \frac{3}{4}}\right)}^{\alpha +2}-{\displaystyle \frac{221}{960}},\hfill & {\displaystyle \frac{ln\left(51/90\right)}{ln(1/2)}}<\alpha \le {\displaystyle \frac{ln(51/90)}{ln(3/4)}},\hfill \\ \hfill & \\ {\displaystyle \frac{17}{192}}+{\displaystyle \frac{1}{\alpha +2}}{\left({\displaystyle \frac{1}{2}}\right)}^{\alpha +2}-{\displaystyle \frac{1}{\alpha +2}}{\left({\displaystyle \frac{3}{4}}\right)}^{\alpha +2},\hfill & \alpha >{\displaystyle \frac{ln(51/90)}{ln(3/4)}},\hfill \end{array}\right.\hfill \\ \hfill & {\mathsf{A}}_8\left(\alpha \right)={\int }_{{\displaystyle \frac{3}{4}}}^1\vartheta \left|{\vartheta }^{\alpha }-{\displaystyle \frac{83}{90}}\right|d\vartheta \hfill \\ \hfill & =\left\{\begin{array}{cc}{\displaystyle \frac{1}{\alpha +2}}-{\displaystyle \frac{1}{\alpha +2}}{\left({\displaystyle \frac{3}{4}}\right)}^{\alpha +2}-{\displaystyle \frac{51}{2880}},\hfill & 0<\alpha \le {\displaystyle \frac{ln(83/90)}{ln(3/4)}},\hfill \\ \hfill & \\ {\displaystyle \frac{\alpha }{\alpha +2}}{\left({\displaystyle \frac{83}{90}}\right)}^{1+{\displaystyle \frac{2}{\alpha }}}+{\displaystyle \frac{1}{\alpha +2}}{\left({\displaystyle \frac{3}{4}}\right)}^{\alpha +2}+{\displaystyle \frac{1}{\alpha +2}}-{\displaystyle \frac{415}{576}},\hfill & \alpha >{\displaystyle \frac{ln(83/90)}{ln(3/4)}}.\hfill \end{array}\right.\hfill \end{array}$$

2.1. Fractional Boole’s Type Inequality

Theorem 3.

Let all the conditions in Lemma 1 be satisfied, and assume that $|{\mathrm{\Theta }}^{\prime }|$ is a convex function. Then, the following inequality holds:

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{90}}\left[7\mathrm{\Theta }\left(\sigma \right)+32\mathrm{\Theta }\left({\displaystyle \frac{3\sigma +\rho }{4}}\right)+12\mathrm{\Theta }\left({\displaystyle \frac{\sigma +\rho }{2}}\right)+32\mathrm{\Theta }\left({\displaystyle \frac{\sigma +3\rho }{4}}\right)+7\mathrm{\Theta }\left(\rho \right)\right]\right.\hfill \\ \hfill & \left.-{\displaystyle \frac{\mathrm{\Gamma }(\alpha +1)}{2{(\rho -\sigma )}^{\alpha }}}\left[J_{\rho -}^{\alpha }\mathrm{\Theta }\left(\sigma \right)+J_{\sigma +}^{\alpha }\mathrm{\Theta }\left(\rho \right)\right]\right|\hfill \\ \hfill & \le {\displaystyle \frac{\rho -\sigma }{2}}\left({\mathsf{A}}_1\left(\alpha \right)+{\mathsf{A}}_2\left(\alpha \right)+{\mathsf{A}}_3\left(\alpha \right)+{\mathsf{A}}_4\left(\alpha \right)\right)\left[|{\mathrm{\Theta }}^{\prime }\left(\sigma \right)|+|{\mathrm{\Theta }}^{\prime }\left(\rho \right)|\right],\hfill \end{array}$$

(15)

where ${\mathsf{A}}_1\left(\alpha \right)-{\mathsf{A}}_4\left(\alpha \right)$ are defined as in (14).

Proof. 

From Lemma 1 and applying the convexity of $|{\mathrm{\Theta }}^{\prime }|$, we observe

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{90}}\left[7\mathrm{\Theta }\left(\sigma \right)+32\mathrm{\Theta }\left({\displaystyle \frac{3\sigma +\rho }{4}}\right)+12\mathrm{\Theta }\left({\displaystyle \frac{\sigma +\rho }{2}}\right)+32\mathrm{\Theta }\left({\displaystyle \frac{\sigma +3\rho }{4}}\right)+7\mathrm{\Theta }\left(\rho \right)\right]\right.\hfill \\ \hfill & \left.-{\displaystyle \frac{\mathrm{\Gamma }(\alpha +1)}{2{(\rho -\sigma )}^{\alpha }}}\left[J_{\rho -}^{\alpha }\mathrm{\Theta }\left(\sigma \right)+J_{\sigma +}^{\alpha }\mathrm{\Theta }\left(\rho \right)\right]\right|\hfill \\ \hfill & \le {\displaystyle \frac{\rho -\sigma }{2}}\left[{\int }_0^{{\displaystyle \frac{1}{4}}}\left|{\vartheta }^{\alpha }-{\displaystyle \frac{7}{90}}\right||{\mathrm{\Theta }}^{\prime }(\vartheta \rho +(1-\vartheta )\sigma )|d\vartheta +{\int }_0^{{\displaystyle \frac{1}{4}}}\left|{\vartheta }^{\alpha }-{\displaystyle \frac{7}{90}}\right|\left|{\mathrm{\Theta }}^{\prime }(\vartheta \sigma +(1-\vartheta )\rho )\right|d\vartheta \right.\hfill \\ \hfill & +{\int }_{{\displaystyle \frac{1}{4}}}^{{\displaystyle \frac{1}{2}}}\left|{\vartheta }^{\alpha }-{\displaystyle \frac{39}{90}}\right||{\mathrm{\Theta }}^{\prime }(\vartheta \rho +(1-\vartheta )\sigma )|d\vartheta +{\int }_{{\displaystyle \frac{1}{4}}}^{{\displaystyle \frac{1}{2}}}\left|{\vartheta }^{\alpha }-{\displaystyle \frac{39}{90}}\right|\left|{\mathrm{\Theta }}^{\prime }(\vartheta \sigma +(1-\vartheta )\rho )\right|d\vartheta \hfill \\ \hfill & +{\int }_{{\displaystyle \frac{1}{2}}}^{{\displaystyle \frac{3}{4}}}\left|{\vartheta }^{\alpha }-{\displaystyle \frac{51}{90}}\right||{\mathrm{\Theta }}^{\prime }(\vartheta \rho +(1-\vartheta )\sigma )|d\vartheta +{\int }_{{\displaystyle \frac{1}{2}}}^{{\displaystyle \frac{3}{4}}}\left|{\vartheta }^{\alpha }-{\displaystyle \frac{51}{90}}\right|\left|{\mathrm{\Theta }}^{\prime }(\vartheta \sigma +(1-\vartheta )\rho )\right|d\vartheta \hfill \\ \hfill & \left.+{\int }_{{\displaystyle \frac{3}{4}}}^1\left|{\vartheta }^{\alpha }-{\displaystyle \frac{83}{90}}\right||{\mathrm{\Theta }}^{\prime }(\vartheta \rho +(1-\vartheta )\sigma )|d\vartheta +{\int }_{{\displaystyle \frac{3}{4}}}^1\left|{\vartheta }^{\alpha }-{\displaystyle \frac{83}{90}}\right|\left|{\mathrm{\Theta }}^{\prime }(\vartheta \sigma +(1-\vartheta )\rho )\right|d\vartheta \right]\hfill \\ \hfill & \le {\displaystyle \frac{\rho -\sigma }{2}}\left[{\int }_0^{{\displaystyle \frac{1}{4}}}\left|{\vartheta }^{\alpha }-{\displaystyle \frac{7}{90}}\right|\left[\vartheta \right|{\mathrm{\Theta }}^{\prime }\left(\rho \right)|+(1-\vartheta )|{\mathrm{\Theta }}^{\prime }\left(\sigma \right)|+\vartheta |{\mathrm{\Theta }}^{\prime }\left(\sigma \right)+(1-\vartheta )|{\mathrm{\Theta }}^{\prime }\left(\rho \right)\left|\right]d\vartheta \right.\hfill \\ \hfill & +{\int }_{{\displaystyle \frac{1}{4}}}^{{\displaystyle \frac{1}{2}}}\left|{\vartheta }^{\alpha }-{\displaystyle \frac{39}{90}}\right|\left[\vartheta \right|{\mathrm{\Theta }}^{\prime }\left(\rho \right)|+(1-\vartheta )|{\mathrm{\Theta }}^{\prime }\left(\sigma \right)|+\vartheta |{\mathrm{\Theta }}^{\prime }\left(\sigma \right)+(1-\vartheta )|{\mathrm{\Theta }}^{\prime }\left(\rho \right)\left|\right]d\vartheta \hfill \\ \hfill & +{\int }_{{\displaystyle \frac{1}{2}}}^{{\displaystyle \frac{3}{4}}}\left|{\vartheta }^{\alpha }-{\displaystyle \frac{51}{90}}\right|\left[\vartheta \right|{\mathrm{\Theta }}^{\prime }\left(\rho \right)|+(1-\vartheta )|{\mathrm{\Theta }}^{\prime }\left(\sigma \right)|+\vartheta |{\mathrm{\Theta }}^{\prime }\left(\sigma \right)+(1-\vartheta )|{\mathrm{\Theta }}^{\prime }\left(\rho \right)\left|\right]d\vartheta \hfill \\ \hfill & \left.+{\int }_{{\displaystyle \frac{3}{4}}}^1\left|{\vartheta }^{\alpha }-{\displaystyle \frac{83}{90}}\right|\left[\vartheta \right|{\mathrm{\Theta }}^{\prime }\left(\rho \right)|+(1-\vartheta )|{\mathrm{\Theta }}^{\prime }\left(\sigma \right)|+\vartheta |{\mathrm{\Theta }}^{\prime }\left(\sigma \right)+(1-\vartheta )|{\mathrm{\Theta }}^{\prime }\left(\rho \right)\left|\right]d\vartheta \right]\hfill \\ \hfill & ={\displaystyle \frac{\rho -\sigma }{2}}\left({\mathsf{A}}_1\left(\alpha \right)+{\mathsf{A}}_2\left(\alpha \right)+{\mathsf{A}}_3\left(\alpha \right)+{\mathsf{A}}_4\left(\alpha \right)\right)\left[|{\mathrm{\Theta }}^{\prime }\left(\sigma \right)|+|{\mathrm{\Theta }}^{\prime }\left(\rho \right)|\right].\hfill \end{array}$$

(16)

Thus, we have concluded the proof. □

Corollary 1.

In Theorem 3, setting $\alpha =1$ yields the following inequality:

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{90}}\left[7\mathrm{\Theta }\left(\sigma \right)+32\mathrm{\Theta }\left({\displaystyle \frac{3\sigma +\rho }{4}}\right)+12\mathrm{\Theta }\left({\displaystyle \frac{\sigma +\rho }{2}}\right)+32\mathrm{\Theta }\left({\displaystyle \frac{\sigma +3\rho }{4}}\right)+7\mathrm{\Theta }\left(\rho \right)\right]\right.\hfill \\ \hfill & \left.-{\displaystyle \frac{1}{\rho -\sigma }}{\int }_{\sigma }^{\rho }\mathrm{\Theta }\left(\varpi \right)d\varpi \right|\le {\displaystyle \frac{239(\rho -\sigma )}{6480}}\left[|{\mathrm{\Theta }}^{\prime }\left(\sigma \right)|+|{\mathrm{\Theta }}^{\prime }\left(\rho \right)|\right],\hfill \end{array}$$

which is proved by Shehzadi et al. in [33] (Theorem 3).

Corollary 2.

In Theorem 3, if $\mathrm{\Theta }\left(\sigma \right)=\mathrm{\Theta }\left({\displaystyle \frac{\sigma +\rho }{2}}\right)=\mathrm{\Theta }\left({\displaystyle \frac{3\sigma +\rho }{4}}\right)=\mathrm{\Theta }\left({\displaystyle \frac{\sigma +3\rho }{4}}\right)=\mathrm{\Theta }\left(\rho \right),$ then we have subsequent new midpoint type inequality for convex function:

$$\begin{array}{cc}\hfill & \left|\mathrm{\Theta }\left({\displaystyle \frac{\sigma +\rho }{2}}\right)-{\displaystyle \frac{1}{\rho -\sigma }}{\int }_{\sigma }^{\rho }\mathrm{\Theta }\left(\varpi \right)d\varpi \right|\le {\displaystyle \frac{239(\rho -\sigma )}{6480}}\left[|{\mathrm{\Theta }}^{\prime }\left(\sigma \right)|+|{\mathrm{\Theta }}^{\prime }\left(\rho \right)|\right].\hfill \end{array}$$

(17)

Remark 1.

It can be observed that the inequality (17) provides better bounds for the midpoint compared to the inequality presented in [34].

Theorem 4.

Assume that all conditions in Lemma 1 are satisfied and that $|{\mathrm{\Theta }}^{\prime }{|}^q$, where $q>1$, is a convex function. Then, the following inequality holds:

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{90}}\left[7\mathrm{\Theta }\left(\sigma \right)+32\mathrm{\Theta }\left({\displaystyle \frac{3\sigma +\rho }{4}}\right)+12\mathrm{\Theta }\left({\displaystyle \frac{\sigma +\rho }{2}}\right)+32\mathrm{\Theta }\left({\displaystyle \frac{\sigma +3\rho }{4}}\right)+7\mathrm{\Theta }\left(\rho \right)\right]\right.\hfill \\ \hfill & \left.-{\displaystyle \frac{\mathrm{\Gamma }(\alpha +1)}{2{(\rho -\sigma )}^{\alpha }}}\left[J_{\rho -}^{\alpha }\mathrm{\Theta }\left(\sigma \right)+J_{\sigma +}^{\alpha }\mathrm{\Theta }\left(\rho \right)\right]\right|\hfill \\ \hfill & \le {\displaystyle \frac{\rho -\sigma }{2}}\left[{\left({\int }_0^{{\displaystyle \frac{1}{4}}}{\left|{\vartheta }^{\alpha }-{\displaystyle \frac{7}{90}}\right|}^{p}d\vartheta \right)}^{{\displaystyle \frac{1}{p}}}\left\{{\left({\displaystyle \frac{|{\mathrm{\Theta }}^{\prime }{\left(\rho \right)|}^q+7{\left|{\mathrm{\Theta }}^{\prime }\left(\sigma \right)\right|}^q}{32}}\right)}^{{\displaystyle \frac{1}{q}}}+{\left({\displaystyle \frac{|{\mathrm{\Theta }}^{\prime }{\left(\sigma \right)|}^q+7{\left|{\mathrm{\Theta }}^{\prime }\left(\rho \right)\right|}^q}{32}}\right)}^{{\displaystyle \frac{1}{q}}}\right\}\right.\hfill \\ \hfill & +{\left({\int }_{{\displaystyle \frac{1}{4}}}^{{\displaystyle \frac{1}{2}}}{\left|{\vartheta }^{\alpha }-{\displaystyle \frac{39}{90}}\right|}^{p}d\vartheta \right)}^{{\displaystyle \frac{1}{p}}}\left\{{\left({\displaystyle \frac{3|{\mathrm{\Theta }}^{\prime }{\left(\rho \right)|}^q+5{\left|{\mathrm{\Theta }}^{\prime }\left(\sigma \right)\right|}^q}{32}}\right)}^{{\displaystyle \frac{1}{q}}}+{\left({\displaystyle \frac{3|{\mathrm{\Theta }}^{\prime }{\left(\sigma \right)|}^q+5{\left|{\mathrm{\Theta }}^{\prime }\left(\rho \right)\right|}^q}{32}}\right)}^{{\displaystyle \frac{1}{q}}}\right\}\hfill \\ \hfill & +{\left({\int }_{{\displaystyle \frac{1}{2}}}^{{\displaystyle \frac{3}{4}}}{\left|{\vartheta }^{\alpha }-{\displaystyle \frac{51}{90}}\right|}^{p}d\vartheta \right)}^{{\displaystyle \frac{1}{p}}}\left\{{\left({\displaystyle \frac{5|{\mathrm{\Theta }}^{\prime }{\left(\rho \right)|}^q+3{\left|{\mathrm{\Theta }}^{\prime }\left(\sigma \right)\right|}^q}{32}}\right)}^{{\displaystyle \frac{1}{q}}}+{\left({\displaystyle \frac{5|{\mathrm{\Theta }}^{\prime }{\left(\sigma \right)|}^q+3{\left|{\mathrm{\Theta }}^{\prime }\left(\rho \right)\right|}^q}{32}}\right)}^{{\displaystyle \frac{1}{q}}}\right\}\hfill \\ \hfill & \left.+{\left({\int }_{{\displaystyle \frac{3}{4}}}^1{\left|{\vartheta }^{\alpha }-{\displaystyle \frac{83}{90}}\right|}^{p}d\vartheta \right)}^{{\displaystyle \frac{1}{p}}}\left\{{\left({\displaystyle \frac{7|{\mathrm{\Theta }}^{\prime }{\left(\rho \right)|}^q+{\left|{\mathrm{\Theta }}^{\prime }\left(\sigma \right)\right|}^q}{32}}\right)}^{{\displaystyle \frac{1}{q}}}+{\left({\displaystyle \frac{7|{\mathrm{\Theta }}^{\prime }{\left(\sigma \right)|}^q+{\left|{\mathrm{\Theta }}^{\prime }\left(\rho \right)\right|}^q}{32}}\right)}^{{\displaystyle \frac{1}{q}}}\right\}\right],\hfill \end{array}$$

(18)

where $p^{-1}+q^{-1}=1.$

Proof. 

By applying Hölder’s inequality (16), it becomes

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{90}}\left[7\mathrm{\Theta }\left(\sigma \right)+32\mathrm{\Theta }\left({\displaystyle \frac{3\sigma +\rho }{4}}\right)+12\mathrm{\Theta }\left({\displaystyle \frac{\sigma +\rho }{2}}\right)+32\mathrm{\Theta }\left({\displaystyle \frac{\sigma +3\rho }{4}}\right)+7\mathrm{\Theta }\left(\rho \right)\right]\right.\hfill \\ \hfill & \left.-{\displaystyle \frac{\mathrm{\Gamma }(\alpha +1)}{2{(\rho -\sigma )}^{\alpha }}}\left[J_{\rho -}^{\alpha }\mathrm{\Theta }\left(\sigma \right)+J_{\sigma +}^{\alpha }\mathrm{\Theta }\left(\rho \right)\right]\right|\hfill \\ \hfill & \le {\displaystyle \frac{\rho -\sigma }{2}}\left[{\left({\int }_0^{{\displaystyle \frac{1}{4}}}{\left|{\vartheta }^{\alpha }-{\displaystyle \frac{7}{90}}\right|}^{p}d\vartheta \right)}^{{\displaystyle \frac{1}{p}}}{\left({\int }_0^{{\displaystyle \frac{1}{4}}}{\left|{\mathrm{\Theta }}^{\prime }(\vartheta \rho +(1-\vartheta )\sigma )\right|}^{q}d\vartheta \right)}^{{\displaystyle \frac{1}{q}}}+{\left({\int }_0^{{\displaystyle \frac{1}{4}}}{\left|{\vartheta }^{\alpha }-{\displaystyle \frac{7}{90}}\right|}^{p}d\vartheta \right)}^{{\displaystyle \frac{1}{p}}}\right.\hfill \\ \hfill & \times {\left({\int }_0^{{\displaystyle \frac{1}{4}}}{\left|{\mathrm{\Theta }}^{\prime }(\vartheta \sigma +(1-\vartheta )\rho )\right|}^{q}d\vartheta \right)}^{{\displaystyle \frac{1}{q}}}+{\left({\int }_{{\displaystyle \frac{1}{4}}}^{{\displaystyle \frac{1}{2}}}{\left|{\vartheta }^{\alpha }-{\displaystyle \frac{39}{90}}\right|}^{p}d\vartheta \right)}^{{\displaystyle \frac{1}{p}}}{\left({\int }_{{\displaystyle \frac{1}{4}}}^{{\displaystyle \frac{1}{2}}}{\left|{\mathrm{\Theta }}^{\prime }(\vartheta \rho +(1-\vartheta )\sigma )\right|}^{q}d\vartheta \right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & +{\left({\int }_{{\displaystyle \frac{1}{4}}}^{{\displaystyle \frac{1}{2}}}{\left|{\vartheta }^{\alpha }-{\displaystyle \frac{39}{90}}\right|}^{p}d\vartheta \right)}^{{\displaystyle \frac{1}{p}}}{\left({\int }_{{\displaystyle \frac{1}{4}}}^{{\displaystyle \frac{1}{2}}}{\left|{\mathrm{\Theta }}^{\prime }(\vartheta \sigma +(1-\vartheta )\rho )\right|}^{q}d\vartheta \right)}^{{\displaystyle \frac{1}{q}}}+{\left({\int }_{{\displaystyle \frac{1}{2}}}^{{\displaystyle \frac{3}{4}}}{\left|{\vartheta }^{\alpha }-{\displaystyle \frac{51}{90}}\right|}^{p}d\vartheta \right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ \hfill & \times {\left({\int }_{{\displaystyle \frac{1}{2}}}^{{\displaystyle \frac{3}{4}}}{\left|{\mathrm{\Theta }}^{\prime }(\vartheta \rho +(1-\vartheta )\sigma )\right|}^{q}d\vartheta \right)}^{{\displaystyle \frac{1}{q}}}+{\left({\int }_{{\displaystyle \frac{1}{2}}}^{{\displaystyle \frac{3}{4}}}{\left|{\vartheta }^{\alpha }-{\displaystyle \frac{51}{90}}\right|}^{p}d\vartheta \right)}^{{\displaystyle \frac{1}{p}}}{\left({\int }_{{\displaystyle \frac{1}{2}}}^{{\displaystyle \frac{3}{4}}}{\left|{\mathrm{\Theta }}^{\prime }(\vartheta \sigma +(1-\vartheta )\rho )\right|}^{q}d\vartheta \right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & +{\left({\int }_{{\displaystyle \frac{3}{4}}}^1{\left|{\vartheta }^{\alpha }-{\displaystyle \frac{83}{90}}\right|}^{p}d\vartheta \right)}^{{\displaystyle \frac{1}{p}}}{\left({\int }_{{\displaystyle \frac{3}{4}}}^1{\left|{\mathrm{\Theta }}^{\prime }(\vartheta \rho +(1-\vartheta )\sigma )\right|}^{q}d\vartheta \right)}^{{\displaystyle \frac{1}{q}}}+{\left({\int }_{{\displaystyle \frac{3}{4}}}^1{\left|{\vartheta }^{\alpha }-{\displaystyle \frac{83}{90}}\right|}^{p}d\vartheta \right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ \hfill & \left.\times {\left({\int }_{{\displaystyle \frac{3}{4}}}^1{\left|{\mathrm{\Theta }}^{\prime }(\vartheta \sigma +(1-\vartheta )\rho )\right|}^{q}d\vartheta \right)}^{{\displaystyle \frac{1}{q}}}\right].\hfill \end{array}$$

By leveraging the convexity of $|{\mathrm{\Theta }}^{\prime }{|}^q$, we can determine

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{90}}\left[7\mathrm{\Theta }\left(\sigma \right)+32\mathrm{\Theta }\left({\displaystyle \frac{3\sigma +\rho }{4}}\right)+12\mathrm{\Theta }\left({\displaystyle \frac{\sigma +\rho }{2}}\right)+32\mathrm{\Theta }\left({\displaystyle \frac{\sigma +3\rho }{4}}\right)+7\mathrm{\Theta }\left(\rho \right)\right]\right.\hfill \\ \hfill & \left.-{\displaystyle \frac{\mathrm{\Gamma }(\alpha +1)}{2{(\rho -\sigma )}^{\alpha }}}\left[J_{\rho -}^{\alpha }\mathrm{\Theta }\left(\sigma \right)+J_{\sigma +}^{\alpha }\mathrm{\Theta }\left(\rho \right)\right]\right|\hfill \\ \hfill & \le {\displaystyle \frac{\rho -\sigma }{2}}\left[{\left({\int }_0^{{\displaystyle \frac{1}{4}}}{\left|{\vartheta }^{\alpha }-{\displaystyle \frac{7}{90}}\right|}^{p}d\vartheta \right)}^{{\displaystyle \frac{1}{p}}}\left\{{\left({\int }_0^{{\displaystyle \frac{1}{4}}}\left[\vartheta \right|{\mathrm{\Theta }}^{\prime }{\left(\rho \right)|}^q+(1-\vartheta )|{\mathrm{\Theta }}^{\prime }\left(\sigma \right){|}^q]d\vartheta \right)}^{{\displaystyle \frac{1}{q}}}\right.\right.\hfill \\ \hfill & \left.+{\left({\int }_0^{{\displaystyle \frac{1}{4}}}\left[\vartheta \right|{\mathrm{\Theta }}^{\prime }{\left(\sigma \right)|}^q+(1-\vartheta )|{\mathrm{\Theta }}^{\prime }\left(\rho \right){|}^q]d\vartheta \right)}^{{\displaystyle \frac{1}{q}}}\right\}+{\left({\int }_{{\displaystyle \frac{1}{4}}}^{{\displaystyle \frac{1}{2}}}{\left|{\vartheta }^{\alpha }-{\displaystyle \frac{39}{90}}\right|}^{p}d\vartheta \right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ \hfill & \times \left\{{\left({\int }_{{\displaystyle \frac{1}{4}}}^{{\displaystyle \frac{1}{2}}}\left[\vartheta \right|{\mathrm{\Theta }}^{\prime }{\left(\rho \right)|}^q+(1-\vartheta )|{\mathrm{\Theta }}^{\prime }\left(\sigma \right){|}^q]d\vartheta \right)}^{{\displaystyle \frac{1}{q}}}+{\left({\int }_{{\displaystyle \frac{1}{4}}}^{{\displaystyle \frac{1}{2}}}\left[\vartheta \right|{\mathrm{\Theta }}^{\prime }{\left(\sigma \right)|}^q+(1-\vartheta )|{\mathrm{\Theta }}^{\prime }\left(\rho \right){|}^q]d\vartheta \right)}^{{\displaystyle \frac{1}{q}}}\right\}\hfill \\ \hfill & +{\left({\int }_{{\displaystyle \frac{1}{2}}}^{{\displaystyle \frac{3}{4}}}{\left|{\vartheta }^{\alpha }-{\displaystyle \frac{51}{90}}\right|}^{p}d\vartheta \right)}^{{\displaystyle \frac{1}{p}}}\left\{{\left({\int }_{{\displaystyle \frac{1}{2}}}^{{\displaystyle \frac{3}{4}}}\left(\vartheta \right|{\mathrm{\Theta }}^{\prime }{\left(\rho \right)|}^q+(1-\vartheta )|{\mathrm{\Theta }}^{\prime }\left(\sigma \right){|}^q)d\vartheta \right)}^{{\displaystyle \frac{1}{q}}}\right.\hfill \\ \hfill & \left.+{\left({\int }_{{\displaystyle \frac{1}{2}}}^{{\displaystyle \frac{3}{4}}}\left[\vartheta \right|{\mathrm{\Theta }}^{\prime }{\left(\sigma \right)|}^q+(1-\vartheta )|{\mathrm{\Theta }}^{\prime }\left(\rho \right){|}^q]d\vartheta \right)}^{{\displaystyle \frac{1}{q}}}\right\}+{\left({\int }_{{\displaystyle \frac{3}{4}}}^1{\left|{\vartheta }^{\alpha }-{\displaystyle \frac{83}{90}}\right|}^{p}d\vartheta \right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ \hfill & \left.\times \left\{{\left({\int }_{{\displaystyle \frac{3}{4}}}^1\left[\vartheta \right|{\mathrm{\Theta }}^{\prime }{\left(\rho \right)|}^q+(1-\vartheta )|{\mathrm{\Theta }}^{\prime }\left(\sigma \right){|}^q]d\vartheta \right)}^{{\displaystyle \frac{1}{q}}}+{\left({\int }_{{\displaystyle \frac{3}{4}}}^1\left[\vartheta \right|{\mathrm{\Theta }}^{\prime }{\left(\sigma \right)|}^q+(1-\vartheta )|{\mathrm{\Theta }}^{\prime }\left(\rho \right){|}^q]d\vartheta \right)}^{{\displaystyle \frac{1}{q}}}\right\}\right]\hfill \\ \hfill & ={\displaystyle \frac{\rho -\sigma }{2}}\left[{\left({\int }_0^{{\displaystyle \frac{1}{4}}}{\left|{\vartheta }^{\alpha }-{\displaystyle \frac{7}{90}}\right|}^{p}d\vartheta \right)}^{{\displaystyle \frac{1}{p}}}\left\{{\left({\displaystyle \frac{|{\mathrm{\Theta }}^{\prime }{\left(\rho \right)|}^q+7{\left|{\mathrm{\Theta }}^{\prime }\left(\sigma \right)\right|}^q}{32}}\right)}^{{\displaystyle \frac{1}{q}}}+{\left({\displaystyle \frac{|{\mathrm{\Theta }}^{\prime }{\left(\sigma \right)|}^q+7{\left|{\mathrm{\Theta }}^{\prime }\left(\rho \right)\right|}^q}{32}}\right)}^{{\displaystyle \frac{1}{q}}}\right\}\right.\hfill \\ \hfill & +{\left({\int }_{{\displaystyle \frac{1}{4}}}^{{\displaystyle \frac{1}{2}}}{\left|{\vartheta }^{\alpha }-{\displaystyle \frac{39}{90}}\right|}^{p}d\vartheta \right)}^{{\displaystyle \frac{1}{p}}}\left\{{\left({\displaystyle \frac{3|{\mathrm{\Theta }}^{\prime }{\left(\rho \right)|}^q+5{\left|{\mathrm{\Theta }}^{\prime }\left(\sigma \right)\right|}^q}{32}}\right)}^{{\displaystyle \frac{1}{q}}}+{\left({\displaystyle \frac{3|{\mathrm{\Theta }}^{\prime }{\left(\sigma \right)|}^q+5{\left|{\mathrm{\Theta }}^{\prime }\left(\rho \right)\right|}^q}{32}}\right)}^{{\displaystyle \frac{1}{q}}}\right\}\hfill \\ \hfill & +{\left({\int }_{{\displaystyle \frac{1}{2}}}^{{\displaystyle \frac{3}{4}}}{\left|{\vartheta }^{\alpha }-{\displaystyle \frac{51}{90}}\right|}^{p}d\vartheta \right)}^{{\displaystyle \frac{1}{p}}}\left\{{\left({\displaystyle \frac{5|{\mathrm{\Theta }}^{\prime }{\left(\rho \right)|}^q+3{\left|{\mathrm{\Theta }}^{\prime }\left(\sigma \right)\right|}^q}{32}}\right)}^{{\displaystyle \frac{1}{q}}}+{\left({\displaystyle \frac{5|{\mathrm{\Theta }}^{\prime }{\left(\sigma \right)|}^q+3{\left|{\mathrm{\Theta }}^{\prime }\left(\rho \right)\right|}^q}{32}}\right)}^{{\displaystyle \frac{1}{q}}}\right\}\hfill \\ \hfill & \left.+{\left({\int }_{{\displaystyle \frac{3}{4}}}^1{\left|{\vartheta }^{\alpha }-{\displaystyle \frac{83}{90}}\right|}^{p}d\vartheta \right)}^{{\displaystyle \frac{1}{p}}}\left\{{\left({\displaystyle \frac{7|{\mathrm{\Theta }}^{\prime }{\left(\rho \right)|}^q+{\left|{\mathrm{\Theta }}^{\prime }\left(\sigma \right)\right|}^q}{32}}\right)}^{{\displaystyle \frac{1}{q}}}+{\left({\displaystyle \frac{7|{\mathrm{\Theta }}^{\prime }{\left(\sigma \right)|}^q+{\left|{\mathrm{\Theta }}^{\prime }\left(\rho \right)\right|}^q}{32}}\right)}^{{\displaystyle \frac{1}{q}}}\right\}\right].\hfill \end{array}$$

The demonstration for Theorem 4 has been finalized. □

Remark 2.

In Theorem 4, setting $\alpha =1$ yields the following inequality:

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{90}}\left[7\mathrm{\Theta }\left(\sigma \right)+32\mathrm{\Theta }\left({\displaystyle \frac{3\sigma +\rho }{4}}\right)+12\mathrm{\Theta }\left({\displaystyle \frac{\sigma +\rho }{2}}\right)+32\mathrm{\Theta }\left({\displaystyle \frac{\sigma +3\rho }{4}}\right)+7\mathrm{\Theta }\left(\rho \right)\right]\right.\hfill \\ \hfill & \left.-{\displaystyle \frac{1}{\rho -\sigma }}{\int }_{\sigma }^{\rho }\mathrm{\Theta }\left(\varpi \right)d\varpi \right|\hfill \\ \hfill & \le {\displaystyle \frac{\rho -\sigma }{2}}\left[{\left({\displaystyle \frac{{\left(\frac{14}{45}\right)}^{p+1}+{\left(\frac{31}{45}\right)}^{p+1}}{4^{p+1}(p+1)}}\right)}^{{\displaystyle \frac{1}{p}}}\left\{{\left({\displaystyle \frac{|{\mathrm{\Theta }}^{\prime }{\left(\rho \right)|}^q+7{\left|{\mathrm{\Theta }}^{\prime }\left(\sigma \right)\right|}^q}{32}}\right)}^{{\displaystyle \frac{1}{q}}}+{\left({\displaystyle \frac{|{\mathrm{\Theta }}^{\prime }{\left(\sigma \right)|}^q+7{\left|{\mathrm{\Theta }}^{\prime }\left(\rho \right)\right|}^q}{32}}\right)}^{{\displaystyle \frac{1}{q}}}\right\}\right.\hfill \\ \hfill & +{\left({\displaystyle \frac{{\left(\frac{4}{15}\right)}^{p+1}+{\left(\frac{11}{15}\right)}^{p+1}}{4^{p+1}(p+1)}}\right)}^{{\displaystyle \frac{1}{p}}}\left\{{\left({\displaystyle \frac{3|{\mathrm{\Theta }}^{\prime }{\left(\rho \right)|}^q+5{\left|{\mathrm{\Theta }}^{\prime }\left(\sigma \right)\right|}^q}{32}}\right)}^{{\displaystyle \frac{1}{q}}}+{\left({\displaystyle \frac{3|{\mathrm{\Theta }}^{\prime }{\left(\sigma \right)|}^q+5{\left|{\mathrm{\Theta }}^{\prime }\left(\rho \right)\right|}^q}{32}}\right)}^{{\displaystyle \frac{1}{q}}}\right\}\hfill \\ \hfill & +{\left({\displaystyle \frac{{\left(\frac{4}{15}\right)}^{p+1}+{\left(\frac{11}{15}\right)}^{p+1}}{4^{p+1}(p+1)}}\right)}^{{\displaystyle \frac{1}{p}}}\left\{{\left({\displaystyle \frac{5|{\mathrm{\Theta }}^{\prime }{\left(\rho \right)|}^q+3{\left|{\mathrm{\Theta }}^{\prime }\left(\sigma \right)\right|}^q}{32}}\right)}^{{\displaystyle \frac{1}{q}}}+{\left({\displaystyle \frac{5|{\mathrm{\Theta }}^{\prime }{\left(\sigma \right)|}^q+3{\left|{\mathrm{\Theta }}^{\prime }\left(\rho \right)\right|}^q}{32}}\right)}^{{\displaystyle \frac{1}{q}}}\right\}\hfill \\ \hfill & \left.+{\left({\displaystyle \frac{{\left(\frac{14}{45}\right)}^{p+1}+{\left(\frac{31}{45}\right)}^{p+1}}{4^{p+1}(p+1)}}\right)}^{{\displaystyle \frac{1}{p}}}\left\{{\left({\displaystyle \frac{7|{\mathrm{\Theta }}^{\prime }{\left(\rho \right)|}^q+{\left|{\mathrm{\Theta }}^{\prime }\left(\sigma \right)\right|}^q}{32}}\right)}^{{\displaystyle \frac{1}{q}}}+{\left({\displaystyle \frac{7|{\mathrm{\Theta }}^{\prime }{\left(\sigma \right)|}^q+{\left|{\mathrm{\Theta }}^{\prime }\left(\rho \right)\right|}^q}{32}}\right)}^{{\displaystyle \frac{1}{q}}}\right\}\right].\hfill \end{array}$$

which is proved by Shehzadi et al. in [33] (Theorem 4).

Theorem 5.

Assume that all conditions in Lemma 1 are satisfied and that $|{\mathrm{\Theta }}^{\prime }{|}^q,$ where $q\ge 1$ is a convex function. Then, the following inequality holds:

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{90}}\left[7\mathrm{\Theta }\left(\sigma \right)+32\mathrm{\Theta }\left({\displaystyle \frac{3\sigma +\rho }{4}}\right)+12\mathrm{\Theta }\left({\displaystyle \frac{\sigma +\rho }{2}}\right)+32\mathrm{\Theta }\left({\displaystyle \frac{\sigma +3\rho }{4}}\right)+7\mathrm{\Theta }\left(\rho \right)\right]\right.\hfill \\ \hfill & \left.-{\displaystyle \frac{\mathrm{\Gamma }(\alpha +1)}{2{(\rho -\sigma )}^{\alpha }}}\left[J_{\rho -}^{\alpha }\mathrm{\Theta }\left(\sigma \right)+J_{\sigma +}^{\alpha }\mathrm{\Theta }\left(\rho \right)\right]\right|\hfill \\ \hfill & \le {\displaystyle \frac{\rho -\sigma }{2}}\left[{\left({\mathsf{A}}_1\left(\alpha \right)\right)}^{1-{\displaystyle \frac{1}{q}}}\left\{{\left({\mathsf{A}}_5\left(\alpha \right)|{\mathrm{\Theta }}^{\prime }{\left(\rho \right)|}^q+\left({\mathsf{A}}_1\left(\alpha \right)-{\mathsf{A}}_5\left(\alpha \right)\right){\left|{\mathrm{\Theta }}^{\prime }\left(\sigma \right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\right.\right.\hfill \\ \hfill & \left.+{\left({\mathsf{A}}_5\left(\alpha \right)|{\mathrm{\Theta }}^{\prime }{\left(\sigma \right)|}^q+\left({\mathsf{A}}_1\left(\alpha \right)-{\mathsf{A}}_5\left(\alpha \right)\right){\left|{\mathrm{\Theta }}^{\prime }\left(\rho \right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\right\}\hfill \\ \hfill & +{\left({\mathsf{A}}_2\left(\alpha \right)\right)}^{1-{\displaystyle \frac{1}{q}}}\left\{{\left({\mathsf{A}}_6\left(\alpha \right)|{\mathrm{\Theta }}^{\prime }{\left(\rho \right)|}^q+\left({\mathsf{A}}_2\left(\alpha \right)-{\mathsf{A}}_6\left(\alpha \right)\right){\left|{\mathrm{\Theta }}^{\prime }\left(\sigma \right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\right.\hfill \\ \hfill & \left.+{\left({\mathsf{A}}_6\left(\alpha \right)|{\mathrm{\Theta }}^{\prime }{\left(\sigma \right)|}^q+\left({\mathsf{A}}_2\left(\alpha \right)-{\mathsf{A}}_6\left(\alpha \right)\right){\left|{\mathrm{\Theta }}^{\prime }\left(\rho \right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\right\}\hfill \\ \hfill & +{\left({\mathsf{A}}_3\left(\alpha \right)\right)}^{1-{\displaystyle \frac{1}{q}}}\left\{{\left({\mathsf{A}}_7\left(\alpha \right)|{\mathrm{\Theta }}^{\prime }{\left(\rho \right)|}^q+\left({\mathsf{A}}_3\left(\alpha \right)-{\mathsf{A}}_7\left(\alpha \right)\right){\left|{\mathrm{\Theta }}^{\prime }\left(\sigma \right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\right.\hfill \\ \hfill & \left.+{\left({\mathsf{A}}_7\left(\alpha \right)|{\mathrm{\Theta }}^{\prime }{\left(\sigma \right)|}^q+\left({\mathsf{A}}_3\left(\alpha \right)-{\mathsf{A}}_7\left(\alpha \right)\right){\left|{\mathrm{\Theta }}^{\prime }\left(\rho \right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\right\}\hfill \\ \hfill & +{\left({\mathsf{A}}_4\left(\alpha \right)\right)}^{1-{\displaystyle \frac{1}{q}}}\left\{{\left({\mathsf{A}}_8\left(\alpha \right)|{\mathrm{\Theta }}^{\prime }{\left(\rho \right)|}^q+\left({\mathsf{A}}_4\left(\alpha \right)-{\mathsf{A}}_8\left(\alpha \right)\right){\left|{\mathrm{\Theta }}^{\prime }\left(\sigma \right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\right.\hfill \\ \hfill & \left.\left.+{\left({\mathsf{A}}_8\left(\alpha \right)|{\mathrm{\Theta }}^{\prime }{\left(\sigma \right)|}^q+\left({\mathsf{A}}_4\left(\alpha \right)-{\mathsf{A}}_8\left(\alpha \right)\right){\left|{\mathrm{\Theta }}^{\prime }\left(\rho \right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\right\}\right].\hfill \end{array}$$

(19)

Proof. 

By applying the power mean inequality (16), we have

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{90}}\left[7\mathrm{\Theta }\left(\sigma \right)+32\mathrm{\Theta }\left({\displaystyle \frac{3\sigma +\rho }{4}}\right)+12\mathrm{\Theta }\left({\displaystyle \frac{\sigma +\rho }{2}}\right)+32\mathrm{\Theta }\left({\displaystyle \frac{\sigma +3\rho }{4}}\right)+7\mathrm{\Theta }\left(\rho \right)\right]\right.\hfill \\ \hfill & \left.-{\displaystyle \frac{\mathrm{\Gamma }(\alpha +1)}{2{(\rho -\sigma )}^{\alpha }}}\left[J_{\rho -}^{\alpha }\mathrm{\Theta }\left(\sigma \right)+J_{\sigma +}^{\alpha }\mathrm{\Theta }\left(\rho \right)\right]\right|\hfill \\ \hfill & \le {\displaystyle \frac{\rho -\sigma }{2}}\left[{\left({\int }_0^{{\displaystyle \frac{1}{4}}}\left|{\vartheta }^{\alpha }-{\displaystyle \frac{7}{90}}\right|d\vartheta \right)}^{1-{\displaystyle \frac{1}{q}}}\left\{{\left({\int }_0^{{\displaystyle \frac{1}{4}}}\left|{\vartheta }^{\alpha }-{\displaystyle \frac{7}{90}}\right|{\left|{\mathrm{\Theta }}^{\prime }(\vartheta \rho +(1-\vartheta )\sigma )\right|}^{q}d\vartheta \right)}^{{\displaystyle \frac{1}{q}}}\right.\right.\hfill \\ \hfill & \left.+{\left({\int }_0^{{\displaystyle \frac{1}{4}}}\left|{\vartheta }^{\alpha }-{\displaystyle \frac{7}{90}}\right|{\left|{\mathrm{\Theta }}^{\prime }(\vartheta \sigma +(1-\vartheta )\rho )\right|}^{q}d\vartheta \right)}^{{\displaystyle \frac{1}{q}}}\right\}+{\left({\int }_{{\displaystyle \frac{1}{4}}}^{{\displaystyle \frac{1}{2}}}\left|{\vartheta }^{\alpha }-{\displaystyle \frac{39}{90}}\right|d\vartheta \right)}^{1-{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & \times \left\{{\left({\int }_{{\displaystyle \frac{1}{4}}}^{{\displaystyle \frac{1}{2}}}\left|{\vartheta }^{\alpha }-{\displaystyle \frac{39}{90}}\right|{\left|{\mathrm{\Theta }}^{\prime }(\vartheta \rho +(1-\vartheta )\sigma )\right|}^{q}d\vartheta \right)}^{{\displaystyle \frac{1}{q}}}+{\left({\int }_{{\displaystyle \frac{1}{4}}}^{{\displaystyle \frac{1}{2}}}\left|{\vartheta }^{\alpha }-{\displaystyle \frac{39}{90}}\right|{\left|{\mathrm{\Theta }}^{\prime }(\vartheta \sigma +(1-\vartheta )\rho )\right|}^{q}d\vartheta \right)}^{{\displaystyle \frac{1}{q}}}\right\}\hfill \\ \hfill & +{\left({\int }_{{\displaystyle \frac{1}{2}}}^{{\displaystyle \frac{3}{4}}}\left|{\vartheta }^{\alpha }-{\displaystyle \frac{51}{90}}\right|d\vartheta \right)}^{1-{\displaystyle \frac{1}{q}}}\left\{{\left({\int }_{{\displaystyle \frac{1}{2}}}^{{\displaystyle \frac{3}{4}}}\left|{\vartheta }^{\alpha }-{\displaystyle \frac{51}{90}}\right|{\left|{\mathrm{\Theta }}^{\prime }(\vartheta \rho +(1-\vartheta )\sigma )\right|}^{q}d\vartheta \right)}^{{\displaystyle \frac{1}{q}}}\right.\hfill \\ \hfill & \left.+{\left({\int }_{{\displaystyle \frac{1}{2}}}^{{\displaystyle \frac{3}{4}}}\left|{\vartheta }^{\alpha }-{\displaystyle \frac{51}{90}}\right|{\left|{\mathrm{\Theta }}^{\prime }(\vartheta \sigma +(1-\vartheta )\rho )\right|}^{q}d\vartheta \right)}^{{\displaystyle \frac{1}{q}}}\right\}+{\left({\int }_{{\displaystyle \frac{3}{4}}}^1\left|{\vartheta }^{\alpha }-{\displaystyle \frac{83}{90}}\right|d\vartheta \right)}^{1-{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & \left.\times \left\{{\left({\int }_{{\displaystyle \frac{3}{4}}}^1\left|{\vartheta }^{\alpha }-{\displaystyle \frac{83}{90}}\right|{\left|{\mathrm{\Theta }}^{\prime }(\vartheta \rho +(1-\vartheta )\sigma )\right|}^{q}d\vartheta \right)}^{{\displaystyle \frac{1}{q}}}+{\left({\int }_{{\displaystyle \frac{3}{4}}}^1\left|{\vartheta }^{\alpha }-{\displaystyle \frac{83}{90}}\right|{\left|{\mathrm{\Theta }}^{\prime }(\vartheta \sigma +(1-\vartheta )\rho )\right|}^{q}d\vartheta \right)}^{{\displaystyle \frac{1}{q}}}\right\}\right].\hfill \end{array}$$

As $|{\mathrm{\Theta }}^{\prime }{|}^q$ is convex on the interval $[\sigma ,\rho ],$ we arrive at

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{90}}\left[7\mathrm{\Theta }\left(\sigma \right)+32\mathrm{\Theta }\left({\displaystyle \frac{3\sigma +\rho }{4}}\right)+12\mathrm{\Theta }\left({\displaystyle \frac{\sigma +\rho }{2}}\right)+32\mathrm{\Theta }\left({\displaystyle \frac{\sigma +3\rho }{4}}\right)+7\mathrm{\Theta }\left(\rho \right)\right]\right.\hfill \\ \hfill & \left.-{\displaystyle \frac{\mathrm{\Gamma }(\alpha +1)}{2{(\rho -\sigma )}^{\alpha }}}\left[J_{\rho -}^{\alpha }\mathrm{\Theta }\left(\sigma \right)+J_{\sigma +}^{\alpha }\mathrm{\Theta }\left(\rho \right)\right]\right|\hfill \\ \hfill & \le {\displaystyle \frac{\rho -\sigma }{2}}\left[{\left({\int }_0^{{\displaystyle \frac{1}{4}}}\left|{\vartheta }^{\alpha }-{\displaystyle \frac{7}{90}}\right|d\vartheta \right)}^{1-{\displaystyle \frac{1}{q}}}\left\{{\left({\int }_0^{{\displaystyle \frac{1}{4}}}\left|{\vartheta }^{\alpha }-{\displaystyle \frac{7}{90}}\right|\left[\vartheta \right|{\mathrm{\Theta }}^{\prime }{\left(\rho \right)|}^q+(1-\vartheta )|{\mathrm{\Theta }}^{\prime }\left(\sigma \right){|}^q]d\vartheta \right)}^{{\displaystyle \frac{1}{q}}}\right.\right.\hfill \\ \hfill & \left.+{\left({\int }_0^{{\displaystyle \frac{1}{4}}}\left|{\vartheta }^{\alpha }-{\displaystyle \frac{7}{90}}\right|\left[\vartheta \right|{\mathrm{\Theta }}^{\prime }{\left(\sigma \right)|}^q+(1-\vartheta )|{\mathrm{\Theta }}^{\prime }\left(\rho \right){|}^q]d\vartheta \right)}^{{\displaystyle \frac{1}{q}}}\right\}+{\left({\int }_{{\displaystyle \frac{1}{4}}}^{{\displaystyle \frac{1}{2}}}\left|{\vartheta }^{\alpha }-{\displaystyle \frac{39}{90}}\right|d\vartheta \right)}^{1-{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & \times \left\{{\left({\int }_{{\displaystyle \frac{1}{4}}}^{{\displaystyle \frac{1}{2}}}\left|{\vartheta }^{\alpha }-{\displaystyle \frac{39}{90}}\right|\left[\vartheta \right|{\mathrm{\Theta }}^{\prime }{\left(\rho \right)|}^q+(1-\vartheta )|{\mathrm{\Theta }}^{\prime }\left(\sigma \right){|}^q]d\vartheta \right)}^{{\displaystyle \frac{1}{q}}}\right.\hfill \\ \hfill & \left.+{\left({\int }_{{\displaystyle \frac{1}{4}}}^{{\displaystyle \frac{1}{2}}}\left|{\vartheta }^{\alpha }-{\displaystyle \frac{39}{90}}\right|\left[\vartheta \right|{\mathrm{\Theta }}^{\prime }{\left(\sigma \right)|}^q+(1-\vartheta )|{\mathrm{\Theta }}^{\prime }\left(\rho \right){|}^q]d\vartheta \right)}^{{\displaystyle \frac{1}{q}}}\right\}\hfill \\ \hfill & +{\left({\int }_{{\displaystyle \frac{1}{2}}}^{{\displaystyle \frac{3}{4}}}\left|{\vartheta }^{\alpha }-{\displaystyle \frac{51}{90}}\right|d\vartheta \right)}^{1-{\displaystyle \frac{1}{q}}}\left\{{\left({\int }_{{\displaystyle \frac{1}{2}}}^{{\displaystyle \frac{3}{4}}}\left|{\vartheta }^{\alpha }-{\displaystyle \frac{51}{90}}\right|\left[\vartheta \right|{\mathrm{\Theta }}^{\prime }{\left(\rho \right)|}^q+(1-\vartheta )|{\mathrm{\Theta }}^{\prime }\left(\sigma \right){|}^q]d\vartheta \right)}^{{\displaystyle \frac{1}{q}}}\right.\hfill \\ \hfill & \left.+{\left({\int }_{{\displaystyle \frac{1}{2}}}^{{\displaystyle \frac{3}{4}}}\left|{\vartheta }^{\alpha }-{\displaystyle \frac{51}{90}}\right|\left[\vartheta \right|{\mathrm{\Theta }}^{\prime }{\left(\sigma \right)|}^q+(1-\vartheta )|{\mathrm{\Theta }}^{\prime }\left(\rho \right){|}^q]d\vartheta \right)}^{{\displaystyle \frac{1}{q}}}\right\}+{\left({\int }_{{\displaystyle \frac{3}{4}}}^1\left|{\vartheta }^{\alpha }-{\displaystyle \frac{83}{90}}\right|d\vartheta \right)}^{1-{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & \times \left\{{\left({\int }_{{\displaystyle \frac{3}{4}}}^1\left|{\vartheta }^{\alpha }-{\displaystyle \frac{83}{90}}\right|\left[\vartheta \right|{\mathrm{\Theta }}^{\prime }{\left(\rho \right)|}^q+(1-\vartheta )|{\mathrm{\Theta }}^{\prime }\left(\sigma \right){|}^q]d\vartheta \right)}^{{\displaystyle \frac{1}{q}}}\right.\hfill \\ \hfill & \left.\left.+{\left({\int }_{{\displaystyle \frac{3}{4}}}^1\left|{\vartheta }^{\alpha }-{\displaystyle \frac{83}{90}}\right|\left[\vartheta \right|{\mathrm{\Theta }}^{\prime }{\left(\sigma \right)|}^q+(1-\vartheta )|{\mathrm{\Theta }}^{\prime }\left(\rho \right){|}^q]d\vartheta \right)}^{{\displaystyle \frac{1}{q}}}\right\}\right]\hfill \\ \hfill & ={\displaystyle \frac{\rho -\sigma }{2}}\left[{\left({\mathsf{A}}_1\left(\alpha \right)\right)}^{1-{\displaystyle \frac{1}{q}}}\left\{{\left({\mathsf{A}}_5\left(\alpha \right)|{\mathrm{\Theta }}^{\prime }{\left(\rho \right)|}^q+\left({\mathsf{A}}_1\left(\alpha \right)-{\mathsf{A}}_5\left(\alpha \right)\right){\left|{\mathrm{\Theta }}^{\prime }\left(\sigma \right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\right.\right.\hfill \\ \hfill & \left.+{\left({\mathsf{A}}_5\left(\alpha \right)|{\mathrm{\Theta }}^{\prime }{\left(\sigma \right)|}^q+\left({\mathsf{A}}_1\left(\alpha \right)-{\mathsf{A}}_5\left(\alpha \right)\right){\left|{\mathrm{\Theta }}^{\prime }\left(\rho \right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\right\}+{\left({\mathsf{A}}_2\left(\alpha \right)\right)}^{1-{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & \times \left\{{\left({\mathsf{A}}_6\left(\alpha \right)|{\mathrm{\Theta }}^{\prime }{\left(\rho \right)|}^q+\left({\mathsf{A}}_2\left(\alpha \right)-{\mathsf{A}}_6\left(\alpha \right)\right){\left|{\mathrm{\Theta }}^{\prime }\left(\sigma \right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}+{\left({\mathsf{A}}_6\left(\alpha \right)|{\mathrm{\Theta }}^{\prime }{\left(\sigma \right)|}^q+\left({\mathsf{A}}_2\left(\alpha \right)-{\mathsf{A}}_6\left(\alpha \right)\right){\left|{\mathrm{\Theta }}^{\prime }\left(\rho \right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\right\}\hfill \\ \hfill & +{\left({\mathsf{A}}_3\left(\alpha \right)\right)}^{1-{\displaystyle \frac{1}{q}}}\left\{{\left({\mathsf{A}}_7\left(\alpha \right)|{\mathrm{\Theta }}^{\prime }{\left(\rho \right)|}^q+\left({\mathsf{A}}_3\left(\alpha \right)-{\mathsf{A}}_7\left(\alpha \right)\right){\left|{\mathrm{\Theta }}^{\prime }\left(\sigma \right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\right.\hfill \\ \hfill & \left.+{\left({\mathsf{A}}_7\left(\alpha \right)|{\mathrm{\Theta }}^{\prime }{\left(\sigma \right)|}^q+\left({\mathsf{A}}_3\left(\alpha \right)-{\mathsf{A}}_7\left(\alpha \right)\right){\left|{\mathrm{\Theta }}^{\prime }\left(\rho \right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\right\}+{\left({\mathsf{A}}_4\left(\alpha \right)\right)}^{1-{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & \left.\times \left\{{\left({\mathsf{A}}_8\left(\alpha \right)|{\mathrm{\Theta }}^{\prime }{\left(\rho \right)|}^q+\left({\mathsf{A}}_4\left(\alpha \right)-{\mathsf{A}}_8\left(\alpha \right)\right){\left|{\mathrm{\Theta }}^{\prime }\left(\sigma \right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}+{\left({\mathsf{A}}_8\left(\alpha \right)|{\mathrm{\Theta }}^{\prime }{\left(\sigma \right)|}^q+\left({\mathsf{A}}_4\left(\alpha \right)-{\mathsf{A}}_8\left(\alpha \right)\right){\left|{\mathrm{\Theta }}^{\prime }\left(\rho \right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\right\}\right].\hfill \end{array}$$

The proof of Theorem 5 has been finalized. □

Remark 3.

In Theorem 5, setting $\alpha =1$ yields the following inequality:

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{90}}\left[7\mathrm{\Theta }\left(\sigma \right)+32\mathrm{\Theta }\left({\displaystyle \frac{3\sigma +\rho }{4}}\right)+12\mathrm{\Theta }\left({\displaystyle \frac{\sigma +\rho }{2}}\right)+32\mathrm{\Theta }\left({\displaystyle \frac{\sigma +3\rho }{4}}\right)+7\mathrm{\Theta }\left(\rho \right)\right]-{\displaystyle \frac{1}{\rho -\sigma }}{\int }_{\sigma }^{\rho }\mathrm{\Theta }\left(\varpi \right)d\varpi \right|\hfill \\ \hfill & \le {\displaystyle \frac{\rho -\sigma }{2880}}\left[{\left({\displaystyle \frac{1157}{45}}\right)}^{1-{\displaystyle \frac{1}{q}}}\left\{{\left({\displaystyle \frac{25672|{\mathrm{\Theta }}^{\prime }{\left(\rho \right)|}^q+130523{\left|{\mathrm{\Theta }}^{\prime }\left(\sigma \right)\right|}^q}{6075}}\right)}^{{\displaystyle \frac{1}{q}}}\right.\right.\hfill \\ \hfill & \left.+{\left({\displaystyle \frac{25672|{\mathrm{\Theta }}^{\prime }{\left(\sigma \right)|}^q+130523{\left|{\mathrm{\Theta }}^{\prime }\left(\rho \right)\right|}^q}{6075}}\right)}^{{\displaystyle \frac{1}{q}}}\right\}+{\left({\displaystyle \frac{137}{5}}\right)}^{1-{\displaystyle \frac{1}{q}}}\left\{{\left({\displaystyle \frac{2038|{\mathrm{\Theta }}^{\prime }{\left(\rho \right)|}^q+4127{\left|{\mathrm{\Theta }}^{\prime }\left(\sigma \right)\right|}^q}{225}}\right)}^{{\displaystyle \frac{1}{q}}}\right.\hfill \\ \hfill & \left.+{\left({\displaystyle \frac{2038|{\mathrm{\Theta }}^{\prime }{\left(\sigma \right)|}^q+4127{\left|{\mathrm{\Theta }}^{\prime }\left(\rho \right)\right|}^q}{225}}\right)}^{{\displaystyle \frac{1}{q}}}\right\}+{\left({\displaystyle \frac{137}{5}}\right)}^{1-{\displaystyle \frac{1}{q}}}\left\{{\left({\displaystyle \frac{4127|{\mathrm{\Theta }}^{\prime }{\left(\rho \right)|}^q+2038{\left|{\mathrm{\Theta }}^{\prime }\left(\sigma \right)\right|}^q}{225}}\right)}^{{\displaystyle \frac{1}{q}}}\right.\hfill \\ \hfill & \left.+{\left({\displaystyle \frac{4127|{\mathrm{\Theta }}^{\prime }{\left(\sigma \right)|}^q+2038{\left|{\mathrm{\Theta }}^{\prime }\left(\rho \right)\right|}^q}{225}}\right)}^{{\displaystyle \frac{1}{q}}}\right\}+{\left({\displaystyle \frac{1157}{45}}\right)}^{1-{\displaystyle \frac{1}{q}}}\left\{{\left({\displaystyle \frac{130523|{\mathrm{\Theta }}^{\prime }{\left(\rho \right)|}^q+25672{\left|{\mathrm{\Theta }}^{\prime }\left(\sigma \right)\right|}^q}{6075}}\right)}^{{\displaystyle \frac{1}{q}}}\right.\hfill \\ \hfill & \left.\left.+{\left({\displaystyle \frac{130523|{\mathrm{\Theta }}^{\prime }{\left(\sigma \right)|}^q+25672{\left|{\mathrm{\Theta }}^{\prime }\left(\rho \right)\right|}^q}{6075}}\right)}^{{\displaystyle \frac{1}{q}}}\right\}\right].\hfill \end{array}$$

which is proved by Shehzadi et al. in [33] (Theorem 5).

2.2. Fractional Boole’s Type Inequality for Bounded Functions

Theorem 6.

Assuming all conditions of Lemma 1 are satisfied, if there exist $m,M\in \mathfrak{R}$ such that $m\le {\mathrm{\Theta }}^{\prime }\left(\vartheta \right)\le M$ for $\vartheta \in [\sigma ,\rho ],$ then we have the subsequent Boole-rule-type inequality

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{90}}\left[7\mathrm{\Theta }\left(\sigma \right)+32\mathrm{\Theta }\left({\displaystyle \frac{3\sigma +\rho }{4}}\right)+12\mathrm{\Theta }\left({\displaystyle \frac{\sigma +\rho }{2}}\right)+32\mathrm{\Theta }\left({\displaystyle \frac{\sigma +3\rho }{4}}\right)+7\mathrm{\Theta }\left(\rho \right)\right]\right.\hfill \\ \hfill & \left.-{\displaystyle \frac{\mathrm{\Gamma }(\alpha +1)}{2{(\rho -\sigma )}^{\alpha }}}\left[J_{\rho -}^{\alpha }\mathrm{\Theta }\left(\sigma \right)+J_{\sigma +}^{\alpha }\mathrm{\Theta }\left(\rho \right)\right]\right|\le {\displaystyle \frac{(\rho -\sigma )}{2}}\left[{\mathsf{A}}_1\left(\alpha \right)+{\mathsf{A}}_2\left(\alpha \right)+{\mathsf{A}}_3\left(\alpha \right)+{\mathsf{A}}_4\left(\alpha \right)\right](M-m),\hfill \end{array}$$

(20)

where ${\mathsf{A}}_1\left(\alpha \right)-{\mathsf{A}}_4\left(\alpha \right)$ are defined in (14).

Proof. 

With the help of Lemma 1, we have

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{90}}\left[7\mathrm{\Theta }\left(\sigma \right)+32\mathrm{\Theta }\left({\displaystyle \frac{3\sigma +\rho }{4}}\right)+12\mathrm{\Theta }\left({\displaystyle \frac{\sigma +\rho }{2}}\right)+32\mathrm{\Theta }\left({\displaystyle \frac{\sigma +3\rho }{4}}\right)+7\mathrm{\Theta }\left(\rho \right)\right]\hfill \\ \hfill & -{\displaystyle \frac{\mathrm{\Gamma }(\alpha +1)}{2{(\rho -\sigma )}^{\alpha }}}\left[J_{\rho -}^{\alpha }\mathrm{\Theta }\left(\sigma \right)+J_{\sigma +}^{\alpha }\mathrm{\Theta }\left(\rho \right)\right]\hfill \\ \hfill & ={\displaystyle \frac{\rho -\sigma }{2}}\left[{\int }_0^{{\displaystyle \frac{1}{4}}}\left({\vartheta }^{\alpha }-{\displaystyle \frac{7}{90}}\right)\left[{\mathrm{\Theta }}^{\prime }(\vartheta \rho +(1-\vartheta )\sigma )-{\displaystyle \frac{m+M}{2}}\right]d\vartheta \right.\hfill \\ \hfill & +{\int }_0^{{\displaystyle \frac{1}{4}}}\left({\vartheta }^{\alpha }-{\displaystyle \frac{7}{90}}\right)\left[{\displaystyle \frac{m+M}{2}}-{\mathrm{\Theta }}^{\prime }(\vartheta \sigma +(1-\vartheta )\rho )\right]d\vartheta \hfill \\ \hfill & +{\int }_{{\displaystyle \frac{1}{4}}}^{{\displaystyle \frac{1}{2}}}\left({\vartheta }^{\alpha }-{\displaystyle \frac{39}{90}}\right)\left[{\mathrm{\Theta }}^{\prime }(\vartheta \rho +(1-\vartheta )\sigma )-{\displaystyle \frac{m+M}{2}}\right]d\vartheta \hfill \\ \hfill & +{\int }_{{\displaystyle \frac{1}{4}}}^{{\displaystyle \frac{1}{2}}}\left({\vartheta }^{\alpha }-{\displaystyle \frac{39}{90}}\right)\left[{\displaystyle \frac{m+M}{2}}-{\mathrm{\Theta }}^{\prime }(\vartheta \sigma +(1-\vartheta )\rho )\right]d\vartheta \hfill \\ \hfill & +{\int }_{{\displaystyle \frac{1}{2}}}^{{\displaystyle \frac{3}{4}}}\left({\vartheta }^{\alpha }-{\displaystyle \frac{51}{90}}\right)\left[{\mathrm{\Theta }}^{\prime }(\vartheta \rho +(1-\vartheta )\sigma )-{\displaystyle \frac{m+M}{2}}\right]d\vartheta \hfill \\ \hfill & +{\int }_{{\displaystyle \frac{1}{2}}}^{{\displaystyle \frac{3}{4}}}\left({\vartheta }^{\alpha }-{\displaystyle \frac{51}{90}}\right)\left[{\displaystyle \frac{m+M}{2}}-{\mathrm{\Theta }}^{\prime }(\vartheta \sigma +(1-\vartheta )\rho )\right]d\vartheta \hfill \\ \hfill & +{\int }_{{\displaystyle \frac{3}{4}}}^1\left({\vartheta }^{\alpha }-{\displaystyle \frac{83}{90}}\right)\left[{\mathrm{\Theta }}^{\prime }(\vartheta \rho +(1-\vartheta )\sigma )-{\displaystyle \frac{m+M}{2}}\right]d\vartheta \hfill \\ \hfill & \left.+{\int }_{{\displaystyle \frac{3}{4}}}^1\left({\vartheta }^{\alpha }-{\displaystyle \frac{83}{90}}\right)\left[{\displaystyle \frac{m+M}{2}}-{\mathrm{\Theta }}^{\prime }(\vartheta \sigma +(1-\vartheta )\rho )\right]d\vartheta \right].\hfill \end{array}$$

(21)

Leveraging the properties of modulus in (21), we derive

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{90}}\left[7\mathrm{\Theta }\left(\sigma \right)+32\mathrm{\Theta }\left({\displaystyle \frac{3\sigma +\rho }{4}}\right)+12\mathrm{\Theta }\left({\displaystyle \frac{\sigma +\rho }{2}}\right)+32\mathrm{\Theta }\left({\displaystyle \frac{\sigma +3\rho }{4}}\right)+7\mathrm{\Theta }\left(\rho \right)\right]\right.\hfill \\ \hfill & \left.-{\displaystyle \frac{\mathrm{\Gamma }(\alpha +1)}{2{(\rho -\sigma )}^{\alpha }}}\left[J_{\rho -}^{\alpha }\mathrm{\Theta }\left(\sigma \right)+J_{\sigma +}^{\alpha }\mathrm{\Theta }\left(\rho \right)\right]\right|\hfill \\ \hfill & \le {\displaystyle \frac{\rho -\sigma }{2}}\left[{\int }_0^{{\displaystyle \frac{1}{4}}}\left|{\vartheta }^{\alpha }-{\displaystyle \frac{7}{90}}\right|\left|{\mathrm{\Theta }}^{\prime }(\vartheta \rho +(1-\vartheta )\sigma )-{\displaystyle \frac{m+M}{2}}\right|d\vartheta \right.\hfill \\ \hfill & +{\int }_0^{{\displaystyle \frac{1}{4}}}\left|{\vartheta }^{\alpha }-{\displaystyle \frac{7}{90}}\right|\left|{\displaystyle \frac{m+M}{2}}-{\mathrm{\Theta }}^{\prime }(\vartheta \sigma +(1-\vartheta )\rho )\right|d\vartheta \hfill \\ \hfill & +{\int }_{{\displaystyle \frac{1}{4}}}^{{\displaystyle \frac{1}{2}}}\left|{\vartheta }^{\alpha }-{\displaystyle \frac{39}{90}}\right|\left|{\mathrm{\Theta }}^{\prime }(\vartheta \rho +(1-\vartheta )\sigma )-{\displaystyle \frac{m+M}{2}}\right|d\vartheta \hfill \\ \hfill & +{\int }_{{\displaystyle \frac{1}{4}}}^{{\displaystyle \frac{1}{2}}}\left|{\vartheta }^{\alpha }-{\displaystyle \frac{39}{90}}\right|\left|{\displaystyle \frac{m+M}{2}}-{\mathrm{\Theta }}^{\prime }(\vartheta \sigma +(1-\vartheta )\rho )\right|d\vartheta \hfill \\ \hfill & +{\int }_{{\displaystyle \frac{1}{2}}}^{{\displaystyle \frac{3}{4}}}\left|{\vartheta }^{\alpha }-{\displaystyle \frac{51}{90}}\right|\left|{\mathrm{\Theta }}^{\prime }(\vartheta \rho +(1-\vartheta )\sigma )-{\displaystyle \frac{m+M}{2}}\right|d\vartheta \hfill \\ \hfill & +{\int }_{{\displaystyle \frac{1}{2}}}^{{\displaystyle \frac{3}{4}}}\left|{\vartheta }^{\alpha }-{\displaystyle \frac{51}{90}}\right|\left|{\displaystyle \frac{m+M}{2}}-{\mathrm{\Theta }}^{\prime }(\vartheta \sigma +(1-\vartheta )\rho )\right|d\vartheta \hfill \\ \hfill & +{\int }_{{\displaystyle \frac{3}{4}}}^1\left|{\vartheta }^{\alpha }-{\displaystyle \frac{83}{90}}\right|\left|{\mathrm{\Theta }}^{\prime }(\vartheta \rho +(1-\vartheta )\sigma )-{\displaystyle \frac{m+M}{2}}\right|d\vartheta \hfill \\ \hfill & \left.+{\int }_{{\displaystyle \frac{3}{4}}}^1\left|{\vartheta }^{\alpha }-{\displaystyle \frac{83}{90}}\right|\left|{\displaystyle \frac{m+M}{2}}-{\mathrm{\Theta }}^{\prime }(\vartheta \sigma +(1-\vartheta )\rho )\right|d\vartheta \right].\hfill \end{array}$$

From the suppositions $m\le {\mathrm{\Theta }}^{\prime }\left(\vartheta \right)\le M$ for $\vartheta \in [\sigma ,\rho ],$ we achieve

$$\left|{\mathrm{\Theta }}^{\prime }(\vartheta \rho +(1-\vartheta )\sigma )-{\displaystyle \frac{m+M}{2}}\right|\le {\displaystyle \frac{M-m}{2}},$$

(22)

and

$$\left|{\displaystyle \frac{m+M}{2}}-{\mathrm{\Theta }}^{\prime }(\vartheta \sigma +(1-\vartheta )\rho )\right|\le {\displaystyle \frac{M-m}{2}}.$$

(23)

By applying inequalities (22) and (23), we conclude

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{90}}\left[7\mathrm{\Theta }\left(\sigma \right)+32\mathrm{\Theta }\left({\displaystyle \frac{3\sigma +\rho }{4}}\right)+12\mathrm{\Theta }\left({\displaystyle \frac{\sigma +\rho }{2}}\right)+32\mathrm{\Theta }\left({\displaystyle \frac{\sigma +3\rho }{4}}\right)+7\mathrm{\Theta }\left(\rho \right)\right]\right.\hfill \\ \hfill & \left.-{\displaystyle \frac{\mathrm{\Gamma }(\alpha +1)}{2{(\rho -\sigma )}^{\alpha }}}\left[J_{\rho -}^{\alpha }\mathrm{\Theta }\left(\sigma \right)+J_{\sigma +}^{\alpha }\mathrm{\Theta }\left(\rho \right)\right]\right|\hfill \\ \hfill & \le {\displaystyle \frac{\rho -\sigma }{2}}(M-m)\left[{\int }_0^{{\displaystyle \frac{1}{4}}}\left|{\vartheta }^{\alpha }-{\displaystyle \frac{7}{90}}\right|d\vartheta +{\int }_{{\displaystyle \frac{1}{4}}}^{{\displaystyle \frac{1}{2}}}\left|{\vartheta }^{\alpha }-{\displaystyle \frac{39}{90}}\right|d\vartheta +{\int }_{{\displaystyle \frac{1}{2}}}^{{\displaystyle \frac{3}{4}}}\left|{\vartheta }^{\alpha }-{\displaystyle \frac{51}{90}}\right|d\vartheta +{\int }_{{\displaystyle \frac{3}{4}}}^1\left|{\vartheta }^{\alpha }-{\displaystyle \frac{83}{90}}\right|d\vartheta \right]\hfill \\ \hfill & ={\displaystyle \frac{(\rho -\sigma )}{2}}\left[{\mathsf{A}}_1\left(\alpha \right)+{\mathsf{A}}_2\left(\alpha \right)+{\mathsf{A}}_3\left(\alpha \right)+{\mathsf{A}}_4\left(\alpha \right)\right](M-m).\hfill \end{array}$$

This completes the proof. □

Remark 4.

In Theorem 6, setting $\alpha =1$ yields the following inequality:

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{90}}\left[7\mathrm{\Theta }\left(\sigma \right)+32\mathrm{\Theta }\left({\displaystyle \frac{3\sigma +\rho }{4}}\right)+12\mathrm{\Theta }\left({\displaystyle \frac{\sigma +\rho }{2}}\right)+32\mathrm{\Theta }\left({\displaystyle \frac{\sigma +3\rho }{4}}\right)+7\mathrm{\Theta }\left(\rho \right)\right]-{\displaystyle \frac{1}{\rho -\sigma }}{\int }_{\sigma }^{\rho }\mathrm{\Theta }\left(\varpi \right)d\varpi \right|\hfill \\ \hfill & \le {\displaystyle \frac{239(\rho -\sigma )}{6480}}(M-m),\hfill \end{array}$$

which is proved by Shehzadi et al. in [33] (Theorem 6).

Corollary 3.

Under the assumption of Theorem 6, if there exist $M\in {\mathfrak{R}}^{+}$ such that $|{\mathrm{\Theta }}^{\prime }\left(\vartheta \right)\le M$ for all $\vartheta \in [\sigma ,\rho ],$ then we attain

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{90}}\left[7\mathrm{\Theta }\left(\sigma \right)+32\mathrm{\Theta }\left({\displaystyle \frac{3\sigma +\rho }{4}}\right)+12\mathrm{\Theta }\left({\displaystyle \frac{\sigma +\rho }{2}}\right)+32\mathrm{\Theta }\left({\displaystyle \frac{\sigma +3\rho }{4}}\right)+7\mathrm{\Theta }\left(\rho \right)\right]\right.\hfill \\ \hfill & \left.-{\displaystyle \frac{\mathrm{\Gamma }(\alpha +1)}{2{(\rho -\sigma )}^{\alpha }}}\left[J_{\rho -}^{\alpha }\mathrm{\Theta }\left(\sigma \right)+J_{\sigma +}^{\alpha }\mathrm{\Theta }\left(\rho \right)\right]\right|\le {\displaystyle \frac{M(\rho -\sigma )}{2}}\left[{\mathsf{A}}_1\left(\alpha \right)+{\mathsf{A}}_2\left(\alpha \right)+{\mathsf{A}}_3\left(\alpha \right)+{\mathsf{A}}_4\left(\alpha \right)\right].\hfill \end{array}$$

(24)

Remark 5.

By taking $\alpha =1$ in inequality (24), then we attain the following inequality:

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{90}}\left[7\mathrm{\Theta }\left(\sigma \right)+32\mathrm{\Theta }\left({\displaystyle \frac{3\sigma +\rho }{4}}\right)+12\mathrm{\Theta }\left({\displaystyle \frac{\sigma +\rho }{2}}\right)+32\mathrm{\Theta }\left({\displaystyle \frac{\sigma +3\rho }{4}}\right)+7\mathrm{\Theta }\left(\rho \right)\right]-{\displaystyle \frac{1}{\rho -\sigma }}{\int }_{\sigma }^{\rho }\mathrm{\Theta }\left(\varpi \right)d\varpi \right|\hfill \\ \hfill & \le {\displaystyle \frac{239(\rho -\sigma )}{3240}}M,\hfill \end{array}$$

which is established in [33] (Corollary 1).

2.3. Fractional Boole’s Type Inequality for Lipschitzian Functions

Theorem 7.

Assuming all conditions of Lemma 1 are satisfied, if ${\mathrm{\Theta }}^{\prime }$ is an $\mathcal{L}$-Lipschitzian function on $[\sigma ,\rho ],$ then we have the following inequality:

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{90}}\left[7\mathrm{\Theta }\left(\sigma \right)+32\mathrm{\Theta }\left({\displaystyle \frac{3\sigma +\rho }{4}}\right)+12\mathrm{\Theta }\left({\displaystyle \frac{\sigma +\rho }{2}}\right)+32\mathrm{\Theta }\left({\displaystyle \frac{\sigma +3\rho }{4}}\right)+7\mathrm{\Theta }\left(\rho \right)\right]\right.\hfill \\ \hfill & \left.-{\displaystyle \frac{\mathrm{\Gamma }(\alpha +1)}{2{(\rho -\sigma )}^{\alpha }}}\left[J_{\rho -}^{\alpha }\mathrm{\Theta }\left(\sigma \right)+J_{\sigma +}^{\alpha }\mathrm{\Theta }\left(\rho \right)\right]\right|\le {\displaystyle \frac{\mathcal{L}{(\rho -\sigma )}^2}{2}}\left[{\mathsf{A}}_5\left(\alpha \right)+{\mathsf{A}}_6\left(\alpha \right)+{\mathsf{A}}_7\left(\alpha \right)+{\mathsf{A}}_8\left(\alpha \right)\right],\hfill \end{array}$$

(25)

where ${\mathsf{A}}_5\left(\alpha \right)-{\mathsf{A}}_8\left(\alpha \right)$ are defined in (14).

Proof. 

By leveraging Lemma 1, and taking the absolute, since ${\mathrm{\Theta }}^{\prime }$ is $\mathcal{L}$-Lipschitzian function, we observe

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{90}}\left[7\mathrm{\Theta }\left(\sigma \right)+32\mathrm{\Theta }\left({\displaystyle \frac{3\sigma +\rho }{4}}\right)+12\mathrm{\Theta }\left({\displaystyle \frac{\sigma +\rho }{2}}\right)+32\mathrm{\Theta }\left({\displaystyle \frac{\sigma +3\rho }{4}}\right)+7\mathrm{\Theta }\left(\rho \right)\right]\right.\hfill \\ \hfill & \left.-{\displaystyle \frac{\mathrm{\Gamma }(\alpha +1)}{2{(\rho -\sigma )}^{\alpha }}}\left[J_{\rho -}^{\alpha }\mathrm{\Theta }\left(\sigma \right)+J_{\sigma +}^{\alpha }\mathrm{\Theta }\left(\rho \right)\right]\right|\hfill \\ \hfill & \le {\displaystyle \frac{\rho -\sigma }{2}}\left[{\int }_0^{{\displaystyle \frac{1}{4}}}\left|{\vartheta }^{\alpha }-{\displaystyle \frac{7}{90}}\right|\left|{\mathrm{\Theta }}^{\prime }(\vartheta \rho +(1-\vartheta )\sigma )-{\mathrm{\Theta }}^{\prime }(\vartheta \sigma +(1-\vartheta )\rho )\right|d\vartheta \right.\hfill \\ \hfill & +{\int }_{{\displaystyle \frac{1}{4}}}^{{\displaystyle \frac{1}{2}}}\left|{\vartheta }^{\alpha }-{\displaystyle \frac{39}{90}}\right|\left|{\mathrm{\Theta }}^{\prime }(\vartheta \rho +(1-\vartheta )\sigma )-{\mathrm{\Theta }}^{\prime }(\vartheta \sigma +(1-\vartheta )\rho )\right|d\vartheta \hfill \\ \hfill & +{\int }_{{\displaystyle \frac{1}{2}}}^{{\displaystyle \frac{3}{4}}}\left|{\vartheta }^{\alpha }-{\displaystyle \frac{51}{90}}\right|\left|{\mathrm{\Theta }}^{\prime }(\vartheta \rho +(1-\vartheta )\sigma )-{\mathrm{\Theta }}^{\prime }(\vartheta \sigma +(1-\vartheta )\rho )\right|d\vartheta \hfill \\ \hfill & \left.+{\int }_{{\displaystyle \frac{3}{4}}}^1\left|{\vartheta }^{\alpha }-{\displaystyle \frac{83}{90}}\right|\left|{\mathrm{\Theta }}^{\prime }(\vartheta \rho +(1-\vartheta )\sigma )-{\mathrm{\Theta }}^{\prime }(\vartheta \sigma +(1-\vartheta )\rho )\right|d\vartheta \right]\hfill \\ \hfill & \le {\displaystyle \frac{\mathcal{L}{(\rho -\sigma )}^2}{2}}\left[\left({\int }_0^{{\displaystyle \frac{1}{4}}}\left|{\vartheta }^{\alpha }-{\displaystyle \frac{7}{90}}\right|+{\int }_{{\displaystyle \frac{1}{4}}}^{{\displaystyle \frac{1}{2}}}\left|{\vartheta }^{\alpha }-{\displaystyle \frac{39}{90}}\right|+{\int }_{{\displaystyle \frac{1}{2}}}^{{\displaystyle \frac{3}{4}}}\left|{\vartheta }^{\alpha }-{\displaystyle \frac{51}{90}}\right|+{\int }_{{\displaystyle \frac{3}{4}}}^1\left|{\vartheta }^{\alpha }-{\displaystyle \frac{83}{90}}\right|\right)\left(2\vartheta -1\right)d\vartheta \right]\hfill \\ \hfill & =\mathcal{L}{(\rho -\sigma )}^2\left[{\mathsf{A}}_5\left(\alpha \right)+{\mathsf{A}}_6\left(\alpha \right)+{\mathsf{A}}_7\left(\alpha \right)+{\mathsf{A}}_8\left(\alpha \right)\right].\hfill \end{array}$$

The proof has been successfully concluded. □

Remark 6.

In Theorem 7, setting $\alpha =1$ yields the following inequality:

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{90}}\left[7\mathrm{\Theta }\left(\sigma \right)+32\mathrm{\Theta }\left({\displaystyle \frac{3\sigma +\rho }{4}}\right)+12\mathrm{\Theta }\left({\displaystyle \frac{\sigma +\rho }{2}}\right)+32\mathrm{\Theta }\left({\displaystyle \frac{\sigma +3\rho }{4}}\right)+7\mathrm{\Theta }\left(\rho \right)\right]-{\displaystyle \frac{1}{\rho -\sigma }}{\int }_{\sigma }^{\rho }\mathrm{\Theta }\left(\varpi \right)d\varpi \right|\hfill \\ \hfill & \le {\displaystyle \frac{80627{(\rho -\sigma )}^2}{4374000}}\mathcal{L},\hfill \end{array}$$

which is proved in [33] (Theorem 7).

2.4. Fractional Boole’s Type Inequality for Functions of Bounded Variation

Theorem 8.

Presume $\mathrm{\Theta }:[\sigma ,\rho ]\to \mathfrak{R}$ is a function of bounded variation on $[\sigma ,\rho ].$ Then, we have the subsequent inequality:

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{90}}\left[7\mathrm{\Theta }\left(\sigma \right)+32\mathrm{\Theta }\left({\displaystyle \frac{3\sigma +\rho }{4}}\right)+12\mathrm{\Theta }\left({\displaystyle \frac{\sigma +\rho }{2}}\right)+32\mathrm{\Theta }\left({\displaystyle \frac{\sigma +3\rho }{4}}\right)+7\mathrm{\Theta }\left(\rho \right)\right]\right.\hfill \\ \hfill & \left.-{\displaystyle \frac{\mathrm{\Gamma }(\alpha +1)}{2{(\rho -\sigma )}^{\alpha }}}\left[J_{\rho -}^{\alpha }\mathrm{\Theta }\left(\sigma \right)+J_{\sigma +}^{\alpha }\mathrm{\Theta }\left(\rho \right)\right]\right|\hfill \\ \hfill & \le {\displaystyle \frac{83}{90}}\underset{\sigma }{\overset{\rho }{\bigvee }}\left(\mathrm{\Theta }\right),\hfill \end{array}$$

where ${\displaystyle \underset{\sigma }{\overset{\rho }{\bigvee }}}\left(\mathrm{\Theta }\right)$ represent the total variation of Θ on $[\sigma ,\rho ].$

Proof. 

Let us define the mappings

$${\mathcal{K}}_{\alpha }\left(\vartheta \right)=\left\{\begin{array}{cc}{\left(\vartheta -\sigma \right)}^{\alpha }-{\displaystyle \frac{7}{90}}{(\rho -\sigma )}^{\alpha },\hfill & \sigma \le \vartheta <{\displaystyle \frac{3\sigma +\rho }{4}},\hfill \\ \hfill & \\ {\left(\vartheta -\sigma \right)}^{\alpha }-{\displaystyle \frac{39}{90}}{(\rho -\sigma )}^{\alpha },\hfill & {\displaystyle \frac{3\sigma +\rho }{4}}\le \vartheta <{\displaystyle \frac{\sigma +\rho }{2}},\hfill \\ \hfill & \\ {\left(\vartheta -\sigma \right)}^{\alpha }-{\displaystyle \frac{51}{90}}{(\rho -\sigma )}^{\alpha },\hfill & {\displaystyle \frac{\sigma +\rho }{2}}\le \vartheta <{\displaystyle \frac{\sigma +3\rho }{4}},\hfill \\ \hfill & \\ {\left(\vartheta -\sigma \right)}^{\alpha }-{\displaystyle \frac{83}{90}}{(\rho -\sigma )}^{\alpha },\hfill & {\displaystyle \frac{\sigma +3\rho }{4}}\le \vartheta \le \rho ,\hfill \end{array}\right.$$

and

$${\mathcal{R}}_{\alpha }\left(\vartheta \right)=\left\{\begin{array}{cc}{\displaystyle \frac{7}{90}}{(\rho -\sigma )}^{\alpha }-{\left(\rho -\vartheta \right)}^{\alpha },\hfill & \sigma \le \vartheta <{\displaystyle \frac{3\sigma +\rho }{4}},\hfill \\ \hfill & \\ {\displaystyle \frac{39}{90}}{(\rho -\sigma )}^{\alpha }-{\left(\rho -\vartheta \right)}^{\alpha },\hfill & {\displaystyle \frac{3\sigma +\rho }{4}}\le \vartheta <{\displaystyle \frac{\sigma +\rho }{2}},\hfill \\ \hfill & \\ {\displaystyle \frac{51}{90}}{(\rho -\sigma )}^{\alpha }-{\left(\rho -\vartheta \right)}^{\alpha },\hfill & {\displaystyle \frac{\sigma +\rho }{2}}\le \vartheta <{\displaystyle \frac{\sigma +3\rho }{4}},\hfill \\ \hfill & \\ {\displaystyle \frac{83}{90}}{(\rho -\sigma )}^{\alpha }-{\left(\rho -\vartheta \right)}^{\alpha },\hfill & {\displaystyle \frac{\sigma +3\rho }{4}}\le \vartheta \le \rho .\hfill \end{array}\right.$$

Integrating by parts, we have

$$\begin{array}{cc}\hfill & {\int }_{\sigma }^{\rho }{\mathcal{K}}_{\alpha }\left(\vartheta \right)d\mathrm{\Theta }\left(\vartheta \right)\hfill \\ \hfill & ={\int }_{\sigma }^{{\displaystyle \frac{3\sigma +\rho }{4}}}\left({\left(\vartheta -\sigma \right)}^{\alpha }-{\displaystyle \frac{7}{90}}{\left(\rho -\sigma \right)}^{\alpha }\right)d\mathrm{\Theta }\left(\vartheta \right)+{\int }_{{\displaystyle \frac{3\sigma +\rho }{4}}}^{{\displaystyle \frac{\sigma +\rho }{2}}}\left({\left(\vartheta -\sigma \right)}^{\alpha }-{\displaystyle \frac{39}{90}}{\left(\rho -\sigma \right)}^{\alpha }\right)d\mathrm{\Theta }\left(\vartheta \right)\hfill \\ \hfill & +{\int }_{{\displaystyle \frac{\sigma +\rho }{2}}}^{{\displaystyle \frac{\sigma +3\rho }{4}}}\left({\left(\vartheta -\sigma \right)}^{\alpha }-{\displaystyle \frac{51}{90}}{\left(\rho -\sigma \right)}^{\alpha }\right)d\mathrm{\Theta }\left(\vartheta \right)+{\int }_{{\displaystyle \frac{\sigma +3\rho }{4}}}^{\rho }\left({\left(\vartheta -\sigma \right)}^{\alpha }-{\displaystyle \frac{83}{90}}{\left(\rho -\sigma \right)}^{\alpha }\right)d\mathrm{\Theta }\left(\vartheta \right)\hfill \\ \hfill & ={\left.\left({\left(\vartheta -\sigma \right)}^{\alpha }-{\displaystyle \frac{7}{90}}{\left(\rho -\sigma \right)}^{\alpha }\right)\mathrm{\Theta }\left(\vartheta \right)\right|}_{\sigma }^{{\displaystyle \frac{3\sigma +\rho }{4}}}-\alpha {\int }_{\sigma }^{{\displaystyle \frac{3\sigma +\rho }{4}}}{\left(\vartheta -\sigma \right)}^{\alpha -1}\mathrm{\Theta }\left(\vartheta \right)d\vartheta \hfill \\ \hfill & +{\left.\left({\left(\vartheta -\sigma \right)}^{\alpha }-{\displaystyle \frac{39}{90}}{\left(\rho -\sigma \right)}^{\alpha }\right)\mathrm{\Theta }\left(\vartheta \right)\right|}_{{\displaystyle \frac{3\sigma +\rho }{4}}}^{{\displaystyle \frac{\sigma +\rho }{2}}}-\alpha {\int }_{{\displaystyle \frac{3\sigma +\rho }{4}}}^{{\displaystyle \frac{\sigma +\rho }{2}}}{\left(\vartheta -\sigma \right)}^{\alpha -1}\mathrm{\Theta }\left(\vartheta \right)d\vartheta \hfill \\ \hfill & {\left.+\left({\left(\vartheta -\sigma \right)}^{\alpha }-{\displaystyle \frac{51}{90}}{\left(\rho -\sigma \right)}^{\alpha }\right)\mathrm{\Theta }\left(\vartheta \right)\right|}_{{\displaystyle \frac{\sigma +\rho }{2}}}^{{\displaystyle \frac{\sigma +3\rho }{4}}}-\alpha {\int }_{{\displaystyle \frac{\sigma +\rho }{2}}}^{{\displaystyle \frac{\sigma +3\rho }{4}}}{\left(\vartheta -\sigma \right)}^{\alpha -1}\mathrm{\Theta }\left(\vartheta \right)d\vartheta \hfill \\ \hfill & +{\left.\left({\left(\vartheta -\sigma \right)}^{\alpha }-{\displaystyle \frac{83}{90}}{\left(\rho -\sigma \right)}^{\alpha }\right)\mathrm{\Theta }\left(\vartheta \right)\right|}_{{\displaystyle \frac{\sigma +3\rho }{4}}}^{\rho }-\alpha {\int }_{{\displaystyle \frac{\sigma +3\rho }{4}}}^{\rho }{\left(\vartheta -\sigma \right)}^{\alpha -1}\mathrm{\Theta }\left(\vartheta \right)d\vartheta \hfill \\ \hfill & =\left({\left({\displaystyle \frac{\rho -\sigma }{4}}\right)}^{\alpha }-{\displaystyle \frac{7}{90}}{\left(\rho -\sigma \right)}^{\alpha }\right)\mathrm{\Theta }\left({\displaystyle \frac{3\sigma +\rho }{4}}\right)+{\displaystyle \frac{7}{90}}{\left(\rho -\sigma \right)}^{\alpha }\mathrm{\Theta }\left(\sigma \right)\hfill \\ \hfill & -\alpha {\int }_{\sigma }^{{\displaystyle \frac{3\sigma +\rho }{4}}}{\left(\vartheta -\sigma \right)}^{\alpha -1}\mathrm{\Theta }\left(\vartheta \right)d\vartheta +\left({\left({\displaystyle \frac{\rho -\sigma }{2}}\right)}^{\alpha }-{\displaystyle \frac{39}{90}}{\left(\rho -\sigma \right)}^{\alpha }\right)\mathrm{\Theta }\left({\displaystyle \frac{\sigma +\rho }{2}}\right)\hfill \\ \hfill & -\left({\left({\displaystyle \frac{\rho -\sigma }{4}}\right)}^{\alpha }-{\displaystyle \frac{39}{90}}{\left(\rho -\sigma \right)}^{\alpha }\right)\mathrm{\Theta }\left({\displaystyle \frac{3\sigma +\rho }{4}}\right)-\alpha {\int }_{{\displaystyle \frac{3\sigma +\rho }{4}}}^{{\displaystyle \frac{\sigma +\rho }{2}}}{\left(\vartheta -\sigma \right)}^{\alpha -1}\mathrm{\Theta }\left(\vartheta \right)d\vartheta \hfill \\ \hfill & +\left({\left({\displaystyle \frac{3(\rho -\sigma )}{4}}\right)}^{\alpha }-{\displaystyle \frac{51}{90}}{\left(\rho -\sigma \right)}^{\alpha }\right)\mathrm{\Theta }\left({\displaystyle \frac{\sigma +3\rho }{4}}\right)-\left({\left({\displaystyle \frac{\rho -\sigma }{2}}\right)}^{\alpha }-{\displaystyle \frac{51}{90}}{\left(\rho -\sigma \right)}^{\alpha }\right)\mathrm{\Theta }\left({\displaystyle \frac{\sigma +\rho }{2}}\right)\hfill \\ \hfill & -\alpha {\int }_{{\displaystyle \frac{\sigma +\rho }{2}}}^{{\displaystyle \frac{\sigma +3\rho }{4}}}{\left(\vartheta -\sigma \right)}^{\alpha -1}\mathrm{\Theta }\left(\vartheta \right)d\vartheta +{\displaystyle \frac{7}{90}}{\left(\rho -\sigma \right)}^{\alpha }\mathrm{\Theta }\left(\rho \right)\hfill \\ \hfill & -\left({\left({\displaystyle \frac{3(\rho -\sigma )}{4}}\right)}^{\alpha }-{\displaystyle \frac{83}{90}}{\left(\rho -\sigma \right)}^{\alpha }\right)\mathrm{\Theta }\left({\displaystyle \frac{\sigma +3\rho }{4}}\right)-\alpha {\int }_{{\displaystyle \frac{\sigma +3\rho }{4}}}^{\rho }{\left(\vartheta -\sigma \right)}^{\alpha -1}\mathrm{\Theta }\left(\vartheta \right)d\vartheta \hfill \\ \hfill & ={\displaystyle \frac{{(\rho -\sigma )}^{\alpha }}{90}}\left[7\mathrm{\Theta }\left(\sigma \right)+32\mathrm{\Theta }\left({\displaystyle \frac{3\sigma +\rho }{4}}\right)+12\mathrm{\Theta }\left({\displaystyle \frac{\sigma +\rho }{2}}\right)+32\mathrm{\Theta }\left({\displaystyle \frac{\sigma +3\rho }{4}}\right)+7\mathrm{\Theta }\left(\rho \right)\right]\hfill \\ \hfill & -\mathrm{\Gamma }(\alpha +1)J_{\rho -}^{\alpha }\mathrm{\Theta }\left(\sigma \right).\hfill \end{array}$$

Similarly, we have

$$\begin{array}{cc}\hfill & {\int }_{\sigma }^{\rho }{\mathcal{R}}_{\alpha }\left(\vartheta \right)d\mathrm{\Theta }\left(\vartheta \right)\hfill \\ \hfill & ={\displaystyle \frac{{(\rho -\sigma )}^{\alpha }}{90}}\left[7\mathrm{\Theta }\left(\sigma \right)+32\mathrm{\Theta }\left({\displaystyle \frac{3\sigma +\rho }{4}}\right)+12\mathrm{\Theta }\left({\displaystyle \frac{\sigma +\rho }{2}}\right)+32\mathrm{\Theta }\left({\displaystyle \frac{\sigma +3\rho }{4}}\right)+7\mathrm{\Theta }\left(\rho \right)\right]\hfill \\ \hfill & -\mathrm{\Gamma }(\alpha +1)J_{\sigma +}^{\alpha }\mathrm{\Theta }\left(\rho \right).\hfill \end{array}$$

Therefore, we attain

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{90}}\left[7\mathrm{\Theta }\left(\sigma \right)+32\mathrm{\Theta }\left({\displaystyle \frac{3\sigma +\rho }{4}}\right)+12\mathrm{\Theta }\left({\displaystyle \frac{\sigma +\rho }{2}}\right)+32\mathrm{\Theta }\left({\displaystyle \frac{\sigma +3\rho }{4}}\right)+7\mathrm{\Theta }\left(\rho \right)\right]\hfill \\ \hfill & -{\displaystyle \frac{\mathrm{\Gamma }(\alpha +1)}{2{(\rho -\sigma )}^{\alpha }}}\left[J_{\rho -}^{\alpha }\mathrm{\Theta }\left(\sigma \right)+J_{\sigma +}^{\alpha }\mathrm{\Theta }\left(\rho \right)\right]={\displaystyle \frac{1}{2{(\rho -\sigma )}^{\alpha }}}{\int }_{\sigma }^{\rho }[{\mathcal{K}}_{\alpha }\left(\vartheta \right)+{\mathcal{R}}_{\alpha }\left(\vartheta \right)]d\mathrm{\Theta }\left(\vartheta \right).\hfill \end{array}$$

It is understood that if $g,\mathrm{\Theta }:\left[\sigma ,\rho \right]\to \mathfrak{R}$ satisfy that g is continuous on $\left[\sigma ,\rho \right]$ and $\mathrm{\Theta }$ possesses bounded variation on $\left[\sigma ,\rho \right]$, then $\underset{\sigma }{\overset{\rho }{\int }}g\left(\vartheta \right)d\mathrm{\Theta }\left(\vartheta \right)$ exist and

$$\left|\underset{\sigma }{\overset{\rho }{\int }}g\left(\vartheta \right)d\mathrm{\Theta }\left(\vartheta \right)\right|\le \underset{\vartheta \in \left[\sigma ,\rho \right]}{sup}\left|g\left(\vartheta \right)\right|\underset{\sigma }{\overset{\rho }{\bigvee }}\left(\mathrm{\Theta }\right).$$

(26)

On the other hand, using (26), we arrive at

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{90}}\left[7\mathrm{\Theta }\left(\sigma \right)+32\mathrm{\Theta }\left({\displaystyle \frac{3\sigma +\rho }{4}}\right)+12\mathrm{\Theta }\left({\displaystyle \frac{\sigma +\rho }{2}}\right)+32\mathrm{\Theta }\left({\displaystyle \frac{\sigma +3\rho }{4}}\right)+7\mathrm{\Theta }\left(\rho \right)\right]\right.\hfill \\ \hfill & \left.-{\displaystyle \frac{\mathrm{\Gamma }(\alpha +1)}{2{(\rho -\sigma )}^{\alpha }}}\left[J_{\rho -}^{\alpha }\mathrm{\Theta }\left(\sigma \right)+J_{\sigma +}^{\alpha }\mathrm{\Theta }\left(\rho \right)\right]\right|\hfill \\ \hfill & \le {\displaystyle \frac{1}{2{(\rho -\sigma )}^{\alpha }}}\left|{\int }_{\sigma }^{\rho }[{\mathcal{K}}_{\alpha }\left(\vartheta \right)d\mathrm{\Theta }\left(\vartheta \right)+{\mathcal{R}}_{\alpha }\left(\vartheta \right)d\mathrm{\Theta }\left(\vartheta \right)]\right|\hfill \\ \hfill & \le {\displaystyle \frac{1}{2{(\rho -\sigma )}^{\alpha }}}\left[\left|{\int }_{\sigma }^{{\displaystyle \frac{3\sigma +\rho }{4}}}\left({\left(\vartheta -\sigma \right)}^{\alpha }-{\displaystyle \frac{7}{90}}{(\rho -\sigma )}^{\alpha }\right)d\mathrm{\Theta }\left(\vartheta \right)\right|\right.\hfill \\ \hfill & +\left|{\int }_{{\displaystyle \frac{3\sigma +\rho }{4}}}^{{\displaystyle \frac{\sigma +\rho }{2}}}\left({\left(\vartheta -\sigma \right)}^{\alpha }-{\displaystyle \frac{39}{90}}{(\rho -\sigma )}^{\alpha }\right)d\mathrm{\Theta }\left(\vartheta \right)\right|+\left|{\int }_{{\displaystyle \frac{\sigma +\rho }{2}}}^{{\displaystyle \frac{\sigma +3\rho }{4}}}\left({\left(\vartheta -\sigma \right)}^{\alpha }-{\displaystyle \frac{51}{90}}{(\rho -\sigma )}^{\alpha }\right)d\mathrm{\Theta }\left(\vartheta \right)\right|\hfill \\ \hfill & +\left|{\int }_{{\displaystyle \frac{\sigma +3\rho }{4}}}^{\rho }\left({\left(\vartheta -\sigma \right)}^{\alpha }-{\displaystyle \frac{83}{90}}{(\rho -\sigma )}^{\alpha }\right)d\mathrm{\Theta }\left(\varpi \right)\right|+\left|{\int }_{\sigma }^{{\displaystyle \frac{3\sigma +\rho }{4}}}\left({\displaystyle \frac{7}{90}}{(\rho -\sigma )}^{\alpha }-{\left(\rho -\vartheta \right)}^{\alpha }\right)d\mathrm{\Theta }\left(\vartheta \right)\right|\hfill \\ \hfill & +\left|{\int }_{{\displaystyle \frac{3\sigma +\rho }{4}}}^{{\displaystyle \frac{\sigma +\rho }{2}}}\left({\displaystyle \frac{39}{90}}{(\rho -\sigma )}^{\alpha }-{\left(\rho -\vartheta \right)}^{\alpha }\right)d\mathrm{\Theta }\left(\vartheta \right)\right|+\left|{\int }_{{\displaystyle \frac{\sigma +\rho }{2}}}^{{\displaystyle \frac{\sigma +3\rho }{4}}}\left({\displaystyle \frac{51}{90}}{(\rho -\sigma )}^{\alpha }-{\left(\rho -\vartheta \right)}^{\alpha }\right)d\mathrm{\Theta }\left(\vartheta \right)\right|\hfill \\ \hfill & \left.+\left|{\int }_{{\displaystyle \frac{\sigma +3\rho }{4}}}^{\rho }\left({\displaystyle \frac{83}{90}}{(\rho -\sigma )}^{\alpha }-{\left(\rho -\vartheta \right)}^{\alpha }\right)d\mathrm{\Theta }\left(\vartheta \right)\right|\right]\hfill \\ \hfill & ={\displaystyle \frac{1}{2{(\rho -\sigma )}^{\alpha }}}\left[\underset{\vartheta \in \left(\sigma ,{\displaystyle \frac{3\sigma +\rho }{4}}\right)}{sup}\left|{\left(\vartheta -\sigma \right)}^{\alpha }-{\displaystyle \frac{7}{90}}{(\rho -\sigma )}^{\alpha }\right|\underset{\sigma }{\overset{{\displaystyle \frac{3\sigma +\rho }{4}}}{\bigvee }}\left(\mathrm{\Theta }\right)\right.\hfill \\ \hfill & +\underset{\vartheta \in \left({\displaystyle \frac{3\sigma +\rho }{4}},{\displaystyle \frac{\sigma +\rho }{2}}\right)}{sup}\left|{\left(\vartheta -\sigma \right)}^{\alpha }-{\displaystyle \frac{39}{90}}{(\rho -\sigma )}^{\alpha }\right|\underset{{\displaystyle \frac{3\sigma +\rho }{4}}}{\overset{{\displaystyle \frac{\sigma +\rho }{2}}}{\bigvee }}\left(\mathrm{\Theta }\right)+\underset{\vartheta \in \left({\displaystyle \frac{\sigma +\rho }{2}},{\displaystyle \frac{\sigma +3\rho }{4}}\right)}{sup}\left|{\left(\vartheta -\sigma \right)}^{\alpha }-{\displaystyle \frac{51}{90}}{(\rho -\sigma )}^{\alpha }\right|\underset{{\displaystyle \frac{\sigma +\rho }{2}}}{\overset{{\displaystyle \frac{\sigma +3\rho }{4}}}{\bigvee }}\hfill \\ \hfill & +\underset{\vartheta \in \left({\displaystyle \frac{\sigma +3\rho }{4}},\rho \right)}{sup}\left|{\left(\vartheta -\sigma \right)}^{\alpha }-{\displaystyle \frac{83}{90}}{(\rho -\sigma )}^{\alpha }\right|\underset{{\displaystyle \frac{\sigma +3\rho }{4}}}{\overset{\rho }{\bigvee }}+\underset{\vartheta \in \left(\sigma ,{\displaystyle \frac{3\sigma +\rho }{4}}\right)}{sup}\left|{\displaystyle \frac{7}{90}}{(\rho -\sigma )}^{\alpha }-{\left(\rho -\vartheta \right)}^{\alpha }\right|\underset{\sigma }{\overset{{\displaystyle \frac{3\sigma +\rho }{4}}}{\bigvee }}\left(\mathrm{\Theta }\right)\hfill \\ \hfill & +\underset{\vartheta \in \left({\displaystyle \frac{3\sigma +\rho }{4}},{\displaystyle \frac{\sigma +\rho }{2}}\right)}{sup}\left|{\displaystyle \frac{39}{90}}{(\rho -\sigma )}^{\alpha }-{\left(\rho -\vartheta \right)}^{\alpha }\right|\underset{{\displaystyle \frac{3\sigma +\rho }{4}}}{\overset{{\displaystyle \frac{\sigma +\rho }{2}}}{\bigvee }}\left(\mathrm{\Theta }\right)+\underset{\vartheta \in \left({\displaystyle \frac{\sigma +\rho }{2}},{\displaystyle \frac{\sigma +3\rho }{4}}\right)}{sup}\left|{\displaystyle \frac{51}{90}}{(\rho -\sigma )}^{\alpha }-{\left(\rho -\vartheta \right)}^{\alpha }\right|\underset{{\displaystyle \frac{\sigma +\rho }{2}}}{\overset{{\displaystyle \frac{\sigma +3\rho }{4}}}{\bigvee }}\left(\mathrm{\Theta }\right)\hfill \\ \hfill & \left.+\underset{\vartheta \in \left({\displaystyle \frac{\sigma +3\rho }{4}},\rho \right)}{sup}\left|{\displaystyle \frac{83}{90}}{(\rho -\sigma )}^{\alpha }-{\left(\rho -\vartheta \right)}^{\alpha }\right|\underset{{\displaystyle \frac{\sigma +3\rho }{4}}}{\overset{\rho }{\bigvee }}\left(\mathrm{\Theta }\right)\right]\hfill \\ \hfill & ={\displaystyle \frac{1}{2{(\rho -\sigma )}^{\alpha }}}\left[{\displaystyle \frac{7{(\rho -\sigma )}^{\alpha }}{90}}\underset{\sigma }{\overset{{\displaystyle \frac{3\sigma +\rho }{4}}}{\bigvee }}\left(\mathrm{\Theta }\right)+{\displaystyle \frac{39{(\rho -\sigma )}^{\alpha }}{90}}\underset{{\displaystyle \frac{3\sigma +\rho }{4}}}{\overset{{\displaystyle \frac{\sigma +\rho }{2}}}{\bigvee }}\left(\mathrm{\Theta }\right)+{\displaystyle \frac{51{(\rho -\sigma )}^{\alpha }}{90}}\underset{{\displaystyle \frac{\sigma +\rho }{2}}}{\overset{{\displaystyle \frac{\sigma +3\rho }{4}}}{\bigvee }}\left(\mathrm{\Theta }\right)\right.\hfill \\ \hfill & +{\displaystyle \frac{83{(\rho -\sigma )}^{\alpha }}{90}}\underset{{\displaystyle \frac{\sigma +3\rho }{4}}}{\overset{\rho }{\bigvee }}\left(\mathrm{\Theta }\right)+{\displaystyle \frac{7{(\rho -\sigma )}^{\alpha }}{90}}\underset{\sigma }{\overset{{\displaystyle \frac{3\sigma +\rho }{4}}}{\bigvee }}\left(\mathrm{\Theta }\right)+{\displaystyle \frac{39{(\rho -\sigma )}^{\alpha }}{90}}\underset{{\displaystyle \frac{3\sigma +\rho }{4}}}{\overset{{\displaystyle \frac{\sigma +\rho }{2}}}{\bigvee }}\left(\mathrm{\Theta }\right)\hfill \\ \hfill & \left.+{\displaystyle \frac{51{(\rho -\sigma )}^{\alpha }}{90}}\underset{{\displaystyle \frac{\sigma +\rho }{2}}}{\overset{{\displaystyle \frac{\sigma +3\rho }{4}}}{\bigvee }}\left(\mathrm{\Theta }\right)+{\displaystyle \frac{83{(\rho -\sigma )}^{\alpha }}{90}}\underset{{\displaystyle \frac{\sigma +3\rho }{4}}}{\overset{\rho }{\bigvee }}\left(\mathrm{\Theta }\right)\right]\hfill \\ \hfill & \le {\displaystyle \frac{83}{90}}\underset{\sigma }{\overset{\rho }{\bigvee }}\left(\mathrm{\Theta }\right).\hfill \end{array}$$

This completes the proof. □

Corollary 4.

In Theorem 8, setting $\alpha =1$ yields the following inequality:

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{90}}\left[7\mathrm{\Theta }\left(\sigma \right)+32\mathrm{\Theta }\left({\displaystyle \frac{3\sigma +\rho }{4}}\right)+12\mathrm{\Theta }\left({\displaystyle \frac{\sigma +\rho }{2}}\right)+32\mathrm{\Theta }\left({\displaystyle \frac{\sigma +3\rho }{4}}\right)+7\mathrm{\Theta }\left(\rho \right)\right]-{\displaystyle \frac{1}{\rho -\sigma }}{\int }_{\sigma }^{\rho }\mathrm{\Theta }\left(\varpi \right)d\varpi \right|\hfill \\ \hfill & \le {\displaystyle \frac{83}{90}}\underset{\sigma }{\overset{\rho }{\bigvee }}\left(\mathrm{\Theta }\right).\hfill \end{array}$$

3. Numerical Analysis with Graphical Representations

This section includes numerical examples and visual illustrations of the recently established inequalities. The graphical depictions of these inequalities offer a clear representation of their numerical behavior, highlighting their accuracy and practical significance. These visualizations demonstrate the feasibility and robustness of the proposed inequalities, emphasizing their utility in both practical and computational scenarios.

Example 1.

Consider a differentiable convex functions $\mathrm{\Theta }\left(\varpi \right)=e^{2\varpi }$ in Theorem 3 for all $\alpha >0$. We observe that the following numerical verification (see

Table 1. Numerical values of the inequality (15) for $\mathrm{\Theta }\left(\varpi \right)=e^{2\varpi }$ when $[\sigma ,\rho ]=[0,1]$.

$\mathit{\alpha }$	Left Term	Right Term
0.1	0.737923	7.43516
0.2	0.542236	5.08644
0.3	0.394662	3.49176
0.4	0.282683	2.41946
0.5	0.197466	1.70182
0.6	0.132648	1.22749
0.7	0.0835601	0.923571
0.8	0.0467237	0.741396
0.9	0.0195095	0.647624
1.0	0.0000931233	0.618822

) and corresponding graphs (see Figure 1, Figure 2 and Figure 3) demonstrates that the left-hand side consistently remains less than the right-hand side. This observation confirms our main findings.

Remark 7.

We provided an exposition of how fractional Boole’s inequalities of Theorem 3 behave. One can see that Figure 1, Figure 2 and Figure 3 check the inequalities of Example 1, a two-dimensional and a three-dimensional plot, to understand its behavior in various dimensions and different parameter spaces. These plots assist one in explaining how inequality persists under certain circumstances and may shed light on where breakpoints or singularities emerge. This means that by comparing the results of the given inequality with those outcomes shown by these graphical representations, one can better understand the inherent fractional phenomena behind this inequality.

Example 2.

Consider a differentiable convex function $\mathrm{\Theta }\left(\varpi \right)=ArcTan\left(\varpi \right)$ in Theorem 6 for all $\alpha >0$; then we observe the following numerical verification (see

Table 2. Numerical values of the inequality (6) for $\mathrm{\Theta }\left(\varpi \right)=ArcTan\left(\varpi \right)$, when $[\sigma ,\rho ]=[0,1]$.

$\mathit{\alpha }$	Left Term	Right Term
0.1	0.0342898	1.39218
0.2	0.0253673	0.952403
0.3	0.0185739	0.65381
0.4	0.013374	0.453028
0.5	0.0093851	0.318655
0.6	0.00632887	0.22984
0.7	0.00399883	0.172933
0.8	0.00223952	0.138822
0.9	0.000932294	0.121263
1.0	0.0000144492	0.11587

) and corresponding graphs (see Figure 4).

Remark 8.

Boole’s formula provides superior absolute error estimates for higher-degree polynomials when compared to Simpson’s and trapezoidal rules, as evidenced by the results presented in

Table 3. Comparison of absolute error, taking $\mathrm{\Theta }\left(\varpi \right)={\varpi }^6$, $\alpha =0.9$ and $[\sigma ,\rho ]=[0,1]$ for different choices of n in quadrature formulae.

n	Quadrature Formulae	Absolute Error
4	Boole’s Rule	0.00597078
2	Simpson’s $1/3$ Rule	0.0278834
1	Trapezoidal Rule	0.3508

. The principal motivation behind this work lies in extending classical calculus to encompass non-integer or fractional values, whereas classical approaches are traditionally confined to integer values. This generalization offers greater flexibility and accuracy in applications involving fractional derivatives and integrals. Furthermore, as observed in Table 1 and Table 2, when the fractional order α approaches 1, the absolute error significantly improves, aligning with the precision of classical calculus methods.

4. Applications to Numerical Integration

This section demonstrates how the newly established inequalities can be applied to improve numerical quadrature formulas, such as Boole’s and midpoint rules. By utilizing these results, error bounds for numerical integration are refined, offering greater precision in approximating definite integrals. These applications are particularly beneficial in scenarios where classical methods encounter limitations due to higher-order derivatives or rapidly varying functions.

4.1. Application to Boole’s Formula

Let $\mathrm{\Omega }$ represent the division of the interval $[\sigma ,\rho ]$, where $\mathrm{\Omega }:\sigma ={\eta }_0<{\eta }_1<{\eta }_2<\cdots <{\eta }_n=\rho$, and $h_j={\displaystyle \frac{{\eta }_{j+1}-{\eta }_j}{4}}$ ($j=1,2,3,\cdots ,n-1$), with n being divisible by 4. It is well known that if the mapping $\mathrm{\Theta }:[\sigma ,\rho ]\to \mathbb{R}$ is differentiable and ${\mathrm{\Theta }}^{\left(6\right)}\left(\eta \right)$ exists on $(\sigma ,\rho )$, with $M={max}_{\eta \in (\sigma ,\rho )}\left|{\mathrm{\Theta }}^{\left(6\right)}\left(\eta \right)\right|<\infty$, then

$$\begin{array}{c}\hfill {\int }_{\sigma }^{\rho }\mathrm{\Theta }\left(\eta \right)d\eta =S_B\left(\mathrm{\Theta },\mathrm{\Omega }\right)+{\mathfrak{R}}_B\left(\mathrm{\Theta },\mathrm{\Omega }\right),\end{array}$$

(27)

where

$$\begin{array}{cc}\hfill S_B\left(\mathrm{\Theta },\mathrm{\Omega }\right)& =\sum _{j=0}^{n-1}{\displaystyle \frac{\left({\eta }_{j+1}-{\eta }_j\right)}{90}}[7\mathrm{\Theta }\left({\eta }_j\right)+12\mathrm{\Theta }\left({\displaystyle \frac{{\eta }_j+{\eta }_{j+1}}{2}}\right)+32\mathrm{\Theta }\left({\displaystyle \frac{3{\eta }_j+{\eta }_{j+1}}{4}}\right)\hfill \\ \hfill & +32\mathrm{\Theta }\left({\displaystyle \frac{{\eta }_j+3{\eta }_{j+1}}{4}}\right)+7\mathrm{\Theta }\left({\eta }_{j+1}\right)],\hfill \end{array}$$

(28)

and the approximation error ${\mathfrak{R}}_w(\mathrm{\Theta },\mathrm{\Omega })$ of the integral ${\int }_{\sigma }^{\rho }\mathrm{\Theta }\left(\eta \right)d\eta$ using Boole’s formula $S_B(\mathrm{\Theta },\mathrm{\Omega })$ satisfies

$$\left|{\mathfrak{R}}_B\left(\mathrm{\Theta },\mathrm{\Omega }\right)\right|\le {\displaystyle \frac{239}{6480}}\sum _{j=0}^{n-1}{\displaystyle \frac{{\left({\eta }_{j+1}-{\eta }_j\right)}^2}{n}}\left[|{\mathrm{\Theta }}^{\prime }\left({\eta }_j\right)|+|{\mathrm{\Theta }}^{\prime }\left({\eta }_{j+1}\right)|\right].$$

(29)

Below, we provide estimates for the remainder term ${\mathfrak{R}}_w(\mathrm{\Theta },\mathrm{\Omega })$ in terms of the first derivative.

Proposition 1.

Let $\mathrm{\Theta }:I\to \mathfrak{R}$ be a differentiable function on $I^{\circ }=[\sigma ,\rho ]$, with ${\mathrm{\Theta }}^{\prime }$ bounded and convex on $[\sigma ,\rho ]$. For any division Ω of $[\sigma ,\rho ]$, the following inequality holds:

$$\left|{\mathfrak{R}}_w\left(\mathrm{\Theta },\mathrm{\Omega }\right)\right|\le {\displaystyle \frac{239}{6480}}\sum _{j=0}^{n-1}{\displaystyle \frac{{({\eta }_{j+1}-{\eta }_j)}^2}{4}}\left[|{\mathrm{\Theta }}^{\prime }\left({\eta }_j\right)|+|{\mathrm{\Theta }}^{\prime }\left({\eta }_{j+1}\right)|\right],$$

(30)

for all $j=1,2,3,\dots ,n-1.$

Proof. 

Relate the Corollary 1 on the subinterval $\left[{\eta }_j,{\eta }_{j+1}\right]$$(j=0,1,2,\dots ,n-1)$, we have

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{90}}\left({\eta }_{j+1}-{\eta }_j\right)\left[7\mathrm{\Theta }\left({\eta }_i\right)+12\mathrm{\Theta }\left({\displaystyle \frac{{\eta }_j+{\eta }_{j+1}}{2}}\right)+32\mathrm{\Theta }\left({\displaystyle \frac{3{\eta }_j+{\eta }_{j+1}}{4}}\right)+32\mathrm{\Theta }\left({\displaystyle \frac{{\eta }_j+3{\eta }_{j+1}}{4}}\right)\right.\right.\hfill \\ \hfill & \left.\left.+7\mathrm{\Theta }\left({\eta }_{j+1}\right)\right]-{\int }_{{\eta }_j}^{{\eta }_{j+1}}\mathrm{\Theta }\left(\eta \right)d\eta \right|\hfill \\ \hfill & \le {\displaystyle \frac{239}{6480}}\left({\displaystyle \frac{{({\eta }_{j+1}-{\eta }_j)}^2}{4}}\right)\left[|{\mathrm{\Theta }}^{\prime }\left({\eta }_j\right)|+|{\mathrm{\Theta }}^{\prime }\left({\eta }_{j+1}\right)|\right].\hfill \end{array}$$

(31)

Summing (31) over $j=0$ to $n-1$ and using the convexity of $\left|{\mathrm{\Theta }}^{\prime }\right|$, the triangle inequality gives

$$\begin{array}{cc}\hfill & \left|S_B\left(\mathrm{\Theta },\mathrm{\Omega }\right)-{\int }_{\sigma }^{\rho }\mathrm{\Theta }\left(\eta \right)d\eta \right|\hfill \\ \hfill & \le {\displaystyle \frac{239}{6480}}\sum _{j=0}^{n-1}{\displaystyle \frac{{({\eta }_{j+1}-{\eta }_j)}^2}{4}}\left[|{\mathrm{\Theta }}^{\prime }\left({\eta }_j\right)|+|{\mathrm{\Theta }}^{\prime }\left({\eta }_{j+1}\right)|\right].\hfill \end{array}$$

This completes the proof. □

4.2. Application to Midpoint Formula:

Let $\mathrm{\Omega }$ denote the division of the interval $\left[\sigma ,\rho \right]$, where $\mathrm{\Omega }:\sigma ={\eta }_0<{\eta }_1<{\eta }_2<\dots <{\eta }_n=\rho$. Consider the midpoint formula:

$$M_B\left(\mathrm{\Theta },\mathrm{\Omega }\right)=\sum _{j=0}^{n-1}\left({\eta }_{j+1}-{\eta }_j\right)\mathrm{\Theta }\left({\displaystyle \frac{{\eta }_j+{\eta }_{j+1}}{2}}\right).$$

(32)

If the function $\mathrm{\Theta }:\left[\sigma ,\rho \right]\to \mathfrak{R},$ is differentiable and its derivative ${\mathrm{\Theta }}^{\prime }\left(\eta \right)$ is bounded on $\left(\sigma ,\rho \right)$, such that $K={max}_{\eta \in \left(\sigma ,\rho \right)}\left|{\mathrm{\Theta }}^{\prime }\left(\eta \right)\right|<\infty$, then

$$I={\int }_{\sigma }^{\rho }\mathrm{\Theta }\left(\eta \right)d\eta =M_B\left(\mathrm{\Theta },\mathrm{\Omega }\right)+{\mathfrak{R}}_M\left(\mathrm{\Theta },\mathrm{\Omega }\right),$$

(33)

where the approximation error ${\mathfrak{R}}_M\left(\mathrm{\Theta },\mathrm{\Omega }\right)$ of the integral I using the midpoint formula satisfies

$$\left|{\mathfrak{R}}_M\left(\mathrm{\Theta },\mathrm{\Omega }\right)\right|\le \sum _{j=0}^{n-1}{\left({\eta }_{j+1}-{\eta }_j\right)}^2{\mathrm{\Theta }}^{\prime }\left({\displaystyle \frac{{\eta }_j+{\eta }_{j+1}}{2}}\right).$$

(34)

In this work, we establish new bounds for the midpoint formula by refining the approximations of the remainder term ${\mathfrak{R}}_M(\mathrm{\Theta },\mathrm{\Omega })$ in terms of the first derivative, which outperform those provided in [34].

Proposition 2.

Let $\mathrm{\Theta }:I\to \mathfrak{R}$ be a differentiable function on $I^{\circ }=[\sigma ,\rho ]$, with ${\mathrm{\Theta }}^{\prime }$ bounded and convex on $[\sigma ,\rho ]$. For any division Ω of $[\sigma ,\rho ]$, the following inequality holds:

$$\begin{array}{c}\hfill \left|{\mathfrak{R}}_M\left(\mathrm{\Theta },\mathrm{\Omega }\right)\right|\le {\displaystyle \frac{239}{6480}}\sum _{j=0}^{n-1}{({\eta }_{j+1}-{\eta }_j)}^2\left[|{\mathrm{\Theta }}^{\prime }\left({\eta }_j\right)|+|{\mathrm{\Theta }}^{\prime }\left({\eta }_{j+1}\right)|\right].\end{array}$$

(35)

Proof. 

By applying the Corollary 2 on the subinterval $\left[{\eta }_j,{\eta }_{j+1}\right]$$(j=0,1,2,\dots ,n-1)$, we have

$$\begin{array}{cc}\hfill & \left|\left({\eta }_{j+1}-{\eta }_j\right)\mathrm{\Theta }\left({\displaystyle \frac{{\eta }_j+{\eta }_{j+1}}{2}}\right)-{\int }_{{\eta }_j}^{{\eta }_{j+1}}\mathrm{\Theta }\left(\eta \right)d\eta \right|\hfill \\ \hfill & \le {\displaystyle \frac{239{({\eta }_{j+1}-{\eta }_j)}^2}{6480}}\left[|{\mathrm{\Theta }}^{\prime }\left({\eta }_j\right)|+|{\mathrm{\Theta }}^{\prime }\left({\eta }_{j+1}\right)|\right].\hfill \end{array}$$

(36)

Summing (36) over $j=0$ to $n-1$ and using the convexity of $\left|{\mathrm{\Theta }}^{\prime }\right|$, the triangle inequality gives

$$\begin{array}{cc}\hfill & \left|{\mathfrak{R}}_M\left(\mathrm{\Theta },\mathrm{\Omega }\right)\right|\hfill \\ \hfill & \le {\displaystyle \frac{239}{6480}}\sum _{j=0}^{n-1}{({\eta }_{j+1}-{\eta }_j)}^2\left[|{\mathrm{\Theta }}^{\prime }\left({\eta }_j\right)|+|{\mathrm{\Theta }}^{\prime }\left({\eta }_{j+1}\right)|\right].\hfill \end{array}$$

Thus, the proof is complete. □

Remark 9.

The midpoint-type inequality estimations presented in this work provide improved bounds compared to those established in [34].

5. Conclusions

The primary objective of this work is to introduce innovative Boole-formula-type inequalities that integrate Riemann–Liouville fractional integrals for single-time differentiable convex functions. The methodology begins with deriving an integral identity using Riemann–Liouville fractional integrals and proceeds to establish new inequalities specifically suited for differentiable convex functions. These results provide enhanced bounds compared to the classical Boole’s formula, which traditionally relies on integer-order integrals. In contrast, the utilization of Riemann–Liouville fractional integrals necessitates generalized quadrature formulas capable of handling non-integer orders. Through the exploration of various values of $\alpha$, this study offers a robust framework adaptable to a broad spectrum of fractional integral problems. This framework facilitates more precise numerical approximations in fractional calculus by tailoring the quadrature method to the specified order of $\alpha$, thereby shedding light on how integration impacts the behavior of the operator. The findings of this research hold substantial significance for advancing studies in areas such as convexity, s-convexity, generalized fractional integrals, and higher-order derivative equations. Moreover, the potential emergence of similar inequalities in quantum calculus, particularly when applying Riemann–Liouville fractional integrals to convex functions, is proposed. The contributions of this research expand the theoretical framework of integral inequalities and open new avenues for exploration in related mathematical fields.