Exploring Advanced Weighted Integral Inequalities via Extended Fractional Calculus Approaches

Abstract

This paper investigates weighted Milne-type ($\mathcal{M}-\mathfrak{t}$) inequalities within the context of Riemann–Liouville ($\mathfrak{R}-\mathfrak{L}$) fractional integrals. We establish multiple versions of these inequalities, applicable to different function categories, such as convex functions with differentiability properties, bounded functions, functions satisfying Lipschitz conditions, and those exhibiting bounded variation behavior. In particular, we present integral equalities that are essential to establish the main results, using non-negative weighted functions. The findings contribute to the extension of existing inequalities in the literature and provide a deeper understanding of their applications in fractional calculus. This work highlights the advantage of the established inequalities in extending classical results by accommodating a broader class of functions and yielding sharper bounds. It also explores potential directions for future research inspired by these findings.

1. Introduction

Numerical integration methods and their associated error bounds have been the subject of extensive research by mathematicians [1]. When obtaining such estimates for inaccuracies, researchers frequently rely on distinct mathematical estimates, commonly tailored to particular function categories like those exhibiting convexity, boundedness, or Lipschitz continuity [2,3]. By investigating the properties of such functions, this paper aims to provide more accurate estimates of the error involved when using numerical integration formulas, particularly under the assumption of certain smoothness conditions, such as convexity in the derivatives.

Let us first introduce a numerical integration method and the upper error bounds of it.

The following is the expression for the Milne quadrature formula:

$$\underset{e}{\overset{u}{\mathit{\int }}}\mathfrak{h}\left(x\right)dx\approx {\displaystyle \frac{u-e}{3}}\left[2\mathfrak{h}\left(e\right)-\mathfrak{h}\left({\displaystyle \frac{e+u}{2}}\right)+2\mathfrak{h}\left(u\right)\right]$$

(1)

A significant volume of work has addressed fresh upper estimates for the gap between a particular formula and its integral, typically referred to as $\mathcal{M}-\mathfrak{t}$ inequalities. Alomari [4] pioneered such results for both bounded functions and functions of bounded variation. In subsequent explorations, numerous researchers investigated these inequalities from different perspectives. For instance, Djenaoui and Meftah [5] relied on convexity arguments to establish certain $\mathcal{M}-\mathfrak{t}$ results, while Budak et al. [6] adapted them to $\mathfrak{R}-\mathfrak{L}$ fractional integrals and examined further function classes, including bounded, Lipschitz, and bounded variations. Benaissa and Sarikaya [7] contributed new fractional $\mathcal{M}-\mathfrak{t}$ inequalities for h-convex functions. Meanwhile, other authors [8,9,10,11] continued to expand $\mathcal{M}-\mathfrak{t}$ inequalities to various settings, such as $\mathfrak{R}-\mathfrak{L}$ fractional integrals with differentiable convex functions, bounded functions, Lipschitz functions, and functions of bounded variation. Desta et al. [12] studied $\mathcal{M}-\mathfrak{t}$ inequalities for twice-differentiable functions, and additional contributions covered scenarios involving higher-order differentiable functions [13].

Several directions have also aimed at extending $\mathcal{M}-\mathfrak{t}$ inequalities to alternative forms of calculus. Some researchers examined quantum integrals [14,15,16,17], while others investigated different fractional integral definitions, such as conformable [18,19], tempered [20,21], local fractional [22,23,24], and Katugampola fractional integrals [25,26], generalized proportional fractional operators [27], among others [28,29,30,31]. Shehzadi et al. [32] introduced $\mathcal{M}-\mathfrak{t}$ inequalities in the framework of coordinated convexity.

The classical $\mathfrak{R}-\mathfrak{L}$ fractional integrals are expressed as follows:

Definition 1

([33,34]). Let $\varrho >0$. The $\mathfrak{R}-\mathfrak{L}$ fractional integrals $J_{e+}^{\varrho }\mathfrak{h}$ and $J_{u-}^{\varrho }\mathfrak{h}$ are defined by

$$J_{e+}^{\varrho }\mathfrak{h}\left(z\right)={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\varrho \right)}}{\int }_e^z{(z-\tau )}^{\varrho -1}\mathfrak{h}\left(\tau \right)d\tau ,z>e,$$

and

$$J_{u-}^{\varrho }\mathfrak{h}\left(z\right)={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\varrho \right)}}{\int }_z^u{(\tau -z)}^{\varrho -1}\mathfrak{h}\left(\tau \right)d\tau ,z<u,$$

respectively. Here, $\mathfrak{h}$ is assumed to be in $L_1[e,u]$, and $\mathrm{\Gamma }\left(\varrho \right)$ represents the Gamma function. In the particular case when $\varrho =1$, the fractional integral coincides with the standard integral.

Definition 2

([34]). The hypergeometric function is defined by

$${}_2\mathcal{F}_1\left(e,u,c;v\right)={\displaystyle \frac{1}{\mathfrak{B}(u,c-u)}}\underset{0}{\overset{1}{\int }}{\tau }^{u-1}{(1-\tau )}^{c-u-1}{(1-v\tau )}^{-e}d\tau$$

with $c>u>0$, $\left|v\right|<1$ and $\mathfrak{B}(.,.)$ is the beta function given by

$$\mathfrak{B}(\varrho ,\vartheta )=\underset{0}{\overset{1}{\int }}{\tau }^{\varrho -1}{(1-\tau )}^{\vartheta -1}dt={\displaystyle \frac{\mathrm{\Gamma }\left(\varrho \right)\mathrm{\Gamma }\left(\vartheta \right)}{\mathrm{\Gamma }(\varrho +\vartheta )}}.$$

This work is devoted to broadening the scope of traditional $\mathcal{M}-\mathfrak{t}$ inequalities by incorporating them into the fractional calculus framework through the derivation of innovative weighted estimates based on $\mathfrak{R}-\mathfrak{L}$ fractional integrals. Our approach rigorously establishes new error bounds that apply to a range of function families, including those that are convex, bounded, Lipschitz-continuous, and of bounded variation. Central to our analysis is the introduction of a symmetric non-negative weighting function alongside sophisticated integration techniques.

The paper is organized into seven sections. We begin with an introductory discussion and a review of the necessary preliminaries. In Section 2, we present a key integral identity derived using a non-negative weight, which forms the basis for our subsequent results. Section 3 is dedicated to developing weighted $\mathcal{M}-\mathfrak{t}$ inequalities for various function classes within the $\mathfrak{R}-\mathfrak{L}$ fractional integral context, with a special focus on differentiable convex functions. Section 4 examines fractional integration results for bounded functions, while Section 5 extends these inequalities to cover Lipschitz functions. In Section 6, we derive weighted fractional estimates for functions exhibiting bounded variation. Finally, Section 7 summarizes our findings on $\mathcal{M}-\mathfrak{t}$ inequalities.

2. Preliminaries and Some Key Identities

In what follows, we suppose that $\psi :[e,u]\to \mathbb{R}$ is a non-negative, continuous function, and is symmetric about the midpoint ${\displaystyle \frac{e+u}{2}}$, meaning $\psi \left(z\right)=\psi (e+u-z)$ for all $z\in [e,u]$. Based on this, we introduce the functions $\mathrm{{\rm Y}}$ and $\mathrm{\Psi }$ by

$$\mathrm{{\rm Y}}(\varrho ,t)={\int }_t^u{(u-s)}^{\varrho -1}\psi \left(s\right)ds,$$

and

$$\mathrm{\Psi }(\varrho ,t)={\int }_e^t{(s-e)}^{\varrho -1}\psi \left(s\right)ds.$$

By the symmetry of $\psi$ around the midpoint ${\displaystyle \frac{e+u}{2}}$, the identities below will play a significant role in later developments.

$$\begin{array}{ccc}\hfill \mathrm{\Phi }\left(\varrho \right)& :=& \mathrm{{\rm Y}}\left(\varrho ,{\displaystyle \frac{e+u}{2}}\right)=\mathrm{\Psi }\left(\varrho ,{\displaystyle \frac{e+u}{2}}\right)\hfill \\ & =& \mathrm{\Gamma }\left(\varrho \right)J_{{\displaystyle \frac{e+u}{2}}-}^{\varrho }\psi \left(e\right)\hfill \\ & =& \mathrm{\Gamma }\left(\varrho \right)J_{{\displaystyle \frac{e+u}{2}}+}^{\varrho }\psi \left(u\right)\hfill \\ & =& {\displaystyle \frac{\mathrm{\Gamma }\left(\varrho \right)}{2}}\left[J_{{\displaystyle \frac{e+u}{2}}+}^{\varrho }\psi \left(u\right)+J_{{\displaystyle \frac{e+u}{2}}-}^{\varrho }\psi \left(e\right)\right],\hfill \end{array}$$

(2)

as well as the relations

$$\mathrm{{\rm Y}}(\varrho ,t)=\mathrm{\Psi }(\varrho ,e+u-t),\mathrm{\Psi }(\varrho ,t)=\mathrm{{\rm Y}}(\varrho ,e+u-t).$$

(3)

In particular, for $\varrho =1$, we obtain

$$\mathrm{\Phi }\left(1\right)={\int }_{{\displaystyle \frac{e+u}{2}}}^u\psi \left(s\right)ds={\int }_e^{{\displaystyle \frac{e+u}{2}}}\psi \left(s\right)ds={\displaystyle \frac{1}{2}}{\int }_e^u\psi \left(s\right)ds.$$

(4)

The symmetry of the function $\psi$ is necessary to establish important relations between the left and right fractional integrals. This allows us to derive simplified and symmetric identities such as (2)–(4).

Lemma 1.

Consider a function $\mathfrak{h}:[e,u]\to \mathbb{R}$ that is absolutely continuous on the interval $(e,u)$, with its derivative ${\mathfrak{h}}^{\prime }$ being integrable over $[e,u]$. Under these conditions, the following identity holds:

$$\begin{array}{ccc}& & {\displaystyle \frac{1}{3}}\left[2\mathfrak{h}\left(e\right)-\mathfrak{h}\left({\displaystyle \frac{e+u}{2}}\right)+2\mathfrak{h}\left(u\right)\right]\mathrm{\Phi }\left(\varrho \right)-{\displaystyle \frac{\mathrm{\Gamma }\left(\varrho \right)}{2}}\left[J_{{\displaystyle \frac{e+u}{2}}+}^{\varrho }\left(\mathfrak{h}\psi \right)\left(u\right)+J_{{\displaystyle \frac{e+u}{2}}-}^{\varrho }\left(\mathfrak{h}\psi \right)\left(e\right)\right]\hfill \\ & =& {\displaystyle \frac{1}{2}}\left({\mathcal{I}}_1+{\mathcal{I}}_2\right),\hfill \end{array}$$

(5)

where Γ denotes the Gamma function, and

$${\mathcal{I}}_1={\int }_e^{{\displaystyle \frac{e+u}{2}}}\left[\mathrm{\Psi }(\varrho ,t)-{\displaystyle \frac{4}{3}}\mathrm{\Phi }\left(\varrho \right)\right]{\mathfrak{h}}^{\prime }\left(t\right)dt,$$

and

$${\mathcal{I}}_2={\int }_{{\displaystyle \frac{e+u}{2}}}^u\left[{\displaystyle \frac{4}{3}}\mathrm{\Phi }\left(\varrho \right)-\mathrm{{\rm Y}}(\varrho ,t)\right]{\mathfrak{h}}^{\prime }\left(t\right)dt.$$

Proof. 

By applying integration by parts, we easily find

$$\begin{array}{cc}\hfill {\mathcal{I}}_1& ={\int }_e^{{\displaystyle \frac{e+u}{2}}}\left[\mathrm{\Psi }(\varrho ,t)-{\displaystyle \frac{4}{3}}\mathrm{\Phi }\left(\varrho \right)\right]{\mathfrak{h}}^{\prime }\left(t\right)dt\hfill \\ \hfill & ={\left.\left[\mathrm{\Psi }(\varrho ,t)-{\displaystyle \frac{4}{3}}\mathrm{\Phi }\left(\varrho \right)\right]\mathfrak{h}\left(t\right)\right|}_e^{{\displaystyle \frac{e+u}{2}}}-{\int }_e^{{\displaystyle \frac{e+u}{2}}}{(t-e)}^{\varrho -1}\psi \left(t\right)\mathfrak{h}\left(t\right)dt\hfill \\ \hfill & =-{\displaystyle \frac{1}{3}}\mathrm{\Phi }\left(\varrho \right)\mathfrak{h}\left({\displaystyle \frac{e+u}{2}}\right)+{\displaystyle \frac{4}{3}}\mathrm{\Phi }\left(\varrho \right)\mathfrak{h}\left(e\right)-{\int }_e^{{\displaystyle \frac{e+u}{2}}}{(t-e)}^{\varrho -1}\psi \left(t\right)\mathfrak{h}\left(t\right)dt\hfill \\ \hfill & =-{\displaystyle \frac{1}{3}}\mathrm{\Phi }\left(\varrho \right)\mathfrak{h}\left({\displaystyle \frac{e+u}{2}}\right)+{\displaystyle \frac{4}{3}}\mathrm{\Phi }\left(\varrho \right)\mathfrak{h}\left(e\right)-\mathrm{\Gamma }\left(\varrho \right)J_{{\displaystyle \frac{e+u}{2}}-}^{\varrho }\left(\mathfrak{h}\psi \right)\left(e\right).\hfill \end{array}$$

(6)

Similarly, for ${\mathcal{I}}_2$, we obtain

$$\begin{array}{cc}\hfill {\mathcal{I}}_2& ={\int }_{{\displaystyle \frac{e+u}{2}}}^u\left[{\displaystyle \frac{4}{3}}\mathrm{\Phi }\left(\varrho \right)-\mathrm{{\rm Y}}(\varrho ,t)\right]{\mathfrak{h}}^{\prime }\left(t\right)dt\hfill \\ \hfill & ={\left.\left[{\displaystyle \frac{4}{3}}\mathrm{\Phi }\left(\varrho \right)-\mathrm{{\rm Y}}(\varrho ,t)\right]\mathfrak{h}\left(t\right)\right|}_{{\displaystyle \frac{e+u}{2}}}^u-{\int }_{{\displaystyle \frac{e+u}{2}}}^u{(u-t)}^{\varrho -1}\psi \left(t\right)\mathfrak{h}\left(t\right)dt\hfill \\ \hfill & ={\displaystyle \frac{4}{3}}\mathrm{\Phi }\left(\varrho \right)\mathfrak{h}\left(u\right)-{\displaystyle \frac{1}{3}}\mathrm{\Phi }\left(\varrho \right)\mathfrak{h}\left({\displaystyle \frac{e+u}{2}}\right)-{\int }_{{\displaystyle \frac{e+u}{2}}}^u{(u-t)}^{\varrho -1}\psi \left(t\right)\mathfrak{h}\left(t\right)dt\hfill \\ \hfill & ={\displaystyle \frac{4}{3}}\mathrm{\Phi }\left(\varrho \right)\mathfrak{h}\left(u\right)-{\displaystyle \frac{1}{3}}\mathrm{\Phi }\left(\varrho \right)\mathfrak{h}\left({\displaystyle \frac{e+u}{2}}\right)-\mathrm{\Gamma }\left(\varrho \right)J_{{\displaystyle \frac{e+u}{2}}+}^{\varrho }\left(\mathfrak{h}\psi \right)\left(u\right).\hfill \end{array}$$

(7)

Adding Equations (6) and (7) and dividing by 2, we find

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{2}}({\mathcal{I}}_1+{\mathcal{I}}_2)& ={\displaystyle \frac{1}{3}}\left[2\mathfrak{h}\left(e\right)-\mathfrak{h}\left({\displaystyle \frac{e+u}{2}}\right)+2\mathfrak{h}\left(u\right)\right]\mathrm{\Phi }\left(\varrho \right)\hfill \\ \hfill & -{\displaystyle \frac{\mathrm{\Gamma }\left(\varrho \right)}{2}}\left[J_{{\displaystyle \frac{e+u}{2}}+}^{\varrho }\left(\mathfrak{h}\psi \right)\left(u\right)+J_{{\displaystyle \frac{e+u}{2}}-}^{\varrho }\left(\mathfrak{h}\psi \right)\left(e\right)\right],\hfill \end{array}$$

which concludes the proof. □

Corollary 1.

Suppose that all conditions stated in Lemma 1 are fulfilled. Then, the following equalities are valid:

$$\begin{array}{ccc}& & {\displaystyle \frac{1}{3}}\left[2\mathfrak{h}\left(e\right)-\mathfrak{h}\left({\displaystyle \frac{e+u}{2}}\right)+2\mathfrak{h}\left(u\right)\right]\mathrm{\Phi }\left(\varrho \right)-{\displaystyle \frac{\mathrm{\Gamma }\left(\varrho \right)}{2}}\left[J_{{\displaystyle \frac{e+u}{2}}+}^{\varrho }\left(\mathfrak{h}\psi \right)\left(u\right)+J_{{\displaystyle \frac{e+u}{2}}-}^{\varrho }\left(\mathfrak{h}\psi \right)\left(e\right)\right]\hfill \\ & =& {\displaystyle \frac{1}{2}}{\int }_e^{{\displaystyle \frac{e+u}{2}}}\left[\mathrm{\Psi }(\varrho ,t)-{\displaystyle \frac{4}{3}}\mathrm{\Phi }\left(\varrho \right)\right]\left[{\mathfrak{h}}^{\prime }\left(t\right)-{\mathfrak{h}}^{\prime }(e+u-t)\right]dt,\hfill \end{array}$$

(8)

and

$$\begin{array}{ccc}& & {\displaystyle \frac{1}{3}}\left[2\mathfrak{h}\left(e\right)-\mathfrak{h}\left({\displaystyle \frac{e+u}{2}}\right)+2\mathfrak{h}\left(u\right)\right]\mathrm{\Phi }\left(\varrho \right)-{\displaystyle \frac{\mathrm{\Gamma }\left(\varrho \right)}{2}}\left[J_{{\displaystyle \frac{e+u}{2}}+}^{\varrho }\left(\mathfrak{h}\psi \right)\left(u\right)+J_{{\displaystyle \frac{e+u}{2}}-}^{\varrho }\left(\mathfrak{h}\psi \right)\left(e\right)\right]\hfill \\ & =& {\displaystyle \frac{1}{2}}{\int }_{{\displaystyle \frac{e+u}{2}}}^u\left[\mathrm{{\rm Y}}(\varrho ,t)-{\displaystyle \frac{4}{3}}\mathrm{\Phi }\left(\varrho \right)\right]\left[{\mathfrak{h}}^{\prime }(e+u-t)-{\mathfrak{h}}^{\prime }\left(t\right)\right]dt.\hfill \end{array}$$

(9)

Proof. 

Due to the symmetry of $\psi$ about the point ${\displaystyle \frac{e+u}{2}}$, it follows that

$$\mathrm{{\rm Y}}(\varrho ,t)=\mathrm{\Psi }(\varrho ,e+u-t)\mathrm{for}\mathrm{all}t\in [e,\frac{e+u}{2}],$$

(10)

and

$$\mathrm{\Psi }(\varrho ,t)={\rm Y}(\mathrm{\varrho },e+u-t)\mathrm{for}\mathrm{all}t\in [\frac{e+u}{2},u].$$

(11)

Substituting relations (10) and (11) into the expressions established in Lemma 1, the stated equalities are obtained directly. □

3. Fractional Weighted Milne-Type Inequalities Under Convexity

This section develops a series of fractional weighted $\mathcal{M}-\mathfrak{t}$ inequalities tailored for convex and differentiable functions. Our strategy is based on bounding the absolute value of a newly formulated integral equality. Additionally, by utilizing classic tools known as Hölder’s and the power-mean inequality, we derive further related results.

Theorem 1.

Suppose the hypotheses of Lemma 1 hold and that the function $\left|{\mathfrak{h}}^{\prime }\right|$ is convex on $[e,u]$. Under these assumptions, we establish the following fractional weighted $\mathcal{M}-\mathfrak{t}$ inequality:

$$\begin{array}{ccc}& & \left|{\displaystyle \frac{1}{3}}\left[2\mathfrak{h}\left(e\right)-\mathfrak{h}\left({\displaystyle \frac{e+u}{2}}\right)+2\mathfrak{h}\left(u\right)\right]\mathrm{\Phi }\left(\varrho \right)-{\displaystyle \frac{\mathrm{\Gamma }\left(\varrho \right)}{2}}\left[J_{{\displaystyle \frac{e+u}{2}}+}^{\varrho }\left(\mathfrak{h}\psi \right)\left(u\right)+J_{{\displaystyle \frac{e+u}{2}}-}^{\varrho }\left(\mathfrak{h}\psi \right)\left(e\right)\right]\right|\hfill \\ & \le & {\displaystyle \frac{1}{2}}{\mathrm{\Omega }}_1\left(\varrho \right)\left(\left|{\mathfrak{h}}^{\prime }\left(e\right)\right|+\left|{\mathfrak{h}}^{\prime }\left(u\right)\right|\right),\hfill \end{array}$$

where

$${\mathrm{\Omega }}_1\left(\varrho \right)={\int }_e^{{\displaystyle \frac{e+u}{2}}}\left|\mathrm{\Psi }(\varrho ,t)-{\displaystyle \frac{4}{3}}\mathrm{\Phi }\left(\varrho \right)\right|dt.$$

Proof. 

Starting from the identity (8), we write

$$\begin{array}{ccc}& & \left|{\displaystyle \frac{1}{3}}\left[2\mathfrak{h}\left(e\right)-\mathfrak{h}\left({\displaystyle \frac{e+u}{2}}\right)+2\mathfrak{h}\left(u\right)\right]\mathrm{\Phi }\left(\varrho \right)-{\displaystyle \frac{\mathrm{\Gamma }\left(\varrho \right)}{2}}\left[J_{{\displaystyle \frac{e+u}{2}}+}^{\varrho }\left(\mathfrak{h}\psi \right)\left(u\right)+J_{{\displaystyle \frac{e+u}{2}}-}^{\varrho }\left(\mathfrak{h}\psi \right)\left(e\right)\right]\right|\hfill \\ & \le & {\displaystyle \frac{1}{2}}{\int }_e^{{\displaystyle \frac{e+u}{2}}}\left|\mathrm{\Psi }(\varrho ,t)-{\displaystyle \frac{4}{3}}\mathrm{\Phi }\left(\varrho \right)\right|\left|{\mathfrak{h}}^{\prime }\left(t\right)-{\mathfrak{h}}^{\prime }(e+u-t)\right|dt\hfill \\ & \le & {\displaystyle \frac{1}{2}}{\int }_e^{{\displaystyle \frac{e+u}{2}}}\left|\mathrm{\Psi }(\varrho ,t)-{\displaystyle \frac{4}{3}}\mathrm{\Phi }\left(\varrho \right)\right|\left(\left|{\mathfrak{h}}^{\prime }\left(t\right)\right|+\left|{\mathfrak{h}}^{\prime }(e+u-t)\right|\right)dt.\hfill \end{array}$$

(12)

Since $\left|{\mathfrak{h}}^{\prime }\right|$ is convex over $[e,u]$, we use the properties of convex functions:

$$\left|{\mathfrak{h}}^{\prime }\left(t\right)\right|\le {\displaystyle \frac{u-t}{u-e}}\left|{\mathfrak{h}}^{\prime }\left(e\right)\right|+{\displaystyle \frac{t-e}{u-e}}\left|{\mathfrak{h}}^{\prime }\left(u\right)\right|,$$

and similarly,

$$\left|{\mathfrak{h}}^{\prime }(e+u-t)\right|\le {\displaystyle \frac{u-t}{u-e}}\left|{\mathfrak{h}}^{\prime }\left(u\right)\right|+{\displaystyle \frac{t-e}{u-e}}\left|{\mathfrak{h}}^{\prime }\left(e\right)\right|.$$

Adding these two inequalities gives

$$\left|{\mathfrak{h}}^{\prime }\left(t\right)\right|+\left|{\mathfrak{h}}^{\prime }(e+u-t)\right|\le \left|{\mathfrak{h}}^{\prime }\left(e\right)\right|+\left|{\mathfrak{h}}^{\prime }\left(u\right)\right|.$$

Thus, continuing from (12), we deduce

$$\begin{array}{ccc}& & \left|{\displaystyle \frac{1}{3}}\left[2\mathfrak{h}\left(e\right)-\mathfrak{h}\left({\displaystyle \frac{e+u}{2}}\right)+2\mathfrak{h}\left(u\right)\right]\mathrm{\Phi }\left(\varrho \right)-{\displaystyle \frac{\mathrm{\Gamma }\left(\varrho \right)}{2}}\left[J_{{\displaystyle \frac{e+u}{2}}+}^{\varrho }\left(\mathfrak{h}\psi \right)\left(u\right)+J_{{\displaystyle \frac{e+u}{2}}-}^{\varrho }\left(\mathfrak{h}\psi \right)\left(e\right)\right]\right|\hfill \\ & \le & {\displaystyle \frac{\left|{\mathfrak{h}}^{\prime }\left(e\right)\right|+\left|{\mathfrak{h}}^{\prime }\left(u\right)\right|}{2}}{\int }_e^{{\displaystyle \frac{e+u}{2}}}\left|\mathrm{\Psi }(\varrho ,t)-{\displaystyle \frac{4}{3}}\mathrm{\Phi }\left(\varrho \right)\right|dt={\displaystyle \frac{1}{2}}{\mathrm{\Omega }}_1\left(\varrho \right)\left(\left|{\mathfrak{h}}^{\prime }\left(e\right)\right|+\left|{\mathfrak{h}}^{\prime }\left(u\right)\right|\right).\hfill \end{array}$$

□

Remark 1.

Taking the function $\psi \left(v\right)=1$$\forall v\in [e,u]$ within Theorem 1, the resulting fractional $\mathcal{M}-\mathfrak{t}$ inequality is expressed as follows:

$$\begin{array}{ccc}& & \left|{\displaystyle \frac{1}{3}}\left[2\mathfrak{h}\left(e\right)-\mathfrak{h}\left({\displaystyle \frac{e+u}{2}}\right)+2\mathfrak{h}\left(u\right)\right]-{\displaystyle \frac{2^{\varrho -1}\mathrm{\Gamma }(\varrho +1)}{{\left(u-e\right)}^{\varrho }}}\left[J_{{\displaystyle \frac{e+u}{2}}+}^{\varrho }\mathfrak{h}\left(u\right)+J_{{\displaystyle \frac{e+u}{2}}-}^{\varrho }\mathfrak{h}\left(e\right)\right]\right|\hfill \\ & \le & {\displaystyle \frac{4\varrho +1}{12\left(\varrho +1\right)}}\left(u-e\right)\left[\left|{\mathfrak{h}}^{\prime }\left(e\right)\right|+\left|{\mathfrak{h}}^{\prime }\left(u\right)\right|\right]\hfill \end{array}$$

which is proved by Budak and Karagözoğlu in [9] (Theorem 2.2).

Proof. 

By special choice of $\psi \left(v\right)=1$ for all $v\in \left[e,u\right]$ and by changing variables, we have

$$\begin{array}{ccc}\hfill \underset{e}{\overset{{\displaystyle \frac{e+u}{2}}}{\int }}\left|\mathrm{\Psi }\left(\varrho ,t\right)-{\displaystyle \frac{4}{3}}\mathrm{\Phi }\left(\varrho \right)\right|dt& =& {\displaystyle \frac{1}{\varrho }}\underset{e}{\overset{{\displaystyle \frac{e+u}{2}}}{\int }}\left|{\left(t-e\right)}^{\varrho }-{\displaystyle \frac{4}{3}}{\left({\displaystyle \frac{u-e}{2}}\right)}^{\varrho }\right|dt\hfill \\ & =& {\displaystyle \frac{1}{\varrho }}{\left({\displaystyle \frac{u-e}{2}}\right)}^{\varrho +1}\underset{0}{\overset{1}{\int }}\left|t^{\varrho }-{\displaystyle \frac{4}{3}}\right|dt\hfill \\ & =& {\left({\displaystyle \frac{u-e}{2}}\right)}^{\varrho +1}{\displaystyle \frac{4\varrho +1}{3\varrho \left(\varrho +1\right)}}.\hfill \end{array}$$

(13)

This completes the proof. □

Corollary 2.

Taking $\varrho =1$ within Theorem 1, The weighted $\mathcal{M}-\mathfrak{t}$ inequality is obtained as follows:

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{3}}\left[2\mathfrak{h}\left(e\right)-\mathfrak{h}\left({\displaystyle \frac{e+u}{2}}\right)+2\mathfrak{h}\left(u\right)\right]{\int }_e^u\psi \left(t\right)dt-{\int }_e^u\psi \left(t\right)\mathfrak{h}\left(t\right)dt\right|\hfill \\ \hfill & \le \left(\left|{\mathfrak{h}}^{\prime }\left(e\right)\right|+\left|{\mathfrak{h}}^{\prime }\left(u\right)\right|\right){\int }_e^{{\displaystyle \frac{e+u}{2}}}\left({\displaystyle \frac{2}{3}}{\int }_e^u\psi \left(s\right)ds-{\int }_e^t\psi \left(s\right)ds\right)dt.\hfill \end{array}$$

(14)

Remark 2.

In particular, by setting $\psi \left(v\right)=1,$$\forall v\in [e,u]$ within Corollary 2, we arrive at the classical $\mathcal{M}-\mathfrak{t}$ inequality:

$$\left|{\displaystyle \frac{1}{3}}\left[2\mathfrak{h}\left(e\right)-\mathfrak{h}\left({\displaystyle \frac{e+u}{2}}\right)+2\mathfrak{h}\left(u\right)\right]-{\displaystyle \frac{1}{u-e}}{\int }_e^u\mathfrak{h}\left(t\right)dt\right|\le {\displaystyle \frac{5(u-e)}{24}}\left(\left|{\mathfrak{h}}^{\prime }\left(e\right)\right|+\left|{\mathfrak{h}}^{\prime }\left(u\right)\right|\right),$$

(15)

which coincides with the result provided by Budak et al. in [6] (Remark 1).

Theorem 2.

Let the conditions outlined in Lemma 1 hold, and assume further that ${\left|{\mathfrak{h}}^{\prime }\right|}^q$ exhibits convexity over the interval $[e,u]$ for some real number $q>1$. Under these circumstances, we establish the fractional weighted $\mathcal{M}-\mathfrak{t}$ inequality presented below:

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{3}}\left[2\mathfrak{h}\left(e\right)-\mathfrak{h}\left({\displaystyle \frac{e+u}{2}}\right)+2\mathfrak{h}\left(u\right)\right]\mathrm{\Phi }\left(\varrho \right)-{\displaystyle \frac{\mathrm{\Gamma }\left(\varrho \right)}{2}}\left[J_{{\displaystyle \frac{e+u}{2}}+}^{\varrho }\left(\mathfrak{h}\psi \right)\left(u\right)+J_{{\displaystyle \frac{e+u}{2}}-}^{\varrho }\left(\mathfrak{h}\psi \right)\left(e\right)\right]\right|\hfill \\ \hfill & \le {\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{u-e}{2}}\right)}^{1/q}{\left({\int }_e^{{\displaystyle \frac{e+u}{2}}}{\left|\mathrm{\Psi }(\varrho ,t)-{\displaystyle \frac{4}{3}}\mathrm{\Phi }\left(\varrho \right)\right|}^{p}dt\right)}^{1/p}\hfill \\ \hfill & \times \left[{\left({\displaystyle \frac{3{\left|{\mathfrak{h}}^{\prime }\left(e\right)\right|}^q+{\left|{\mathfrak{h}}^{\prime }\left(u\right)\right|}^q}{4}}\right)}^{1/q}+{\left({\displaystyle \frac{3{\left|{\mathfrak{h}}^{\prime }\left(u\right)\right|}^q+{\left|{\mathfrak{h}}^{\prime }\left(e\right)\right|}^q}{4}}\right)}^{1/q}\right],\hfill \end{array}$$

where p and q satisfy ${\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{q}}=1$.

Proof. 

Applying Hölder’s inequality to the estimate in (12), we obtain:

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{3}}\left[2\mathfrak{h}\left(e\right)-\mathfrak{h}\left({\displaystyle \frac{e+u}{2}}\right)+2\mathfrak{h}\left(u\right)\right]\mathrm{\Phi }\left(\varrho \right)-{\displaystyle \frac{\mathrm{\Gamma }\left(\varrho \right)}{2}}\left[J_{{\displaystyle \frac{e+u}{2}}+}^{\varrho }\left(\mathfrak{h}\psi \right)\left(u\right)+J_{{\displaystyle \frac{e+u}{2}}-}^{\varrho }\left(\mathfrak{h}\psi \right)\left(e\right)\right]\right|\hfill \\ \hfill & \le {\displaystyle \frac{1}{2}}{\left({\int }_e^{{\displaystyle \frac{e+u}{2}}}{\left|\mathrm{\Psi }(\varrho ,t)-{\displaystyle \frac{4}{3}}\mathrm{\Phi }\left(\varrho \right)\right|}^{p}dt\right)}^{{\displaystyle \frac{1}{p}}}\left({\left({\int }_e^{{\displaystyle \frac{e+u}{2}}}{\left|{\mathfrak{h}}^{\prime }\left(t\right)\right|}^{q}dt\right)}^{{\displaystyle \frac{1}{q}}}+{\left({\int }_e^{{\displaystyle \frac{e+u}{2}}}{\left|{\mathfrak{h}}^{\prime }(e+u-t)\right|}^{q}dt\right)}^{{\displaystyle \frac{1}{q}}}\right).\hfill \end{array}$$

Since ${\left|{\mathfrak{h}}^{\prime }\right|}^q$ is convex on $[e,u]$, we can estimate that

$$\begin{array}{cc}\hfill {\left|{\mathfrak{h}}^{\prime }\left(t\right)\right|}^q& \le {\displaystyle \frac{u-t}{u-e}}{\left|{\mathfrak{h}}^{\prime }\left(e\right)\right|}^q+{\displaystyle \frac{t-e}{u-e}}{\left|{\mathfrak{h}}^{\prime }\left(u\right)\right|}^q,\hfill \\ \hfill {\left|{\mathfrak{h}}^{\prime }(e+u-t)\right|}^q& \le {\displaystyle \frac{u-t}{u-e}}{\left|{\mathfrak{h}}^{\prime }\left(u\right)\right|}^q+{\displaystyle \frac{t-e}{u-e}}{\left|{\mathfrak{h}}^{\prime }\left(e\right)\right|}^q.\hfill \end{array}$$

Thus, inserting these into the previous inequality, we have

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{3}}\left[2\mathfrak{h}\left(e\right)-\mathfrak{h}\left({\displaystyle \frac{e+u}{2}}\right)+2\mathfrak{h}\left(u\right)\right]\mathrm{\Phi }\left(\varrho \right)-{\displaystyle \frac{\mathrm{\Gamma }\left(\varrho \right)}{2}}\left[J_{{\displaystyle \frac{e+u}{2}}+}^{\varrho }\left(\mathfrak{h}\psi \right)\left(u\right)+J_{{\displaystyle \frac{e+u}{2}}-}^{\varrho }\left(\mathfrak{h}\psi \right)\left(e\right)\right]\right|\hfill \\ \hfill & \le {\displaystyle \frac{1}{2}}{\left({\int }_e^{{\displaystyle \frac{e+u}{2}}}{\left|\mathrm{\Psi }(\varrho ,t)-{\displaystyle \frac{4}{3}}\mathrm{\Phi }\left(\varrho \right)\right|}^{p}dt\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ \hfill & \times \left({\left({\int }_e^{{\displaystyle \frac{e+u}{2}}}\left({\displaystyle \frac{u-t}{u-e}}{\left|{\mathfrak{h}}^{\prime }\left(e\right)\right|}^q+{\displaystyle \frac{t-e}{u-e}}{\left|{\mathfrak{h}}^{\prime }\left(u\right)\right|}^q\right)dt\right)}^{{\displaystyle \frac{1}{q}}}+{\left({\int }_e^{{\displaystyle \frac{e+u}{2}}}\left({\displaystyle \frac{u-t}{u-e}}{\left|{\mathfrak{h}}^{\prime }\left(u\right)\right|}^q+{\displaystyle \frac{t-e}{u-e}}{\left|{\mathfrak{h}}^{\prime }\left(e\right)\right|}^q\right)dt\right)}^{{\displaystyle \frac{1}{q}}}\right).\hfill \end{array}$$

Simplifying the integrals gives

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{3}}\left[2\mathfrak{h}\left(e\right)-\mathfrak{h}\left({\displaystyle \frac{e+u}{2}}\right)+2\mathfrak{h}\left(u\right)\right]\mathrm{\Phi }\left(\varrho \right)-{\displaystyle \frac{\mathrm{\Gamma }\left(\varrho \right)}{2}}\left[J_{{\displaystyle \frac{e+u}{2}}+}^{\varrho }\left(\mathfrak{h}\psi \right)\left(u\right)+J_{{\displaystyle \frac{e+u}{2}}-}^{\varrho }\left(\mathfrak{h}\psi \right)\left(e\right)\right]\right|\hfill \\ \hfill & \le {\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{u-e}{2}}\right)}^{1/q}{\left({\int }_e^{{\displaystyle \frac{e+u}{2}}}{\left|\mathrm{\Psi }(\varrho ,t)-{\displaystyle \frac{4}{3}}\mathrm{\Phi }\left(\varrho \right)\right|}^{p}dt\right)}^{1/p}\hfill \\ \hfill & \times \left[{\left({\displaystyle \frac{3{\left|{\mathfrak{h}}^{\prime }\left(e\right)\right|}^q+{\left|{\mathfrak{h}}^{\prime }\left(u\right)\right|}^q}{4}}\right)}^{1/q}+{\left({\displaystyle \frac{3{\left|{\mathfrak{h}}^{\prime }\left(u\right)\right|}^q+{\left|{\mathfrak{h}}^{\prime }\left(e\right)\right|}^q}{4}}\right)}^{1/q}\right].\hfill \end{array}$$

This concludes the proof of Theorem 2. □

Corollary 3.

Taking $\psi \left(v\right)=1,$$\forall v\in [e,u]$ within Theorem 2, the result reduces to the following inequality:

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{8}}\left[\mathfrak{h}\left(e\right)+3\mathfrak{h}\left({\displaystyle \frac{2e+u}{3}}\right)+3\mathfrak{h}\left({\displaystyle \frac{e+2u}{3}}\right)+\mathfrak{h}\left(u\right)\right]-{\displaystyle \frac{2^{\varrho -1}\mathrm{\Gamma }(\varrho +1)}{{(u-e)}^{\varrho }}}\left[J_{{\displaystyle \frac{e+u}{2}}+}^{\varrho }\mathfrak{h}\left(u\right)+J_{{\displaystyle \frac{e+u}{2}}-}^{\varrho }\mathfrak{h}\left(e\right)\right]\right|\hfill \\ \hfill & \le {\displaystyle \frac{u-e}{3}}{\left({}_2\mathcal{F}_1\left(-p,{\displaystyle \frac{1}{\varrho }},{\displaystyle \frac{1}{\varrho }}+1;{\displaystyle \frac{3}{4}}\right)\right)}^{1/p}\left[{\left({\displaystyle \frac{3|{\mathfrak{h}}^{\prime }{\left(e\right)|}^q+{\left|{\mathfrak{h}}^{\prime }\left(u\right)\right|}^q}{4}}\right)}^{1/q}+{\left({\displaystyle \frac{3|{\mathfrak{h}}^{\prime }{\left(u\right)|}^q+{\left|{\mathfrak{h}}^{\prime }\left(e\right)\right|}^q}{4}}\right)}^{1/q}\right],\hfill \end{array}$$

(16)

where ${}_2\mathcal{F}_1$ denotes the classical hypergeometric function.

Proof. 

By selecting $\psi \left(t\right)=1$ throughout the interval $[e,u]$ and performing an appropriate substitution, we compute

$$\begin{array}{cc}\hfill {\int }_e^{{\displaystyle \frac{e+u}{2}}}{\left|\mathrm{\Psi }(\varrho ,t)-{\displaystyle \frac{4}{3}}\mathrm{\Phi }\left(\varrho \right)\right|}^{p}dt& ={\displaystyle \frac{1}{{\varrho }^p}}{\int }_e^{{\displaystyle \frac{e+u}{2}}}{\left|{\displaystyle \frac{4}{3}}{\left({\displaystyle \frac{u-e}{2}}\right)}^{\varrho }-{(t-e)}^{\varrho }\right|}^{p}dt\hfill \\ \hfill & ={\displaystyle \frac{1}{{\varrho }^p}}{\left({\displaystyle \frac{u-e}{2}}\right)}^{p(\varrho +1)}{\int }_0^1{\left({\displaystyle \frac{4}{3}}-t^{\varrho }\right)}^{p}dt\hfill \\ \hfill & ={\displaystyle \frac{1}{{\varrho }^p}}{\left({\displaystyle \frac{u-e}{2}}\right)}^{p(\varrho +1)}{\left({\displaystyle \frac{4}{3}}\right)}^p{}_2\mathcal{F}_1\left(-p,{\displaystyle \frac{1}{\varrho }},{\displaystyle \frac{1}{\varrho }}+1;{\displaystyle \frac{3}{4}}\right).\hfill \end{array}$$

The proof is complete. □

Corollary 4.

By assigning $\varrho =1$ within Theorem 2, we derive the following weighted $\mathcal{M}-\mathfrak{t}$ inequality:

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{3}}\left[2\mathfrak{h}\left(e\right)-\mathfrak{h}\left({\displaystyle \frac{e+u}{2}}\right)+2\mathfrak{h}\left(u\right)\right]{\int }_e^u\psi \left(t\right)dt-{\int }_e^u\psi \left(t\right)\mathfrak{h}\left(t\right)dt\right|\hfill \\ \hfill & \le {\left({\displaystyle \frac{u-e}{2}}\right)}^{1/q}{\left({\int }_e^{{\displaystyle \frac{e+u}{2}}}{\left({\displaystyle \frac{2}{3}}{\int }_e^u\psi \left(s\right)ds-{\int }_e^t\psi \left(s\right)ds\right)}^{p}dt\right)}^{1/p}\hfill \\ \hfill & \times \left[{\left({\displaystyle \frac{3|{\mathfrak{h}}^{\prime }{\left(e\right)|}^q+{\left|{\mathfrak{h}}^{\prime }\left(u\right)\right|}^q}{4}}\right)}^{1/q}+{\left({\displaystyle \frac{3|{\mathfrak{h}}^{\prime }{\left(u\right)|}^q+{\left|{\mathfrak{h}}^{\prime }\left(e\right)\right|}^q}{4}}\right)}^{1/q}\right].\hfill \end{array}$$

Corollary 5.

If we further specify $\psi \left(v\right)=1,$$\forall v\in [e,u]$ within Corollary 4, the inequality simplifies as follows:

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{3}}\left[2\mathfrak{h}\left(e\right)-\mathfrak{h}\left({\displaystyle \frac{e+u}{2}}\right)+2\mathfrak{h}\left(u\right)\right]-{\displaystyle \frac{1}{u-e}}{\int }_e^u\mathfrak{h}\left(t\right)dt\right|\hfill \\ \hfill & \le {\displaystyle \frac{u-e}{2{(p+1)}^{\frac{1}{p}}}}{\left(\left[{\left({\displaystyle \frac{5}{6}}\right)}^{p+1}+{\left({\displaystyle \frac{4}{3}}\right)}^{p+1}\right]\right)}^{1/p}\hfill \\ \hfill & \times \left[{\left({\displaystyle \frac{3|{\mathfrak{h}}^{\prime }{\left(e\right)|}^q+{\left|{\mathfrak{h}}^{\prime }\left(u\right)\right|}^q}{4}}\right)}^{1/q}+{\left({\displaystyle \frac{3|{\mathfrak{h}}^{\prime }{\left(u\right)|}^q+{\left|{\mathfrak{h}}^{\prime }\left(e\right)\right|}^q}{4}}\right)}^{1/q}\right].\hfill \end{array}$$

Theorem 3.

Suppose that the conditions stated in Lemma 1 are satisfied. If the function ${\left|{\mathfrak{h}}^{\prime }\right|}^q$ is convex over $[e,u]$ for some $q\ge 1$, then the coming weighted fractional $\mathcal{M}-\mathfrak{t}$ inequality holds:

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{3}}\left[2\mathfrak{h}\left(e\right)-\mathfrak{h}\left({\displaystyle \frac{e+u}{2}}\right)+2\mathfrak{h}\left(u\right)\right]\mathrm{\Phi }\left(\varrho \right)-{\displaystyle \frac{\mathrm{\Gamma }\left(\varrho \right)}{2}}\left[J_{{\displaystyle \frac{e+u}{2}}+}^{\varrho }\left(\mathfrak{h}\psi \right)\left(u\right)+J_{{\displaystyle \frac{e+u}{2}}-}^{\varrho }\left(\mathfrak{h}\psi \right)\left(e\right)\right]\right|\hfill \\ \hfill & \le {\displaystyle \frac{1}{4{(u-e)}^{1/q}}}{\left({\mathrm{\Omega }}_1\left(\varrho \right)\right)}^{1-1/q}\left[{\left({\mathrm{\Omega }}_2\left(\varrho \right)|{\mathfrak{h}}^{\prime }{\left(e\right)|}^q+{\mathrm{\Omega }}_3\left(\varrho \right){\left|{\mathfrak{h}}^{\prime }\left(u\right)\right|}^q\right)}^{1/q}+{\left({\mathrm{\Omega }}_2\left(\varrho \right)|{\mathfrak{h}}^{\prime }{\left(u\right)|}^q+{\mathrm{\Omega }}_3\left(\varrho \right){\left|{\mathfrak{h}}^{\prime }\left(e\right)\right|}^q\right)}^{1/q}\right],\hfill \end{array}$$

where the quantity ${\mathrm{\Omega }}_1\left(\varrho \right)$ is introduced as in Theorem 1, and we set

$$\begin{array}{cc}\hfill {\mathrm{\Omega }}_2\left(\varrho \right)& ={\int }_e^{{\displaystyle \frac{e+u}{2}}}\left|\mathrm{\Psi }(\varrho ,\nu )-{\displaystyle \frac{4}{3}}\mathrm{\Phi }\left(\varrho \right)\right|(u-\nu )d\nu ,\hfill \\ \hfill {\mathrm{\Omega }}_3\left(\varrho \right)& ={\int }_e^{{\displaystyle \frac{e+u}{2}}}\left|\mathrm{\Psi }(\varrho ,\nu )-{\displaystyle \frac{4}{3}}\mathrm{\Phi }\left(\varrho \right)\right|(\nu -e)d\nu .\hfill \end{array}$$

Proof. 

Starting from inequality (12) and employing the power mean inequality, we derive

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{3}}\left[2\mathfrak{h}\left(e\right)-\mathfrak{h}\left({\displaystyle \frac{e+u}{2}}\right)+2\mathfrak{h}\left(u\right)\right]\mathrm{\Phi }\left(\varrho \right)-{\displaystyle \frac{\mathrm{\Gamma }\left(\varrho \right)}{2}}\left[J_{{\displaystyle \frac{e+u}{2}}+}^{\varrho }\left(\mathfrak{h}\psi \right)\left(u\right)+J_{{\displaystyle \frac{e+u}{2}}-}^{\varrho }\left(\mathfrak{h}\psi \right)\left(e\right)\right]\right|\hfill \\ \hfill & \le {\displaystyle \frac{1}{2}}{\left({\int }_e^{{\displaystyle \frac{e+u}{2}}}\left|\mathrm{\Psi }(\varrho ,\nu )-{\displaystyle \frac{4}{3}}\mathrm{\Phi }\left(\varrho \right)\right|d\nu \right)}^{1-{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & \times \left[{\left({\int }_e^{{\displaystyle \frac{e+u}{2}}}\left|\mathrm{\Psi }(\varrho ,\nu )-{\displaystyle \frac{4}{3}}\mathrm{\Phi }\left(\varrho \right)\right|{\left|{\mathfrak{h}}^{\prime }\left(\nu \right)\right|}^{q}d\nu \right)}^{{\displaystyle \frac{1}{q}}}+{\left({\int }_e^{{\displaystyle \frac{e+u}{2}}}\left|\mathrm{\Psi }(\varrho ,\nu )-{\displaystyle \frac{4}{3}}\mathrm{\Phi }\left(\varrho \right)\right|{\left|{\mathfrak{h}}^{\prime }(e+u-\nu )\right|}^{q}d\nu \right)}^{{\displaystyle \frac{1}{q}}}\right].\hfill \end{array}$$

Due to the convexity assumption on ${\left|{\mathfrak{h}}^{\prime }\right|}^q$, the following bounds hold:

$$\begin{array}{cc}\hfill |{\mathfrak{h}}^{\prime }{\left(\nu \right)|}^q& \le {\displaystyle \frac{u-\nu }{u-e}}|{\mathfrak{h}}^{\prime }{\left(e\right)|}^q+{\displaystyle \frac{\nu -e}{u-e}}{\left|{\mathfrak{h}}^{\prime }\left(u\right)\right|}^q,\hfill \\ \hfill |{\mathfrak{h}}^{\prime }{(e+u-\nu )|}^q& \le {\displaystyle \frac{u-\nu }{u-e}}|{\mathfrak{h}}^{\prime }{\left(u\right)|}^q+{\displaystyle \frac{\nu -e}{u-e}}{\left|{\mathfrak{h}}^{\prime }\left(e\right)\right|}^q.\hfill \end{array}$$

Inserting these estimates into the previous inequality gives

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{3}}\left[2\mathfrak{h}\left(e\right)-\mathfrak{h}\left({\displaystyle \frac{e+u}{2}}\right)+2\mathfrak{h}\left(u\right)\right]\mathrm{\Phi }\left(\varrho \right)-{\displaystyle \frac{\mathrm{\Gamma }\left(\varrho \right)}{2}}\left[J_{{\displaystyle \frac{e+u}{2}}+}^{\varrho }\left(\mathfrak{h}\psi \right)\left(u\right)+J_{{\displaystyle \frac{e+u}{2}}-}^{\varrho }\left(\mathfrak{h}\psi \right)\left(e\right)\right]\right|\hfill \\ \hfill & \le {\displaystyle \frac{1}{4{(u-e)}^{\frac{1}{q}}}}{\left({\mathrm{\Omega }}_1\left(\varrho \right)\right)}^{1-{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & \times \left[{\left({\mathrm{\Omega }}_2\left(\varrho \right)|{\mathfrak{h}}^{\prime }{\left(e\right)|}^q+{\mathrm{\Omega }}_3\left(\varrho \right){\left|{\mathfrak{h}}^{\prime }\left(u\right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}+{\left({\mathrm{\Omega }}_2\left(\varrho \right)|{\mathfrak{h}}^{\prime }{\left(u\right)|}^q+{\mathrm{\Omega }}_3\left(\varrho \right){\left|{\mathfrak{h}}^{\prime }\left(e\right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\right],\hfill \end{array}$$

thus finalizing the proof. □

4. Fractional Milne-Type Estimates with Weights Involving Functions of Limited Magnitude

This section is devoted to deriving weighted fractional $\mathcal{M}-\mathfrak{t}$ inequalities for functions whose derivatives are bounded.

Theorem 4.

Let the conditions mentioned in Lemma 1 be valid. Suppose further that real constants $\mathfrak{m},\mathfrak{M}\in \mathbb{R}$ exist, fulfilling the inequality $\mathfrak{m}\le {\mathfrak{h}}^{\prime }\left(z\right)\le \mathfrak{M}$ for every z within $[e,u]$. Under these assumptions, we derive the subsequent weighted fractional $\mathcal{M}-\mathfrak{t}$ inequality:

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{3}}\left[2\mathfrak{h}\left(e\right)-\mathfrak{h}\left({\displaystyle \frac{e+u}{2}}\right)+2\mathfrak{h}\left(u\right)\right]\mathrm{\Phi }\left(\varrho \right)-{\displaystyle \frac{\mathrm{\Gamma }\left(\varrho \right)}{2}}\left[J_{e+}^{\varrho }\left(\mathfrak{h}\psi \right)\left({\displaystyle \frac{e+u}{2}}\right)+J_{u-}^{\varrho }\left(\mathfrak{h}\psi \right)\left({\displaystyle \frac{e+u}{2}}\right)\right]\right|\hfill \\ \hfill & \le {\displaystyle \frac{1}{2}}{\mathrm{\Omega }}_1\left(\varrho \right)(\mathfrak{M}-\mathfrak{m}),\hfill \end{array}$$

where ${\mathrm{\Omega }}_1\left(\varrho \right)$ is given as shown in Theorem 1.

Proof. 

Starting from equality (8), we have

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{3}}\left[2\mathfrak{h}\left(e\right)-\mathfrak{h}\left({\displaystyle \frac{e+u}{2}}\right)+2\mathfrak{h}\left(u\right)\right]\mathrm{\Phi }\left(\varrho \right)-{\displaystyle \frac{\mathrm{\Gamma }\left(\varrho \right)}{2}}\left[J_{{\displaystyle \frac{e+u}{2}}+}^{\varrho }\left(\mathfrak{h}\psi \right)\left(u\right)+J_{{\displaystyle \frac{e+u}{2}}-}^{\varrho }\left(\mathfrak{h}\psi \right)\left(e\right)\right]\hfill \\ \hfill & ={\displaystyle \frac{1}{2}}\left[{\int }_e^{{\displaystyle \frac{e+u}{2}}}\left(\mathrm{\Psi }(\varrho ,t)-{\displaystyle \frac{4}{3}}\mathrm{\Phi }\left(\varrho \right)\right)\left({\mathfrak{h}}^{\prime }\left(t\right)-{\displaystyle \frac{\mathfrak{m}+\mathfrak{M}}{2}}\right)dt\right.\hfill \\ \hfill & \left.+{\int }_e^{{\displaystyle \frac{e+u}{2}}}\left(\mathrm{\Psi }(\varrho ,t)-{\displaystyle \frac{4}{3}}\mathrm{\Phi }\left(\varrho \right)\right)\left({\displaystyle \frac{\mathfrak{m}+\mathfrak{M}}{2}}-{\mathfrak{h}}^{\prime }(e+u-t)\right)dt\right].\hfill \end{array}$$

(17)

Taking the modulus of both sides of (17), it follows that:

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{3}}\left[2\mathfrak{h}\left(e\right)-\mathfrak{h}\left({\displaystyle \frac{e+u}{2}}\right)+2\mathfrak{h}\left(u\right)\right]\mathrm{\Phi }\left(\varrho \right)-{\displaystyle \frac{\mathrm{\Gamma }\left(\varrho \right)}{2}}\left[J_{{\displaystyle \frac{e+u}{2}}+}^{\varrho }\left(\mathfrak{h}\psi \right)\left(u\right)+J_{{\displaystyle \frac{e+u}{2}}-}^{\varrho }\left(\mathfrak{h}\psi \right)\left(e\right)\right]\right|\hfill \\ \hfill & \le {\displaystyle \frac{1}{2}}\left[{\int }_e^{{\displaystyle \frac{e+u}{2}}}\left|\mathrm{\Psi }(\varrho ,t)-{\displaystyle \frac{4}{3}}\mathrm{\Phi }\left(\varrho \right)\right|\left|{\mathfrak{h}}^{\prime }\left(t\right)-{\displaystyle \frac{\mathfrak{m}+\mathfrak{M}}{2}}\right|dt+{\int }_e^{{\displaystyle \frac{e+u}{2}}}\left|\mathrm{\Psi }(\varrho ,t)-{\displaystyle \frac{4}{3}}\mathrm{\Phi }\left(\varrho \right)\right|\left|{\displaystyle \frac{\mathfrak{m}+\mathfrak{M}}{2}}-{\mathfrak{h}}^{\prime }(e+u-t)\right|dt\right].\hfill \end{array}$$

Since, by assumption, $\mathfrak{m}\le {\mathfrak{h}}^{\prime }\left(z\right)\le \mathfrak{M}$ on $[e,u]$, it follows that:

$$\left|{\mathfrak{h}}^{\prime }\left(z\right)-{\displaystyle \frac{\mathfrak{m}+\mathfrak{M}}{2}}\right|\le {\displaystyle \frac{\mathfrak{M}-\mathfrak{m}}{2}},\forall z\in [e,u].$$

(18)

Applying this bound gives

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{3}}\left[2\mathfrak{h}\left(e\right)-\mathfrak{h}\left({\displaystyle \frac{e+u}{2}}\right)+2\mathfrak{h}\left(u\right)\right]\mathrm{\Phi }\left(\varrho \right)-{\displaystyle \frac{\mathrm{\Gamma }\left(\varrho \right)}{2}}\left[J_{e+}^{\varrho }\left(\mathfrak{h}\psi \right)\left({\displaystyle \frac{e+u}{2}}\right)+J_{u-}^{\varrho }\left(\mathfrak{h}\psi \right)\left({\displaystyle \frac{e+u}{2}}\right)\right]\right|\hfill \\ \hfill & \le {\displaystyle \frac{\mathfrak{M}-\mathfrak{m}}{2}}{\int }_e^{{\displaystyle \frac{e+u}{2}}}\left|\mathrm{\Psi }(\varrho ,t)-{\displaystyle \frac{4}{3}}\mathrm{\Phi }\left(\varrho \right)\right|dt.\hfill \end{array}$$

This finishes the proof. □

Remark 3.

If we select $\psi \left(v\right)=1,$$\forall v\in [e,u]$ within Theorem 4, we derive the coming fractional $\mathcal{M}-\mathfrak{t}$ inequality:

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{3}}\left[2\mathfrak{h}\left(e\right)-\mathfrak{h}\left({\displaystyle \frac{e+u}{2}}\right)+2\mathfrak{h}\left(u\right)\right]-{\displaystyle \frac{2^{\varrho -1}\mathrm{\Gamma }(\varrho +1)}{{(u-e)}^{\varrho }}}\left[J_{{\displaystyle \frac{e+u}{2}}+}^{\varrho }\mathfrak{h}\left(u\right)+J_{{\displaystyle \frac{e+u}{2}}-}^{\varrho }\mathfrak{h}\left(e\right)\right]\right|\hfill \\ \hfill & \le {\displaystyle \frac{4\varrho +1}{12(\varrho +1)}}(u-e)(\mathfrak{M}-\mathfrak{m}),\hfill \end{array}$$

as established by Budak and Karagözoğlu in [9] (Theorem 3.1).

Corollary 6.

If we take $\varrho =1$ in Theorem 4, the following weighted $\mathcal{M}-\mathfrak{t}$ inequality is achieved:

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{3}}\left[2\mathfrak{h}\left(e\right)-\mathfrak{h}\left({\displaystyle \frac{e+u}{2}}\right)+2\mathfrak{h}\left(u\right)\right]{\int }_e^u\psi \left(t\right)dt-{\int }_e^u\psi \left(t\right)\mathfrak{h}\left(t\right)dt\right|\hfill \\ \hfill & \le \left({\displaystyle \frac{\mathfrak{M}-\mathfrak{m}}{2}}\right){\int }_e^{{\displaystyle \frac{e+u}{2}}}\left[{\displaystyle \frac{2}{3}}{\int }_e^u\psi \left(s\right)ds-{\int }_e^t\psi \left(s\right)ds\right]dt.\hfill \end{array}$$

Remark 4.

Choosing $\psi \left(t\right)=1$ on $[e,u]$ in Corollary 6, we retrieve the classical $\mathcal{M}-\mathfrak{t}$ inequality:

$$\left|{\displaystyle \frac{1}{3}}\left[2\mathfrak{h}\left(e\right)-\mathfrak{h}\left({\displaystyle \frac{e+u}{2}}\right)+2\mathfrak{h}\left(u\right)\right]-{\displaystyle \frac{1}{u-e}}{\int }_e^u\mathfrak{h}\left(t\right)dt\right|\le {\displaystyle \frac{5(u-e)}{24}}(\mathfrak{M}-\mathfrak{m}),$$

as presented by Budak et al. in [6] (Corollary 2).

Corollary 7.

Suppose the conditions outlined in Theorem 4 are valid, and consider the existence of a positive real number $\mathfrak{M}\in {\mathbb{R}}^{+}$ fulfilling $\left|\mathfrak{h}\right(t\left)\right|\le \mathfrak{M}$, $\forall t\in [e,u]$. Then, we obtain the inequality given below:

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{3}}\left[2\mathfrak{h}\left(e\right)-\mathfrak{h}\left({\displaystyle \frac{e+u}{2}}\right)+2\mathfrak{h}\left(u\right)\right]\mathrm{\Phi }\left(\varrho \right)-{\displaystyle \frac{\mathrm{\Gamma }\left(\varrho \right)}{2}}\left[J_{{\displaystyle \frac{e+u}{2}}+}^{\varrho }\left(\mathfrak{h}\psi \right)\left(u\right)+J_{{\displaystyle \frac{e+u}{2}}-}^{\varrho }\left(\mathfrak{h}\psi \right)\left(e\right)\right]\right|\hfill \\ \hfill & =\mathfrak{M}{\int }_e^{{\displaystyle \frac{e+u}{2}}}\left|\mathrm{\Psi }(\varrho ,t)-{\displaystyle \frac{4}{3}}\mathrm{\Phi }\left(\varrho \right)\right|dt.\hfill \end{array}$$

Remark 5.

If we additionally set $\psi \left(t\right)=1$ and $\varrho =1$ in Corollary 7, we find

$$\left|{\displaystyle \frac{1}{3}}\left[2\mathfrak{h}\left(e\right)-\mathfrak{h}\left({\displaystyle \frac{e+u}{2}}\right)+2\mathfrak{h}\left(u\right)\right]-{\displaystyle \frac{1}{u-e}}{\int }_e^u\mathfrak{h}\left(t\right)dt\right|\le {\displaystyle \frac{5(u-e)}{12}}\mathfrak{M},$$

which coincides with the result established by Alomari in [4] (Theorem 3.1).

5. Milne-Inspired Fractional Weighted Inequalities for Lipschitz Families

Now, we establish several weighted fractional $\mathcal{M}-\mathfrak{t}$ inequalities for the scenario where functions are Lipschitz continuous.

Theorem 5.

Assume the conditions in Lemma 1 hold, and let ${\mathfrak{h}}^{\prime }$ be L-Lipschitz function on $[e,u]$. Then, the following weighted fractional $\mathcal{M}-\mathfrak{t}$ inequality is valid:

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{3}}\left[2\mathfrak{h}\left(e\right)-\mathfrak{h}\left(\frac{e+u}{2}\right)+2\mathfrak{h}\left(u\right)\right]\mathrm{\Phi }\left(\varrho \right)-{\displaystyle \frac{\mathrm{\Gamma }\left(\varrho \right)}{2}}\left[J_{{\displaystyle \frac{e+u}{2}}+}^{\varrho }\left(\mathfrak{h}\psi \right)\left(u\right)+J_{{\displaystyle \frac{e+u}{2}}-}^{\varrho }\left(\mathfrak{h}\psi \right)\left(e\right)\right]\right|\hfill \\ \hfill & \le L{\int }_e^{{\displaystyle \frac{e+u}{2}}}\left|\mathrm{\Psi }(\varrho ,t)-\frac{4}{3}\mathrm{\Phi }\left(\varrho \right)\right|\left(\frac{e+u}{2}-t\right)dt.\hfill \end{array}$$

Proof. 

From (8) and the fact that ${\mathfrak{h}}^{\prime }$ is L-Lipschitz on $[e,u]$, we find

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{3}}\left[2\mathfrak{h}\left(e\right)-\mathfrak{h}\left(\frac{e+u}{2}\right)+2\mathfrak{h}\left(u\right)\right]\mathrm{\Phi }\left(\varrho \right)-{\displaystyle \frac{\mathrm{\Gamma }\left(\varrho \right)}{2}}\left[J_{{\displaystyle \frac{e+u}{2}}+}^{\varrho }\left(\mathfrak{h}\psi \right)\left(u\right)+J_{{\displaystyle \frac{e+u}{2}}-}^{\varrho }\left(\mathfrak{h}\psi \right)\left(e\right)\right]\right|\hfill \\ \hfill & \le \frac{1}{2}{\int }_e^{{\displaystyle \frac{e+u}{2}}}\left|\mathrm{\Psi }(\varrho ,t)-\frac{4}{3}\mathrm{\Phi }\left(\varrho \right)\right|\left|{\mathfrak{h}}^{\prime }\left(t\right)-{\mathfrak{h}}^{\prime }(e+u-t)\right|dt\hfill \\ \hfill & \le \frac{1}{2}{\int }_e^{{\displaystyle \frac{e+u}{2}}}\left|\mathrm{\Psi }(\varrho ,t)-\frac{4}{3}\mathrm{\Phi }\left(\varrho \right)\right|L\left|2t-(e+u)\right|dt.\hfill \end{array}$$

The proof is complete. □

Remark 6.

By setting $\psi \left(t\right)=1$ on $[e,u]$ in Theorem 5, we obtain this fractional $\mathcal{M}-\mathfrak{t}$ inequality:

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{3}}\left[2\mathfrak{h}\left(e\right)-\mathfrak{h}\left(\frac{e+u}{2}\right)+2\mathfrak{h}\left(u\right)\right]-{\displaystyle \frac{2^{\varrho -1}\mathrm{\Gamma }(\varrho +1)}{{(u-e)}^{\varrho }}}\left[J_{{\displaystyle \frac{e+u}{2}}+}^{\varrho }\mathfrak{h}\left(u\right)+J_{{\displaystyle \frac{e+u}{2}}-}^{\varrho }\mathfrak{h}\left(e\right)\right]\right|\hfill \\ \hfill & \le {\displaystyle \frac{{(u-e)}^2}{4}}\left({\displaystyle \frac{2}{3}}-{\displaystyle \frac{1}{(\varrho +1)(\varrho +2)}}\right)L,\hfill \end{array}$$

which corresponds to a result of Budak and Karagözoğlu in [9] (Theorem 4.1).

Proof. 

Taking $\psi \left(t\right)=1$ on $[e,u]$ and employing a change of variables, one obtains

$$\begin{array}{cc}\hfill & {\int }_e^{{\displaystyle \frac{e+u}{2}}}\left|\mathrm{\Psi }(\varrho ,t)-\frac{4}{3}\mathrm{\Phi }\left(\varrho \right)\right|\left(\frac{e+u}{2}-t\right)dt\hfill \\ \hfill & ={\displaystyle \frac{1}{\varrho }}{\int }_e^{{\displaystyle \frac{e+u}{2}}}\left|{(t-e)}^{\varrho }-\frac{4}{3}{\left(\frac{u-e}{2}\right)}^{\varrho }\right|\left(\frac{e+u}{2}-t\right)dt\hfill \\ \hfill & ={\displaystyle \frac{1}{\varrho }}{\left(\frac{u-e}{2}\right)}^{\varrho +2}{\int }_0^1(1-\tau )\left|{\tau }^{\varrho }-\frac{4}{3}\right|d\tau \hfill \\ \hfill & ={\displaystyle \frac{1}{\varrho }}{\left(\frac{u-e}{2}\right)}^{\varrho +2}\left(\frac{2}{3}-\frac{1}{(\varrho +1)(\varrho +2)}\right).\hfill \end{array}$$

This concludes the proof. □

Corollary 8.

Taking $\varrho =1$ in Theorem 5 yields the weighted $\mathcal{M}-\mathfrak{t}$ inequality

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{3}}\left[2\mathfrak{h}\left(e\right)-\mathfrak{h}\left(\frac{e+u}{2}\right)+2\mathfrak{h}\left(u\right)\right]{\int }_e^u\psi \left(t\right)dt-{\int }_e^u\psi \left(t\right)\mathfrak{h}\left(t\right)dt\right|\hfill \\ \hfill & \le 2L\left[{\displaystyle \frac{{(u-e)}^2}{12}}{\int }_e^u\psi \left(s\right)ds-\frac{1}{2}{\int }_e^{{\displaystyle \frac{e+u}{2}}}{\left(\frac{e+u}{2}-t\right)}^2\psi \left(t\right)dt\right].\hfill \end{array}$$

Proof. 

Observe that

$$\begin{array}{cc}\hfill & {\int }_e^{{\displaystyle \frac{e+u}{2}}}\left|\mathrm{\Psi }(\varrho ,t)-\frac{4}{3}\mathrm{\Phi }\left(\varrho \right)\right|\left(\frac{e+u}{2}-t\right)dt\hfill \\ \hfill & ={\int }_e^{{\displaystyle \frac{e+u}{2}}}\left(\frac{2}{3}{\int }_e^u\psi \left(s\right)ds-{\int }_e^t\psi \left(s\right)ds\right)\left(\frac{e+u}{2}-t\right)dt\hfill \\ \hfill & ={\displaystyle \frac{2}{3}}\left({\int }_e^u\psi \left(s\right)ds\right){\int }_e^{{\displaystyle \frac{e+u}{2}}}\left(\frac{e+u}{2}-t\right)dt-{\int }_e^{{\displaystyle \frac{e+u}{2}}}\left({\int }_e^t\psi \left(s\right)ds\right)\left(\frac{e+u}{2}-t\right)dt\hfill \\ \hfill & ={\displaystyle \frac{{(u-e)}^2}{12}}{\int }_e^u\psi \left(s\right)ds-\frac{1}{2}{\int }_e^{{\displaystyle \frac{e+u}{2}}}{\left(\frac{e+u}{2}-t\right)}^2\psi \left(t\right)dt.\hfill \end{array}$$

□

Remark 7.

By taking $\psi \left(t\right)=1$ for all $t\in [e,u]$ in Corollary 8, one obtains the $\mathcal{M}-\mathfrak{t}$ result

$$\left|{\displaystyle \frac{1}{3}}\left(2\mathfrak{h}\left(e\right)-\mathfrak{h}\left(\frac{e+u}{2}\right)+2\mathfrak{h}\left(u\right)\right)-{\displaystyle \frac{1}{u-e}}{\int }_e^u\mathfrak{h}\left(t\right)dt\right|\le {\displaystyle \frac{{(u-e)}^2}{8}}L,$$

which was previously shown by Budak et al. in [6] (Corollary 4).

6. Fractionally Weighted Milne-Form Results for Bounded-Variation Functions

In the following, we assume that the function has bounded variation and develop a number of weighted fractional $\mathcal{M}-\mathfrak{t}$ estimates.

Theorem 6.

Suppose $\mathfrak{h}:[e,u]\to \mathbb{R}$ has bounded variation on $[e,u]$. Then the following weighted fractional $\mathcal{M}-\mathfrak{t}$ inequality holds:

$$\begin{array}{c}\hfill \left|\frac{1}{3}\left[2\mathfrak{h}\left(e\right)-\mathfrak{h}\left(\frac{e+u}{2}\right)+2\mathfrak{h}\left(u\right)\right]\mathrm{\Phi }\left(\varrho \right)-\frac{\mathrm{\Gamma }\left(\varrho \right)}{2}\left(J_{\frac{e+u}{2}+}^{\varrho }\left(\mathfrak{h}\psi \right)\left(u\right)+J_{\frac{e+u}{2}-}^{\varrho }\left(\mathfrak{h}\psi \right)\left(e\right)\right)\right|\le {\displaystyle \frac{2}{3}}\mathrm{\Phi }\left(\varrho \right)\underset{e}{\overset{u}{\bigvee }}\left(\mathfrak{h}\right),\end{array}$$

where ${\bigvee }_e^u\left(\mathfrak{h}\right)$ stands for the total variation of $\mathfrak{h}$ on $[e,u]$.

Proof. 

Define the function

$${\mathfrak{K}}_{\psi }(\varrho ,x)=\left\{\begin{array}{cc}\mathrm{\Psi }(\varrho ,x)-\frac{4}{3}\mathrm{\Phi }\left(\varrho \right),\hfill & e\le x<\frac{e+u}{2},\hfill \\ \frac{4}{3}\mathrm{\Phi }\left(\varrho \right)-\mathrm{{\rm Y}}(\varrho ,x),\hfill & \frac{e+u}{2}\le x\le u.\hfill \end{array}\right.$$

A straightforward integration by parts argument implies:

$${\int }_e^u{\mathfrak{K}}_{\psi }(\varrho ,x)d\mathfrak{h}\left(x\right)={\displaystyle \frac{2}{3}}\left[2\mathfrak{h}\left(e\right)-\mathfrak{h}\left(\frac{e+u}{2}\right)+2\mathfrak{h}\left(u\right)\right]\mathrm{\Phi }\left(\varrho \right)-\mathrm{\Gamma }\left(\varrho \right)\left[J_{\frac{e+u}{2}+}^{\varrho }\left(\mathfrak{h}\psi \right)\left(u\right)+J_{\frac{e+u}{2}-}^{\varrho }\left(\mathfrak{h}\psi \right)\left(e\right)\right].$$

(19)

It is well known that if $\mathfrak{h}$ has bounded variation on $[e,u]$ and g is continuous on the same interval, then

$$\left|{\int }_e^{u}g\left(x\right)d\mathfrak{h}\left(x\right)\right|\le \left(\underset{x\in [e,u]}{sup}\left|g\left(x\right)\right|\right)\underset{e}{\overset{u}{\bigvee }}\left(\mathfrak{h}\right).$$

From (19), we deduce

$$\begin{array}{cc}\hfill & \left|\frac{1}{3}\left[2\mathfrak{h}\left(e\right)-\mathfrak{h}\left(\frac{e+u}{2}\right)+2\mathfrak{h}\left(u\right)\right]\mathrm{\Phi }\left(\varrho \right)-\frac{\mathrm{\Gamma }\left(\varrho \right)}{2}\left[J_{\frac{e+u}{2}+}^{\varrho }\left(\mathfrak{h}\psi \right)\left(u\right)+J_{\frac{e+u}{2}-}^{\varrho }\left(\mathfrak{h}\psi \right)\left(e\right)\right]\right|=\frac{1}{2}\left|{\int }_e^u{\mathfrak{K}}_{\psi }(\varrho ,x)d\mathfrak{h}\left(x\right)\right|.\hfill \end{array}$$

(20)

Using the integral bound above yields

$$\begin{array}{cc}\hfill \left|{\int }_e^u{\mathfrak{K}}_{\psi }(\varrho ,x)d\mathfrak{h}\left(x\right)\right|& \le \left|{\int }_e^{\frac{e+u}{2}}\left(\mathrm{\Psi }(\varrho ,x)-\frac{4}{3}\mathrm{\Phi }\left(\varrho \right)\right)d\mathfrak{h}\left(x\right)\right|+\left|{\int }_{\frac{e+u}{2}}^u\left(\frac{4}{3}\mathrm{\Phi }\left(\varrho \right)-\mathrm{{\rm Y}}(\varrho ,x)\right)d\mathfrak{h}\left(x\right)\right|\hfill \\ \hfill & \le \left(\underset{x\in [e,\frac{e+u}{2}]}{sup}\left|\mathrm{\Psi }(\varrho ,x)-\frac{4}{3}\mathrm{\Phi }\left(\varrho \right)\right|\right)\underset{e}{\overset{\frac{e+u}{2}}{\bigvee }}\left(\mathfrak{h}\right)+\left(\underset{x\in [\frac{e+u}{2},u]}{sup}\left|\frac{4}{3}\mathrm{\Phi }\left(\varrho \right)-\mathrm{{\rm Y}}(\varrho ,x)\right|\right)\underset{\frac{e+u}{2}}{\overset{u}{\bigvee }}\left(\mathfrak{h}\right)\hfill \\ \hfill & =\frac{4}{3}\mathrm{\Phi }\left(\varrho \right)\underset{e}{\overset{\frac{e+u}{2}}{\bigvee }}\left(\mathfrak{h}\right)+\frac{4}{3}\mathrm{\Phi }\left(\varrho \right)\underset{\frac{e+u}{2}}{\overset{u}{\bigvee }}\left(\mathfrak{h}\right)\le \frac{4}{3}\mathrm{\Phi }\left(\varrho \right)\underset{e}{\overset{u}{\bigvee }}\left(\mathfrak{h}\right).\hfill \end{array}$$

(21)

Combining (20) and (21) completes the argument. □

Remark 8.

If $\psi \left(t\right)=1$ on $[e,u]$ in Theorem 6, then

$$\left|\frac{1}{3}\left[2\mathfrak{h}\left(e\right)-\mathfrak{h}\left(\frac{e+u}{2}\right)+2\mathfrak{h}\left(u\right)\right]-{\displaystyle \frac{2^{\varrho -1}\mathrm{\Gamma }(\varrho +1)}{{(u-e)}^{\varrho }}}\left(J_{\frac{e+u}{2}+}^{\varrho }\mathfrak{h}\left(u\right)+J_{\frac{e+u}{2}-}^{\varrho }\mathfrak{h}\left(e\right)\right]\right|\le {\displaystyle \frac{2}{3}}\underset{e}{\overset{u}{\bigvee }}\left(\mathfrak{h}\right),$$

matching the conclusion of Budak and Karagözoğlu in [9] (Theorem 5.1).

Corollary 9.

Taking $\varrho =1$ in Theorem 6 implies:

$$\left|\frac{1}{3}\left[2\mathfrak{h}\left(e\right)-\mathfrak{h}\left(\frac{e+u}{2}\right)+2\mathfrak{h}\left(u\right)\right]{\int }_e^u\psi \left(t\right)dt-{\int }_e^u\psi \left(t\right)\mathfrak{h}\left(t\right)dt\right|\le {\displaystyle \frac{2}{3}}\left({\int }_e^u\psi \left(t\right)dt\right)\underset{e}{\overset{u}{\bigvee }}\left(\mathfrak{h}\right).$$

Remark 9.

When $\psi \left(t\right)=1$ on $[e,u]$ and $\varrho =1$ in Corollary 9, one finds

$$\left|\frac{1}{3}\left[2\mathfrak{h}\left(e\right)-\mathfrak{h}\left(\frac{e+u}{2}\right)+2\mathfrak{h}\left(u\right)\right]-{\displaystyle \frac{1}{u-e}}{\int }_e^u\mathfrak{h}\left(t\right)dt\right|\le {\displaystyle \frac{2}{3}}\underset{e}{\overset{u}{\bigvee }}\left(\mathfrak{h}\right),$$

as shown by Alomari in [4] (Theorem 2.1).

7. Concluding Remarks

In this work, we introduced several new weighted $\mathcal{M}-\mathfrak{t}$ inequalities for various classes of functions by utilizing Riemann–Liouville fractional integrals. The obtained inequalities are valid for several function classes, including those with differentiable convexity, boundedness, Lipschitz continuity, and bounded variation properties. The integral identities in Section 2 played a fundamental role in developing the presented inequalities. By expanding on existing research, our findings offer new perspectives on the performance of these inequalities across different function categories. Future studies could explore additional extensions of $\mathcal{M}-\mathfrak{t}$ inequalities, particularly those involving other forms of fractional integrals and higher-order differentiability. Moreover, the results may have potential applications in numerical methods, error estimation, and other areas where such inequalities are useful. Such advances are likely to deepen our understanding of fractional calculus and broaden its practical relevance in both mathematics and scientific applications.