Comprehensive Weighted Newton Inequalities for Broad Function Classes via Generalized Proportional Fractional Operators

Abstract

In this work, weighted Newton-type inequalities $(\mathcal{WN}-\mathcal{TI})$ for multiple classes of functions by employing generalized proportional fractional $\left(\mathbb{GPF}\right)$ integrals are established. The core step involves establishing a key integral identity under a positive weighting scheme, which underlies all subsequent results. By integrating this identity with $\mathbb{GPF}$ operators, we derive various forms of $(\mathcal{WN}-\mathcal{TI})$ suitable for several function types, including differentiable convex, bounded, and Lipschitz functions.

1. Introduction

Inequalities hold a pivotal role in mathematics, offering extensive applicability in diverse areas. Research has particularly refined classical results such as the Hermite–Hadamard, Simpson, and Newton inequalities, motivated by the need to adapt these approaches to broader function families (e.g., s-convex, quasi-convex, log-convex) [1,2,3,4]. In recent years, fractional calculus has attracted considerable interest for its capacity to generalize and strengthen inequality frameworks in convex analysis. By leveraging Hermite–Hadamard-, Simpson-, and Newton-type inequalities in a fractional context, novel results can be produced, thus opening new avenues of investigation [5,6,7,8]. Specifically, Simpson-type inequalities—originating from Simpson’s quadrature rules—can be stated in multiple equivalent forms, illustrated as follows:

i.

Simpson’s quadrature formula:

$$\begin{array}{c}\hfill {\int }_{\theta }^{\rho }\mathcal{H}\left(\tau \right)d\tau \approx \frac{\rho -\theta }{6}\left[\mathcal{H}\left(\theta \right)+4\mathcal{H}\left(\frac{\theta +\rho }{2}\right)+\mathcal{H}\left(\rho \right)\right].\end{array}$$

ii.

The Newton–Cotes quadrature formula:

$$\begin{array}{c}\hfill {\int }_{\theta }^{\rho }\mathcal{H}\left(\tau \right)d\tau \approx \frac{\rho -\theta }{8}\left[\mathcal{H}\left(\theta \right)+3\mathcal{H}\left(\frac{2\theta +\rho }{3}\right)+3\mathcal{H}\left(\frac{\theta +2\rho }{3}\right)+\mathcal{H}\left(\rho \right)\right].\end{array}$$

A frequently used Newton–Cotes quadrature approach, derived from a three-point Simpson-type inequality, is presented as follows:

Theorem 1

([6]). Assume $\mathcal{H}:[\theta ,\rho ]\to \mathbb{R}$ is a function that is four times continuously differentiable on the interval $(\theta ,\rho )$, with the fourth derivative ${\mathcal{H}}^{\left(4\right)}$ bounded such that $\left|\right|{\mathcal{H}}^{\left(4\right)}{\left|\right|}_{\infty }={sup}_{\tau \in (\theta ,\rho )}\left|{\mathcal{H}}^{\left(4\right)}\left(\tau \right)\right|<\infty$. Then, the following inequality holds:

$$\left|\frac{1}{6}\left[\mathcal{H}\left(\theta \right)+4\mathcal{H}\left(\frac{\theta +\rho }{2}\right)+\mathcal{H}\left(\rho \right)\right]-\frac{1}{\rho -\theta }{\int }_{\theta }^{\rho }\mathcal{H}\left(\tau \right)d\tau \right|\le \frac{1}{2880}{∥{\mathcal{H}}^{\left(4\right)}∥}_{\infty }{\left(\rho -\theta \right)}^4.$$

The Simpson $3/8$ rule is a well-known closed quadrature method formulated using the principles of the Simpson $3/8$ inequality.

Theorem 2

([6]). Assume $\mathcal{H}:[\theta ,\rho ]\to \mathbb{R}$ is a function that is four times continuously differentiable on the interval $(\theta ,\rho )$, with the fourth derivative ${\mathcal{H}}^{\left(4\right)}$ bounded such that $\left|\right|{\mathcal{H}}^{\left(4\right)}{\left|\right|}_{\infty }={sup}_{\tau \in (\theta ,\rho )}\left|{\mathcal{H}}^{\left(4\right)}\left(\tau \right)\right|<\infty$. Then, the following inequality holds:

$$\left|\frac{1}{8}\left[\mathcal{H}\left(\theta \right)+3\mathcal{H}\left(\frac{2\theta +\rho }{3}\right)+3\mathcal{H}\left(\frac{\theta +2\rho }{3}\right)+\mathcal{H}\left(\rho \right)\right]-\frac{1}{\rho -\theta }{\int }_{\theta }^{\rho }\mathcal{H}\left(\tau \right)d\tau \right|\le \frac{1}{6480}{∥{\mathcal{H}}^{\left(4\right)}∥}_{\infty }{\left(\rho -\theta \right)}^4.$$

Definition 1

([9]). Let a and b be real numbers, such that $a<b$. If a function $\mathcal{H}:[a,b]\to \mathbb{R}$ meets the following inequality, it is considered convex:

$$\begin{array}{c}\hfill \mathcal{H}\left(\eta \nu +\left(1-\eta \right)\theta \right)\le \eta \mathcal{H}\left(\nu \right)+\left(1-\eta \right)\mathcal{H}\left(\theta \right),\end{array}$$

for all $\nu ,\theta \in [a,b]$ and $\eta \in [0,1]$.

Newton-type inequalities, which arise from computations involving three-step quadratic kernels, have been a prominent area of study in mathematical analysis. These results, rooted in the three-point Newton–Cotes quadrature rule, serve as a foundation for Simpson’s second rule, a fundamental technique in numerical integration. The field has attracted considerable attention from researchers who have contributed significantly to its advancement.

For instance, Ref. [10] focuses on error estimates for Newton-type quadrature formulas, emphasizing functions with bounded variation and Lipschitz continuity, which are crucial in numerical applications. Ref. [11] explores inequalities for functions with convex second derivatives, providing theoretical insights into their structure. Extensions to harmonic convex and p-harmonic convex functions are discussed in [12,13], respectively, broadening the scope of these inequalities. Further contributions include [14], where Newton-type inequalities are developed in the context of post-quantum integrals, and [15], which examines their applicability to quantum differentiable convex functions.

For additional developments and recent breakthroughs, the works [16,17,18,19] present innovative methodologies and theoretical advancements. Collectively, these studies underscore the importance and versatility of Newton-type inequalities in both theoretical research and practical applications.

Definition 2

([20]). For $1\ge \vartheta >0$ and $r\in (0,1]$, the ϑ-order proportional Caputo fractional integral operators are defined as:

$${}_{\mathit{PC}}\mathcal{J}^{\vartheta ,r}\mathcal{H}\left(\nu \right)=\frac{1}{r^{\vartheta }\Gamma \left(\vartheta \right)}{\int }_0^{\nu }{e}^{\frac{r-1}{r}(\nu -\tau )}{(\nu -\tau )}^{\vartheta -1}\mathcal{H}\left(\tau \right)d\tau ,$$

(1)

$${}_{\mathit{PC}}\mathcal{J}_{\theta +}^{\vartheta ,r}\mathcal{H}\left(\nu \right)=\frac{1}{r^{\vartheta }\Gamma \left(\vartheta \right)}{\int }_{\theta }^{\nu }{e}^{\frac{r-1}{r}(\nu -\tau )}{(\nu -\tau )}^{\vartheta -1}\mathcal{H}\left(\tau \right)d\tau ,\nu >\theta ,$$

(2)

and

$${}_{\mathit{PC}}\mathcal{J}_{\rho -}^{\vartheta ,r}\mathcal{H}\left(\nu \right)=\frac{1}{r^{\vartheta }\Gamma \left(\vartheta \right)}{\int }_{\nu }^{\rho }{e}^{\frac{r-1}{r}(\tau -\nu )}{(\tau -\nu )}^{\vartheta -1}\mathcal{H}\left(\tau \right)d\tau ,\nu <\rho ,$$

(3)

where $\Gamma \left(\vartheta \right)$ is the Gamma function, given by:

$$\Gamma \left(\vartheta \right)={\int }_0^{\infty }{e}^{-u}{u}^{\vartheta -1}du.$$

Employing Riemann–Liouville fractional integrals has significantly advanced the theory of Newton-type inequalities, particularly for differentiable convex functions, yielding numerous new results. Moreover, the growing influence of fractional calculus in various scientific domains has prompted the introduction of multiple novel fractional integral operators, each offering potential for further exploration.

For instance, in [21], Riemann–Liouville fractional integrals are used to derive several Newton-type inequalities in the context of differentiable convex functions. Likewise, Ref. [22] explores Newton-type inequalities for arithmetic–harmonic convexity, with a special focus on the absolute value of the first derivative raised to specific powers, and further investigates fractional Newton-type inequalities for functions of bounded variation.

Further research into this domain is detailed in [23,24,25,26,27,28,29,30,31], which explore related topics and offer additional insights. Collectively, these contributions highlight the expanding role of fractional calculus and its integral operators in both theoretical and practical contexts.

The central aim of this work is to formulate fresh $(\mathcal{WN}-\mathcal{TI})$ for a range of function classes by employing $\mathbb{GPF}$ integrals. The investigation begins by establishing a key integral equality, derived using a positive weighted function, which serves as the foundation for the main results. By integrating this equality with $\mathbb{GPF}$ integrals, several $(\mathcal{WN}-\mathcal{TI})$ are derived for function classes such as differentiable convex functions, bounded functions, Lipschitz functions, and functions of bounded variation. These findings enhance the understanding of Newton-type inequalities and highlight possible avenues for further research.

In what follows, the paper is divided into four main sections. The first provides an overview and introduction. Section 2 presents a key integral identity established using a positively weighted function which forms the basis for subsequent developments. In Section 3, we introduce weighted Newton inequalities for three function categories: differentiable convex, bounded derivative, and Lipschitz—through $\mathbb{GPF}$ integrals. Finally, Section 4 discusses the general insights on Newton-type inequalities and highlights potential directions for further inquiry.

2. A Key Integral Equality

In this section, we develop an integral identity that underpins our central results. By exploiting the symmetry of a positive weighting function in conjunction with proportional Caputo fractional integrals, we establish a foundational relationship that will guide the derivation of $(\mathcal{WN}-\mathcal{TI})$ in the sections that follow.

Definition 3.

Let $b:[\theta ,\rho ]\to \mathbb{R}$ be a positive integrable function, symmetric with respect to $\frac{\theta +\rho }{2}$, i.e.,

$$b\left(l\right)=b(\theta +\rho -l),\forall l\in [\theta ,\rho ].$$

The function $\mathcal{B}\left(\vartheta \right)$ is defined as:

$$\mathcal{B}\left(\vartheta \right)={\int }_0^1{e}^{\frac{(r-1)u(\rho -\theta )}{r}}{u}^{\vartheta -1}b\left(u\rho +(1-u)\theta \right)du.$$

Using the proportional Caputo fractional integral operator, $\mathcal{B}\left(\vartheta \right)$ can be expressed as:

$$\mathcal{B}\left(\vartheta \right)=\frac{r^{\vartheta }\Gamma \left(\vartheta \right)}{{(\rho -\theta )}^{\vartheta }}{}_{\mathit{PC}}\mathcal{J}_{\rho -}^{\vartheta ,r}b\left(\theta \right)=\frac{r^{\vartheta }\Gamma \left(\vartheta \right)}{{(\rho -\theta )}^{\vartheta }}{}_{\mathit{PC}}\mathcal{J}_{\theta +}^{\vartheta ,r}b\left(\rho \right),$$

where ${}_{\mathit{PC}}\mathcal{J}_{\rho -}^{\vartheta ,r}$ and ${}_{\mathit{PC}}\mathcal{J}_{\theta +}^{\vartheta ,r}$ are the proportional Caputo fractional integral operators.

To account for symmetry, $\mathcal{B}\left(\vartheta \right)$ can also be written as:

$$\mathcal{B}\left(\vartheta \right)=\frac{r^{\vartheta }\Gamma \left(\vartheta \right)}{2{(\rho -\theta )}^{\vartheta }}\left[{}_{\mathit{PC}}\mathcal{J}_{\rho -}^{\vartheta ,r}b\left(\theta \right)+{}_{\mathit{PC}}\mathcal{J}_{\theta +}^{\vartheta ,r}b\left(\rho \right)\right].$$

In the special case where $\vartheta =1$, $\mathcal{B}\left(\vartheta \right)$ simplifies to:

$$\mathcal{B}\left(1\right)=\frac{1}{\rho -\theta }{\int }_{\theta }^{\rho }{e}^{\frac{(r-1)}{r}(l-\theta )}b\left(l\right)dl.$$

Lemma 1.

Let $\mathcal{H}$ be a real function on $[\theta ,\rho ],$ and absolutely continuous on $(\theta ,\rho )$, with ${\mathcal{H}}^{\prime }$ belonging to the space $L_1[\theta ,\rho ]$. Then, the following relation holds:

$$\begin{array}{cc}\hfill & \frac{1}{8}\left[\mathcal{H}\left(\theta \right)+3\mathcal{H}\left(\frac{2\theta +\rho }{3}\right)+3\mathcal{H}\left(\frac{\theta +2\rho }{3}\right)+\mathcal{H}\left(\rho \right)\right]\mathcal{B}\left(\vartheta \right)\hfill \\ \hfill & -\frac{r^{\vartheta }\Gamma \left(\vartheta \right)}{2{(\rho -\theta )}^{\vartheta }}\left[{}_{\mathit{PC}}\mathcal{J}_{\rho -}^{\vartheta ,r}\mathcal{H}\left(\theta \right)b\left(\theta \right)+{}_{\mathit{PC}}\mathcal{J}_{\theta +}^{\vartheta ,r}\mathcal{H}\left(\rho \right)b\left(\rho \right)\right]=\frac{(\rho -\theta )({\mathcal{A}}_1+{\mathcal{A}}_2+{\mathcal{A}}_3)}{2},\hfill \end{array}$$

(4)

where $\Gamma \left(\vartheta \right)$ represents the Gamma function, and

$$\left\{\begin{array}{c}{\mathcal{A}}_1=\underset{0}{\overset{\frac{1}{3}}{\int }}{\varphi }_1(\theta ,u)\left[{\mathcal{H}}^{\prime }\left(u\rho +\left(1-u\right)\theta \right)-{\mathcal{H}}^{\prime }\left(u\theta +\left(1-u\right)\rho \right)\right]du,\hfill \\ {\mathcal{A}}_2=\underset{\frac{1}{3}}{\overset{\frac{2}{3}}{\int }}{\varphi }_2(\vartheta ,u)\left[{\mathcal{H}}^{\prime }\left(u\rho +\left(1-u\right)\theta \right)-{\mathcal{H}}^{\prime }\left(u\theta +\left(1-u\right)\rho \right)\right]du,\hfill \\ {\mathcal{A}}_3=\underset{\frac{2}{3}}{\overset{1}{\int }}{\varphi }_3(\theta ,\rho )\left[{\mathcal{H}}^{\prime }\left(u\rho +\left(1-u\right)\theta \right)-{\mathcal{H}}^{\prime }\left(u\theta +\left(1-u\right)\rho \right)\right]du,\hfill \end{array}\right.$$

where

$$\begin{array}{ccc}\hfill {\varphi }_1(\vartheta ,u)& =& \underset{0}{\overset{u}{\int }}{e}^{\frac{r-1}{r}u(\rho -\theta )}{u}^{\vartheta -1}b\left(u\rho +\left(1-u\right)\theta \right)du-\frac{1}{8}\mathcal{B}\left(\vartheta \right),\hfill \\ \hfill {\varphi }_2(\vartheta ,u)& =& \underset{0}{\overset{u}{\int }}{e}^{\frac{r-1}{r}u(\rho -\theta )}{u}^{\vartheta -1}b\left(u\rho +\left(1-u\right)\theta \right)du-\frac{1}{2}\mathcal{B}\left(\vartheta \right),\hfill \\ \hfill {\varphi }_3(\vartheta ,u)& =& \underset{0}{\overset{u}{\int }}{e}^{\frac{r-1}{r}u(\rho -\theta )}{u}^{\vartheta -1}b\left(u\rho +\left(1-u\right)\theta \right)du-\frac{7}{8}\mathcal{B}\left(\vartheta \right).\hfill \end{array}$$

Proof. 

We start by considering the integral:

$$A_1={\int }_0^{\frac{1}{3}}{\varphi }_1(\vartheta ,u)\left[H^{\prime }\left(u\rho +(1-u)\theta \right)-H^{\prime }\left(u\theta +(1-u)\rho \right)\right]du,$$

where:

$${\varphi }_1(\vartheta ,u)={\int }_0^u{e}^{\frac{(r-1)\zeta (\rho -\theta )}{r}}{\zeta }^{\vartheta -1}b\left(\zeta \rho +(1-\zeta )\theta \right)d\zeta -\frac{1}{8}\mathcal{B}\left(\vartheta \right).$$

Using integration by parts, we rewrite $A_1$ as:

$$A_1=\frac{{\varphi }_1(\vartheta ,u)}{\rho -\theta }[H\left(u\rho +(1-u)\theta \right)+H\left(u\theta +(1-u)\rho \right)]{|}_0^{\frac{1}{3}}$$

$$-\frac{1}{\rho -\theta }{\int }_0^{\frac{1}{3}}{e}^{\frac{(r-1)u(\rho -\theta )}{r}}{u}^{\vartheta -1}b\left(u\rho +(1-u)\theta \right)\left[H\left(u\rho +(1-u)\theta \right)+H\left(u\theta +(1-u)\rho \right)\right]du.$$

Thus:

$$\begin{array}{ccc}& A_1& =\frac{1}{\rho -\theta }[{\int }_0^{\frac{1}{3}}{e}^{\frac{(r-1)u(\rho -\theta )}{r}}{u}^{\vartheta -1}b\left(u\rho +(1-u)\theta \right)du-\frac{1}{8}\mathcal{B}\left(\vartheta \right)]\left[\mathcal{H}\left(\frac{2\theta +\rho }{3}\right)\right. \hfill \\ & & \left. +\mathcal{H}\left(\frac{\theta +2\rho }{3}\right)\right]+\frac{\mathcal{B}\left(\vartheta \right)}{8(\rho -\theta )}[\mathcal{H}(\theta )+\mathcal{H}(\rho )]\hfill \\ & & -\frac{1}{\rho -\theta }{\int }_0^{\frac{1}{3}}{e}^{\frac{(r-1)u(\rho -\theta )}{r}}{u}^{\vartheta -1}b\left(u\rho +(1-u)\theta \right)\left[H\left(u\rho +(1-u)\theta \right)+H\left(u\theta +(1-u)\rho \right)\right]du.\hfill \end{array}$$

Similarly, for $A_2$ and $A_3$, we have:

$$\begin{array}{cc}\hfill {\mathcal{A}}_2& ={\displaystyle \frac{1}{\rho -\theta }}\left[{\int }_{\frac{1}{3}}^{\frac{2}{3}}{e}^{\frac{(r-1)u(\rho -\theta )}{r}}{u}^{\vartheta -1}b\left(u\rho +(1-u)\theta \right)du\right][\mathcal{H}\left({\displaystyle \frac{2\theta +\rho }{3}}\right)+\mathcal{H}\left({\displaystyle \frac{\theta +2\rho }{3}}\right)]\hfill \\ \hfill & -{\displaystyle \frac{1}{\rho -\theta }}{\int }_{\frac{1}{3}}^{\frac{2}{3}}{e}^{\frac{r-1}{r}u(\rho -\theta )}{u}^{\vartheta -1}b\left(u\rho +(1-u)\theta \right)\left[\mathcal{H}\left(u\rho +(1-u)\theta \right)+\mathcal{H}\left(u\theta +(1-u)\rho \right)\right]du.\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\mathcal{A}}_3& ={\displaystyle \frac{\left[\mathcal{H}\right(\theta )+\mathcal{H}(\rho \left)\right]\mathcal{B}\left(\vartheta \right)}{8(\rho -\theta )}}+{\displaystyle \frac{1}{\rho -\theta }}[{\int }_{\frac{2}{3}}^1{e}^{\frac{(r-1)u(\rho -\theta )}{r}}{u}^{\vartheta -1}b\left(u\rho +(1-u)\theta \right)du-{\displaystyle \frac{1}{8}}\mathcal{B}\left(\vartheta \right)]\left[\mathcal{H}\left({\displaystyle \frac{2\theta +\rho }{3}}\right)+\mathcal{H}\left({\displaystyle \frac{\theta +2\rho }{3}}\right)\right]\hfill \\ \hfill & -{\displaystyle \frac{1}{\rho -\theta }}{\int }_{\frac{2}{3}}^1{e}^{\frac{r-1}{r}u(\rho -\theta )}{u}^{\vartheta -1}b\left(u\rho +(1-u)\theta \right)\left[\mathcal{H}\left(u\rho +(1-u)\theta \right)+\mathcal{H}\left(u\theta +(1-u)\rho \right)\right]du.\hfill \end{array}$$

Adding $A_1$, $A_2$, and $A_3$, we obtain:

$$\begin{array}{cc}\hfill A_1+A_2+A_3& =\frac{1}{4(\rho -\theta )}\mathcal{B}\left(\vartheta \right)[\mathcal{H}(\theta )+\mathcal{H}(\rho )]+\frac{3}{4(\rho -\theta )}\mathcal{B}\left(\vartheta \right)\left[\mathcal{H}\left(\frac{2\theta +\rho }{3}\right)+\mathcal{H}\left(\frac{\theta +2\rho }{3}\right)\right]\hfill \\ & -\frac{1}{\rho -\theta }{\int }_0^1{e}^{\frac{r-1}{r}u(\rho -\theta )}{u}^{\vartheta -1}b\left(u\rho +(1-u)\theta \right)[\mathcal{H}\left(u\rho +(1-u)\theta \right)+\mathcal{H}\left(u\theta +(1-u)\rho \right)]du.\hfill \end{array}$$

Finally, multiplying both sides by $(\rho -\theta )/2$, we conclude:

$$\begin{array}{ccc}& \frac{1}{8}& \left[H\left(\theta \right)+3\mathcal{H}\left(\frac{2\theta +\rho }{3}\right)+3\mathcal{H}\left(\frac{\theta +2\rho }{3}\right)+H\left(\rho \right)\right]\mathcal{B}\left(\vartheta \right)-\frac{r^{\vartheta }\Gamma \left(\vartheta \right)}{2{(\rho -\theta )}^{\vartheta }}\left[{}_{\mathit{PC}}\mathcal{J}_{\rho -}^{\vartheta ,r}\mathcal{H}\left(\theta \right)b\left(\theta \right))+{}_{\mathit{PC}}\mathcal{J}_{\theta +}^{\vartheta ,r}\mathcal{H}\left(\rho \right)b\left(\rho \right)\right]\hfill \\ & & =\frac{(\rho -\theta )(A_1+A_2+A_3)}{2}.\hfill \end{array}$$

This completes the proof. □

3. Fractional Newton-Based Inequalities for Distinct Function Families

In this part, we formulate $(\mathcal{WN}-\mathcal{TI})$ for three categories of functions: differential convex, bounded derivative, and Lipschitz by applying $\mathbb{GPF}$ integrals. These developments are based on a central identity from Lemma 1, used alongside the structure of the derivative (${\mathcal{H}}^{\prime }$) and the symmetry properties of the weight b. Each function category is handled in its own subsection for clarity.

3.1. Fractional Weighted Newton-Type Results for Differentiable Convex Functions

In this subsection, we focus on deriving fractional $(\mathcal{WN}-\mathcal{TI})$ tailored to differentiable convex functions. We begin by considering the modulus of the integral identity established earlier, which yields the fundamental estimates. Subsequently, we employ Hölder’s inequality and the power mean approach to refine these results further.

Theorem 3.

Assume the conditions in Lemma 1 hold, and let $\left|{\mathcal{H}}^{\prime }\right|$ be a convex function over the interval $[\theta ,\rho ]$. Then, the following fractional $(\mathcal{WN}-\mathcal{TI})$ is valid:

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{8}}\left[\mathcal{H}\left(\theta \right)+3\mathcal{H}\left({\displaystyle \frac{2\theta +\rho }{3}}\right)+3\mathcal{H}\left({\displaystyle \frac{\theta +2\rho }{3}}\right)+\mathcal{H}\left(\rho \right)\right]\mathcal{B}\left(\vartheta \right)\right.\hfill \\ \hfill & \left.-{\displaystyle \frac{r^{\vartheta }\Gamma \left(\vartheta \right)}{2{(\rho -\theta )}^{\vartheta }}}[{}_{\mathit{PC}}\mathcal{J}_{\rho -}^{\vartheta ,r}\mathcal{H}\left(\theta \right)b\left(\theta \right)+{}_{\mathit{PC}}\mathcal{J}_{\theta +}^{\vartheta ,r}\mathcal{H}\left(\rho \right)b\left(\rho \right)]\right|\hfill \\ \hfill & \le {\displaystyle \frac{\rho -\theta }{2}}({\mathrm{{\rm Y}}}_1\left(\vartheta \right)+{\mathrm{{\rm Y}}}_2\left(\vartheta \right)+{\mathrm{{\rm Y}}}_3\left(\vartheta \right))[\left|{\mathcal{H}}^{\prime }\left(\theta \right)\right|+\left|{\mathcal{H}}^{\prime }\left(\rho \right)\right|].\hfill \end{array}$$

(5)

where:

$$\begin{array}{cc}\hfill {\mathrm{{\rm Y}}}_1\left(\vartheta \right)& ={\int }_0^{\frac{1}{3}}\left|{\mathrm{\varphi }}_1(\vartheta ,\zeta )\right|d\zeta ,\hfill \\ \hfill {\mathrm{{\rm Y}}}_2\left(\vartheta \right)& ={\int }_{\frac{1}{3}}^{\frac{2}{3}}\left|{\mathrm{\varphi }}_2(\vartheta ,\zeta )\right|d\zeta ,\hfill \\ \hfill {\mathrm{{\rm Y}}}_3\left(\vartheta \right)& ={\int }_{\frac{2}{3}}^1\left|{\varphi }_3(\vartheta ,\zeta )\right|d\zeta .\hfill \end{array}$$

Proof. 

Starting from the integral identity in Lemma 1, we have:

$$\begin{array}{cc}\hfill & \frac{1}{8}\left[\mathcal{H}\left(\theta \right)+3\mathcal{H}\left(\frac{2\theta +\rho }{3}\right)+3\mathcal{H}\left(\frac{\theta +2\rho }{3}\right)+\mathcal{H}\left(\rho \right)\right]\mathcal{B}\left(\vartheta \right)\hfill \\ \hfill & -\frac{r^{\vartheta }\Gamma \left(\vartheta \right)}{2{(\rho -\theta )}^{\vartheta }}\left[{}_{\mathit{PC}}\mathcal{J}_{\rho -}^{\vartheta ,r}\mathcal{H}\left(\theta \right)b\left(\theta \right)+{}_{\mathit{PC}}\mathcal{J}_{\theta +}^{\vartheta ,r}\mathcal{H}\left(\rho \right)b\left(\rho \right)\right]=\frac{\rho -\theta }{2}\left({\mathcal{A}}_1+{\mathcal{A}}_2+{\mathcal{A}}_3\right),\hfill \end{array}$$

where:

$${\mathcal{A}}_1={\int }_0^{\frac{1}{3}}{\varphi }_1(\vartheta ,u)\left[{\mathcal{H}}^{\prime }\left(u\rho +(1-u)\theta \right)-{\mathcal{H}}^{\prime }\left(u\theta +(1-u)\rho \right)\right]du,$$

$${\mathcal{A}}_2={\int }_{\frac{1}{3}}^{\frac{2}{3}}{\varphi }_2(\vartheta ,u)\left[{\mathcal{H}}^{\prime }\left(u\rho +(1-u)\theta \right)-{\mathcal{H}}^{\prime }\left(u\theta +(1-u)\rho \right)\right]du,$$

$${\mathcal{A}}_3={\int }_{\frac{2}{3}}^1{\varphi }_3(\vartheta ,u)\left[{\mathcal{H}}^{\prime }\left(u\rho +(1-u)\theta \right)-{\mathcal{H}}^{\prime }\left(u\theta +(1-u)\rho \right)\right]du.$$

Taking the modulus on both sides, we have:

$$\begin{array}{cc}\hfill & |\frac{1}{8}\left[\mathcal{H}\left(\theta \right)+3\mathcal{H}\left(\frac{2\theta +\rho }{3}\right)+3\mathcal{H}\left(\frac{\theta +2\rho }{3}\right)+\mathcal{H}\left(\rho \right)\right]\mathcal{B}\left(\vartheta \right)\hfill \\ \hfill & -\frac{r^{\vartheta }\Gamma \left(\vartheta \right)}{2{(\rho -\theta )}^{\vartheta }}\left[{}_{\mathit{PC}}\mathcal{J}_{\rho -}^{\vartheta ,r}\mathcal{H}\left(\theta \right)b\left(\theta \right)+{}_{\mathit{PC}}\mathcal{J}_{\theta +}^{\vartheta ,r}\mathcal{H}\left(\rho \right)b\left(\rho \right)\right]|\le \frac{\rho -\theta }{2}|{\mathcal{A}}_1+{\mathcal{A}}_2+{\mathcal{A}}_3|.\hfill \end{array}$$

Using the triangle inequality for the modulus, we can write:

$$|{\mathcal{A}}_1+{\mathcal{A}}_2+{\mathcal{A}}_3|\le \left|{\mathcal{A}}_1\right|+\left|{\mathcal{A}}_2\right|+\left|{\mathcal{A}}_3\right|.$$

For each term ${\mathcal{A}}_i$, we substitute the definition of ${\varphi }_i$:

$${\varphi }_1(\vartheta ,u)={\int }_0^u{e}^{\frac{r-1}{r}v(\rho -\theta )}{v}^{\vartheta -1}b\left(v\rho +(1-v)\theta \right)dv-\frac{1}{8}\mathcal{B}\left(\vartheta \right),$$

$${\varphi }_2(\vartheta ,u)={\int }_0^u{e}^{\frac{r-1}{r}v(\rho -\theta )}{v}^{\vartheta -1}b\left(v\rho +(1-v)\theta \right)dv-\frac{1}{2}\mathcal{B}\left(\vartheta \right),$$

$${\varphi }_3(\vartheta ,u)={\int }_0^u{e}^{\frac{r-1}{r}v(\rho -\theta )}{v}^{\vartheta -1}b\left(v\rho +(1-v)\theta \right)dv-\frac{7}{8}\mathcal{B}\left(\vartheta \right).$$

The modulus of each ${\mathcal{A}}_i$ can be bounded by:

$$\left|{\mathcal{A}}_1\right|\le {\int }_0^{\frac{1}{3}}\left|{\varphi }_1(\vartheta ,u)\right|\left[\right|{\mathcal{H}}^{\prime }\left(u\rho +(1-u)\theta \right)|+|{\mathcal{H}}^{\prime }\left(u\theta +(1-u)\rho \right)\left|\right]du,$$

$$\left|{\mathcal{A}}_2\right|\le {\int }_{\frac{1}{3}}^{\frac{2}{3}}\left|{\varphi }_2(\vartheta ,u)\right|\left[\right|{\mathcal{H}}^{\prime }\left(u\rho +(1-u)\theta \right)|+|{\mathcal{H}}^{\prime }\left(u\theta +(1-u)\rho \right)\left|\right]du,$$

$$\left|{\mathcal{A}}_3\right|\le {\int }_{\frac{2}{3}}^1\left|{\varphi }_3(\vartheta ,u)\right|\left[\right|{\mathcal{H}}^{\prime }\left(u\rho +(1-u)\theta \right)|+|{\mathcal{H}}^{\prime }\left(u\theta +(1-u)\rho \right)\left|\right]du.$$

Using the convexity of $\left|{\mathcal{H}}^{\prime }\right|$, we know:

$$\left|{\mathcal{H}}^{\prime }\left(u\rho +(1-u)\theta \right)\right|+\left|{\mathcal{H}}^{\prime }\left(u\theta +(1-u)\rho \right)\right|\le \left|{\mathcal{H}}^{\prime }\left(\theta \right)\right|+\left|{\mathcal{H}}^{\prime }\left(\rho \right)\right|.$$

Thus:

$$\left|{\mathcal{A}}_1\right|\le {\mathrm{{\rm Y}}}_1\left(\vartheta \right)\left[\left|{\mathcal{H}}^{\prime }\left(\theta \right)\right|+\left|{\mathcal{H}}^{\prime }\left(\rho \right)\right|\right],$$

$$\left|{\mathcal{A}}_2\right|\le {\mathrm{{\rm Y}}}_2\left(\vartheta \right)\left[\left|{\mathcal{H}}^{\prime }\left(\theta \right)\right|+\left|{\mathcal{H}}^{\prime }\left(\rho \right)\right|\right],$$

$$\left|{\mathcal{A}}_3\right|\le {\mathrm{{\rm Y}}}_3\left(\vartheta \right)\left[\left|{\mathcal{H}}^{\prime }\left(\theta \right)\right|+\left|{\mathcal{H}}^{\prime }\left(\rho \right)\right|\right].$$

Here:

$${\mathrm{{\rm Y}}}_1\left(\vartheta \right)={\int }_0^{\frac{1}{3}}\left|{\varphi }_1(\vartheta ,u)\right|du,{\mathrm{{\rm Y}}}_2\left(\vartheta \right)={\int }_{\frac{1}{3}}^{\frac{2}{3}}\left|{\mathrm{\varphi }}_2(\vartheta ,u)\right|du,{\mathrm{{\rm Y}}}_3\left(\vartheta \right)={\int }_{\frac{2}{3}}^1\left|{\varphi }_3(\vartheta ,u)\right|du.$$

Adding the bounds:

$$|{\mathcal{A}}_1+{\mathcal{A}}_2+{\mathcal{A}}_3|\le ({\mathrm{{\rm Y}}}_1\left(\vartheta \right)+{\mathrm{{\rm Y}}}_2\left(\vartheta \right)+{\mathrm{{\rm Y}}}_3\left(\vartheta \right))\left[\left|{\mathcal{H}}^{\prime }\left(\theta \right)\right|+\left|{\mathcal{H}}^{\prime }\left(\rho \right)\right|\right].$$

Substituting back:

$$|\frac{1}{8}\left[\mathcal{H}\left(\theta \right)+3\mathcal{H}\left(\frac{2\theta +\rho }{3}\right)+3\mathcal{H}\left(\frac{\theta +2\rho }{3}\right)+\mathcal{H}\left(\rho \right)\right]\mathcal{B}\left(\vartheta \right)-\frac{r^{\vartheta }\Gamma \left(\vartheta \right)}{2{(\rho -\theta )}^{\vartheta }}[{}_{\mathit{PC}}\mathcal{J}_{\rho -}^{\vartheta ,r}\mathcal{H}\left(\theta \right)b\left(\theta \right)+{}_{\mathit{PC}}\mathcal{J}_{\theta +}^{\vartheta ,r}\mathcal{H}\left(\rho \right)b\left(\rho \right)\left]\right|$$

$$\le \frac{\rho -\theta }{2}({\mathrm{{\rm Y}}}_1\left(\vartheta \right)+{\mathrm{{\rm Y}}}_2\left(\vartheta \right)+{\mathrm{{\rm Y}}}_3\left(\vartheta \right))\left[\right|{\mathcal{H}}^{\prime }\left(\theta \right)|+|{\mathcal{H}}^{\prime }\left(\rho \right)\left|\right].$$

This completes the proof. □

The following result demonstrates a fractional Milne-type inequality for convex functions based on the framework of $\mathbb{GPF}$ integrals. By leveraging the convexity of $|{\mathcal{H}}^{\prime }{|}^{\varrho }$, along with the previously established results, we derive an inequality that quantifies the relationship between weighted fractional integrals and the convex properties of the derivative. This result extends the application of fractional calculus to a broader class of convex functions.

Theorem 4.

Let the assumptions of Lemma 1 be satisfied, and suppose that $|{\mathcal{H}}^{\prime }{|}^{\varrho }$ is a convex function on $[\theta ,\rho ]$, where $\varrho >1$. Then, the following fractional Milne-type inequality holds:

$$\begin{array}{cc}\hfill & |\frac{1}{8}\left[\mathcal{H}\left(\theta \right)+3\mathcal{H}\left(\frac{2\theta +\rho }{3}\right)+3\mathcal{H}\left(\frac{\theta +2\rho }{3}\right)+\mathcal{H}\left(\rho \right)\right]\mathcal{B}\left(\vartheta \right)\hfill \\ \hfill & -\frac{r^{\vartheta }\Gamma \left(\vartheta \right)}{2{(\rho -\theta )}^{\vartheta }}\left[{}_{\mathit{PC}}\mathcal{J}_{\rho -}^{\vartheta ,r}\mathcal{H}b\left(\theta \right)+{}_{\mathit{PC}}\mathcal{J}_{\theta +}^{\vartheta ,r}\mathcal{H}b\left(\rho \right)\right]|\hfill \\ \hfill & \le \frac{\rho -\theta }{2}\left(\right[{\left({\mathrm{\Lambda }}_1\left(\vartheta \right)\right)}^{\frac{1}{t}}+{\left({\mathrm{\Lambda }}_3\left(\vartheta \right)\right)}^{\frac{1}{t}}\left]\right[{\left(\frac{|{\mathcal{H}}^{\prime }\left(\rho \right){|}^{\varrho }+5|{\mathcal{H}}^{\prime }\left(\theta \right){|}^{\varrho }}{18}\right)}^{\frac{1}{\varrho }}+{\left(\frac{5|{\mathcal{H}}^{\prime }\left(\rho \right){|}^{\varrho }+|{\mathcal{H}}^{\prime }\left(\theta \right){|}^{\varrho }}{18}\right)}^{\frac{1}{\varrho }}]\hfill \\ \hfill & +2{\left({\mathrm{\Lambda }}_2\left(\vartheta \right)\right)}^{\frac{1}{t}}\left(\frac{|{\mathcal{H}}^{\prime }\left(\rho \right){|}^{\varrho }+|{\mathcal{H}}^{\prime }\left(\theta \right){|}^{\varrho }}{6}\right)).\hfill \end{array}$$

Here, $\frac{1}{t}+\frac{1}{\varrho }=1$, and the components ${\mathrm{\Lambda }}_i\left(\vartheta \right)$ are defined as:

$${\mathrm{\Lambda }}_1\left(\vartheta \right)={\int }_0^{\frac{1}{3}}|{\varphi }_1(\vartheta ,u){|}^{t}du,{\mathrm{\Lambda }}_2\left(\vartheta \right)={\int }_{\frac{1}{3}}^{\frac{2}{3}}|{\varphi }_2(\vartheta ,u){|}^{t}du,{\mathrm{\Lambda }}_3\left(\vartheta \right)={\int }_{\frac{2}{3}}^1|{\varphi }_3(\vartheta ,u){|}^{t}du.$$

Proof. 

To prove the theorem, we start with the integral identity from Lemma 1:

$$\begin{array}{cc}\hfill & \frac{1}{8}\left[\mathcal{H}\left(\theta \right)+3\mathcal{H}\left(\frac{2\theta +\rho }{3}\right)+3\mathcal{H}\left(\frac{\theta +2\rho }{3}\right)+\mathcal{H}\left(\rho \right)\right]\mathcal{B}\left(\vartheta \right)\hfill \\ \hfill & -\frac{r^{\vartheta }\Gamma \left(\vartheta \right)}{2{(\rho -\theta )}^{\vartheta }}\left[{}_{\mathit{PC}}\mathcal{J}_{\rho -}^{\vartheta ,r}\mathcal{H}b\left(\theta \right)+{}_{\mathit{PC}}\mathcal{J}_{\theta +}^{\vartheta ,r}\mathcal{H}b\left(\rho \right)\right]=\frac{\rho -\theta }{2}\left({\mathcal{A}}_1+{\mathcal{A}}_2+{\mathcal{A}}_3\right),\hfill \end{array}$$

Using the convexity of $|{\mathcal{H}}^{\prime }{|}^{\varrho }$, we apply Hölder’s inequality:

$$\begin{array}{cc}\hfill & \left|\frac{1}{8}\left[\mathcal{H}\left(\theta \right)+3\mathcal{H}\left(\frac{2\theta +\rho }{3}\right)+3\mathcal{H}\left(\frac{\theta +2\rho }{3}\right)+\mathcal{H}\left(\rho \right)\right]\mathcal{B}\left(\vartheta \right)-\frac{r^{\vartheta }\Gamma \left(\vartheta \right)}{2{(\rho -\theta )}^{\vartheta }}\left[{}_{\mathit{PC}}\mathcal{J}_{\rho -}^{\vartheta ,r}\mathcal{H}b\left(\theta \right)+{}_{\mathit{PC}}\mathcal{J}_{\theta +}^{\vartheta ,r}\mathcal{H}b\left(\rho \right)\right]\right|\hfill \\ & \le \frac{\rho -\theta }{2}({\left[{\int }_0^{\frac{1}{3}}{\left|{\varphi }_1(\vartheta ,\zeta )\right|}^{t}d\zeta \right]}^{\frac{1}{t}}\left({\left[{\int }_0^{\frac{1}{3}}\left(\zeta |{\mathcal{H}}^{\prime }{\left(\rho \right)|}^{\varrho }+(1-\zeta ){\left|{\mathcal{H}}^{\prime }\left(\theta \right)\right|}^{\varrho }\right)d\zeta \right]}^{\frac{1}{\varrho }}+{\left[{\int }_0^{\frac{1}{3}}\left(\zeta |{\mathcal{H}}^{\prime }{\left(\theta \right)|}^{\varrho }+(1-\zeta ){\left|{\mathcal{H}}^{\prime }\left(\rho \right)\right|}^{\varrho }\right)d\zeta \right]}^{\frac{1}{\varrho }}\right)\hfill \\ \hfill & +{\left[{\int }_{\frac{1}{3}}^{\frac{2}{3}}{\left|{\varphi }_2(\vartheta ,\zeta )\right|}^{t}d\zeta \right]}^{\frac{1}{t}}\left({\left[{\int }_{\frac{1}{3}}^{\frac{2}{3}}\left(\zeta |{\mathcal{H}}^{\prime }{\left(\rho \right)|}^{\varrho }+(1-\zeta ){\left|{\mathcal{H}}^{\prime }\left(\theta \right)\right|}^{\varrho }\right)d\zeta \right]}^{\frac{1}{\varrho }}+{\left[{\int }_{\frac{1}{3}}^{\frac{2}{3}}\left(\zeta |{\mathcal{H}}^{\prime }{\left(\theta \right)|}^{\varrho }+(1-\zeta ){\left|{\mathcal{H}}^{\prime }\left(\rho \right)\right|}^{\varrho }\right)d\zeta \right]}^{\frac{1}{\varrho }}\right)\hfill \\ \hfill & +{\left[{\int }_{\frac{2}{3}}^1{\left|{\varphi }_3(\vartheta ,\zeta )\right|}^{t}d\zeta \right]}^{\frac{1}{t}}\left({\left[{\int }_{\frac{2}{3}}^1\left(\zeta |{\mathcal{H}}^{\prime }{\left(\rho \right)|}^{\varrho }+(1-\zeta ){\left|{\mathcal{H}}^{\prime }\left(\theta \right)\right|}^{\varrho }\right)d\zeta \right]}^{\frac{1}{\varrho }}+{\left[{\int }_{\frac{2}{3}}^1\left(\zeta |{\mathcal{H}}^{\prime }{\left(\theta \right)|}^{\varrho }+(1-\zeta ){\left|{\mathcal{H}}^{\prime }\left(\rho \right)\right|}^{\varrho }\right)d\zeta \right]}^{\frac{1}{\varrho }}\right)).\hfill \end{array}$$

By evaluating these integrals, we obtain:

$$\begin{array}{cc}\hfill & \le \frac{\rho -\theta }{2}({\left[{\int }_0^{\frac{1}{3}}{\left|{\varphi }_1(\vartheta ,\zeta )\right|}^{t}d\zeta \right]}^{\frac{1}{t}}\left[{\left(\frac{|{\mathcal{H}}^{\prime }{\left(\rho \right)|}^{\varrho }+5{\left|{\mathcal{H}}^{\prime }\left(\theta \right)\right|}^{\varrho }}{18}\right)}^{\frac{1}{\varrho }}+{\left(\frac{5|{\mathcal{H}}^{\prime }{\left(\rho \right)|}^{\varrho }+{\left|{\mathcal{H}}^{\prime }\left(\theta \right)\right|}^{\varrho }}{18}\right)}^{\frac{1}{\varrho }}\right]\hfill \\ \hfill & +2{\left[{\int }_{\frac{1}{3}}^{\frac{2}{3}}{\left|{\varphi }_2(\vartheta ,\zeta )\right|}^{t}d\zeta \right]}^{\frac{1}{t}}{\left[\left(\frac{|{\mathcal{H}}^{\prime }{\left(\rho \right)|}^{\varrho }+{\left|{\mathcal{H}}^{\prime }\left(\theta \right)\right|}^{\varrho }}{6}\right)\right]}^{\frac{1}{\varrho }}\hfill \\ \hfill & +{\left[{\int }_{\frac{2}{3}}^1{\left|{\varphi }_3(\vartheta ,\zeta )\right|}^{t}d\zeta \right]}^{\frac{1}{t}}\left[{\left(\frac{|{\mathcal{H}}^{\prime }{\left(\rho \right)|}^{\varrho }+5{\left|{\mathcal{H}}^{\prime }\left(\theta \right)\right|}^{\varrho }}{18}\right)}^{\frac{1}{\varrho }}+{\left(\frac{5|{\mathcal{H}}^{\prime }{\left(\rho \right)|}^{\varrho }+{\left|{\mathcal{H}}^{\prime }\left(\theta \right)\right|}^{\varrho }}{18}\right)}^{\frac{1}{\varrho }}\right]).\hfill \end{array}$$

We conclude that:

$$\begin{array}{cc}\hfill & |\frac{1}{8}\left[\mathcal{H}\left(\theta \right)+3\mathcal{H}\left(\frac{2\theta +\rho }{3}\right)+3\mathcal{H}\left(\frac{\theta +2\rho }{3}\right)+\mathcal{H}\left(\rho \right)\right]\mathcal{B}\left(\vartheta \right)-\frac{r^{\vartheta }\Gamma \left(\vartheta \right)}{2{(\rho -\theta )}^{\vartheta }}\left[{}_{\mathit{PC}}\mathcal{J}_{\rho -}^{\vartheta ,r}\mathcal{H}b\left(\theta \right)+{}_{\mathit{PC}}\mathcal{J}_{\theta +}^{\vartheta ,r}\mathcal{H}b\left(\rho \right)\right]|\hfill \\ \hfill & \le \frac{\rho -\theta }{2}\left(\right[{\left({\mathrm{\Lambda }}_1\left(\vartheta \right)\right)}^{\frac{1}{t}}+{\left({\mathrm{\Lambda }}_3\left(\vartheta \right)\right)}^{\frac{1}{t}}]\left[{\left(\frac{|{\mathcal{H}}^{\prime }{\left(\rho \right)|}^{\varrho }+5{\left|{\mathcal{H}}^{\prime }\left(\theta \right)\right|}^{\varrho }}{18}\right)}^{\frac{1}{\varrho }}+{\left(\frac{5|{\mathcal{H}}^{\prime }{\left(\rho \right)|}^{\varrho }+{\left|{\mathcal{H}}^{\prime }\left(\theta \right)\right|}^{\varrho }}{18}\right)}^{\frac{1}{\varrho }}\right]\hfill \\ \hfill & +2{\left({\mathrm{\Lambda }}_2\left(\vartheta \right)\right)}^{\frac{1}{t}}{\left(\frac{|{\mathcal{H}}^{\prime }{\left(\rho \right)|}^{\varrho }+{\left|{\mathcal{H}}^{\prime }\left(\theta \right)\right|}^{\varrho }}{6}\right)}^{\frac{1}{\varrho }}).\hfill \end{array}$$

This completes the proof. □

The following result establishes a weighted fractional Milne-type inequality for convex functions using the proportional Caputo fractional integral, extending Newton-type quadrature bounds in fractional calculus.

Theorem 5.

Under the assumptions outlined in Lemma 1, if $|{\mathcal{H}}^{\prime }{|}^{\varrho }$ is convex on $[\theta ,\rho ]$ for some $\varrho \ge 1$, then the subsequent inequality is valid:

$$\begin{array}{cc}\hfill & |\frac{1}{8}\left[\mathcal{H}\left(\theta \right)+3\mathcal{H}\left(\frac{2\theta +\rho }{3}\right)+3\mathcal{H}\left(\frac{\theta +2\rho }{3}\right)+\mathcal{H}\left(\rho \right)\right]\mathcal{B}\left(\vartheta \right)\hfill \\ \hfill & -\frac{r^{\vartheta }\Gamma \left(\vartheta \right)}{2{(\rho -\theta )}^{\vartheta }}\left[{}_{\mathit{PC}}\mathcal{J}_{\rho -}^{\vartheta ,r}\mathcal{H}b\left(\theta \right)+{}_{\mathit{PC}}\mathcal{J}_{\theta +}^{\vartheta ,r}\mathcal{H}b\left(\rho \right)\right]|\hfill \\ \hfill & \le \frac{\rho -\theta }{2}\left[{\left({\mathrm{{\rm Y}}}_1\left(\vartheta \right)\right)}^{1-\frac{1}{\varrho }}\right({\left({\mathrm{{\rm Y}}}_4\left(\vartheta \right)|{\mathcal{H}}^{\prime }{\left(\rho \right)|}^{\varrho }+\left({\mathrm{{\rm Y}}}_1\left(\vartheta \right)-{\mathrm{{\rm Y}}}_4\left(\vartheta \right)\right){\left|{\mathcal{H}}^{\prime }\left(\theta \right)\right|}^{\varrho }\right)}^{\frac{1}{\varrho }}\hfill \\ \hfill & +{\left({\mathrm{{\rm Y}}}_4\left(\vartheta \right)|{\mathcal{H}}^{\prime }{\left(\theta \right)|}^{\varrho }+\left({\mathrm{{\rm Y}}}_1\left(\vartheta \right)-{\mathrm{{\rm Y}}}_4\left(\vartheta \right)\right){\left|{\mathcal{H}}^{\prime }\left(\rho \right)\right|}^{\varrho }\right)}^{\frac{1}{\varrho }})\hfill \\ \hfill & +{\left({\mathrm{{\rm Y}}}_2\left(\vartheta \right)\right)}^{1-\frac{1}{\varrho }}({\left({\mathrm{{\rm Y}}}_5\left(\vartheta \right)|{\mathcal{H}}^{\prime }{\left(\rho \right)|}^{\varrho }+\left({\mathrm{{\rm Y}}}_2\left(\vartheta \right)-{\mathrm{{\rm Y}}}_5\left(\vartheta \right)\right){\left|{\mathcal{H}}^{\prime }\left(\theta \right)\right|}^{\varrho }\right)}^{\frac{1}{\varrho }}\hfill \\ \hfill & +{\left({\mathrm{{\rm Y}}}_5\left(\vartheta \right)|{\mathcal{H}}^{\prime }{\left(\theta \right)|}^{\varrho }+\left({\mathrm{{\rm Y}}}_2\left(\vartheta \right)-{\mathrm{{\rm Y}}}_5\left(\vartheta \right)\right){\left|{\mathcal{H}}^{\prime }\left(\rho \right)\right|}^{\varrho }\right)}^{\frac{1}{\varrho }})\hfill \\ \hfill & +{\left({\mathrm{{\rm Y}}}_3\left(\vartheta \right)\right)}^{1-\frac{1}{\varrho }}({\left({\mathrm{{\rm Y}}}_6\left(\vartheta \right)|{\mathcal{H}}^{\prime }{\left(\rho \right)|}^{\varrho }+\left({\mathrm{{\rm Y}}}_3\left(\vartheta \right)-{\mathrm{{\rm Y}}}_6\left(\vartheta \right)\right){\left|{\mathcal{H}}^{\prime }\left(\theta \right)\right|}^{\varrho }\right)}^{\frac{1}{\varrho }}\hfill \\ \hfill & +{\left({\mathrm{{\rm Y}}}_6\left(\vartheta \right)|{\mathcal{H}}^{\prime }{\left(\theta \right)|}^{\varrho }+\left({\mathrm{{\rm Y}}}_3\left(\vartheta \right)-{\mathrm{{\rm Y}}}_6\left(\vartheta \right)\right){\left|{\mathcal{H}}^{\prime }\left(\rho \right)\right|}^{\varrho }\right)}^{\frac{1}{\varrho }}\left)\right].\hfill \end{array}$$

Here, the terms ${\mathrm{{\rm Y}}}_1\left(\vartheta \right)$, ${\mathrm{{\rm Y}}}_2\left(\vartheta \right)$, and ${\mathrm{{\rm Y}}}_3\left(\vartheta \right)$ follow from Theorem 3, while the new quantities ${\mathrm{{\rm Y}}}_4\left(\vartheta \right)$, ${\mathrm{{\rm Y}}}_5\left(\vartheta \right)$, and ${\mathrm{{\rm Y}}}_6\left(\vartheta \right)$ are defined as:

$$\left\{\begin{array}{c}{\mathrm{{\rm Y}}}_4\left(\vartheta \right)={\int }_0^{\frac{1}{3}}\zeta \left|{\varphi }_1(\vartheta ,\zeta )\right|d\zeta ,\hfill \\ {\mathrm{{\rm Y}}}_5\left(\vartheta \right)={\int }_{\frac{1}{3}}^{\frac{2}{3}}\zeta \left|{\varphi }_2(\vartheta ,\zeta )\right|d\zeta ,\hfill \\ {\mathrm{{\rm Y}}}_6\left(\vartheta \right)={\int }_{\frac{2}{3}}^1\zeta \left|{\varphi }_3(\vartheta ,\zeta )\right|d\zeta .\hfill \end{array}\right.$$

Proof. 

Applying the power mean inequality to the integral representation in (5), we obtain:

$$\begin{array}{cc}\hfill & |\frac{1}{8}\left[\mathcal{H}\left(\theta \right)+3\mathcal{H}\left(\frac{2\theta +\rho }{3}\right)+3\mathcal{H}\left(\frac{\theta +2\rho }{3}\right)+\mathcal{H}\left(\rho \right)\right]\mathcal{B}\left(\vartheta \right)\hfill \\ \hfill & -\frac{r^{\vartheta }\Gamma \left(\vartheta \right)}{2{(\rho -\theta )}^{\vartheta }}\left[{}_{\mathit{PC}}\mathcal{J}_{\rho -}^{\vartheta ,r}\mathcal{H}b\left(\theta \right)+{}_{\mathit{PC}}\mathcal{J}_{\theta +}^{\vartheta ,r}\mathcal{H}b\left(\rho \right)\right]|\hfill \\ \hfill & \le \frac{\rho -\theta }{2}\left[{\left({\int }_0^{\frac{1}{3}}\left|{\varphi }_1(\vartheta ,u)\right|du\right)}^{1-\frac{1}{\varrho }}\right({\left({\int }_0^{\frac{1}{3}}|{\varphi }_1(\vartheta ,u)\left|\right|{\mathcal{H}}^{\prime }{(u\rho +(1-u)\theta )|}^{\varrho }du\right)}^{\frac{1}{\varrho }}\hfill \\ \hfill & +{\left({\int }_0^{\frac{1}{3}}|{\varphi }_1(\vartheta ,u)\left|\right|{\mathcal{H}}^{\prime }{(u\theta +(1-u)\rho )|}^{\varrho }du\right)}^{\frac{1}{\varrho }})\hfill \\ \hfill & +{\left({\int }_{\frac{1}{3}}^{\frac{2}{3}}\left|{\varphi }_2(\vartheta ,u)\right|du\right)}^{1-\frac{1}{\varrho }}({\left({\int }_{\frac{1}{3}}^{\frac{2}{3}}|{\varphi }_2(\vartheta ,u)\left|\right|{\mathcal{H}}^{\prime }{(u\rho +(1-u)\theta )|}^{\varrho }du\right)}^{\frac{1}{\varrho }}\hfill \\ \hfill & +{\left({\int }_{\frac{1}{3}}^{\frac{2}{3}}|{\varphi }_2(\vartheta ,u)\left|\right|{\mathcal{H}}^{\prime }{(u\theta +(1-u)\rho )|}^{\varrho }du\right)}^{\frac{1}{\varrho }})\hfill \\ \hfill & +{\left({\int }_{\frac{2}{3}}^1\left|{\varphi }_3(\vartheta ,u)\right|du\right)}^{1-\frac{1}{\varrho }}({\left({\int }_{\frac{2}{3}}^1|{\varphi }_3(\vartheta ,u)\left|\right|{\mathcal{H}}^{\prime }{(u\rho +(1-u)\theta )|}^{\varrho }du\right)}^{\frac{1}{\varrho }}\hfill \\ \hfill & +{\left({\int }_{\frac{2}{3}}^1|{\varphi }_3(\vartheta ,u)\left|\right|{\mathcal{H}}^{\prime }{(u\theta +(1-u)\rho )|}^{\varrho }du\right)}^{\frac{1}{\varrho }}\left)\right].\hfill \end{array}$$

Applying the convexity of $|{\mathcal{H}}^{\prime }{|}^{\varrho }$, we rewrite the integrals using weighted convex combinations:

$$\begin{array}{cc}\hfill & |\frac{1}{8}\left[\mathcal{H}\left(\theta \right)+3\mathcal{H}\left(\frac{2\theta +\rho }{3}\right)+3\mathcal{H}\left(\frac{\theta +2\rho }{3}\right)+\mathcal{H}\left(\rho \right)\right]\mathcal{B}\left(\vartheta \right)\hfill \\ \hfill & -\frac{r^{\vartheta }\Gamma \left(\vartheta \right)}{2{(\rho -\theta )}^{\vartheta }}\left[{}_{\mathit{PC}}\mathcal{J}_{\rho -}^{\vartheta ,r}\mathcal{H}b\left(\theta \right)+{}_{\mathit{PC}}\mathcal{J}_{\theta +}^{\vartheta ,r}\mathcal{H}b\left(\rho \right)\right]|\hfill \\ \hfill & \le \frac{\rho -\theta }{2}\left[{\left({\mathrm{{\rm Y}}}_1\left(\vartheta \right)\right)}^{1-\frac{1}{\varrho }}\right({\left({\mathrm{{\rm Y}}}_4\left(\vartheta \right)|{\mathcal{H}}^{\prime }{\left(\rho \right)|}^{\varrho }+\left({\mathrm{{\rm Y}}}_1\left(\vartheta \right)-{\mathrm{{\rm Y}}}_4\left(\vartheta \right)\right){\left|{\mathcal{H}}^{\prime }\left(\theta \right)\right|}^{\varrho }\right)}^{\frac{1}{\varrho }}\hfill \\ \hfill & +{\left({\mathrm{{\rm Y}}}_4\left(\vartheta \right)|{\mathcal{H}}^{\prime }{\left(\theta \right)|}^{\varrho }+\left({\mathrm{{\rm Y}}}_1\left(\vartheta \right)-{\mathrm{{\rm Y}}}_4\left(\vartheta \right)\right){\left|{\mathcal{H}}^{\prime }\left(\rho \right)\right|}^{\varrho }\right)}^{\frac{1}{\varrho }})\hfill \\ \hfill & +{\left({\mathrm{{\rm Y}}}_2\left(\vartheta \right)\right)}^{1-\frac{1}{\varrho }}({\left({\mathrm{{\rm Y}}}_5\left(\vartheta \right)|{\mathcal{H}}^{\prime }{\left(\rho \right)|}^{\varrho }+\left({\mathrm{{\rm Y}}}_2\left(\vartheta \right)-{\mathrm{{\rm Y}}}_5\left(\vartheta \right)\right){\left|{\mathcal{H}}^{\prime }\left(\theta \right)\right|}^{\varrho }\right)}^{\frac{1}{\varrho }}\hfill \\ \hfill & +{\left({\mathrm{{\rm Y}}}_5\left(\vartheta \right)|{\mathcal{H}}^{\prime }{\left(\theta \right)|}^{\varrho }+\left({\mathrm{{\rm Y}}}_2\left(\vartheta \right)-{\mathrm{{\rm Y}}}_5\left(\vartheta \right)\right){\left|{\mathcal{H}}^{\prime }\left(\rho \right)\right|}^{\varrho }\right)}^{\frac{1}{\varrho }})\hfill \\ \hfill & +{\left({\mathrm{{\rm Y}}}_3\left(\vartheta \right)\right)}^{1-\frac{1}{\varrho }}({\left({\mathrm{{\rm Y}}}_6\left(\vartheta \right)|{\mathcal{H}}^{\prime }{\left(\rho \right)|}^{\varrho }+\left({\mathrm{{\rm Y}}}_3\left(\vartheta \right)-{\mathrm{{\rm Y}}}_6\left(\vartheta \right)\right){\left|{\mathcal{H}}^{\prime }\left(\theta \right)\right|}^{\varrho }\right)}^{\frac{1}{\varrho }}\hfill \\ \hfill & +{\left({\mathrm{{\rm Y}}}_6\left(\vartheta \right)|{\mathcal{H}}^{\prime }{\left(\theta \right)|}^{\varrho }+\left({\mathrm{{\rm Y}}}_3\left(\vartheta \right)-{\mathrm{{\rm Y}}}_6\left(\vartheta \right)\right){\left|{\mathcal{H}}^{\prime }\left(\rho \right)\right|}^{\varrho }\right)}^{\frac{1}{\varrho }}\left)\right].\hfill \end{array}$$

This concludes the proof. □

3.2. Weighted Newton-Type Inequalities for Functions with Bounded Derivatives

This subsection develops weighted fractional Newton-type inequalities for functions whose derivatives are restricted within fixed bounds. The results are established using the proportional Caputo fractional integral approach.

Theorem 6.

Suppose that the conditions of Lemma 1 hold. If there exist two real constants $l,L,$ and the function $\mathcal{H}$ that satisfies

$$l\le {\mathcal{H}}^{\prime }\left(u\right)\le L,\forall u\in [\theta ,\rho ]$$

then the following weighted $\mathbb{GPF}$ Newton-type inequality is valid:

$$\begin{array}{cc}\hfill & |\frac{1}{8}\left[\mathcal{H}\left(\theta \right)+3\mathcal{H}\left(\frac{2\theta +\rho }{3}\right)+3\mathcal{H}\left(\frac{\theta +2\rho }{3}\right)+\mathcal{H}\left(\rho \right)\right]\mathcal{B}\left(\vartheta \right)\hfill \\ \hfill & -\frac{r^{\vartheta }\Gamma \left(\vartheta \right)}{2{(\rho -\theta )}^{\vartheta }}\left[{}_{\mathit{PC}}\mathcal{J}_{\rho -}^{\vartheta ,r}\mathcal{H}b\left(\theta \right)+{}_{\mathit{PC}}\mathcal{J}_{\theta +}^{\vartheta ,r}\mathcal{H}b\left(\rho \right)\right]|\hfill \\ \hfill & \le \frac{\rho -\theta }{2}[{\mathrm{{\rm Y}}}_1\left(\vartheta \right)+{\mathrm{{\rm Y}}}_2\left(\vartheta \right)+{\mathrm{{\rm Y}}}_3\left(\vartheta \right)](L-l).\hfill \end{array}$$

Proof. 

Applying Lemma 1, we obtain:

$$\begin{array}{cc}\hfill & \frac{1}{8}\left[\mathcal{H}\left(\theta \right)+3\mathcal{H}\left(\frac{2\theta +\rho }{3}\right)+3\mathcal{H}\left(\frac{\theta +2\rho }{3}\right)+\mathcal{H}\left(\rho \right)\right]\mathcal{B}\left(\vartheta \right)\hfill \\ \hfill & -\frac{r^{\vartheta }\Gamma \left(\vartheta \right)}{2{(\rho -\theta )}^{\vartheta }}\left[{}_{\mathit{PC}}\mathcal{J}_{\rho -}^{\vartheta ,r}\mathcal{H}b\left(\theta \right)+{}_{\mathit{PC}}\mathcal{J}_{\theta +}^{\vartheta ,r}\mathcal{H}b\left(\rho \right)\right]\hfill \\ \hfill & =\frac{\rho -\theta }{2}\{{\int }_0^{\frac{1}{3}}{\varphi }_1(\vartheta ,u)\left[{\mathcal{H}}^{\prime }\left(u\rho +(1-u)\theta \right)-\frac{l+L}{2}\right]du\hfill \\ \hfill & +{\int }_0^{\frac{1}{3}}{\varphi }_1(\vartheta ,u)\left[\frac{l+L}{2}-{\mathcal{H}}^{\prime }\left(u\theta +(1-u)\rho \right)\right]du\hfill \\ \hfill & +{\int }_{\frac{1}{3}}^{\frac{2}{3}}{\varphi }_2(\vartheta ,u)\left[{\mathcal{H}}^{\prime }\left(u\rho +(1-u)\theta \right)-\frac{l+L}{2}\right]du\hfill \\ \hfill & +{\int }_{\frac{1}{3}}^{\frac{2}{3}}{\varphi }_2(\vartheta ,u)\left[\frac{l+L}{2}-{\mathcal{H}}^{\prime }\left(u\theta +(1-u)\rho \right)\right]du\hfill \\ \hfill & +{\int }_{\frac{2}{3}}^1{\varphi }_3(\vartheta ,u)\left[{\mathcal{H}}^{\prime }\left(u\rho +(1-u)\theta \right)-\frac{l+L}{2}\right]du\hfill \\ \hfill & +{\int }_{\frac{2}{3}}^1{\varphi }_3(\vartheta ,u)\left[\frac{l+L}{2}-{\mathcal{H}}^{\prime }\left(u\theta +(1-u)\rho \right)\right]du\}.\hfill \end{array}$$

(6)

Taking the absolute value on both sides of (6), we obtain:

$$\begin{array}{cc}\hfill & |\frac{1}{8}\left[\mathcal{H}\left(\theta \right)+3\mathcal{H}\left(\frac{2\theta +\rho }{3}\right)+3\mathcal{H}\left(\frac{\theta +2\rho }{3}\right)+\mathcal{H}\left(\rho \right)\right]\mathcal{B}\left(\vartheta \right)\hfill \\ \hfill & -\frac{r^{\vartheta }\Gamma \left(\vartheta \right)}{2{(\rho -\theta )}^{\vartheta }}\left[{}_{\mathit{PC}}\mathcal{J}_{\rho -}^{\vartheta ,r}\mathcal{H}b\left(\theta \right)+{}_{\mathit{PC}}\mathcal{J}_{\theta +}^{\vartheta ,r}\mathcal{H}b\left(\rho \right)\right]|\hfill \\ \hfill & \le \frac{\rho -\theta }{2}\{{\int }_0^{\frac{1}{3}}|{\varphi }_1(\vartheta ,u)||{\mathcal{H}}^{\prime }\left(u\rho +(1-u)\theta \right)-\frac{l+L}{2}|du\hfill \\ \hfill & +{\int }_0^{\frac{1}{3}}|{\varphi }_1(\vartheta ,u)||\frac{l+L}{2}-{\mathcal{H}}^{\prime }\left(u\theta +(1-u)\rho \right)|du\hfill \\ \hfill & +{\int }_{\frac{1}{3}}^{\frac{2}{3}}|{\varphi }_2(\vartheta ,u)||{\mathcal{H}}^{\prime }\left(u\rho +(1-u)\theta \right)-\frac{l+L}{2}|du\hfill \\ \hfill & +{\int }_{\frac{1}{3}}^{\frac{2}{3}}|{\varphi }_2(\vartheta ,u)||\frac{l+L}{2}-{\mathcal{H}}^{\prime }\left(u\theta +(1-u)\rho \right)|du\hfill \\ \hfill & +{\int }_{\frac{2}{3}}^1|{\varphi }_3(\vartheta ,u)||{\mathcal{H}}^{\prime }\left(u\rho +(1-u)\theta \right)-\frac{l+L}{2}|du\hfill \\ \hfill & +{\int }_{\frac{2}{3}}^1|{\varphi }_3(\vartheta ,u)||\frac{l+L}{2}-{\mathcal{H}}^{\prime }\left(u\theta +(1-u)\rho \right)|du\}.\hfill \end{array}$$

Since it is given that $l\le {\mathcal{H}}^{\prime }\left(u\right)\le L$ for all $u\in [\theta ,\rho ]$, we deduce:

$$|{\mathcal{H}}^{\prime }\left(u\rho +(1-u)\theta \right)-\frac{l+L}{2}|\le \frac{L-l}{2},$$

(7)

$$|\frac{l+L}{2}-{\mathcal{H}}^{\prime }\left(u\theta +(1-u)\rho \right)|\le \frac{L-l}{2}.$$

(8)

Using (7) and (8), we obtain:

$$\begin{array}{cc}\hfill & |\frac{1}{8}\left[\mathcal{H}\left(\theta \right)+3\mathcal{H}\left(\frac{2\theta +\rho }{3}\right)+3\mathcal{H}\left(\frac{\theta +2\rho }{3}\right)+\mathcal{H}\left(\rho \right)\right]\mathcal{B}\left(\vartheta \right)\hfill \\ \hfill & -\frac{r^{\vartheta }\Gamma \left(\vartheta \right)}{2{(\rho -\theta )}^{\vartheta }}\left[{}_{\mathit{PC}}\mathcal{J}_{\rho -}^{\vartheta ,r}\mathcal{H}b\left(\theta \right)+{}_{\mathit{PC}}\mathcal{J}_{\theta +}^{\vartheta ,r}\mathcal{H}b\left(\rho \right)\right]|\hfill \\ \hfill & \le \frac{\rho -\theta }{2}\left[{\int }_0^{\frac{1}{3}}|{\varphi }_1(\vartheta ,u)|du+{\int }_{\frac{1}{3}}^{\frac{2}{3}}|{\varphi }_2(\vartheta ,u)|du+{\int }_{\frac{2}{3}}^1|{\varphi }_3(\vartheta ,u)|du\right](L-l)\hfill \\ \hfill & \le \frac{\rho -\theta }{2}[{\mathrm{{\rm Y}}}_1\left(\vartheta \right)+{\mathrm{{\rm Y}}}_2\left(\vartheta \right)+{\mathrm{{\rm Y}}}_3\left(\vartheta \right)](L-l).\hfill \end{array}$$

□

3.3. Fractional Fractional Newton-Based Inequalities for Lipschitz Functions

Here, we derive a fractional $(\mathcal{WN}-\mathcal{TI})$ under a Lipschitz condition on the derivative of the function, extending the results obtained in previous subsections.

Theorem 7.

Assume that the hypotheses of Lemma 1 hold and that ${\mathcal{H}}^{\prime }$ is L-Lipschitz on the interval $[\theta ,\rho ]$. Then, the following weighted fractional Newton-type inequality is satisfied:

$$\begin{array}{cc}\hfill & \left|\frac{1}{8}\left[\mathcal{H}\left(\theta \right)+3\mathcal{H}\left(\frac{2\theta +\rho }{3}\right)+3\mathcal{H}\left(\frac{\theta +2\rho }{3}\right)+\mathcal{H}\left(\rho \right)\right]\mathcal{B}\left(\vartheta \right)-\frac{r^{\vartheta }\Gamma \left(\vartheta \right)}{2{(\rho -\theta )}^{\vartheta }}\left[{}_{\mathit{PC}}\mathcal{J}_{\rho -}^{\vartheta ,r}\left(\mathcal{H}b\right)\left(\theta \right)+{}_{\mathit{PC}}\mathcal{J}_{\theta +}^{\vartheta ,r}\left(\mathcal{H}b\right)\left(\rho \right)\right]\right|\hfill \\ \hfill & \le \frac{L{(\rho -\theta )}^2}{2}[{\mathrm{{\rm Y}}}_1\left(\vartheta \right)-2{\mathrm{{\rm Y}}}_4\left(\vartheta \right)+{\mathcal{M}}_1\left(\vartheta \right)+2{\mathrm{{\rm Y}}}_6\left(\vartheta \right)-{\mathrm{{\rm Y}}}_3\left(\vartheta \right)],\hfill \end{array}$$

where

$${\mathcal{M}}_1\left(\vartheta \right)={\int }_{\frac{1}{3}}^{\frac{2}{3}}\left|{\varphi }_2\left(\vartheta ,\zeta \right)\right|\left|1-2\zeta \right|d\zeta .$$

Proof. 

By Lemma 1, we have the identity

$$\begin{array}{ccc}\hfill |& \frac{1}{8}& \left[\mathcal{H}\left(\theta \right)+3\mathcal{H}\left(\frac{2\theta +\rho }{3}\right)+3\mathcal{H}\left(\frac{\theta +2\rho }{3}\right)+\mathcal{H}\left(\rho \right)\right]\mathcal{B}\left(\vartheta \right)-\frac{r^{\vartheta }\Gamma \left(\vartheta \right)}{2{(\rho -\theta )}^{\vartheta }}\left[{}_{\mathit{PC}}\mathcal{J}_{\rho -}^{\vartheta ,r}\left(\mathcal{H}b\right)\left(\theta \right)+{}_{\mathit{PC}}\mathcal{J}_{\theta +}^{\vartheta ,r}\left(\mathcal{H}b\right)\left(\rho \right)\right]|\hfill \\ & & \le \frac{\rho -\theta }{2}\left|{\mathcal{A}}_1+{\mathcal{A}}_2+{\mathcal{A}}_3\right|,\hfill \end{array}$$

(9)

Since ${\mathcal{H}}^{\prime }$ is L-Lipschitz, we have

$$\left|{\mathcal{H}}^{\prime }\left(x\right)-{\mathcal{H}}^{\prime }\left(y\right)\right|\le L\left|x-y\right|\mathrm{for}\mathrm{all}x,y\in [\theta ,\rho ].$$

For $x=u\rho +(1-u)\theta$ and $y=u\theta +(1-u)\rho$, we observe

$$x-y=(u\rho +(1-u\left)\theta \right)-(u\theta +(1-u\left)\rho \right)=(2u-1)(\rho -\theta ).$$

Hence,

$$\left|{\mathcal{H}}^{\prime }\left(x\right)-{\mathcal{H}}^{\prime }\left(y\right)\right|\le L\left|2u-1\right|\left|\rho -\theta \right|$$

We then bound each integral ${\mathcal{A}}_i$:

$$\begin{array}{ccc}\hfill \left|{\mathcal{A}}_1\right|& \le & {\int }_0^{\frac{1}{3}}\left|{\varphi }_1\left(\vartheta ,u\right)\right|\left|{\mathcal{H}}^{\prime }\left(x\right)-{\mathcal{H}}^{\prime }\left(y\right)\right|du\le L\left|\rho -\theta \right|{\int }_0^{\frac{1}{3}}\left|{\varphi }_1\left(\vartheta ,u\right)\right|(1-2u)du\hfill \\ & & \le L\left|\rho -\theta \right|({\mathrm{{\rm Y}}}_1-2{\mathrm{{\rm Y}}}_4).\hfill \end{array}$$

(10)

Similarly,

$$\begin{array}{ccc}\hfill \left|{\mathcal{A}}_2\right|& \le & {\int }_{\frac{1}{3}}^{\frac{2}{3}}\left|{\varphi }_2\left(\vartheta ,u\right)\right|\left|{\mathcal{H}}^{\prime }\left(x\right)-{\mathcal{H}}^{\prime }\left(y\right)\right|du\le L\left|\rho -\theta \right|{\int }_{\frac{1}{3}}^{\frac{2}{3}}\left|{\varphi }_2\left(\vartheta ,u\right)\right|(1-2u)du\hfill \\ & & \le L{\mathcal{M}}_1\left|\rho -\theta \right|,\hfill \end{array}$$

(11)

$$\begin{array}{ccc}\hfill \left|{\mathcal{A}}_1\right|& \le & {\int }_{\frac{2}{3}}^1\left|{\varphi }_3\left(\vartheta ,u\right)\right|\left|{\mathcal{H}}^{\prime }\left(x\right)-{\mathcal{H}}^{\prime }\left(y\right)\right|du\le L\left|\rho -\theta \right|{\int }_{\frac{2}{3}}^1\left|{\varphi }_3\left(\vartheta ,u\right)\right|(2u-1)du\hfill \\ & & \le L\left|\rho -\theta \right|(2{\mathrm{{\rm Y}}}_6-{\mathrm{{\rm Y}}}_3).\hfill \end{array}$$

(12)

Substituting in (9) from (10)–(12), we obtain

$$\begin{array}{cc}\hfill & \left|\frac{1}{8}\left[\mathcal{H}\left(\theta \right)+3\mathcal{H}\left(\frac{2\theta +\rho }{3}\right)+3\mathcal{H}\left(\frac{\theta +2\rho }{3}\right)+\mathcal{H}\left(\rho \right)\right]\mathcal{B}\left(\vartheta \right)-\frac{r^{\vartheta }\Gamma \left(\vartheta \right)}{2{(\rho -\theta )}^{\vartheta }}\left[{}_{\mathit{PC}}\mathcal{J}_{\rho -}^{\vartheta ,r}\left(\mathcal{H}b\right)\left(\theta \right)+{}_{\mathit{PC}}\mathcal{J}_{\theta +}^{\vartheta ,r}\left(\mathcal{H}b\right)\left(\rho \right)\right]\right|\hfill \\ \hfill & \le \frac{L{(\rho -\theta )}^2}{2}[{\mathrm{{\rm Y}}}_1\left(\vartheta \right)-2{\mathrm{{\rm Y}}}_4\left(\vartheta \right)+{\mathcal{M}}_1\left(\vartheta \right)+2{\mathrm{{\rm Y}}}_6\left(\vartheta \right)-{\mathrm{{\rm Y}}}_3\left(\vartheta \right)].\hfill \end{array}$$

This completes the proof. □

4. Conclusions

In this work, we developed a unified approach to $(\mathcal{WN}-\mathcal{TI})$ for multiple classes of functions, including differentiable convex functions, functions with bounded derivatives and Lipschitz functions, by exploiting generalized proportional fractional ($\mathbb{GPF}$) integrals. The key ingredient was a core integral identity, formulated via a positive weighted function, which enabled the derivation of diverse fractional weighted inequalities. This method highlights how judicious weighting, coupled with generalized fractional operators, can provide deeper insights into the convexity, boundedness, and Lipschitz-type properties of the underlying functions. An interesting special case emerges when $r=1$. In that scenario, the $\mathbb{GPF}$ operators reduce to well-known classical fractional integrals (see [20]), simplifying the resulting inequalities to more familiar forms of fractional Newton-type inequalities given in [32].