More General Weighted-Type Fractional Integral Inequalities via Chebyshev Functionals

Abstract

The purpose of this research paper is first to propose the generalized weighted-type fractional integrals. Then, we investigate some novel inequalities for a class of differentiable functions related to Chebyshev’s functionals by utilizing the proposed modified weighted-type fractional integral incorporating another function in the kernel $\mathcal{F}\left(\theta \right)$. For the weighted and extended Chebyshev’s functionals, we also propose weighted fractional integral inequalities. With specific choices of $\varpi \left(\theta \right)$ and $\mathcal{F}\left(\theta \right)$ as stated in the literature, one may easily study certain new inequalities involving all other types of weighted fractional integrals related to Chebyshev’s functionals. Furthermore, the inequalities for all other type of fractional integrals associated with Chebyshev’s functionals with certain choices of $\varpi \left(\theta \right)$ and $\mathcal{F}\left(\theta \right)$ are covered from the obtained generalized weighted-type fractional integral inequalities.

1. Introduction

In [1], for two integrable functions ${\mathcal{Z}}_1$ and ${\mathcal{Z}}_2$ on $[v_1,v_2]$, the Chebyshev functional and the weighted Chebyshev functional are respectively proposed as:

$$\begin{array}{c}\hfill T({\mathcal{Z}}_1,{\mathcal{Z}}_2)=\frac{1}{v_1-v_2}{\int }_{v_1}^{v_2}{\mathcal{Z}}_1\left(\varrho \right){\mathcal{Z}}_2\left(\varrho \right)d\varrho -\frac{1}{v_1-v_2}\left({\int }_{v_1}^{v_2}{\mathcal{Z}}_1\left(\varrho \right)d\varrho \right)\frac{1}{v_1-v_2}\left({\int }_{v_1}^{v_2}{\mathcal{Z}}_2\left(\varrho \right)d\varrho \right),\end{array}$$

(1)

and:

$$\begin{array}{c}\hfill T({\mathcal{Z}}_1,{\mathcal{Z}}_2,{\hslash }_1)={\int }_{v_1}^{v_2}{\hslash }_1\left(\varrho \right)d\varrho {\int }_{v_1}^{v_2}{\hslash }_1\left(\varrho \right){\mathcal{Z}}_1\left(\varrho \right){\mathcal{Z}}_2\left(\varrho \right)d\varrho -{\int }_{v_1}^{v_2}{\hslash }_1\left(\varrho \right){\mathcal{Z}}_1\left(\varrho \right)d\varrho {\int }_{v_1}^{v_2}{\hslash }_1\left(\varrho \right){\mathcal{Z}}_2\left(\varrho \right)d\varrho ,\end{array}$$

(2)

where the function ${\hslash }_1$ is positive and integrable on $[v_1,v_2]$. In the study of probability and statistical problems, (2) has several applications. In addition, the functional (2) has applications in the domain of integral and differential equations. Readers may refer to [2,3,4].

For two differentiable functions ${\mathcal{Z}}_1$ and ${\mathcal{Z}}_2$, Dragomir [5] defined the inequality below as:

$$\begin{array}{c}\hfill \mid T({\mathcal{Z}}_1,{\mathcal{Z}}_2,{\hslash }_1)\mid \le \Vert {\mathcal{Z}}_1^{\prime }\Vert \Vert {\mathcal{Z}}_2^{\prime }\Vert \left[{\int }_{v_1}^{v_2}{\hslash }_1\left(\varrho \right)d\varrho {\int }_{v_1}^{v_2}{\varrho }^2{\hslash }_1\left(\varrho \right)d\varrho -{\left({\int }_{v_1}^{v_2}\varrho {\hslash }_1\left(\varrho \right)d\varrho \right)}^2\right],\end{array}$$

where ${\mathcal{Z}}_1^{\prime },{\mathcal{Z}}_2^{\prime }\in L_{\infty }(v_1,v_2)$ and ${\hslash }_1$ is integrable and a positive function on $[v_1,v_2]$. Using various methodologies, the researchers investigated the functionals (1) and (2) and discovered some notable inequalities. Readers are advised to see the works of [6,7,8,9,10,11,12]. Very recently, Srivastava et al. [13] investigated the Chebyshev inequality via the general family of fractional integral operators.

Elezovic et al. [14] proposed the inequality below for the weighted Chebyshev functional:

$$\begin{array}{cc}\hfill \mid T({\mathcal{Z}}_1,{\mathcal{Z}}_2,{\hslash }_1)\mid & \le \frac{1}{2}{\left({\int }_{v_1}^{v_2}{\int }_{v_1}^{v_2}{\hslash }_1\left(\xi \right){\hslash }_1\left(\zeta \right)\mid \xi -\zeta {\mid }^{\frac{1}{p^{\prime }}+\frac{1}{q^{\prime }}}\mid {\int }_{\zeta }^{\xi }\mid {\mathcal{Z}}_1^{\prime }\left(\varrho \right){\mid }^{p}d\varrho \mid \frac{r}{p}d\xi d\zeta \right)}^{\frac{1}{r}}\hfill \\ \hfill \times & {\left({\int }_{v_1}^{v_2}{\int }_{v_1}^{v_2}{\hslash }_1\left(\xi \right){\hslash }_1\left(\zeta \right)\mid \xi -\zeta {\mid }^{\frac{1}{p^{\prime }}+\frac{1}{q^{\prime }}}\mid {\int }_{\zeta }^{\xi }\mid {\mathcal{Z}}_2^{\prime }\left(\varrho \right){\mid }^{q}d\varrho {\mid }^{\frac{r}{q}}d\xi d\zeta \right)}^{\frac{1}{r^{\prime }}}\hfill \\ \hfill \le & \frac{1}{2}\Vert {\mathcal{Z}}_1^{\prime }\Vert \Vert {\mathcal{Z}}_2^{\prime }\Vert \left({\int }_{v_1}^{v_2}{\int }_{v_1}^{v_2}{\hslash }_1\left(\xi \right){\hslash }_1\left(\zeta \right)\mid \xi -\zeta {\mid }^{\frac{1}{p^{\prime }}+\frac{1}{q^{\prime }}}d\xi d\zeta \right),\hfill \end{array}$$

(3)

where ${\mathcal{Z}}_1^{\prime }\in L^p\left([v_1,v_2]\right)$, ${\mathcal{Z}}_2^{\prime }\in L^q\left([v_1,v_2]\right)$, $p,q,r>1$, $\frac{1}{p}+\frac{1}{p^{\prime }}=1$, $\frac{1}{q}+\frac{1}{q^{\prime }}=1$, and $\frac{1}{r}+\frac{1}{r^{\prime }}=1$.

In [9], the authors established the following fractional integral inequality for the Chebyshev functional (2) by:

$$\begin{array}{cc}\hfill & 2\mid {\mathcal{I}}^{\alpha }{\hslash }_1\left(\tau \right){\mathcal{I}}^{\alpha }{\hslash }_1{\mathcal{Z}}_1{\mathcal{Z}}_2\left(\theta \right)-{\mathcal{I}}^{\alpha }{\hslash }_1{\mathcal{Z}}_1\left(\theta \right){\mathcal{I}}^{\alpha }{\hslash }_1{\mathcal{Z}}_2\left(\theta \right)\mid \hfill \\ \hfill \le & \frac{\Vert {\mathcal{Z}}_1^{\prime }{\Vert }_p{\Vert {\mathcal{Z}}_2^{\prime }\Vert }_q}{{\mathsf{\Gamma }}^2\left(\alpha \right)}{\int }_0^{\theta }{\int }_0^{\theta }{(\theta -\varrho )}^{\alpha -1}{(\theta -\zeta )}^{\alpha -1}|\varrho -\zeta |{\hslash }_1\left(\varrho \right){\hslash }_1\left(\zeta \right)d\varrho d\zeta ,\hfill \end{array}$$

where ${\mathcal{Z}}_1^{\prime }\in L^p\left(\right[0,\infty \left[\right)$, ${\mathcal{Z}}_2^{\prime }\in L^q\left(\right[0,\infty \left[\right)$, $p,q>1$, $\frac{1}{p}+\frac{1}{q}=1$.

In [15,16], the extended Chebyshev functional was presented as:

$$\begin{array}{cc}\hfill \tilde{T}({\mathcal{Z}}_1,{\mathcal{Z}}_2,{\hslash }_1,{\hslash }_1^{\prime })& ={\int }_{v_1}^{v_2}{\hslash }_1^{\prime }\left(\varrho \right)d\varrho {\int }_{v_1}^{v_2}{\hslash }_1\left(\varrho \right){\mathcal{Z}}_1\left(\varrho \right){\mathcal{Z}}_2\left(\varrho \right)d\varrho +{\int }_{v_1}^{v_2}{\hslash }_1\left(\varrho \right)d\varrho {\int }_{v_1}^{v_2}{\hslash }_1^{\prime }\left(\varrho \right){\mathcal{Z}}_1\left(\varrho \right){\mathcal{Z}}_2\left(\varrho \right)d\varrho \hfill \\ \hfill -& {\int }_{v_1}^{v_2}{\hslash }_1\left(\varrho \right){\mathcal{Z}}_1\left(\varrho \right)d\varrho {\int }_{v_1}^{v_2}{\hslash }_1^{\prime }\left(\varrho \right){\mathcal{Z}}_2\left(\varrho \right)d\varrho -{\int }_{v_1}^{v_2}{\hslash }_1^{\prime }\left(\varrho \right){\mathcal{Z}}_1\left(\varrho \right)d\varrho {\int }_{v_1}^{v_2}{\hslash }_1\left(\varrho \right){\mathcal{Z}}_2\left(\varrho \right)d\varrho .\hfill \end{array}$$

(4)

This paper is organized as follows:

The generalized weighted-type fractional integral inequalities connected to the functionals (1) and (2) are discussed in Section 2. We propose some generalized weighted-type fractional integral inequalities connected to (3) and (4) in Section 3. Finally, in Section 4, we give the concluding remarks.

We recall the following results from [17] as follows:

Definition 1.

Suppose that the function $\mathsf{\Psi }:[0,\infty )\to [0,\infty )$ satisfies the conditions given below:

$$\begin{array}{c}\hfill {\int }_0^1\frac{\mathsf{\Psi }\left(\varrho \right)}{\varrho }d\varrho ,\end{array}$$

(5)

$$\begin{array}{c}\hfill \frac{1}{P}\le \frac{\mathsf{\Psi }\left(\mu \right)}{\mathsf{\Psi }\left(\nu \right)}\le P,\frac{1}{2}\le \frac{\mu }{\nu }\le 2,\end{array}$$

(6)

$$\begin{array}{c}\hfill \frac{\mathsf{\Psi }\left(\nu \right)}{{\nu }^2}\le Q\frac{\mathsf{\Psi }\left(\mu \right)}{{\mu }^2},\mu \le \nu ,\end{array}$$

(7)

$$\begin{array}{c}\hfill |\frac{\mathsf{\Psi }\left(\nu \right)}{{\nu }^2}-\frac{\mathsf{\Psi }\left(\mu \right)}{{\mu }^2}|\le S|\nu -\mu |\frac{\mathsf{\Psi }\left(\nu \right)}{{\nu }^2},\frac{1}{2}\le \frac{\mu }{\nu }\le 2,\end{array}$$

(8)

where P, Q, $S>0$, independent of $\mu ,\nu >0$. If $\mathsf{\Psi }\left(\nu \right){\nu }^{\alpha }$ is increasing for some $\alpha >0$ and $\frac{\mathsf{\Psi }\left(\nu \right)}{{\nu }^{\beta }}$ is decreasing for some $\beta >0$, then Ψ satisfies (5)–(8).

Here, we define the following generalized weighted-type fractional integral operators.

Definition 2.

The generalized weighted-type fractional integral operators, both left and right sided, are respectively defined by:

$$\begin{array}{c}\hfill \left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\mathcal{Z}}_1\right)\left(\theta \right)={\varpi }^{-1}\left(\theta \right){\int }_{v_1}^{\theta }\frac{\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right){\mathcal{Z}}_1\left(\varrho \right)d\varrho ,v_1<\theta ,\end{array}$$

(9)

and:

$$\begin{array}{c}\hfill \left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_2-}^{\mathsf{\Psi }}{\mathcal{Z}}_1\right)\left(\theta \right)={\varpi }^{-1}\left(\theta \right){\int }_{\theta }^{v_2}\frac{\mathsf{\Psi }\left(\mathcal{F}\left(\varrho \right)-\mathcal{F}\left(\theta \right)\right)}{\mathcal{F}\left(\varrho \right)-\mathcal{F}\left(\theta \right)}\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right){\mathcal{Z}}_1\left(\varrho \right)d\varrho ,v_2>\theta .\end{array}$$

(10)

Remark 1.

1. If we consider $\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)\right)=\mathcal{F}\left(\theta \right)$, the fractional integrals (9) and (10) reduce to the following:

$$\begin{array}{c}\hfill \left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}{\mathcal{Z}}_1\right)\left(\theta \right)={\varpi }^{-1}\left(\theta \right){\int }_{v_1}^{\theta }\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right){\mathcal{Z}}_1\left(\varrho \right)d\varrho ,v_1<\theta ,\end{array}$$

and:

$$\begin{array}{c}\hfill \left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_2-}{\mathcal{Z}}_1\right)\left(\theta \right)={\varpi }^{-1}\left(\theta \right){\int }_{\theta }^{v_2}\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right){\mathcal{Z}}_1\left(\varrho \right)d\varrho ,v_2>\theta ,\end{array}$$

respectively.

2. If we consider $\mathcal{F}\left(\theta \right)=\theta$, the fractional integrals (9) and (10) reduce to the following, respectively:

$$\begin{array}{c}\hfill \left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}{\mathcal{Z}}_1\right)\left(\theta \right)={\varpi }^{-1}\left(\theta \right){\int }_{v_1}^{\theta }\frac{\mathsf{\Psi }\left(\theta -\varrho \right)}{\theta -\varrho }\varpi \left(\varrho \right){\mathcal{Z}}_1\left(\varrho \right)d\varrho ,v_1<\theta ,\end{array}$$

and:

$$\begin{array}{c}\hfill \left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_2-}{\mathcal{Z}}_1\right)\left(\theta \right)={\varpi }^{-1}\left(\theta \right){\int }_{\theta }^{v_2}\frac{\mathsf{\Psi }\left(\varrho -\theta \right)}{\varrho -\theta }\varpi \left(\varrho \right){\mathcal{Z}}_1\left(\varrho \right)d\varrho ,v_2>\theta .\end{array}$$

3. If we consider $\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)\right)=\frac{\mathcal{F}{\left(\theta \right)}^{\kappa }}{\mathsf{\Gamma }\left(\kappa \right)}$, the fractional integrals (9) and (10) reduce to the following, respectively (see [18]):

$$\begin{array}{c}\hfill \left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1}^{\kappa }{\mathcal{Z}}_1\right)\left(\theta \right)=\frac{{\varpi }^{-1}\left(\theta \right)}{\mathsf{\Gamma }\left(\kappa \right)}{\int }_{v_1}^{\theta }{\left(\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)\right)}^{\kappa -1}\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right){\mathcal{Z}}_1\left(\varrho \right)d\varrho ,v_1<\theta ,\end{array}$$

and:

$$\begin{array}{c}\hfill \left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_2-}^{\kappa }{\mathcal{Z}}_1\right)\left(\theta \right)=\frac{{\varpi }^{-1}\left(\theta \right)}{\mathsf{\Gamma }\left(\kappa \right)}{\int }_{\theta }^{v_2}{\left(\mathcal{F}\left(\varrho \right)-\mathcal{F}\left(\theta \right)\right)}^{\kappa -1}\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right){\mathcal{Z}}_1\left(\varrho \right)d\varrho ,v_2>\theta ,\end{array}$$

where $\kappa ,\in \mathbb{C}$ with $\Re \left(\kappa \right)>0$.

4. If we consider $\mathcal{F}\left(\theta \right)=\theta$ and $\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)\right)=\frac{{\theta }^{\kappa }}{\mathsf{\Gamma }\left(\kappa \right)}$, the fractional integrals (9) and (10) reduce to the following:

$$\begin{array}{c}\hfill \left({}_{\varpi }{\mathfrak{I}}_{v_1+}^{\kappa }{\mathcal{Z}}_1\right)\left(\theta \right)=\frac{{\varpi }^{-1}\left(\theta \right)}{\mathsf{\Gamma }\left(\kappa \right)}{\int }_{v_1}^{\theta }{(\theta -\varrho )}^{\kappa -1}\varpi \left(\varrho \right){\mathcal{Z}}_1\left(\varrho \right)d\varrho ,v_1<\theta ,\end{array}$$

and:

$$\begin{array}{c}\hfill \left({}_{\varpi }{\mathfrak{I}}_{v_2-}^{\kappa }{\mathcal{Z}}_1\right)\left(\theta \right)=\frac{{\varpi }^{-1}\left(\theta \right)}{\mathsf{\Gamma }\left(\kappa \right)}{\int }_{\theta }^{v_2}{(\varrho -\theta )}^{\kappa -1}\varpi \left(\varrho \right){\mathcal{Z}}_1\left(\varrho \right)d\varrho ,v_2>\theta ,\end{array}$$

respectively.

5. If we consider $\mathcal{F}\left(\theta \right)=ln\theta$ and $\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)\right)=\frac{{(ln\theta )}^{\kappa }}{\mathsf{\Gamma }\left(\kappa \right)}$, the fractional integrals (9) and (10) reduce to the following weighted Hadamard fractional integrals:

$$\begin{array}{c}\hfill \left({}_{\varpi }{\mathcal{I}}_{v_1+}^{\kappa }{\mathcal{Z}}_1\right)\left(\theta \right)=\frac{{\varpi }^{-1}\left(\theta \right)}{\mathsf{\Gamma }\left(\kappa \right)}{\int }_{v_1}^{\theta }{(ln\theta -ln\varrho )}^{\kappa -1}\varpi \left(\varrho \right){\mathcal{Z}}_1\left(\varrho \right)\frac{d\varrho }{\varrho },v_1<\theta ,\end{array}$$

and:

$$\begin{array}{c}\hfill \left({}_{\varpi }{\mathcal{I}}_{v_2-}^{\kappa }{\mathcal{Z}}_1\right)\left(\theta \right)=\frac{{\varpi }^{-1}\left(\theta \right)}{\mathsf{\Gamma }\left(\kappa \right)}{\int }_{\theta }^{v_2}{(ln\varrho -ln\theta )}^{\kappa -1}\varpi \left(\varrho \right){\mathcal{Z}}_1\left(\varrho \right)\frac{d\varrho }{\varrho },v_2>\theta .\end{array}$$

6. If we consider $\mathcal{F}\left(\theta \right)={\theta }^{\eta }$ and $\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)\right)=\frac{{\theta }^{\eta }}{\eta }$, $\eta >0$, the fractional integrals (9) and (10) reduce to the following weighted Katugampola fractional integrals,

$$\begin{array}{c}\hfill \left({}_{\varpi }{\mathcal{I}}_{v_1}^{\kappa }{\mathcal{Z}}_1\right)\left(\theta \right)=\frac{{\varpi }^{-1}\left(\theta \right)}{\mathsf{\Gamma }\left(\kappa \right)}{\int }_{v_1}^{\theta }{\left(\frac{{\theta }^{\eta }-{\varrho }^{\eta }}{\eta }\right)}^{\kappa -1}\varpi \left(\varrho \right){\mathcal{Z}}_1\left(\varrho \right)\frac{d\varrho }{{\varrho }^{1-\eta }},v_1<\theta ,\end{array}$$

and:

$$\begin{array}{c}\hfill \left({}_{\varpi }{\mathcal{I}}_{v_2}^{\kappa }{\mathcal{Z}}_1\right)\left(\theta \right)=\frac{{\varpi }^{-1}\left(\theta \right)}{\mathsf{\Gamma }\left(\kappa \right)}{\int }_{\theta }^{v_2}{\left(\frac{{\varrho }^{\eta }-{\theta }^{\eta }}{\eta }\right)}^{\kappa -1}\varpi \left(\varrho \right){\mathcal{Z}}_1\left(\varrho \right)\frac{d\varrho }{{\varrho }^{1-\eta }},v_2>\theta .\end{array}$$

7. If we consider $\mathcal{F}\left(\theta \right)=\theta$ and $\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)\right)=\frac{\theta }{\eta }exp\left(-\frac{1-\eta }{\eta }\theta \right)$, $\eta \in (0,1)$, the fractional integrals (9) and (10) reduce to the following weighted fractional integrals,

$$\begin{array}{c}\hfill \left({}_{\varpi }{\mathcal{I}}_{v_1+}^{\eta }{\mathcal{Z}}_1\right)\left(\theta \right)=\frac{{\varpi }^{-1}\left(\theta \right)}{\eta }{\int }_{v_1}^{\theta }exp\left(-\frac{1-\eta }{\eta }(\theta -\varrho )\right)\varpi \left(\varrho \right){\mathcal{Z}}_1\left(\varrho \right),v_1<\theta ,\end{array}$$

and:

$$\begin{array}{c}\hfill \left({}_{\varpi }{\mathcal{I}}_{v_2-}^{\eta }{\mathcal{Z}}_1\right)\left(\theta \right)=\frac{{\varpi }^{-1}\left(\theta \right)}{\eta }{\int }_{\theta }^{v_2}exp\left(-\frac{1-\eta }{\eta }(\varrho -\theta )\right)\varpi \left(\varrho \right){\mathcal{Z}}_1\left(\varrho \right)d\varrho ,v_2>\theta .\end{array}$$

Furthermore, one can derive the weighted form of conformable fractional integrals introduced by [19,20,21,22].

The following special cases can be easily obtained by applying the conditions on $\varpi \left(\theta \right)$ and $\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)\right)$.

Remark 2.

1. If we consider $\varpi \left(\theta \right)=1$ and $\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)\right)=\mathcal{F}\left(\theta \right)$, the fractional integrals (9) and (10) reduce to the following:

$$\begin{array}{c}\hfill \left({}^{\mathcal{F}}{\mathcal{I}}_{v_1+}{\mathcal{Z}}_1\right)\left(\theta \right)={\int }_{v_1}^{\theta }{\mathcal{F}}^{\prime }\left(\varrho \right){\mathcal{Z}}_1\left(\varrho \right)d\varrho ,v_1<\theta ,\end{array}$$

and:

$$\begin{array}{c}\hfill \left({}^{\mathcal{F}}{\mathcal{I}}_{v_2-}{\mathcal{Z}}_1\right)\left(\theta \right)={\int }_{\theta }^{v_2}{\mathcal{F}}^{\prime }\left(\varrho \right){\mathcal{Z}}_1\left(\varrho \right)d\varrho ,v_2>\theta ,\end{array}$$

respectively.

2. If we consider $\varpi \left(\theta \right)=1$ and $\mathcal{F}\left(\theta \right)=\theta$, the fractional integrals (9) and (10) reduce to the following, respectively (see [23]):

$$\begin{array}{c}\hfill \left({}^{\mathcal{F}}{\mathcal{I}}_{v_1+}{\mathcal{Z}}_1\right)\left(\theta \right)={\int }_{v_1}^{\theta }\frac{\mathsf{\Psi }\left(\theta -\varrho \right)}{\theta -\varrho }{\mathcal{Z}}_1\left(\varrho \right)d\varrho ,v_1<\theta ,\end{array}$$

and:

$$\begin{array}{c}\hfill \left({}^{\mathcal{F}}{\mathcal{I}}_{v_2-}{\mathcal{Z}}_1\right)\left(\theta \right)={\int }_{\theta }^{v_2}\frac{\mathsf{\Psi }\left(\varrho -\theta \right)}{\varrho -\theta }{\mathcal{Z}}_1\left(\varrho \right)d\varrho ,v_2>\theta .\end{array}$$

3. If we consider $\varpi \left(\theta \right)=1$ and $\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)\right)=\frac{\mathcal{F}{\left(\theta \right)}^{\kappa }}{\mathsf{\Gamma }\left(\kappa \right)}$, the fractional integrals (9) and (10) reduce to the following, respectively (see [24,25]):

$$\begin{array}{c}\hfill \left({}^{\mathcal{F}}{\mathcal{I}}_{v_1}^{\kappa }{\mathcal{Z}}_1\right)\left(\theta \right)=\frac{1}{\mathsf{\Gamma }\left(\kappa \right)}{\int }_{v_1}^{\theta }{\left(\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)\right)}^{\kappa -1}{\mathcal{F}}^{\prime }\left(\varrho \right){\mathcal{Z}}_1\left(\varrho \right)d\varrho ,v_1<\theta ,\end{array}$$

and:

$$\begin{array}{c}\hfill \left({}^{\mathcal{F}}{\mathcal{I}}_{v_2-}^{\kappa }{\mathcal{Z}}_1\right)\left(\theta \right)=\frac{1}{\mathsf{\Gamma }\left(\kappa \right)}{\int }_{\theta }^{v_2}{\left(\mathcal{F}\left(\varrho \right)-\mathcal{F}\left(\theta \right)\right)}^{\kappa -1}{\mathcal{F}}^{\prime }\left(\varrho \right){\mathcal{Z}}_1\left(\varrho \right)d\varrho ,v_2>\theta ,\end{array}$$

where $\kappa ,\in \mathbb{C}$ with $\Re \left(\kappa \right)>0$.

4. If we consider $\varpi \left(\theta \right)=1$, $\mathcal{F}\left(\theta \right)=\theta$ and $\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)\right)=\frac{{\theta }^{\kappa }}{\mathsf{\Gamma }\left(\kappa \right)}$, the fractional integrals (9) and (10) reduce to the following (see [24,25]):

$$\begin{array}{c}\hfill \left({\mathfrak{I}}_{v_1+}^{\kappa }{\mathcal{Z}}_1\right)\left(\theta \right)=\frac{1}{\mathsf{\Gamma }\left(\kappa \right)}{\int }_{v_1}^{\theta }{(\theta -\varrho )}^{\kappa -1}{\mathcal{Z}}_1\left(\varrho \right)d\varrho ,v_1<\theta ,\end{array}$$

and:

$$\begin{array}{c}\hfill \left({\mathfrak{I}}_{v_2-}^{\kappa }{\mathcal{Z}}_1\right)\left(\theta \right)=\frac{1}{\mathsf{\Gamma }\left(\kappa \right)}{\int }_{\theta }^{v_2}{(\varrho -\theta )}^{\kappa -1}{\mathcal{Z}}_1\left(\varrho \right)d\varrho ,v_2>\theta ,\end{array}$$

respectively.

5. If we consider $\varpi \left(\theta \right)=1$, $\mathcal{F}\left(\theta \right)=ln\theta$ and $\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)\right)=\frac{{(ln\theta )}^{\kappa }}{\mathsf{\Gamma }\left(\kappa \right)}$, the fractional integrals (9) and (10) reduce to the following weighted Hadamard fractional integrals (see [24,25]):

$$\begin{array}{c}\hfill \left({\mathcal{I}}_{v_1+}^{\kappa }{\mathcal{Z}}_1\right)\left(\theta \right)=\frac{1}{\mathsf{\Gamma }\left(\kappa \right)}{\int }_{v_1}^{\theta }{(ln\theta -ln\varrho )}^{\kappa -1}{\mathcal{Z}}_1\left(\varrho \right)\frac{d\varrho }{\varrho },v_1<\theta ,\end{array}$$

and:

$$\begin{array}{c}\hfill \left({\mathcal{I}}_{v_2-}^{\kappa }{\mathcal{Z}}_1\right)\left(\theta \right)=\frac{1}{\mathsf{\Gamma }\left(\kappa \right)}{\int }_{\theta }^{v_2}{(ln\varrho -ln\theta )}^{\kappa -1}{\mathcal{Z}}_1\left(\varrho \right)\frac{d\varrho }{\varrho },v_2>\theta .\end{array}$$

6. If we consider $\varpi \left(\theta \right)=1$, $\mathcal{F}\left(\theta \right)={\theta }^{\eta }$, and $\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)\right)=\frac{{\theta }^{\eta }}{\eta }$, $\eta >0$, the fractional integrals (9) and (10) reduce to the following Katugampola [26] fractional integrals, respectively,

$$\begin{array}{c}\hfill \left({\mathcal{I}}_{v_1}^{\kappa }{\mathcal{Z}}_1\right)\left(\theta \right)=\frac{1}{\mathsf{\Gamma }\left(\kappa \right)}{\int }_{v_1}^{\theta }{\left(\frac{{\theta }^{\eta }-{\varrho }^{\eta }}{\eta }\right)}^{\kappa -1}{\mathcal{Z}}_1\left(\varrho \right)\frac{d\varrho }{{\varrho }^{1-\eta }},v_1<\theta ,\end{array}$$

and:

$$\begin{array}{c}\hfill \left({\mathcal{I}}_{v_2}^{\kappa }{\mathcal{Z}}_1\right)\left(\theta \right)=\frac{1}{\mathsf{\Gamma }\left(\kappa \right)}{\int }_{\theta }^{v_2}{\left(\frac{{\varrho }^{\eta }-{\theta }^{\eta }}{\eta }\right)}^{\kappa -1}{\mathcal{Z}}_1\left(\varrho \right)\frac{d\varrho }{{\varrho }^{1-\eta }},v_2>\theta .\end{array}$$

7. If we consider $\varpi \left(\theta \right)=1$, $\mathcal{F}\left(\theta \right)=\theta$, and $\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)\right)=\frac{\theta }{\eta }exp\left(-\frac{1-\eta }{\eta }\theta \right)$, $\eta \in (0,1)$, the fractional integrals (9) and (10) reduce to the following weighted fractional integrals,

$$\begin{array}{c}\hfill \left({\mathcal{I}}_{v_1+}^{\eta }{\mathcal{Z}}_1\right)\left(\theta \right)=\frac{1}{\eta }{\int }_{v_1}^{\theta }exp\left(-\frac{1-\eta }{\eta }(\theta -\varrho )\right){\mathcal{Z}}_1\left(\varrho \right),v_1<\theta ,\end{array}$$

and:

$$\begin{array}{c}\hfill \left({\mathcal{I}}_{v_2-}^{\eta }{\mathcal{Z}}_1\right)\left(\theta \right)=\frac{1}{\eta }{\int }_{\theta }^{v_2}exp\left(-\frac{1-\eta }{\eta }(\varrho -\theta )\right){\mathcal{Z}}_1\left(\varrho \right)d\varrho ,v_2>\theta .\end{array}$$

Similarly, (9) and (10) will lead to the fractional integrals defined by [19,20,21,22].

2. Generalized Weighted-Type Fractional Integral Inequalities via Chebyshev’s Functional

Here, we develop weighted-type generalized fractional integral inequalities via Chebyshev’s functional.

Theorem 1.

If the two functions ${\mathcal{Z}}_1$ and ${\mathcal{Z}}_2$ are differentiable on $[0,\infty )$ with ${\mathcal{Z}}_1^{\prime },{\mathcal{Z}}_2^{\prime }\in {\mathcal{L}}_{\infty }\left(\right[0,\infty \left[\right)$ and we suppose $\mathcal{F}$ is positive and increasing on $[0,\infty [$ and its derivative is continuous on $[0,\infty [$, then the following inequality holds:

$$\begin{array}{cc}\hfill & \mid \left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}1\right)\left(\theta \right)\left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\mathcal{Z}}_1{\mathcal{Z}}_2\right)\left(\theta \right)-\left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\mathcal{Z}}_1\right)\left(\theta \right)\left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\mathcal{Z}}_2\left(\theta \right)\right)\mid \hfill \\ \hfill \le & \Vert {\mathcal{Z}}_1^{\prime }{\Vert }_{\infty }{\Vert {\mathcal{Z}}_2^{\prime }\Vert }_{\infty }\left[\left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}1\right)\left(\theta \right)\left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\theta }^2\right)-{\left(\left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\theta \right)\right)}^2\right],\hfill \end{array}$$

(11)

where $\left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}1\right)\left(\theta \right)$ is defined by:

$$\begin{array}{c}\hfill \left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}1\right)\left(\theta \right)={\varpi }^{-1}\left(\theta \right){\int }_{v_1}^{\theta }\frac{\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right)d\varrho ,v_1<\theta \end{array}$$

Proof. 

Let us define:

$$\begin{array}{c}\hfill H(\varrho ,\zeta )=({\mathcal{Z}}_1\left(\varrho \right)-{\mathcal{Z}}_1\left(\zeta \right))({\mathcal{Z}}_2\left(\varrho \right)-{\mathcal{Z}}_2\left(\zeta \right));\varrho ,\zeta \in \left(v_1\theta \right).\end{array}$$

(12)

The product of (12) by ${\displaystyle {\varpi }^{-1}\left(\theta \right)\frac{\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right)}$ and then integrating with respect to $\varrho$ over $(v_1,\theta )$ and employing (9), we have:

$$\begin{array}{cc}\hfill & {\varpi }^{-1}\left(\theta \right){\int }_{v_1}^{\theta }\frac{\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right)H(\varrho ,\zeta )d\varrho \hfill \\ \hfill =& \left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\mathcal{Z}}_1{\mathcal{Z}}_2\right)\left(\theta \right)-{\mathcal{Z}}_1\left(\zeta \right)\left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\mathcal{Z}}_2\right)\left(\theta \right)-{\mathcal{Z}}_2\left(\zeta \right)\left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\mathcal{Z}}_1\right)\left(\theta \right)+{\mathcal{Z}}_1\left(\zeta \right){\mathcal{Z}}_2\left(\zeta \right)\left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1}^{\mathsf{\Psi }}1\right)\left(\theta \right).\hfill \end{array}$$

(13)

Again, conducting the product (13) by ${\displaystyle {\varpi }^{-1}\left(\theta \right)\frac{\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\zeta \right)\right)}{\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\zeta \right)}\varpi \left(\zeta \right){\mathcal{F}}^{\prime }\left(\zeta \right)}$ and then integrating with respect to $\zeta$ over $(v_1,\theta )$, we have:

$$\begin{array}{cc}\hfill & {\varpi }^{-2}\left(\theta \right){\int }_{v_1}^{\theta }{\int }_{v_1}^{\theta }\frac{\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)}\frac{\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\zeta \right)\right)}{\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\zeta \right)}{\mathcal{F}}^{\prime }\left(\varrho \right)\varpi \left(\varrho \right)\varpi \left(\zeta \right){\mathcal{F}}^{\prime }\left(\zeta \right)H(\varrho ,\zeta )d\varrho d\zeta \hfill \\ \hfill =& 2\left(\left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}1\right)\left(\theta \right)\left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\mathcal{Z}}_1{\mathcal{Z}}_2\right)\left(\theta \right)-\left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\mathcal{Z}}_1\right)\left(\theta \right)\left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\mathcal{Z}}_2\right)\left(\theta \right)\right).\hfill \end{array}$$

(14)

On the other side, we also have:

$$\begin{array}{c}\hfill H(\varrho ,\zeta )={\int }_{\varrho }^{\zeta }{\int }_{\varrho }^{\zeta }{\mathcal{Z}}_1^{\prime }\left(x\right){\mathcal{Z}}_2^{\prime }\left(y\right)dxdy.\end{array}$$

(15)

Since ${\mathcal{Z}}_1^{\prime }\left(x\right),{\mathcal{Z}}_2^{\prime }\left(y\right)\in L_{\infty }\left(\right[0,\infty \left[\right)$, therefore we have:

$$\begin{array}{c}\hfill \mid H(\varrho ,\zeta )\mid \le \mid {\int }_{\varrho }^{\zeta }{\mathcal{Z}}_1^{\prime }\left(x\right)dx\mid \mid {\int }_{\varrho }^{\zeta }{\mathcal{Z}}_2^{\prime }\left(y\right)dy\mid \le \Vert {\mathcal{Z}}_1^{\prime }{\Vert }_{\infty }{\Vert {\mathcal{Z}}_2^{\prime }\Vert }_{\infty }{(\varrho -\zeta )}^2.\end{array}$$

(16)

Therefore, it can be written as:

$$\begin{array}{cc}\hfill & {\varpi }^{-2}\left(\theta \right){\int }_{v_1}^{\theta }{\int }_{v_1}^{\theta }\frac{\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)}\frac{\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\zeta \right)\right)}{\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\zeta \right)}{\mathcal{F}}^{\prime }\left(\varrho \right)\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\zeta \right)\varpi \left(\zeta \right)\mid H(\varrho ,\zeta )\mid d\varrho d\zeta \hfill \\ \hfill \le & \Vert {\mathcal{Z}}_1^{\prime }{\Vert }_{\infty }{\Vert {\mathcal{Z}}_2^{\prime }\Vert }_{\infty }{\varpi }^{-2}\left(\theta \right){\int }_{v_1}^{\theta }{\int }_{v_1}^{\theta }\frac{\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)}\frac{\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\zeta \right)\right)}{\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\zeta \right)}{\mathcal{F}}^{\prime }\left(\varrho \right)\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\zeta \right)\varpi \left(\zeta \right)\hfill \\ \hfill \times & ({\varrho }^2-2\varrho \zeta +{\zeta }^2)d\varrho d\zeta .\hfill \end{array}$$

(17)

From (17), we obtain:

$$\begin{array}{cc}\hfill & {\varpi }^{-2}\left(\theta \right){\int }_{v_1}^{\theta }{\int }_{v_1}^{\theta }\frac{\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)}\frac{\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\zeta \right)\right)}{\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\zeta \right)}{\mathcal{F}}^{\prime }\left(\varrho \right)\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\zeta \right)\varpi \left(\zeta \right)\mid H(\varrho ,\zeta )\mid d\varrho d\zeta \hfill \\ \hfill \le & 2\Vert {\mathcal{Z}}_1^{\prime }{\Vert }_{\infty }{\Vert {\mathcal{Z}}_2^{\prime }\Vert }_{\infty }\left[\left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1}^{\mathsf{\Psi }}1\right)\left(\theta \right)\left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\theta }^2\right)-{\left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\theta \right)}^2\right].\hfill \end{array}$$

(18)

Hence, from (14) and (18), we obtain the required proof. □

Corollary 1.

If the two functions ${\mathcal{Z}}_1$ and ${\mathcal{Z}}_2$ are differentiable on $[0,\infty )$ with ${\mathcal{Z}}_1^{\prime },{\mathcal{Z}}_2^{\prime }\in {\mathcal{L}}_{\infty }\left(\right[0,\infty \left[\right)$ and we let $\mathcal{F}$ be a positive and increasing function on $[0,\infty [$ and its derivative be continuous on $[0,\infty [$, then the following inequality holds:

$$\begin{array}{cc}\hfill & \mid \mathsf{\Pi }\left(1\right)\left({}^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\mathcal{Z}}_1{\mathcal{Z}}_2\right)\left(\theta \right)-\left({}^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\mathcal{Z}}_1\right)\left(\theta \right)\left({}^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\mathcal{Z}}_2\right)\left(\theta \right)\mid \hfill \\ \hfill \le & \Vert {\mathcal{Z}}_1^{\prime }{\Vert }_{\infty }{\Vert {\mathcal{Z}}_2^{\prime }\Vert }_{\infty }\left[\mathsf{\Pi }\left(1\right)\left({}^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\theta }^2\right)-{\left({}^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\theta \right)}^2\right],\hfill \end{array}$$

where $\mathsf{\Pi }\left(1\right)$ is defined by:

$$\begin{array}{cc}\hfill \mathsf{\Pi }\left(1\right)& ={\int }_{v_1}^{\theta }\frac{\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)}{\mathcal{F}}^{\prime }\left(\varrho \right)d\varrho .\hfill \end{array}$$

Theorem 2.

If the two functions ${\mathcal{Z}}_1$ and ${\mathcal{Z}}_2$ are differentiable, both have variations in same sense on $[0,\infty )$, and we let ${\hslash }_1$ be a positive function on $[0,\infty )$. Suppose that $\mathcal{F}$ is positive and increasing on $[0,\infty [$ and its derivative is continuous on $[0,\infty [$. Let ${\mathcal{Z}}_1^{\prime },{\mathcal{Z}}_2^{\prime }\in {\mathcal{L}}_{\infty }\left(\right[0,\infty \left[\right)$, then the following inequality holds:

$$\begin{array}{cc}\hfill & 0\le \left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1\right)\left(\theta \right)\left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1{\mathcal{Z}}_1{\mathcal{Z}}_2\right)\left(\theta \right)-\left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1{\mathcal{Z}}_1\right)\left(\theta \right)\left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1{\mathcal{Z}}_2\right)\left(\theta \right)\hfill \\ \hfill \le & \Vert {\mathcal{Z}}_1^{\prime }{\Vert }_{\infty }{\Vert {\mathcal{Z}}_2^{\prime }\Vert }_{\infty }\left[\left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1\right)\left(\theta \right)\left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\theta }^2{\hslash }_1\right)\left(\theta \right)-{\left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_0^{\mathsf{\Psi }}\theta {\hslash }_1\right)}^2\left(\theta \right)\right].\hfill \end{array}$$

(19)

Proof. 

Define:

$$\begin{array}{cc}\hfill H(\varrho ,\zeta )& =({\mathcal{Z}}_1\left(\varrho \right)-{\mathcal{Z}}_1\left(\zeta \right))({\mathcal{Z}}_2\left(\zeta \right)-{\mathcal{Z}}_2\left(\zeta \right));\varrho ,\zeta \in \left(v_1\theta \right),\theta >0\hfill \\ \hfill =& {\mathcal{Z}}_1\left(\varrho \right){\mathcal{Z}}_2\left(\varrho \right)-{\mathcal{Z}}_1\left(\varrho \right){\mathcal{Z}}_2\left(\zeta \right)-{\mathcal{Z}}_1\left(\zeta \right){\mathcal{Z}}_2\left(\varrho \right)+{\mathcal{Z}}_1\left(\zeta \right){\mathcal{Z}}_2\left(\zeta \right).\hfill \end{array}$$

(20)

By Theorem 2, ${\mathcal{Z}}_1$ and ${\mathcal{Z}}_2$ fulfil the hypothesis; therefore, we have:

$$\begin{array}{c}\hfill H(\varrho ,\zeta )\ge 0.\end{array}$$

Taking the product on both sides of (20) by ${\displaystyle {\varpi }^{-1}\left(\theta \right)\frac{\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right){\hslash }_1\left(\varrho \right)}$ and, then, taking the integration of both sides with respect to $\varrho$ over $(v_1,\theta )$ and:

$$\begin{array}{cc}\hfill & {\varpi }^{-1}\left(\theta \right){\int }_{v_1}^{\theta }\frac{\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right){\hslash }_1\left(\varrho \right)H(\varrho ,\zeta )d\varrho \hfill \\ \hfill =& \left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1{\mathcal{Z}}_1{\mathcal{Z}}_2\right)\left(\theta \right)-{\mathcal{Z}}_2\left(\zeta \right)\left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1{\mathcal{Z}}_1\right)\left(\theta \right)-{\mathcal{Z}}_2\left(\zeta \right)\left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1{\mathcal{Z}}_2\right)\left(\theta \right)\hfill \\ \hfill +& {\mathcal{Z}}_1\left(\zeta \right){\mathcal{Z}}_2\left(\zeta \right)\left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1\right)\left(\theta \right)\ge 0.\hfill \end{array}$$

(21)

Again, taking the product of (21) by ${\displaystyle {\varpi }^{-1}\left(\theta \right)\frac{\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\zeta \right)\right)}{\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\zeta \right)}\varpi \left(\zeta \right){\mathcal{F}}^{\prime }\left(\zeta \right){\hslash }_1\left(\zeta \right)}$, then taking integration with respect to $\zeta$ over $(v_1,\theta )$ and using (9), we obtain:

$$\begin{array}{cc}\hfill & {\varpi }^{-2}\left(\theta \right){\int }_{v_1}^{\theta }{\int }_{v_1}^{\theta }\frac{\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)}\frac{\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\zeta \right)\right)}{\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\zeta \right)}{\mathcal{F}}^{\prime }\left(\varrho \right)\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\zeta \right)\varpi \left(\zeta \right){\hslash }_1\left(\varrho \right){\hslash }_1\left(\zeta \right)H(\varrho ,\zeta )d\varrho d\zeta \hfill \\ \hfill =& \left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1\right)\left(\theta \right)\left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1{\mathcal{Z}}_1{\mathcal{Z}}_2\right)\left(\theta \right)-\left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1{\mathcal{Z}}_1\right)\left(\theta \right)\left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1{\mathcal{Z}}_2\right)\left(\theta \right)\ge 0.\hfill \end{array}$$

(22)

From (16), it becomes:

$$\begin{array}{cc}\hfill & \frac{{\varpi }^{-2}\left(\theta \right)}{2}{\int }_{v_1}^{\theta }{\int }_{v_1}^{\theta }\frac{\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)}\frac{\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\zeta \right)\right)}{\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\zeta \right)}{\mathcal{F}}^{\prime }\left(\varrho \right)\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\zeta \right)\varpi \left(\zeta \right){\hslash }_1\left(\varrho \right){\hslash }_1\left(\zeta \right)\mid H(\varrho ,\zeta )\mid d\varrho d\zeta \hfill \\ \hfill \le & \frac{\Vert {\mathcal{Z}}_1^{\prime }{\Vert }_{\infty }{\Vert {\mathcal{Z}}_2^{\prime }\Vert }_{\infty }{\varpi }^{-2}\left(\theta \right)}{2}{\int }_{v_1}^{\theta }{\int }_{v_1}^{\theta }\frac{\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)}\frac{\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\zeta \right)\right)}{\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\zeta \right)}{\mathcal{F}}^{\prime }\left(\varrho \right)\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\zeta \right)\varpi \left(\zeta \right){\hslash }_1\left(\varrho \right){\hslash }_1\left(\zeta \right)\hfill \\ \hfill \times & \left({\varrho }^2-2\varrho \zeta +{\zeta }^2\right)d\varrho d\zeta .\hfill \end{array}$$

(23)

Consequently, it can be written as:

$$\begin{array}{cc}\hfill & \frac{{\varpi }^{-2}\left(\theta \right)}{2}{\int }_{v_1}^{\theta }{\int }_{v_1}^{\theta }\frac{\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)}\frac{\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\zeta \right)\right)}{\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\zeta \right)}{\mathcal{F}}^{\prime }\left(\varrho \right)\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\zeta \right)\varpi \left(\zeta \right){\hslash }_1\left(\varrho \right){\hslash }_1\left(\zeta \right)\mid H(\varrho ,\zeta )\mid d\varrho d\zeta \hfill \\ \hfill \le & 2\Vert {\mathcal{Z}}_1^{\prime }{\Vert }_{\infty }{\Vert {\mathcal{Z}}_2^{\prime }\Vert }_{\infty }\left[\left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1\right)\left(\theta \right)\left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\theta }^2{\hslash }_1\right)\left(\theta \right)-{\left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_0^{\kappa }\theta {\hslash }_1\right)}^2\left(\theta \right)\right].\hfill \end{array}$$

(24)

According to (22) and (24), we obtain the desired proof. □

Setting Theorem 2 for $\varpi =1$, we obtain the following new result.

Corollary 2.

If the two functions ${\mathcal{Z}}_1$ and ${\mathcal{Z}}_2$ are differentiable, both have variations in the same sense on $[0,\infty )$ and ${\hslash }_1$ is a positive function on $[0,\infty )$. Suppose that $\mathcal{F}$ is a positive and increasing function on $[0,\infty [$ and its derivative is continuous on $[0,\infty [$. If ${\mathcal{Z}}_1^{\prime },{\mathcal{Z}}_2^{\prime }\in {\mathcal{L}}_{\infty }\left(\right[0,\infty \left[\right)$, then the following inequality holds:

$$\begin{array}{cc}\hfill & 0\le \left({}^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1\right)\left(\theta \right)\left({}^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1{\mathcal{Z}}_1{\mathcal{Z}}_2\right)\left(\theta \right)-\left({}^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1{\mathcal{Z}}_1\right)\left(\tau \right)\left({}^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1{\mathcal{Z}}_2\right)\left(\theta \right)\hfill \\ \hfill \le & \Vert {\mathcal{Z}}_1^{\prime }{\Vert }_{\infty }{\Vert {\mathcal{Z}}_2^{\prime }\Vert }_{\infty }\left[\left({}^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1\right)\left(\theta \right)\left({}^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\theta }^2{\hslash }_1\right)\left(\tau \right)-{\left({}^{\mathcal{F}}{\mathcal{I}}_0^{\kappa }\theta {\hslash }_1\left(\theta \right)\right)}^2\right].\hfill \end{array}$$

Remark 3.

By considering ${\hslash }_1\left(\theta \right)=1$ in Theorem 2, we obtain Theorem 1. Similarly, taking $\varpi \left(\theta \right)=1$, $\mathcal{F}\left(\theta \right)=\theta$ and $\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)\right)=\frac{{\theta }^{\kappa }}{\mathsf{\Gamma }\left(\kappa \right)}$, we obtain the result of Dahmani [27].

3. Generalized Weighted-Type Integral Inequalities Associated with Weighted and Extended Chebyshev Functionals

In this section, we construct certain weighted-type generalized fractional integral inequalities.

Theorem 3.

If the two functions ${\mathcal{Z}}_1$ and ${\mathcal{Z}}_2$ are differentiable on $[0,\infty )$, ${\hslash }_1$ is a positive and integrable function on $[0,\infty )$. Let $\mathcal{F}$ be positive and increasing on $[0,\infty [$ and its derivative be continuous on $[0,\infty [$. If ${\mathcal{Z}}_1^{\prime }\in {\mathcal{L}}^p\left(\right[0,\infty \left[\right)$, ${\mathcal{Z}}_2^{\prime }\in {\mathcal{L}}^q\left(\right[0,\infty \left[\right)$, $p,q,r>1$ with $\frac{1}{p}+\frac{1}{p^{\prime }}=1$, $\frac{1}{q}+\frac{1}{q^{\prime }}=1$ and $\frac{1}{r}+\frac{1}{r^{\prime }}=1$, then the following weighted fractional integral inequality holds:

$$\begin{array}{cc}\hfill & 2\mid \left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1\right)\left(\theta \right)\left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1{\mathcal{Z}}_1{\mathcal{Z}}_2\right)\left(\theta \right)-\left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1{\mathcal{Z}}_1\right)\left(\theta \right)\left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1{\mathcal{Z}}_2\right)\left(\theta \right)\mid \hfill \\ \hfill \le & (\Vert {\mathcal{Z}}_1^{\prime }{\Vert }_p^r{\varpi }^{-r}\left(\theta \right){\int }_{v_1}^{\theta }{\int }_{v_1}^{\theta }\frac{\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)}\frac{\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\zeta \right)\right)}{\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\zeta \right)}\varpi \left(\zeta \right){\mathcal{F}}^{\prime }\left(\zeta \right){\mathcal{F}}^{\prime }\left(\zeta \right)\varpi \left(\zeta \right){\hslash }_1\left(\varrho \right){\hslash }_1\left(\zeta \right)\hfill \\ \hfill \times & \mid \varrho -\zeta {\mid }^{\frac{1}{p^{\prime }}+\frac{1}{q^{\prime }}}d\varrho d\zeta {)}^{\frac{1}{r}}\hfill \\ \hfill \times & (\Vert {\mathcal{Z}}_2^{\prime }{\Vert }_q^{r^{\prime }}{\varpi }^{-r^{\prime }}\left(\theta \right){\int }_{v_1}^{\theta }{\int }_{v_1}^{\theta }\frac{\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)}\frac{\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\zeta \right)\right)}{\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\zeta \right)}{\mathcal{F}}^{\prime }\left(\varrho \right)\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\zeta \right)\varpi \left(\zeta \right){\hslash }_1\left(\varrho \right){\hslash }_1\left(\zeta \right)\hfill \\ \hfill \times & \mid \varrho -\zeta {\mid }^{\frac{1}{p^{\prime }}+\frac{1}{q^{\prime }}}d\varrho d\zeta {)}^{\frac{1}{r^{\prime }}}\hfill \\ \hfill \le & \Vert {\mathcal{Z}}_1^{\prime }{\Vert }_p{\Vert {\mathcal{Z}}_2^{\prime }\Vert }_q{\varpi }^{-2}\left(\theta \right)({\int }_{v_1}^{\theta }{\int }_{v_1}^{\theta }\frac{\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)}\frac{\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\zeta \right)\right)}{\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\zeta \right)}{\mathcal{F}}^{\prime }\left(\varrho \right)\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\zeta \right)\varpi \left(\zeta \right){\hslash }_1\left(\varrho \right){\hslash }_1\left(\zeta \right)\hfill \\ \hfill \times & \mid \varrho -\zeta {\mid }^{\frac{1}{p^{\prime }}+\frac{1}{q^{\prime }}}d\varrho d\zeta ).\hfill \end{array}$$

(25)

Proof. 

Let us define

$$\begin{array}{cc}\hfill H(\varrho ,\zeta )& =({\mathcal{Z}}_1\left(\varrho \right)-{\mathcal{Z}}_1\left(\zeta \right))({\mathcal{Z}}_2\left(\varrho \right)-{\mathcal{Z}}_1\left(\zeta \right));\varrho ,\zeta \in \left(v_1\theta \right)\hfill \\ \hfill =& {\mathcal{Z}}_1\left(\varrho \right){\mathcal{Z}}_2\left(\varrho \right)-{\mathcal{Z}}_1\left(\varrho \right){\mathcal{Z}}_2\left(\zeta \right)-{\mathcal{Z}}_1\left(\zeta \right){\mathcal{Z}}_2\left(\varrho \right)+{\mathcal{Z}}_1\left(\zeta \right){\mathcal{Z}}_2\left(\zeta \right).\hfill \end{array}$$

(26)

Conducting the product of (26) by ${\displaystyle {\varpi }^{-1}\left(\theta \right)\frac{\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)}{\mathcal{F}}^{\prime }\left(\varrho \right)\varpi \left(\varrho \right){\hslash }_1\left(\varrho \right)}$, then integrating with respect to $\varrho$ over $(v_1,\theta )$ and using (9), we obtain:

$$\begin{array}{cc}\hfill & {\varpi }^{-1}\left(\theta \right){\int }_{v_1}^{\theta }\frac{\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)}{\mathcal{F}}^{\prime }\left(\varrho \right)\varpi \left(\varrho \right){\hslash }_1\left(\varrho \right)H(\varrho ,\zeta )d\varrho \hfill \\ \hfill =& \left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1{\mathcal{Z}}_1{\mathcal{Z}}_2\right)\left(\theta \right)-{\mathcal{Z}}_2\left(\zeta \right)\left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1{\mathcal{Z}}_1\right)\left(\theta \right)-{\mathcal{Z}}_1\left(\zeta \right)\left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1{\mathcal{Z}}_2\right)\left(\theta \right)+{\mathcal{Z}}_1\left(\zeta \right){\mathcal{Z}}_2\left(\zeta \right)\left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1\right)\left(\theta \right).\hfill \end{array}$$

(27)

Again, taking the product of (27) by ${\displaystyle {\varpi }^{-1}\left(\theta \right)\frac{\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\zeta \right)\right)}{\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\zeta \right)}{\mathcal{F}}^{\prime }\left(\zeta \right)\varpi \left(\zeta \right){\hslash }_1\left(\zeta \right)}$, then integrating with respect to $\zeta$ over $(v_1,\theta )$ and using (9), we have:

$$\begin{array}{cc}\hfill & {\varpi }^{-2}\left(\theta \right){\int }_{v_1}^{\theta }{\int }_{v_1}^{\theta }\frac{\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)}\frac{\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\zeta \right)\right)}{\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\zeta \right)}{\mathcal{F}}^{\prime }\left(\varrho \right)\varpi \left(\varrho \right){\hslash }_1\left(\varrho \right){\mathcal{F}}^{\prime }\left(\zeta \right)\varpi \left(\zeta \right){\hslash }_1\left(\zeta \right)H(\varrho ,\zeta )d\varrho d\zeta \hfill \\ \hfill =& 2\left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1\right)\left(\theta \right)\left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1{\mathcal{Z}}_1{\mathcal{Z}}_2\right)\left(\theta \right)-\left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1{\mathcal{Z}}_1\right)\left(\theta \right)\left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1{\mathcal{Z}}_2\right)\left(\theta \right).\hfill \end{array}$$

(28)

On the other side, we also have:

$$\begin{array}{c}\hfill H(\varrho ,\zeta )={\int }_{\zeta }^{\varrho }{\int }_{\zeta }^{\varrho }{\mathcal{Z}}_1^{\prime }\left(u\right){\mathcal{Z}}_2^{\prime }\left(v\right)dudv.\end{array}$$

(29)

By employing the Hölder inequality, we have:

$$\begin{array}{c}\hfill \mid {\mathcal{Z}}_1\left(\varrho \right)-{\mathcal{Z}}_2\left(\zeta \right)\mid \le \mid \varrho -\zeta {\mid }^{\frac{1}{p^{\prime }}}\mid {\int }_{\zeta }^{\varrho }\mid {\mathcal{Z}}_1^{\prime }\left(u\right){\mid }^{p}du{\mid }^{\frac{1}{p}}\end{array}$$

(30)

and:

$$\begin{array}{c}\hfill \mid {\mathcal{Z}}_2\left(\varrho \right)-{\mathcal{Z}}_2\left(\zeta \right)\mid \le \mid \varrho -\zeta {\mid }^{\frac{1}{q^{\prime }}}\mid {\int }_{\zeta }^{\varrho }\mid {\mathcal{Z}}_2^{\prime }\left(v\right){\mid }^{q}dv{\mid }^{\frac{1}{q}}.\end{array}$$

(31)

Thus, H can be estimated as:

$$\begin{array}{c}\hfill \mid H(\varrho ,\zeta )\mid \le \mid \varrho -\zeta {\mid }^{\frac{1}{p^{\prime }}+\frac{1}{q^{\prime }}}\mid {\int }_{\zeta }^{\varrho }\mid {\mathcal{Z}}_1^{\prime }\left(u\right){\mid }^{p}du{\mid }^{\frac{1}{p}}\mid {\int }_{\zeta }^{\varrho }\mid {\mathcal{Z}}_2^{\prime }\left(v\right){\mid }^{q}dv{\mid }^{\frac{1}{q}}.\end{array}$$

(32)

Hence, from (28) and (32), it follows that:

$$\begin{array}{cc}\hfill & 2\mid \left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1\right)\left(\theta \right)\left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1{\mathcal{Z}}_1{\mathcal{Z}}_2\right)\left(\theta \right)-\left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1{\mathcal{Z}}_1\right)\left(\theta \right)\left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1{\mathcal{Z}}_2\right)\left(\theta \right)\mid \hfill \\ \hfill =& {\varpi }^{-2}\left(\theta \right){\int }_{v_1}^{\theta }{\int }_{v_1}^{\theta }\frac{\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)}\frac{\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\zeta \right)\right)}{\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\zeta \right)}{\mathcal{F}}^{\prime }\left(\varrho \right)\varpi \left(\varrho \right){\hslash }_1\left(\varrho \right){\mathcal{F}}^{\prime }\left(\zeta \right)\varpi \left(\zeta \right){\hslash }_1\left(\zeta \right)\mid H(\varrho ,\zeta )\mid d\varrho d\zeta \hfill \\ \hfill \le & {\varpi }^{-2}\left(\theta \right){\int }_{v_1}^{\theta }{\int }_{v_1}^{\theta }\frac{\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)}\frac{\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\zeta \right)\right)}{\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\zeta \right)}{\mathcal{F}}^{\prime }\left(\varrho \right)\varpi \left(\varrho \right){\hslash }_1\left(\varrho \right){\mathcal{F}}^{\prime }\left(\zeta \right)\varpi \left(\zeta \right){\hslash }_1\left(\zeta \right)\hfill \\ \hfill \times & \mid \varrho -\zeta {\mid }^{\frac{1}{p^{\prime }}+\frac{1}{q^{\prime }}}\mid {\int }_{\zeta }^{\varrho }\mid {\mathcal{Z}}_1^{\prime }\left(u\right){\mid }^{p}du{\mid }^{\frac{1}{p}}\mid {\int }_{\zeta }^{\varrho }\mid {\mathcal{Z}}_2^{\prime }\left(v\right){\mid }^{q}dv{\mid }^{\frac{1}{q}}d\varrho d\zeta .\hfill \end{array}$$

(33)

By employing the Hölder inequality for the double integral for (33), we obtain:

$$\begin{array}{cc}\hfill & 2\mid \left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1\right)\left(\theta \right)\left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1{\mathcal{Z}}_1{\mathcal{Z}}_2\right)\left(\theta \right)-\left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1{\mathcal{Z}}_1\right)\left(\theta \right)\left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1{\mathcal{Z}}_2\right)\left(\theta \right)\mid \hfill \\ \hfill \le & {\varpi }^{-2}\left(\theta \right)({\int }_{v_1}^{\theta }{\int }_{v_1}^{\theta }\frac{\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)}\frac{\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\zeta \right)\right)}{\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\zeta \right)}{\mathcal{F}}^{\prime }\left(\varrho \right)\varpi \left(\varrho \right){\hslash }_1\left(\varrho \right){\mathcal{F}}^{\prime }\left(\zeta \right)\varpi \left(\zeta \right){\hslash }_1\left(\zeta \right)\hfill \\ \hfill \times & \mid \varrho -\zeta {\mid }^{\frac{1}{p^{\prime }}+\frac{1}{q^{\prime }}}\mid {\int }_{\zeta }^{\varrho }\mid {\mathcal{Z}}_1^{\prime }\left(u\right){\mid }^{p}du{\mid }^{\frac{r}{p}}d\varrho d\zeta {)}^{\frac{1}{r}}\hfill \\ \hfill \times & ({\int }_{v_1}^{\theta }{\int }_{v_1}^{\theta }\frac{\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)}\frac{\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\zeta \right)\right)}{\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\zeta \right)}{\mathcal{F}}^{\prime }\left(\varrho \right)\varpi \left(\varrho \right){\hslash }_1\left(\varrho \right){\mathcal{F}}^{\prime }\left(\zeta \right)\varpi \left(\zeta \right){\hslash }_1\left(\zeta \right)\hfill \\ \hfill \times & \mid \varrho -\zeta {\mid }^{\frac{1}{p^{\prime }}+\frac{1}{q^{\prime }}}\mid {\int }_{\zeta }^{\varrho }\mid {\mathcal{Z}}_2^{\prime }\left(v\right){\mid }^{q}dv{\mid }^{\frac{r^{\prime }}{q}}d\varrho d\zeta {)}^{\frac{1}{r^{\prime }}}.\hfill \end{array}$$

(34)

Now, utilizing the following relations:

$$\begin{array}{c}\hfill \mid {\int }_{\zeta }^{\varrho }\mid {\mathcal{Z}}_1^{\prime }\left(u\right){\mid }^{p}du\mid \le \Vert {\mathcal{Z}}_1^{\prime }{\Vert }_p^{p}and\mid {\int }_{\zeta }^{\varrho }\mid {\mathcal{Z}}_2^{\prime }\left(v\right){\mid }^{q}dv\mid \le {\Vert {\mathcal{Z}}_2^{\prime }\Vert }_q^q,\end{array}$$

(35)

then (34) becomes,

$$\begin{array}{cc}\hfill & 2\mid \left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1\right)\left(\theta \right)\left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1{\mathcal{Z}}_1{\mathcal{Z}}_2\right)\left(\theta \right)-\left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1{\mathcal{Z}}_1\right)\left(\theta \right)\left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1{\mathcal{Z}}_2\right)\left(\theta \right)\mid \hfill \\ \hfill \le & (\Vert {\mathcal{Z}}_1^{\prime }{\Vert }_p^r{\varpi }^{-r}\left(\theta \right){\int }_{v_1}^{\theta }{\int }_{v_1}^{\theta }\frac{\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)}\frac{\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\zeta \right)\right)}{\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\zeta \right)}{\mathcal{F}}^{\prime }\left(\varrho \right)\varpi \left(\varrho \right){\hslash }_1\left(\varrho \right){\mathcal{F}}^{\prime }\left(\zeta \right)\varpi \left(\zeta \right){\hslash }_1\left(\zeta \right)\hfill \\ \hfill \times & \mid \varrho -\zeta {\mid }^{\frac{1}{p^{\prime }}+\frac{1}{q^{\prime }}}d\varrho d\zeta {)}^{\frac{1}{r}}({\Vert {\mathcal{Z}}_2^{\prime }\Vert }_p^{r^{\prime }}{\varpi }^{-r^{\prime }}\left(\theta \right){\int }_{v_1}^{\theta }{\int }_{v_1}^{\theta }\hfill \\ \hfill \times & {\mathcal{F}}^{\prime }\left(\varrho \right)\varpi \left(\varrho \right){\hslash }_1\left(\varrho \right){\mathcal{F}}^{\prime }\left(\zeta \right)\varpi \left(\zeta \right){\hslash }_1\left(\zeta \right)\mid \varrho -\zeta {\mid }^{\frac{1}{p^{\prime }}+\frac{1}{q^{\prime }}}d\varrho d\zeta {)}^{\frac{1}{r^{\prime }}}.\hfill \end{array}$$

(36)

From (36), we have:

$$\begin{array}{cc}\hfill & 2\mid \left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1\right)\left(\theta \right)\left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1{\mathcal{Z}}_1{\mathcal{Z}}_2\right)\left(\theta \right)-\left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1{\mathcal{Z}}_1\right)\left(\theta \right)\left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1{\mathcal{Z}}_2\right)\left(\theta \right)\mid \hfill \\ \hfill \le & \Vert {\mathcal{Z}}_1^{\prime }{\Vert }_p{\Vert {\mathcal{Z}}_2^{\prime }\Vert }_q{\varpi }^{-2}\left(\theta \right){\int }_{v_1}^{\theta }{\int }_{v_1}^{\theta }\frac{\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)}\frac{\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\zeta \right)\right)}{\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\zeta \right)}\hfill \\ \hfill \times & {\mathcal{F}}^{\prime }\left(\varrho \right)\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\zeta \right)\varpi \left(\zeta \right){\hslash }_1\left(\varrho \right){\hslash }_1\left(\zeta \right)\mid \varrho -\zeta {\mid }^{\frac{1}{p^{\prime }}+\frac{1}{q^{\prime }}}d\varrho d\zeta ,\hfill \end{array}$$

which completes the required result. □

If we consider $\varpi \left(\theta \right)=1$ in Theorem 3, the following new result can be obtained.

Corollary 3.

If the two functions ${\mathcal{Z}}_1$ and ${\mathcal{Z}}_2$ are differentiable on $[0,\infty )$ and if ${\hslash }_1$ is integrable and a positive function on $[0,\infty )$, and we let $\mathcal{F}$ be a positive and increasing function on $[0,\infty [$ and its derivative be continuous on $[0,\infty [$, if ${\mathcal{Z}}_1^{\prime }\in {\mathcal{L}}^p\left(\right[0,\infty \left[\right)$, ${\mathcal{Z}}_2^{\prime }\in {\mathcal{L}}^q\left(\right[0,\infty \left[\right)$, $p,q,r>1$ with $\frac{1}{p}+\frac{1}{p^{\prime }}=1$, $\frac{1}{q}+\frac{1}{q^{\prime }}=1$, and $\frac{1}{r}+\frac{1}{r^{\prime }}=1$, then the following inequality holds:

$$\begin{array}{cc}\hfill & 2\mid \left({}^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1\right)\left(\theta \right)\left({}^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1{\mathcal{Z}}_1{\mathcal{Z}}_2\right)\left(\theta \right)-\left({}^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1{\mathcal{Z}}_1\right)\left(\theta \right)\left({}^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1{\mathcal{Z}}_2\right)\left(\theta \right)\mid \hfill \\ \hfill \le & (\Vert {\mathcal{Z}}_1^{\prime }{\Vert }_p^r{\int }_{v_1}^{\theta }{\int }_{v_1}^{\theta }\frac{\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)}\frac{\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\zeta \right)\right)}{\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\zeta \right)}{\mathcal{F}}^{\prime }\left(\varrho \right){\mathcal{F}}^{\prime }\left(\zeta \right){\hslash }_1\left(\varrho \right){\hslash }_1\left(\zeta \right)\hfill \\ \hfill \times & \mid \varrho -\zeta {\mid }^{\frac{1}{p^{\prime }}+\frac{1}{q^{\prime }}}d\varrho d\zeta {)}^{\frac{1}{r}}\hfill \\ \hfill \times & (\Vert {\mathcal{Z}}_2^{\prime }{\Vert }_q^{r^{\prime }}{\int }_{v_1}^{\theta }{\int }_{v_1}^{\theta }\frac{\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)}\frac{\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\zeta \right)\right)}{\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\zeta \right)}{\mathcal{F}}^{\prime }\left(\varrho \right){\mathcal{F}}^{\prime }\left(\zeta \right){\hslash }_1\left(\varrho \right){\hslash }_1\left(\zeta \right)\hfill \\ \hfill \times & \mid \varrho -\zeta {\mid }^{\frac{1}{p^{\prime }}+\frac{1}{q^{\prime }}}d\varrho d\zeta {)}^{\frac{1}{r^{\prime }}}\hfill \\ \hfill \le & \Vert {\mathcal{Z}}_1^{\prime }{\Vert }_p{\Vert {\mathcal{Z}}_2^{\prime }\Vert }_q({\int }_{v_1}^{\theta }{\int }_{v_1}^{\theta }\frac{\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)}\frac{\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\zeta \right)\right)}{\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\zeta \right)}{\mathcal{F}}^{\prime }\left(\varrho \right){\mathcal{F}}^{\prime }\left(\zeta \right){\hslash }_1\left(\varrho \right){\hslash }_1\left(\zeta \right)\hfill \\ \hfill \times & \mid \varrho -\zeta {\mid }^{\frac{1}{p^{\prime }}+\frac{1}{q^{\prime }}}d\varrho d\zeta ).\hfill \end{array}$$

Remark 4.

If we consider $\varpi \left(\theta \right)=1$, $\mathcal{F}\left(\theta \right)=\theta$ and $\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)\right)=\frac{{\theta }^{\kappa }}{\mathsf{\Gamma }\left(\kappa \right)}$ in Theorem 3, we arrive at the inequality established by Dahmani et al. [28].

Remark 5.

Furthermore, if we consider $\varpi =1$, $\mathcal{F}\left(\theta \right)=\theta$ and $\mathsf{\Psi }\left(\mathcal{F}\right(\theta \left)\right)=\theta$ in Theorem 3, then we obtain the inequality (3) on $[0,\theta ]$.

Theorem 4.

If the two functions ${\mathcal{Z}}_1$ and ${\mathcal{Z}}_2$ are differentiable on $[0,\infty )$ and if ${\hslash }_1$ and ${\hslash }_1^{\prime }$ are integrable and positive functions on $[0,\infty )$, we let $\mathcal{F}$ be positive and increasing on $[0,\infty [$ and its derivative be continuous on $[0,\infty [$, and iff ${\mathcal{Z}}_1^{\prime }\in {\mathcal{L}}^p\left(\right[0,\infty \left[\right)$, ${\mathcal{Z}}_2^{\prime }\in {\mathcal{L}}^q\left(\right[0,\infty \left[\right)$, $p,q,r>1$ such that $\frac{1}{p}+\frac{1}{p^{\prime }}=1$, $\frac{1}{q}+\frac{1}{q^{\prime }}=1$ and $\frac{1}{r}+\frac{1}{r^{\prime }}=1$, then the following weighted fractional integral inequality holds:

$$\begin{array}{cc}\hfill & 2\mid \left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1^{\prime }\right)\left(\theta \right)\left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1{\mathcal{Z}}_1{\mathcal{Z}}_2\right)\left(\theta \right)-\left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1{\mathcal{Z}}_1\right)\left(\theta \right)\left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1^{\prime }{\mathcal{Z}}_2\right)\left(\theta \right)\hfill \\ \hfill -& \left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1^{\prime }{\mathcal{Z}}_1\right)\left(\theta \right)\left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1{\mathcal{Z}}_2\right)\left(\theta \right)+\left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1^{\prime }\right)\left(\theta \right)\left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1^{\prime }{\mathcal{Z}}_1{\mathcal{Z}}_2\right)\left(\theta \right)\mid \hfill \\ \hfill \le & (\Vert {\mathcal{Z}}_1^{\prime }{\Vert }_p^r{\varpi }^{-r}\left(\theta \right){\int }_{v_1}^{\theta }{\int }_{v_1}^{\theta }\frac{\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)}\frac{\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\zeta \right)\right)}{\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\zeta \right)}{\mathcal{F}}^{\prime }\left(\varrho \right)\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\zeta \right)\varpi \left(\zeta \right){\hslash }_1\left(\varrho \right){\hslash }_1^{\prime }\left(\zeta \right)\hfill \\ \hfill \times & \mid \varrho -\zeta {\mid }^{\frac{1}{p^{\prime }}+\frac{1}{q^{\prime }}}d\varrho d\zeta {)}^{\frac{1}{r}}\hfill \\ \hfill \times & (\Vert {\mathcal{Z}}_2^{\prime }{\Vert }_q^{r^{\prime }}{\varpi }^{-r^{\prime }}\left(\theta \right){\int }_{v_1}^{\theta }{\int }_{v_1}^{\theta }\frac{\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)}\frac{\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\zeta \right)\right)}{\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\zeta \right)}{\mathcal{F}}^{\prime }\left(\varrho \right)\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\zeta \right)\varpi \left(\zeta \right){\hslash }_1\left(\varrho \right){\hslash }_1^{\prime }\left(\zeta \right)\hfill \\ \hfill \times & \mid \varrho -\zeta {\mid }^{\frac{1}{p^{\prime }}+\frac{1}{q^{\prime }}}d\varrho d\zeta {)}^{\frac{1}{r^{\prime }}}\hfill \\ \hfill \le & \Vert {\mathcal{Z}}_1^{\prime }{\Vert }_p{\Vert {\mathcal{Z}}_2^{\prime }\Vert }_q{\varpi }^{-2}\left(\theta \right)({\int }_{v_1}^{\theta }{\int }_{v_1}^{\theta }\frac{\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)}\frac{\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\zeta \right)\right)}{\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\zeta \right)}{\mathcal{F}}^{\prime }\left(\varrho \right)\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\zeta \right)\varpi \left(\zeta \right){\hslash }_1\left(\varrho \right){\hslash }_1^{\prime }\left(\zeta \right)\hfill \\ \hfill \times & \mid \varrho -\zeta {\mid }^{\frac{1}{p^{\prime }}+\frac{1}{q^{\prime }}}d\varrho d\zeta ).\hfill \end{array}$$

Proof. 

Conducting the product of (27) by ${\displaystyle {\varpi }^{-1}\left(\theta \right)\frac{\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\zeta \right)\right)}{\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\zeta \right)}{\mathcal{F}}^{\prime }\left(\zeta \right)\varpi \left(\zeta \right){\hslash }_1^{\prime }\left(\zeta \right)}$, then integrating with respect to $\zeta$ over $(v_1,\theta )$ and using (9), we obtain:

$$\begin{array}{cc}\hfill & {\varpi }^{-2}\left(\theta \right){\int }_{v_1}^{\theta }{\int }_{v_1}^{\theta }\frac{\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)}\frac{\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\zeta \right)\right)}{\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\zeta \right)}{\mathcal{F}}^{\prime }\left(\varrho \right)\varpi \left(\varrho \right){\hslash }_1\left(\varrho \right){\mathcal{F}}^{\prime }\left(\zeta \right)\varpi \left(\zeta \right){\hslash }_1^{\prime }\left(\zeta \right)H(\varrho ,\zeta )d\varrho d\zeta \hfill \\ \hfill =& \left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1^{\prime }\right)\left(\theta \right)\left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1{\mathcal{Z}}_1{\mathcal{Z}}_2\right)\left(\theta \right)-\left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1{\mathcal{Z}}_1\right)\left(\theta \right)\left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1^{\prime }{\mathcal{Z}}_2\right)\left(\theta \right)\hfill \\ \hfill -& \left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1^{\prime }{\mathcal{Z}}_1\right)\left(\theta \right)\left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1{\mathcal{Z}}_2\right)\left(\theta \right)+\left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1^{\prime }\right)\left(\theta \right)\left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1^{\prime }{\mathcal{Z}}_1{\mathcal{Z}}_2\right)\left(\theta \right).\hfill \end{array}$$

(37)

Using (32) in (37), we obtain:

$$\begin{array}{cc}\hfill & \mid \left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1^{\prime }\right)\left(\theta \right)\left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1{\mathcal{Z}}_1{\mathcal{Z}}_2\right)\left(\theta \right)-\left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1{\mathcal{Z}}_1\right)\left(\theta \right)\left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1^{\prime }{\mathcal{Z}}_2\right)\left(\theta \right)\hfill \\ \hfill -& \left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1^{\prime }{\mathcal{Z}}_1\right)\left(\theta \right)\left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1{\mathcal{Z}}_2\right)\left(\theta \right)+\left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1^{\prime }\right)\left(\theta \right)\left({}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1^{\prime }{\mathcal{Z}}_1{\mathcal{Z}}_2\right)\left(\theta \right)\mid \hfill \\ \hfill =& {\varpi }^{-2}\left(\theta \right){\int }_{v_1}^{\theta }{\int }_{v_1}^{\theta }\frac{\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)}\frac{\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\zeta \right)\right)}{\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\zeta \right)}{\mathcal{F}}^{\prime }\left(\varrho \right)\varpi \left(\varrho \right){\hslash }_1\left(\varrho \right){\mathcal{F}}^{\prime }\left(\zeta \right)\varpi \left(\zeta \right){\hslash }_1\left(\zeta \right)\mid H(\varrho ,\zeta )\mid d\varrho d\zeta \hfill \\ \hfill \le & {\varpi }^{-2}\left(\theta \right){\int }_{v_1}^{\theta }{\int }_{v_1}^{\theta }\frac{\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)}\frac{\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\zeta \right)\right)}{\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\zeta \right)}{\mathcal{F}}^{\prime }\left(\varrho \right)\varpi \left(\varrho \right){\hslash }_1\left(\varrho \right){\mathcal{F}}^{\prime }\left(\zeta \right)\varpi \left(\zeta \right){\hslash }_1^{\prime }\left(\zeta \right)\hfill \\ \hfill \times & \mid \varrho -\zeta {\mid }^{\frac{1}{p^{\prime }}+\frac{1}{q^{\prime }}}\mid {\int }_{\zeta }^{\varrho }\mid {\mathcal{Z}}_1^{\prime }\left(u\right){\mid }^{p}du{\mid }^{\frac{1}{p}}\mid {\int }_{\zeta }^{\varrho }\mid {\mathcal{Z}}_2^{\prime }\left(v\right){\mid }^{q}dv{\mid }^{\frac{1}{q}}d\varrho d\zeta .\hfill \end{array}$$

(38)

The desired proof can be easily obtained by applying a similar procedure as used in the proof of Theorem 3. □

If we consider $\varpi =1$ in Theorem 4, then we obtain the following new result.

Corollary 4.

If the two functions ${\mathcal{Z}}_1$ and ${\mathcal{Z}}_2$ are differentiable on $[0,\infty )$ and if ${\hslash }_1$ and ${\hslash }_1^{\prime }$ are integrable and positive functions on $[0,\infty )$, we let $\mathcal{F}$ be an increasing and positive function on $[0,\infty [$ and its derivative be continuous on $[0,\infty [$, and if ${\mathcal{Z}}_1^{\prime }\in {\mathcal{L}}^p\left(\right[0,\infty \left[\right)$, ${\mathcal{Z}}_2^{\prime }\in {\mathcal{L}}^q\left(\right[0,\infty \left[\right)$, $p,q,r>1$ such that $\frac{1}{p}+\frac{1}{p^{\prime }}=1$, $\frac{1}{q}+\frac{1}{q^{\prime }}=1$ and $\frac{1}{r}+\frac{1}{r^{\prime }}=1$, then the following fractional integral inequality holds:

$$\begin{array}{cc}\hfill & 2\mid \left({}^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1^{\prime }\right)\left(\theta \right)\left({}^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1{\mathcal{Z}}_1{\mathcal{Z}}_2\right)\left(\theta \right)-\left({}^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1{\mathcal{Z}}_1\right)\left(\theta \right)\left({}^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1^{\prime }{\mathcal{Z}}_2\right)\left(\theta \right)\hfill \\ \hfill -& \left({}^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1^{\prime }{\mathcal{Z}}_1\right)\left(\theta \right)\left({}^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1{\mathcal{Z}}_2\right)\left(\theta \right)+\left({}^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1^{\prime }\right)\left(\theta \right)\left({}^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\hslash }_1^{\prime }{\mathcal{Z}}_1{\mathcal{Z}}_2\right)\left(\theta \right)\mid \hfill \\ \hfill \le & (\Vert {\mathcal{Z}}_1^{\prime }{\Vert }_p^r{\int }_{v_1}^{\theta }{\int }_{v_1}^{\theta }\frac{\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)}\frac{\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\zeta \right)\right)}{\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\zeta \right)}{\mathcal{F}}^{\prime }\left(\varrho \right){\mathcal{F}}^{\prime }\left(\zeta \right){\hslash }_1\left(\varrho \right){\hslash }_1^{\prime }\left(\zeta \right)\hfill \\ \hfill \times & \mid \varrho -\zeta {\mid }^{\frac{1}{p^{\prime }}+\frac{1}{q^{\prime }}}d\varrho d\zeta {)}^{\frac{1}{r}}\hfill \\ \hfill \times & (\Vert {\mathcal{Z}}_2^{\prime }{\Vert }_q^{r^{\prime }}{\int }_{v_1}^{\theta }{\int }_{v_1}^{\theta }\frac{\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)}\frac{\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\zeta \right)\right)}{\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\zeta \right)}{\mathcal{F}}^{\prime }\left(\varrho \right){\mathcal{F}}^{\prime }\left(\zeta \right){\hslash }_1\left(\varrho \right){\hslash }_1^{\prime }\left(\zeta \right)\hfill \\ \hfill \times & \mid \varrho -\zeta {\mid }^{\frac{1}{p^{\prime }}+\frac{1}{q^{\prime }}}d\varrho d\zeta {)}^{\frac{1}{r^{\prime }}}\hfill \\ \hfill \le & \Vert {\mathcal{Z}}_1^{\prime }{\Vert }_p{\Vert {\mathcal{Z}}_2^{\prime }\Vert }_q({\int }_{v_1}^{\theta }{\int }_{v_1}^{\theta }\frac{\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\varrho \right)}\frac{\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\zeta \right)\right)}{\mathcal{F}\left(\theta \right)-\mathcal{F}\left(\zeta \right)}{\mathcal{F}}^{\prime }\left(\varrho \right){\mathcal{F}}^{\prime }\left(\zeta \right){\hslash }_1\left(\varrho \right){\hslash }_1^{\prime }\left(\zeta \right)\hfill \\ \hfill \times & \mid \varrho -\zeta {\mid }^{\frac{1}{p^{\prime }}+\frac{1}{q^{\prime }}}d\varrho d\zeta ).\hfill \end{array}$$

Remark 6.

By considering ${\hslash }_1^{\prime }\left(\theta \right)={\hslash }_1\left(\theta \right)$ in Theorem 4, we obtain Theorem 3.

Remark 7.

If we consider $\varpi \left(\theta \right)=1$, $\mathcal{F}\left(\theta \right)=\theta$ and $\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)\right)=\frac{{\theta }^{\kappa }}{\mathsf{\Gamma }\left(\kappa \right)}$ in Theorem 4, then we are led to the result of Dahmani [28].

4. Concluding Remarks

By utilizing the proposed weighted-type generalized fractional integral operator, we established a class of new integral inequalities for differentiable functions related to Chebyshev’s, weighted Chebyshev’s, and extended Chebyshev’s functionals. The obtained inequalities are in more general form than the existing inequalities, which have been published earlier in the literature. Our result’s exceptional cases can be found in [5,11,12,27,28,29,30]. Furthermore, for other types of operators addressed in Remarks 1 and 2, certain new integral inequalities connected to Chebyshev’s functional and its extensions given in the literature can be easily obtained. One may investigate certain other types of integral inequalities by employing the proposed operators in the near future.