Some Double Generalized Weighted Fractional Integral Inequalities Associated with Monotone Chebyshev Functionals

Abstract

In this manuscript, we study the unified integrals recently defined by Rahman et al. and present some new double generalized weighted type fractional integral inequalities associated with increasing, positive, monotone and measurable function $\mathcal{F}$. Also, we establish some new double-weighted inequalities, which are particular cases of the main result and are represented by corollaries. These inequalities are further refinement of all other inequalities associated with increasing, positive, monotone and measurable function existing in literature. The existing inequalities associated with increasing, positive, monotone and measurable function are also restored by applying specific conditions as given in Remarks. Many other types of fractional integral inequalities can be obtained by applying certain conditions on $\mathcal{F}$ and $\mathsf{\Psi }$ given in the literature.

1. Introduction

In the context of fractional differential equations, integral inequalities are very significant. This field has gained popularity during the last few decades. Various researchers, such as [1,2,3], have investigated the significant developments in this domain. By employing Riemann-Liouville (R-L) fractional integrals, the authors presented Grüss type and several other new inequalities in [4,5]. Certain inequalities for the generalised $(k,\rho )$-fractional integral operator are proposed in [6]. In [7], the modified Hermite-Hadamard type inequalities can be found. Dahmani [8] discovered various fractional integral inequalities employing a family of n positive functions. In [9], Srivastava et al. presented the Chebyshev inequality by employing general family of fractional integral operators. Some remarkable inequalities and their applications can be found in nthe work of [10,11,12,13,14,15].

In [16,17], the Chebyshev functional for the integrable functions ${\mathcal{Z}}_1$ and ${\mathcal{Z}}_1$ on $[v_1,v_2]$, is given by

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

(1)

where the function ${ħ}_1$ is a positive and integrable on $[v_1,v_2]$.

The following extended Chebyshev functional for the integrable functions ${\mathcal{Z}}_1$ and ${\mathcal{Z}}_1$ on $[v_1,v_2]$ can be found in [5,18] by

$$\begin{array}{cc}\hfill \mathcal{H}({\mathcal{Z}}_1,{\mathcal{Z}}_2,{ħ}_1,{ħ}_2)& ={\int }_{v_1}^{v_2}{ħ}_2\left(\theta \right)d\theta {\int }_{v_1}^{v_2}{ħ}_1\left(\theta \right){\mathcal{Z}}_1\left(\theta \right){\mathcal{Z}}_2\left(\theta \right)d\theta +{\int }_{v_1}^{v_2}{ħ}_1\left(\theta \right)d\theta {\int }_{v_1}^{v_2}{ħ}_2\left(\theta \right){\mathcal{Z}}_1\left(\theta \right){\mathcal{Z}}_2\left(\theta \right)d\theta \hfill \\ \hfill -& {\int }_{v_1}^{v_2}{ħ}_1\left(\theta \right){\mathcal{Z}}_1\left(\theta \right)d\theta {\int }_{v_1}^{v_2}{ħ}_2\left(\theta \right){\mathcal{Z}}_2\left(\theta \right)d\theta -{\int }_{v_1}^{v_2}{ħ}_2\left(\theta \right){\mathcal{Z}}_1\left(\theta \right)d\theta {\int }_{v_1}^{v_2}{ħ}_1\left(\theta \right){\mathcal{Z}}_2\left(\theta \right)d\theta .\hfill \end{array}$$

(2)

where the two functions ${ħ}_1$ and ${ħ}_2$ are positive and integrable on $[v_1,v_2]$.

Kuang [19] and Mitrinovic [18] proved that $\mathcal{H}\left({\mathcal{Z}}_1,{\mathcal{Z}}_2,{ħ}_1\right)\ge 0$ and $\mathcal{H}\left({\mathcal{Z}}_1,{\mathcal{Z}}_2,{ħ}_1,{ħ}_2\right)\ge 0$ if the functions ${\mathcal{Z}}_1$ and ${\mathcal{Z}}_2$ are synchronous on $[v_1,v_2]$.

Remark 1.

If we take ${ħ}_1\left(\theta \right)={ħ}_2\left(\theta \right)=1,\theta \in [v_1,v_2]$, then

$$\begin{array}{c}\hfill \mathcal{H}\left({\mathcal{Z}}_1,{\mathcal{Z}}_2,{ħ}_1\right)=\frac{1}{2}\mathcal{H}\left({\mathcal{Z}}_1,{\mathcal{Z}}_2,{ħ}_1,{ħ}_2\right).\end{array}$$

Certain remarkable integral inequalities associated with the Chebyshev’s functionals (1) and (2) can be found in the work of [20,21,22,23,24,25].

Awan et al. [26] proposed the following inequality by:

Theorem 1.

Let the function g be an absolutely continuous on $[v_1,v_2]$, and ${ħ}_1$ be integrable and positive function on $[v_1,v_2]$ and ${\left(g^{\prime }\right)}^2\in L_1[v_1,v_2]$, then the following inequality holds;

$$\begin{array}{c}\hfill \mathcal{H}\left(\mathsf{\Phi },\mathsf{\Phi },{ħ}_1\right)\le \frac{1}{G^2\left(v_2\right)}{\int }_{v_1}^{v_2}\left[{\int }_{v_1}^{\theta }{ħ}_1\left(\varrho \right)d\varrho {\int }_{v_1}^{v_2}\varrho {ħ}_1\left(\varrho \right)d\varrho -{\int }_{v_1}^{v_2}{ħ}_1\left(\varrho \right)d\varrho {\int }_{v_1}^{\theta }\varrho {ħ}_1\left(\varrho \right)d\varrho \right]{\left[{\mathsf{\Phi }}^{\prime }\left(\theta \right)\right]}^{2}d\theta ,\end{array}$$

where $G\left(v_2\right)={\int }_{v_1}^{v_2}{ħ}_1\left(\varrho \right)d\varrho$.

In [27], Bezziou et al. proposed the below result for Riemann-Liouville fractional integral as follows:

Theorem 2.

Assume that the function $g:[v_1,v_2]\to \mathbb{R}$ be an absolutely continuous function, and the function ${ħ}_1:[v_1,v_2]\to {\mathbb{R}}^{+}$ be an integrable, and ${\left(g^{\prime }\right)}^2\in L_1[v_1,v_2]$. Then the following inequality for $\kappa >0$ holds:

$$\begin{array}{c}\hfill {\mathcal{I}}_{v_1+}^{\kappa }{ħ}_1\left(v_2\right){\mathcal{I}}_{v_1+}^{\kappa }{ħ}_1{g}^2\left(v_2\right)-{\left({\mathcal{I}}_{v_1+}^{\kappa }{ħ}_{1}g\left(v_2\right)\right)}^2\le {\int }_{v_1}^{v_2}\mathsf{{\rm Y}}\left(\theta \right){\left[g^{\prime }\left(\theta \right)\right]}^{2}d\theta ,\end{array}$$

where

$$\begin{array}{c}\hfill \mathsf{{\rm Y}}\left(\theta \right)=\frac{1}{2}\left[{\mathcal{I}}_{v_1+}^{\kappa }{v}_2{ħ}_1\left(v_2\right){\int }_{v_1}^{\theta }{(v_2-\varrho )}^{\kappa -1}{ħ}_1\left(\varrho \right)d\varrho -{\mathcal{I}}_{v_1+}^{\kappa }{ħ}_1\left(v_2\right){\int }_{v_1}^{\theta }\varrho {(v_2-\varrho )}^{\kappa -1}{ħ}_1\left(\varrho \right)d\varrho \right].\end{array}$$

Dahmani and Bounoua [28] proposed the following inequality for Riemann-Liouville fractional integral by:

Theorem 3.

If the function $g:[v_1,v_2]\to \mathbb{R}$ be an absolutely continuous and let ${ħ}_1:[v_1,v_2]\to {\mathbb{R}}^{+}$ be an integrable function. If ${\left({\mathsf{\Phi }}^{\prime }\right)}^2\in L_1[v_1,v_2]$, then for all $\kappa >0$, and $\theta \in [v_1,v_2]$, the following inequality holds;

$$\begin{array}{c}\hfill \frac{1}{{\mathcal{I}}_{v_1}^{\kappa }{ħ}_1\left(\theta \right)}{\mathcal{I}}_{v_1+}^{\theta }\left({ħ}_1{g}^2\right)\left(\theta \right)-{\left[\frac{1}{{\mathcal{I}}_{v_1+}^{\kappa }{ħ}_1\left(\theta \right)}{\mathcal{I}}_{v_1+}^{\kappa }\left({ħ}_{1}g\right)\left(\theta \right)\right]}^2\le \frac{1}{{\left[{\mathcal{I}}_{v_1+}^{\kappa }{ħ}_1\left(\theta \right)\right]}^2}{\int }_{v_1}^{\theta }{P}_x\left(\theta \right){\left[g^{\prime }\left(\theta \right)\right]}^{2}d\theta ,\end{array}$$

with

$$\begin{array}{c}\hfill P_x\left(\theta \right)=\frac{1}{\mathsf{\Gamma }\left(\kappa \right)}\left[{\mathcal{I}}_{v_1+}^{\kappa }\left(x{ħ}_1\left(x\right)\right){\int }_{v_1}^{\theta }{ħ}_1\left(\varrho \right){(x-\varrho )}^{\kappa -1}d\varrho -{\mathcal{J}}_{v_1+}^{\kappa }{ħ}_1\left(x\right){\int }_{v_1}^{\theta }\varrho {ħ}_1\left(\varrho \right){(x-\varrho )}^{\kappa -1}d\varrho \right].\end{array}$$

Definition 1

([29]). Suppose that the function $\mathsf{\Psi }:[0,\infty )\to [0,\infty )$ be satisfying the conditions given below:

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

(3)

$$\begin{array}{c}\hfill \frac{1}{P}\le \frac{\mathsf{\Psi }\left({ħ}_1\right)}{\mathsf{\Psi }\left({ħ}_2\right)}\le P,\frac{1}{2}\le \frac{{ħ}_1}{{ħ}_2}\le 2,\end{array}$$

(4)

$$\begin{array}{c}\hfill \frac{\mathsf{\Psi }\left({ħ}_2\right)}{{ħ}_2^2}\le Q\frac{\mathsf{\Psi }\left({ħ}_1\right)}{{ħ}_1^2},{ħ}_1\le {ħ}_2,\end{array}$$

(5)

$$\begin{array}{c}\hfill |\frac{\mathsf{\Psi }\left({ħ}_2\right)}{{ħ}_2^2}-\frac{\mathsf{\Psi }\left({ħ}_1\right)}{{ħ}_1^2}|\le S|{ħ}_2-{ħ}_1|\frac{\mathsf{\Psi }\left({ħ}_2\right)}{{ħ}_2^2},\frac{1}{2}\le \frac{{ħ}_1}{{ħ}_2}\le 2,\end{array}$$

(6)

where P, Q, $S>0$ and are independent of ${ħ}_1,{ħ}_2>0$. If $\mathsf{\Psi }\left({ħ}_2\right){ħ}_2^{\alpha }$ is increasing for some $\alpha >0$ and $\frac{\mathsf{\Psi }\left({ħ}_2\right)}{{ħ}_2^{\beta }}$ is decreasing for some $\beta >0$, then Ψ satisfies (3)–(6).

Next, we recall the following generalized weighted type fractional integral operators recently proposed by Rahman et al. [30].

Definition 2.

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

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

(7)

and

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

(8)

Remark 2.

1. If we consider $\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)\right)=\mathcal{F}\left(\theta \right)$, the fractional integrals (7) and (8) 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 (7) and (8) 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 (7) and (8) reduce to the following respectively (see [31]):

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

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 (7) and (8) 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 (7) and (8) 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 (7) and (8) 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 (7) and (8) 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

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

Also, one can derive the weighted form of conformable fractional integrals introduced by [32,33,34,35].

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 3.

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 (7) and (8) 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 (7) and (8) reduce to the following respectively (see [36]) as follows:

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

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 (7) and (8) reduce to the following respectively (see [37,38]):

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

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 (7) and (8) reduce to the following (see [37,38]):

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

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 (7) and (8) reduce to the following weighted Hadamard fractional integrals (see [37,38]):

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

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 (7) and (8) reduce to the following Katugampola [39] 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 (7) and (8) 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, (7) and (8) will lead to the fractional integrals defined by [32,33,34,35].

2. Some Double-Weighted Generalized Fractional Integral Inequalities

In this section, some double-weighted generalized fractional integral inequalities are presented. To this end, we begin by proving the following Lemma.

Lemma 1.

Let the function $\mathcal{F}$ be measurable, increasing, positive and monotone function on $(v_1,v_2)$, and has a continuous derivative ${\mathcal{F}}^{\prime }$ on $[v_1,v_2]$. If $\mathsf{\Phi }:[v_1,v_2]\to \mathbb{R}$ is continuous on $[v_1,v_2]$, and ${ħ}_1,{ħ}_2:[u,v]\to {\mathbb{R}}^{+}$ are positive integrable. Then, we have

$$\begin{array}{cc}\hfill & \left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{ħ}_1\left(v_2\right)\right]\left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{v}_2\left({ħ}_2\mathsf{\Phi }\right)\left(v_2\right)\right]+\left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\mathsf{\Phi }\left(v_2\right)\right]\left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{v}_2\left({ħ}_1\mathsf{\Phi }\right)\left(v_2\right)\right]\hfill \\ \hfill -& \left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{v}_2{ħ}_1\left(v_2\right)\right]\left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_2\mathsf{\Phi }\right)\left(v_2\right)\right]-\left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{v}_2{ħ}_2\left(v_2\right)\right]\left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_1\mathsf{\Phi }\right)\left(v_2\right)\right]\hfill \\ \hfill =& \frac{1}{\varpi \left(\theta \right)}{\int }_{v_1}^{v_2}[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}{v}_2{ħ}_1\left(v_2\right){\int }_{v_1}^{\theta }\frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right){ħ}_2\left(\varrho \right)d\varrho \hfill \\ \hfill -& {}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{ħ}_1\left(v_2\right){\int }_{v_1}^{\theta }\varrho \frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right){ħ}_2\left(\varrho \right)d\varrho ]\left({\mathsf{\Phi }}^{\prime }\left(\theta \right)\right)d\theta .\hfill \end{array}$$

(9)

Proof. 

Assume that ${\mathcal{Z}}_1:[v_1,v_2]\to \mathbb{R}$ is a continuous function on $[v_1,v_2]$. Then, one may gets

$$\begin{array}{cc}\hfill & \left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{ħ}_2\left(v_2\right)\right]\left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_2{\mathcal{Z}}_1\mathsf{\Phi }\right)\left(v_2\right)\right]+\left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\mathsf{\Phi }\left(v_2\right)\right]\left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_1{\mathcal{Z}}_1\mathsf{\Phi }\right)\left(v_2\right)\right]\hfill \\ \hfill -& \left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_1{\mathcal{Z}}_1\right)\left(v_2\right)\right]\left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_2\mathsf{\Phi }\right)\left(v_2\right)\right]-\left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_2{\mathcal{Z}}_1\right)\left(v_2\right)\right]\left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_1\mathsf{\Phi }\right)\left(v_2\right)\right]\hfill \\ \hfill =& \frac{1}{{\varpi }^2\left(\theta \right)}{\int }_{v_1}^{v_2}{\int }_{v_1}^{v_2}\frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\xi \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\xi \right)}\varpi \left(\xi \right){\mathcal{F}}^{\prime }\left(\xi \right)\hfill \\ \hfill \times & \frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right){ħ}_1\left(\xi \right){ħ}_2\left(\varrho \right)\left[\left({\mathcal{Z}}_1\left(\xi \right)-{\mathcal{Z}}_1\left(\varrho \right)\right)\left(\mathsf{\Phi }\left(\xi \right)-\mathsf{\Phi }\left(\varrho \right)\right)\right]d\xi d\varrho .\hfill \end{array}$$

Consequently, it follows

$$\begin{array}{cc}\hfill & \left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{ħ}_1\left(v_2\right)\right]\left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_2{\mathcal{Z}}_1\mathsf{\Phi }\right)\left(v_2\right)\right]+\left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\mathsf{\Phi }\left(v_2\right)\right]\left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_1{\mathcal{Z}}_1\mathsf{\Phi }\right)\left(v_2\right)\right]\hfill \\ \hfill -& \left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_1{\mathcal{Z}}_1\right)\left(v_2\right)\right]\left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_2\mathsf{\Phi }\right)\left(v_2\right)\right]-\left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_2{\mathcal{Z}}_1\right)\left(v_2\right)\right]\left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_1\mathsf{\Phi }\right)\left(v_2\right)\right]\hfill \\ \hfill =& \frac{1}{{\varpi }^2\left(\theta \right)}{\int }_{v_1}^{v_2}{\int }_{v_1}^{v_2}\frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\xi \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\xi \right)}\varpi \left(\xi \right){\mathcal{F}}^{\prime }\left(\xi \right)\hfill \\ \hfill \times & \frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right){ħ}_1\left(\xi \right){ħ}_2\left(\varrho \right)\left({\mathcal{Z}}_1\left(\xi \right)-{\mathcal{Z}}_1\left(\varrho \right)\right)\left({\int }_{\varrho }^{\xi }{\mathsf{\Phi }}^{\prime }\left(\vartheta \right)d\vartheta \right)d\xi d\varrho .\hfill \end{array}$$

(10)

By utilizing the given condition $v_1\le \varrho \le \theta \le \xi \le v_2$, we get

$$\begin{array}{cc}\hfill & \left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{ħ}_1\left(v_2\right)\right]\left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_2{\mathcal{Z}}_1\mathsf{\Phi }\right)\left(v_2\right)\right]+\left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\mathsf{\Phi }\left(v_2\right)\right]\left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_1{\mathcal{Z}}_1\mathsf{\Phi }\right)\left(v_2\right)\right]\hfill \\ \hfill -& \left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_1{\mathcal{Z}}_1\right)\left(v_2\right)\right]\left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_2\mathsf{\Phi }\right)\left(v_2\right)\right]-\left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_2{\mathcal{Z}}_1\right)\left(x_2\right)\right]\left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_1\mathsf{\Phi }\right)\left(x_2\right)\right]\hfill \\ \hfill =& \frac{1}{{\varpi }^2\left(\theta \right)}{\int }_{v_1}^{v_2}[{\int }_{v_1}^{\theta }\frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right){ħ}_2\left(\varrho \right)\hfill \\ \hfill \times & {\int }_{v_1}^{v_2}\frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\xi \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\xi \right)}\left({\mathcal{Z}}_1\left(\xi \right)-{\mathcal{Z}}_1\left(\varrho \right)\right)\varpi \left(\xi \right){\mathcal{F}}^{\prime }\left(\xi \right){ħ}_1\left(\xi \right)d\xi d\varrho ]\left({\mathsf{\Phi }}^{\prime }\left(\theta \right)\right)d\theta .\hfill \end{array}$$

(11)

Applying (11) for the particular case when ${\mathcal{Z}}_1\left(v\right)=v$, then we can write

$$\begin{array}{cc}\hfill & \left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{ħ}_1\left(v_2\right)\right]\left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{v}_2\left({ħ}_2\mathsf{\Phi }\right)\left(v_2\right)\right]+\left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\mathsf{\Phi }\left(v_2\right)\right]\left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{v}_2\left({ħ}_1\mathsf{\Phi }\right)\left(v_2\right)\right]\hfill \\ \hfill -& \left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{v}_2{ħ}_1\left(v_2\right)\right]\left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_2\mathsf{\Phi }\right)\left(v_2\right)\right]-\left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{v}_2{ħ}_2\left(v_2\right)\right]\left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_1\mathsf{\Phi }\right)\left(v_2\right)\right]\hfill \\ \hfill =& \frac{1}{{\varpi }^2\left(\theta \right)}{\int }_{v_1}^{v_2}[{\int }_{v_1}^{\theta }\frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right){ħ}_2\left(\varrho \right)\hfill \\ \hfill \times & {\int }_{v_1}^{v_2}\frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\xi \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\xi \right)}\left(\xi -\varrho \right)\varpi \left(\xi \right){\mathcal{F}}^{\prime }\left(\xi \right){ħ}_1\left(\xi \right)d\xi d\varrho ]\left({\mathsf{\Phi }}^{\prime }\left(\vartheta \right)\right)d\vartheta \hfill \\ \hfill =& \frac{1}{\varpi \left(\theta \right)}{\int }_{v_1}^{v_2}[\frac{1}{\varpi \left(\theta \right)}{\int }_{v_1}^{v_2}\frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\xi \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\xi \right)}\varpi \left(\xi \right){\mathcal{F}}^{\prime }\left(\xi \right)\xi {ħ}_1\left(\xi \right)d\xi \hfill \\ \hfill \times & {\int }_{v_1}^{\theta }\frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right){ħ}_2\left(\varrho \right)d\varrho -\frac{1}{\varpi \left(\theta \right)}\hfill \\ \hfill \times & {\int }_{v_1}^{v_2}\frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\xi \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\xi \right)}\varpi \left(\xi \right){\mathcal{F}}^{\prime }\left(\xi \right){ħ}_1\left(\xi \right)d\xi \times {\int }_{v_1}^{\theta }\hfill \\ \hfill \times & \frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right)\varrho {ħ}_2\left(\varrho \right)d\varrho ]\left({\mathsf{\Phi }}^{\prime }\left(\theta \right)\right)d\theta .\hfill \end{array}$$

Thus with the aid of (7), the above equation gives,

$$\begin{array}{cc}\hfill & \left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{ħ}_1\left(v_2\right)\right]\left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{v}_2\left({ħ}_2\mathsf{\Phi }\right)\left(v_2\right)\right]+\left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\mathsf{\Phi }\left(v_2\right)\right]\left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{v}_2\left({ħ}_1\mathsf{\Phi }\right)\left(v_2\right)\right]\hfill \\ \hfill -& \left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{v}_2{ħ}_1\left(v_2\right)\right]\left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_2\mathsf{\Phi }\right)\left(v_2\right)\right]-\left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{v}_2{ħ}_2\left(v_2\right)\right]\left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_1\mathsf{\Phi }\right)\left(v_2\right)\right]\hfill \\ \hfill =& \frac{1}{\varpi \left(\theta \right)}{\int }_{v_1}^{v_2}[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{v}_2{ħ}_1\left(v_2\right){\int }_{v_1}^{\theta }\frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right)\varrho {ħ}_2\left(\varrho \right)d\varrho \hfill \\ \hfill -& {}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{ħ}_1\left(v_2\right){\int }_{x_1}^{\vartheta }\varrho \frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right)\varrho {ħ}_2\left(\varrho \right)d\varrho ]\left({\mathsf{\Phi }}^{\prime }\left(\theta \right)\right)d\theta ,\hfill \end{array}$$

which completes the proof. □

Based on Lemma 1, we prove the following theorem.

Theorem 4.

Suppose that the function $\mathcal{F}$ be measurable, increasing, positive and monotone function on $(v_1,v_2)$, and has a continuous derivative ${\mathcal{F}}^{\prime }$ on $(v_1,v_2)$. Assume that $\mathsf{\Phi }:[v_1,v_2]\to \mathbb{R}$ is absolutely continuous on $[v_1,v_2]$, and ${ħ}_1,{ħ}_2:[v_1,v_2]\to {\mathbb{R}}^{+}$ are positive integrable functions. If ${\left({\mathsf{\Phi }}^{\prime }\right)}^2\in L_1[v_1,v_2]$. Then, we have

$$\begin{array}{cc}\hfill & \left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{ħ}_1\left(v_2\right)\right]\left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_2{\mathsf{\Phi }}^2\right)\left(v_2\right)\right]+\left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{ħ}_2\left(v_2\right)\right]\left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_1{\mathsf{\Phi }}^2\right)\left(v_2\right)\right]\hfill \\ \hfill -& 2\left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_1\mathsf{\Phi }\right)\left(v_2\right)\right]\left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_2\mathsf{\Phi }\right)\left(v_2\right)\right]\hfill \\ \hfill \le & \frac{1}{\varpi \left(\theta \right)}{\int }_{v_1}^{v_2}[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{v}_2{ħ}_1\left(v_2\right){\int }_{v_1}^{\theta }\frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right){ħ}_2\left(\varrho \right)d\varrho \hfill \\ \hfill -& {}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{ħ}_1\left(v_2\right){\int }_{v_1}^{\theta }\varrho \frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right){ħ}_2\left(\varrho \right)d\varrho ]{\left({\mathsf{\Phi }}^{\prime }\left(\theta \right)\right)}^{2}d\theta .\hfill \end{array}$$

(12)

Proof. 

By employing the definition (7) and Lemma 1, we obtain

$$\begin{array}{cc}\hfill & \left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{ħ}_1\left(v_2\right)\right]\left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_2{\mathsf{\Phi }}^2\right)\left(v_2\right)\right]+\left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{ħ}_2\left(v_2\right)\right]\left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_1{\mathsf{\Phi }}^2\right)\left(v_2\right)\right]\hfill \\ \hfill -& 2\left[{}_{x_1}^{\mathsf{\Psi }}{\mathcal{I}}_{\kappa }^{\theta }\left({ħ}_1\mathsf{\Phi }\right)\left(x_2\right)\right]\left[{}_{x_1}^{\mathsf{\Psi }}{\mathcal{I}}_{\kappa }^{\theta }\left({ħ}_2\mathsf{\Phi }\right)\left(x_2\right)\right]\hfill \\ \hfill =& \frac{1}{{\varpi }^2\left(\theta \right)}{\int }_{v_1}^{v_2}{\int }_{v_1}^{v_2}\frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\xi \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\xi \right)}\hfill \\ \hfill \times & \frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\xi \right){\mathcal{F}}^{\prime }\left(\xi \right){ħ}_1\left(\xi \right)\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right){ħ}_2\left(\varrho \right){\left(\mathsf{\Phi }\left(\xi \right)-\mathsf{\Phi }\left(\varrho \right)\right)}^{2}d\xi d\varrho \hfill \\ \hfill =& \frac{1}{{\varpi }^2\left(\theta \right)}{\int }_{v_1}^{v_2}{\int }_{v_1}^{v_2}\frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\xi \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\xi \right)}\hfill \\ \hfill \times & \frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\xi \right){\mathcal{F}}^{\prime }\left(\xi \right){ħ}_1\left(\xi \right)\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right){ħ}_2\left(\varrho \right){\left(\xi -\varrho \right)}^2{\left(\frac{\mathsf{\Phi }\left(\xi \right)-\mathsf{\Phi }\left(\varrho \right)}{\xi -\varrho }\right)}^{2}d\xi d\varrho .\hfill \end{array}$$

Consequently, it follows that

$$\begin{array}{cc}\hfill & \left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{ħ}_1\left(v_2\right)\right]\left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_2{\mathsf{\Phi }}^2\right)\left(v_2\right)\right]+\left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{ħ}_2\left(v_2\right)\right]\left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_1{\mathsf{\Phi }}^2\right)\left(v_2\right)\right]\hfill \\ \hfill -& 2\left[{}_{x_1}^{\mathsf{\Psi }}{\mathcal{I}}_{\kappa }^{\theta }\left({ħ}_1\mathsf{\Phi }\right)\left(x_2\right)\right]\left[{}_{x_1}^{\mathsf{\Psi }}{\mathcal{I}}_{\kappa }^{\theta }\left({ħ}_2\mathsf{\Phi }\right)\left(x_2\right)\right]\hfill \\ \hfill =& \frac{1}{{\varpi }^2\left(\theta \right)}{\int }_{v_1}^{v_2}{\int }_{v_1}^{v_2}\frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\xi \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\xi \right)}\hfill \\ \hfill \times & \frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\xi \right){\mathcal{F}}^{\prime }\left(\xi \right){ħ}_1\left(\xi \right)\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right){ħ}_2\left(\varrho \right){\left(\xi -\varrho \right)}^2{\left(\frac{{\int }_{\varrho }^{\xi }{\mathsf{\Phi }}^{\prime }\left(\theta \right)d\theta }{\xi -\varrho }\right)}^{2}d\xi d\varrho .\hfill \end{array}$$

By applying Cauchy-Schwartz inequality [40], we get

$$\begin{array}{cc}\hfill & \left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{ħ}_1\left(v_2\right)\right]\left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_2{\mathsf{\Phi }}^2\right)\left(v_2\right)\right]+\left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{ħ}_2\left(v_2\right)\right]\left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_1{\mathsf{\Phi }}^2\right)\left(v_2\right)\right]\hfill \\ \hfill -& 2\left[{}_{x_1}^{\mathsf{\Psi }}{\mathcal{I}}_{\kappa }^{\theta }\left({ħ}_1\mathsf{\Phi }\right)\left(x_2\right)\right]\left[{}_{x_1}^{\mathsf{\Psi }}{\mathcal{I}}_{\kappa }^{\theta }\left({ħ}_2\mathsf{\Phi }\right)\left(x_2\right)\right]\hfill \\ \hfill =& \frac{1}{{\varpi }^2\left(\theta \right)}{\int }_{v_1}^{v_2}{\int }_{v_1}^{v_2}\frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\xi \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\xi \right)}\hfill \\ \hfill \times & \frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\xi \right){\mathcal{F}}^{\prime }\left(\xi \right){ħ}_1\left(\xi \right)\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right){ħ}_2\left(\varrho \right){\left(\xi -\varrho \right)}^2{\left(\frac{{\int }_{\varrho }^{\xi }{\mathsf{\Phi }}^{\prime }\left(\theta \right)d\theta }{\xi -\varrho }\right)}^{2}d\xi d\varrho \hfill \\ \hfill \le & \frac{1}{{\varpi }^2\left(\theta \right)}{\int }_{v_1}^{v_2}{\int }_{v_1}^{v_2}\frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\xi \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\xi \right)}\hfill \\ \hfill \times & \frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\xi \right){\mathcal{F}}^{\prime }\left(\xi \right){ħ}_1\left(\xi \right)\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right){ħ}_2\left(\varrho \right){\left(\xi -\varrho \right)}^2{\left(\frac{{\left({\int }_{\varrho }^{\xi }d\theta \right)}^{\frac{1}{2}}{\left({\int }_{\varrho }^{\xi }{\left({\mathsf{\Phi }}^{\prime }\left(\theta \right)\right)}^{2}d\theta \right)}^{\frac{1}{2}}}{\xi -\varrho }\right)}^{2}d\xi d\varrho \hfill \\ \hfill \le & \frac{1}{{\varpi }^2\left(\theta \right)}{\int }_{v_1}^{v_2}{\int }_{v_1}^{v_2}\frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\xi \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\xi \right)}\hfill \\ \hfill \times & \frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\xi \right){\mathcal{F}}^{\prime }\left(\xi \right){ħ}_1\left(\xi \right)\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right){ħ}_2\left(\varrho \right){\left(\xi -\varrho \right)}^2\left(\frac{\left({\int }_{\varrho }^{\xi }{\left({\mathsf{\Phi }}^{\prime }\left(\theta \right)\right)}^{2}d\theta \right)}{\xi -\varrho }\right)d\xi d\varrho \hfill \\ \hfill \le & \frac{1}{{\varpi }^2\left(\theta \right)}{\int }_{v_1}^{v_2}{\int }_{v_1}^{v_2}\frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\xi \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\xi \right)}\hfill \\ \hfill \times & \frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\xi \right){\mathcal{F}}^{\prime }\left(\xi \right){ħ}_1\left(\xi \right)\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right){ħ}_2\left(\varrho \right)\left(\xi -\varrho \right)\left({\int }_{\varrho }^{\xi }{\left({\mathsf{\Phi }}^{\prime }\left(\theta \right)\right)}^{2}d\theta \right)d\xi d\varrho \hfill \end{array}$$

(13)

Hence, using (11) and (13) concludes the proof. □

The following new particular results of Theorem 4 can be easily obtained.

Corollary 1.

Suppose that the function $\mathcal{F}$ be measurable, increasing, positive and monotone function on $(v_1,v_2)$, and has a continuous derivative ${\mathcal{F}}^{\prime }$ on $(v_1,v_2)$. Assume that $\mathsf{\Phi }:[v_1,v_2]\to \mathbb{R}$ is absolutely continuous on $[v_1,v_2]$, and ${ħ}_1,{ħ}_2:[v_1,v_2]\to {\mathbb{R}}^{+}$ are positive integrable functions. If ${\left({\mathsf{\Phi }}^{\prime }\right)}^2\in L_1[v_1,v_2]$. Then, we have

$$\begin{array}{cc}\hfill & \left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}1\right]\left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_2{\mathsf{\Phi }}^2\right)\left(v_2\right)\right]+\left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{ħ}_2\left(v_2\right)\right]\left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({\mathsf{\Phi }}^2\right)\left(v_2\right)\right]\hfill \\ \hfill -& 2\left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left(\mathsf{\Phi }\right)\left(v_2\right)\right]\left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_2\mathsf{\Phi }\right)\left(v_2\right)\right]\hfill \\ \hfill \le & \frac{1}{\varpi \left(\theta \right)}{\int }_{v_1}^{v_2}[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{v}_2{\int }_{v_1}^{\theta }\frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right){ħ}_2\left(\varrho \right)d\varrho \hfill \\ \hfill -& {}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}1{\int }_{v_1}^{\theta }\varrho \frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right){ħ}_2\left(\varrho \right)d\varrho ]{\left({\mathsf{\Phi }}^{\prime }\left(\theta \right)\right)}^{2}d\theta .\hfill \end{array}$$

Proof. 

By considering ${ħ}_1\left(\theta \right)=1$, $\theta \in [v_1,v_2]$ in Theorem 4, the desired result is obtained. □

Corollary 2.

Suppose that the function $\mathcal{F}$ be measurable, increasing, positive and monotone function on $(v_1,v_2)$, and has a continuous derivative ${\mathcal{F}}^{\prime }$ on $(v_1,v_2)$. Assume that $\mathsf{\Phi }:[v_1,v_2]\to \mathbb{R}$ is absolutely continuous on $[v_1,v_2]$, and ${ħ}_1,{ħ}_2:[v_1,v_2]\to {\mathbb{R}}^{+}$ are positive integrable functions. If ${\left({\mathsf{\Phi }}^{\prime }\right)}^2\in L_1[v_1,v_2]$. Then, we have

$$\begin{array}{cc}\hfill & \left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{ħ}_1\left(v_2\right)\right]\left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({\mathsf{\Phi }}^2\right)\left(v_2\right)\right]+\left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}1\right]\left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_1{\mathsf{\Phi }}^2\right)\left(v_2\right)\right]\hfill \\ \hfill -& 2\left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_1\mathsf{\Phi }\right)\left(v_2\right)\right]\left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left(\mathsf{\Phi }\right)\left(v_2\right)\right]\hfill \\ \hfill \le & \frac{1}{\varpi \left(\theta \right)}{\int }_{v_1}^{v_2}[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{v}_2{ħ}_1\left(v_2\right){\int }_{v_1}^{\theta }\frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right)d\varrho \hfill \\ \hfill -& {}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{ħ}_1\left(v_2\right){\int }_{v_1}^{\theta }\varrho \frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right)d\varrho ]{\left({\mathsf{\Phi }}^{\prime }\left(\theta \right)\right)}^{2}d\theta .\hfill \end{array}$$

Proof. 

By considering ${ħ}_2\left(\theta \right)=1$, $\theta \in [v_1,v_2]$ in Theorem 4, desired corollary is proven. □

Corollary 3.

Suppose that the function $\mathcal{F}$ be measurable, increasing, positive and monotone function on $(v_1,v_2)$, and has a continuous derivative ${\mathcal{F}}^{\prime }$ on $(v_1,v_2)$. Assume that $\mathsf{\Phi }:[v_1,v_2]\to \mathbb{R}$ is absolutely continuous on $[v_1,v_2]$, and ${ħ}_1,{ħ}_2:[v_1,v_2]\to {\mathbb{R}}^{+}$ are positive integrable functions. If ${\left({\mathsf{\Phi }}^{\prime }\right)}^2\in L_1[v_1,v_2]$. Then, we have

$$\begin{array}{cc}\hfill & \left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}1\right]\left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({\mathsf{\Phi }}^2\right)\left(v_2\right)\right]-{\left[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left(\mathsf{\Phi }\right)\left(v_2\right)\right]}^2\hfill \\ \hfill \le & \frac{1}{\varpi \left(\theta \right)}{\int }_{v_1}^{v_2}[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{v}_2{\int }_{v_1}^{\theta }\frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right)d\varrho \hfill \\ \hfill -& {}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}1{\int }_{v_1}^{\theta }\varrho \frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right)d\varrho ]{\left({\mathsf{\Phi }}^{\prime }\left(\theta \right)\right)}^{2}d\theta .\hfill \end{array}$$

Proof. 

Taking ${ħ}_1\left(\theta \right)={ħ}_2\left(\theta \right)=1$, $\vartheta \in [v_1,v_2]$ in Theorem 4, the desired result is obtained. □

Remark 4.

If we consider $\varpi \left(\theta \right)=1$ and $\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)\right)=\frac{\mathcal{F}{\left(\theta \right)}^{\kappa }}{\mathsf{\Gamma }\left(\kappa \right)}$, then Theorem 4 and Corollaries 1–3 will reduce to the work of Bezziou et al. [27].

Remark 5.

If we consider $\varpi \left(\theta \right)=1$ and $\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)\right)=\frac{\mathcal{F}{\left(\theta \right)}^{\frac{\kappa }{\eta }}}{{\mathsf{\Gamma }}_{\eta }\left(\kappa \right)}$, then Theorem 4 and Corollaries 1–3 will reduce to the work of Rahman et al. [41].

Theorem 5.

Let $\mathcal{F}$ be measurable, increasing, positive and monotone function on $(v_1,v_2)$, and having a continuous derivative ${\mathcal{F}}^{\prime }$ on $(v_1,v_2)$. Assume that $f_1,f_2:[v_1,v_2]\to \mathbb{R}$ are absolutely continuous on $[v_1,v_2]$, and ${ħ}_1,{ħ}_2:[v_1,v_2]\to {\mathbb{R}}^{+}$ are positive integrable. If ${\left(f_1^{\prime }\right)}^2\in L_1[v_1,v_2]$ and ${\left(f_2^{\prime }\right)}^2\in L_1[v_1,v_2]$. Then, we have

$$\begin{array}{cc}\hfill & |{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{ħ}_1\left(v_2\right){}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_2{f}_1{f}_2\right)\left(v_2\right)+{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{ħ}_2\left(v_2\right){}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_1{f}_1{f}_2\right)\left(v_2\right)\hfill \\ \hfill -& {}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_2{f}_1\right)\left(v_2\right){}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_1{f}_2\right)\left(v_2\right)-{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_1{f}_1\right)\left(v_2\right){}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_2{f}_2\right)\left(v_2\right)|\hfill \\ \hfill \le & \frac{1}{\varpi \left(\theta \right)}\left({\int }_{v_1}^{v_2}\right[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{v}_2{ħ}_1\left(v_2\right){\int }_{v_1}^{\theta }\frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right){ħ}_2\left(\varrho \right)d\varrho \hfill \\ \hfill -& {}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{ħ}_1\left(v_2\right){\int }_{v_1}^{\theta }\varrho \frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right){ħ}_2\left(\varrho \right)d\varrho ]{\left(f_1^{\prime }\left(\theta \right)\right)}^{2}d\theta {)}^{\frac{1}{2}}\hfill \\ \hfill \times & \left({\int }_{v_1}^{v_2}\right[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{v}_2{ħ}_1\left(v_2\right){\int }_{v_1}^{\theta }\frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right){ħ}_2\left(\varrho \right)d\varrho \hfill \\ \hfill -& {}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{ħ}_1\left(v_2\right){\int }_{v_1}^{\theta }\varrho \frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right){ħ}_2\left(\varrho \right)d\varrho ]{\left(f_2^{\prime }\left(\theta \right)\right)}^{2}d\theta {)}^{\frac{1}{2}}\hfill \end{array}$$

(14)

Proof. 

Consider the left-hand side of (14), we have

$$\begin{array}{cc}\hfill & |{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{ħ}_2\left(v_2\right){}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_2{f}_1{f}_2\right)\left(v_2\right)+{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{ħ}_2\left(v_2\right){}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_1{f}_1{f}_2\right)\left(v_2\right)\hfill \\ \hfill -& {}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_2{f}_1\right)\left(v_2\right){}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_1{f}_2\right)\left(v_2\right)-{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_1{f}_1\right)\left(v_2\right){}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_2{f}_2\right)\left(v_2\right)|\hfill \\ \hfill \le & \frac{1}{{\varpi }^2\left(\theta \right)}({\int }_{v_1}^{v_2}{\int }_{v_1}^{v_2}\frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\xi \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\xi \right)}\frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\hfill \\ \hfill \times & \varpi \left(\xi \right){\mathcal{F}}^{\prime }\left(\xi \right)\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right){ħ}_1\left(\xi \right){ħ}_2\left(\varrho \right){\left(f_1\left(\xi \right)-f_1\left(\varrho \right)\right)}^{2}d\xi d\varrho {)}^{\frac{1}{2}}\times ({\int }_{v_1}^{v_2}{\int }_{v_1}^{v_2}\frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\xi \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\xi \right)}\hfill \\ \hfill \times & \frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\xi \right){\mathcal{F}}^{\prime }\left(\xi \right)\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right){ħ}_1\left(\xi \right){ħ}_2\left(\varrho \right){\left(f_2\left(\xi \right)-f_2\left(\varrho \right)\right)}^{2}d\xi d\varrho {)}^{\frac{1}{2}}\hfill \\ \hfill \le & \frac{1}{{\varpi }^2\left(\theta \right)}({\int }_{v_1}^{v_2}{\int }_{v_1}^{v_2}\frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\xi \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\xi \right)}\frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\xi \right){\mathcal{F}}^{\prime }\left(\xi \right)\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right){ħ}_1\left(\xi \right){ħ}_2\left(\varrho \right)\hfill \\ \hfill \times & {\left({\int }_{\varrho }^{\xi }{f}_1^{\prime }\left(\theta \right)d\theta \right)}^{2}d\xi d\varrho {)}^{\frac{1}{2}}\times ({\int }_{v_1}^{v_2}{\int }_{v_1}^{v_2}\frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\xi \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\xi \right)}\hfill \\ \hfill \times & \frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\xi \right){\mathcal{F}}^{\prime }\left(\xi \right)\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right){ħ}_1\left(\xi \right){ħ}_2\left(\varrho \right){\left({\int }_{\varrho }^{\xi }{f}_2^{\prime }\left(\theta \right)d\theta \right)}^{2}d\xi d\varrho {)}^{\frac{1}{2}}\hfill \end{array}$$

Applying Cauchy-Schwartz inequality [40] to the above equation yields,

$$\begin{array}{cc}\hfill & |{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{ħ}_2\left(v_2\right){}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_2{f}_1{f}_2\right)\left(v_2\right)+{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{ħ}_2\left(v_2\right){}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_1{f}_1{f}_2\right)\left(v_2\right)\hfill \\ \hfill -& {}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_2{f}_1\right)\left(v_2\right){}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_1{f}_2\right)\left(v_2\right)-{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_1{f}_1\right)\left(v_2\right){}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_2{f}_2\right)\left(v_2\right)|\hfill \\ \hfill \le & \frac{1}{{\varpi }^2\left(\theta \right)}\{{\int }_{v_1}^{v_2}{\int }_{v_1}^{v_2}\frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\xi \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\xi \right)}\frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\xi \right){\mathcal{F}}^{\prime }\left(\xi \right)\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right){ħ}_1\left(\xi \right){ħ}_2\left(\varrho \right)\hfill \\ \hfill \times & {\left({\left({\int }_{\varrho }^{\xi }d\theta \right)}^{\frac{1}{2}}{\left({\int }_{\varrho }^{\xi }{\left(f_1^{\prime }\left(\theta \right)\right)}^{2}d\theta \right)}^{\frac{1}{2}}\right)}^{2}d\xi d\varrho {\}}^{\frac{1}{2}}\times \{{\int }_{v_1}^{v_2}{\int }_{v_1}^{v_2}\frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\xi \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\xi \right)}\hfill \\ \hfill \times & \frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\xi \right){\mathcal{F}}^{\prime }\left(\xi \right)\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right){ħ}_1\left(\xi \right){ħ}_2\left(\varrho \right){\left({\left({\int }_{\varrho }^{\xi }d\theta \right)}^{\frac{1}{2}}{\left({\int }_{\varrho }^{\xi }{\left(f_2^{\prime }\left(\theta \right)\right)}^{2}d\theta \right)}^{\frac{1}{2}}\right)}^{2}d\xi d\varrho {\}}^{\frac{1}{2}}\hfill \\ \hfill \le & \frac{1}{{\varpi }^2\left(\theta \right)}\{{\int }_{v_1}^{v_2}{\int }_{v_1}^{v_2}\frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\xi \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\xi \right)}\frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\xi \right){\mathcal{F}}^{\prime }\left(\xi \right)\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right){ħ}_1\left(\xi \right){ħ}_2\left(\varrho \right)\hfill \\ \hfill \times & \varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right){ħ}_1\left(\xi \right){ħ}_2\left(\varrho \right)\left(\xi -\varrho \right)\left({\int }_{\varrho }^{\xi }{\left(f_1^{\prime }\left(\theta \right)\right)}^{2}d\theta \right)d\xi d\varrho {\}}^{\frac{1}{2}}\times \{{\int }_{v_1}^{v_2}{\int }_{v_1}^{v_2}\frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\xi \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\xi \right)}\hfill \\ \hfill \times & \frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\xi \right){\mathcal{F}}^{\prime }\left(\xi \right)\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right){ħ}_1\left(\xi \right){ħ}_2\left(\varrho \right)\left(\xi -\varrho \right)\left({\int }_{\varrho }^{\xi }{\left(f_2^{\prime }\left(\theta \right)\right)}^{2}d\theta \right)d\xi d\varrho {\}}^{\frac{1}{2}}\hfill \\ \hfill \le & \frac{1}{{\varpi }^2\left(\theta \right)}\left\{{\int }_{v_1}^{v_2}\right({\int }_{v_1}^{v_2}\xi \frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\xi \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\xi \right)}\varpi \left(\xi \right){\mathcal{F}}^{\prime }\left(\xi \right){ħ}_1\left(\xi \right)d\xi \hfill \\ \hfill \times & {\int }_{v_1}^{\theta }\frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right){ħ}_2\left(\varrho \right)d\varrho \hfill \\ \hfill -& {\int }_{v_1}^{v_2}\frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\xi \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\xi \right)}\varpi \left(\xi \right){\mathcal{F}}^{\prime }\left(\xi \right){ħ}_1\left(\xi \right)d\xi \hfill \\ \hfill \times & {\int }_{x_1}^{\theta }\varrho \frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right){ħ}_2\left(\varrho \right)d\varrho ){\left(f_1^{\prime }\left(\theta \right)\right)}^2{\}}^{\frac{1}{2}}\hfill \\ \hfill -& \left\{{\int }_{v_1}^{v_2}\right({\int }_{v_1}^{v_2}\xi \frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\xi \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\xi \right)}\varpi \left(\xi \right){\mathcal{F}}^{\prime }\left(\xi \right){ħ}_1\left(\xi \right)d\xi \hfill \\ \hfill \times & {\int }_{v_1}^{\theta }\frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right){ħ}_2\left(\varrho \right)d\varrho \hfill \\ \hfill -& {\int }_{v_1}^{v_2}\frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\xi \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\xi \right)}\varpi \left(\xi \right){\mathcal{F}}^{\prime }\left(\xi \right){ħ}_1\left(\xi \right)d\xi \hfill \\ \hfill \times & {\int }_{x_1}^{\theta }\varrho \frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right){ħ}_2\left(\varrho \right)d\varrho ){\left(f_2^{\prime }\left(\theta \right)\right)}^2{]}^{\frac{1}{2}}.\hfill \end{array}$$

In view of (7), we get the desired proof of (14). □

Corollary 4.

Let $\mathcal{F}$ be measurable, increasing, positive and monotone function on $(v_1,v_2)$, and having a continuous derivative ${\mathcal{F}}^{\prime }$ on $(v_1,v_2)$. Assume that $f_1,f_2:[v_1,v_2]\to \mathbb{R}$ are absolutely continuous on $[v_1,v_2]$, and ${ħ}_1,{ħ}_2:[v_1,v_2]\to {\mathbb{R}}^{+}$ are positive integrable. If ${\left(f_1^{\prime }\right)}^2\in L_1[v_1,v_2]$ and ${\left(f_2^{\prime }\right)}^2\in L_1[v_1,v_2]$. Then, we have

$$\begin{array}{cc}\hfill & |{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left(1\right){}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_2{f}_1{f}_2\right)\left(v_2\right)+{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{ħ}_2\left(v_2\right){}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left(f_1{f}_2\right)\left(v_2\right)\hfill \\ \hfill -& {}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_2{f}_1\right)\left(v_2\right){}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left(f_2\right)\left(v_2\right)-{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left(f_1\right)\left(v_2\right){}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_2{f}_2\right)\left(v_2\right)|\hfill \\ \hfill \le & \frac{1}{\varpi \left(\theta \right)}\left({\int }_{v_1}^{v_2}\right[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{v}_2{\int }_{v_1}^{\theta }\frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right){ħ}_2\left(\varrho \right)d\varrho \hfill \\ \hfill -& {}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left(1\right){\int }_{v_1}^{\theta }\varrho \frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right){ħ}_2\left(\varrho \right)d\varrho ]{\left(f_1^{\prime }\left(\theta \right)\right)}^{2}d\theta {)}^{\frac{1}{2}}\hfill \\ \hfill \times & \left({\int }_{v_1}^{v_2}\right[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{v}_2{\int }_{v_1}^{\theta }\frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right){ħ}_2\left(\varrho \right)d\varrho \hfill \\ \hfill -& {}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left(1\right){\int }_{v_1}^{\theta }\varrho \frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right){ħ}_2\left(\varrho \right)d\varrho ]{\left(f_2^{\prime }\left(\theta \right)\right)}^{2}d\theta {)}^{\frac{1}{2}}\hfill \end{array}$$

Proof. 

Applying Theorem 5 for ${ħ}_1\left(\theta \right)=1$, $\theta \in [v_1,v_2]$, the desired result is obtained. □

Corollary 5.

Let $\mathcal{F}$ be measurable, increasing, positive and monotone function on $(v_1,v_2)$, and having a continuous derivative ${\mathcal{F}}^{\prime }$ on $(v_1,v_2)$. Assume that $f_1,f_2:[v_1,v_2]\to \mathbb{R}$ are absolutely continuous on $[v_1,v_2]$, and ${ħ}_1,{ħ}_2:[v_1,v_2]\to {\mathbb{R}}^{+}$ are positive integrable. If ${\left(f_1^{\prime }\right)}^2\in L_1[v_1,v_2]$ and ${\left(f_2^{\prime }\right)}^2\in L_1[v_1,v_2]$. Then, we have

$$\begin{array}{cc}\hfill & |{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{ħ}_1\left(v_2\right){}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left(f_1{f}_2\right)\left(v_2\right)+{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left(1\right){}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_1{f}_1{f}_2\right)\left(v_2\right)\hfill \\ \hfill -& {}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left(f_1\right)\left(v_2\right){}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_1{f}_2\right)\left(v_2\right)-{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_1{f}_1\right)\left(v_2\right){}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left(f_2\right)\left(v_2\right)|\hfill \\ \hfill \le & \frac{1}{\varpi \left(\theta \right)}\left({\int }_{v_1}^{v_2}\right[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{v}_2{ħ}_1\left(v_2\right){\int }_{v_1}^{\theta }\frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right)d\varrho \hfill \\ \hfill -& {}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{ħ}_1\left(v_2\right){\int }_{v_1}^{\theta }\varrho \frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right)d\varrho ]{\left(f_1^{\prime }\left(\theta \right)\right)}^{2}d\theta {)}^{\frac{1}{2}}\hfill \\ \hfill \times & \left({\int }_{v_1}^{v_2}\right[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{v}_2{ħ}_1\left(v_2\right){\int }_{v_1}^{\theta }\frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right)d\varrho \hfill \\ \hfill -& {}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{ħ}_1\left(v_2\right){\int }_{v_1}^{\theta }\varrho \frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right)d\varrho ]{\left(f_2^{\prime }\left(\theta \right)\right)}^{2}d\theta {)}^{\frac{1}{2}}\hfill \end{array}$$

Proof. 

Applying Theorem 5 for ${ħ}_2\left(\theta \right)=1$, $\theta \in [v_1,v_2]$, the desired result is obtained. □

Corollary 6.

Let $\mathcal{F}$ be measurable, increasing, positive and monotone function on $(v_1,v_2)$, and having a continuous derivative ${\mathcal{F}}^{\prime }$ on $(v_1,v_2)$. Assume that $f_1,f_2:[v_1,v_2]\to \mathbb{R}$ are absolutely continuous on $[v_1,v_2]$, and ${ħ}_1,{ħ}_2:[v_1,v_2]\to {\mathbb{R}}^{+}$ are positive integrable. If ${\left(f_1^{\prime }\right)}^2\in L_1[v_1,v_2]$ and ${\left(f_2^{\prime }\right)}^2\in L_1[v_1,v_2]$. Then, we have

$$\begin{array}{cc}\hfill & |{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left(1\right){}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left(f_1{f}_2\right)\left(v_2\right)-{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left(f_1\right)\left(v_2\right){}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left(f_2\right)\left(v_2\right)|\hfill \\ \hfill \le & \frac{1}{\varpi \left(\theta \right)}\left({\int }_{v_1}^{v_2}\right[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{v}_2{\int }_{v_1}^{\theta }\frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right)d\varrho \hfill \\ \hfill -& {}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left(1\right){\int }_{v_1}^{\theta }\varrho \frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right)d\varrho ]{\left(f_1^{\prime }\left(\theta \right)\right)}^{2}d\theta {)}^{\frac{1}{2}}\hfill \\ \hfill \times & \left({\int }_{v_1}^{v_2}\right[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{v}_2{\int }_{v_1}^{\theta }\frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right)d\varrho \hfill \\ \hfill -& {}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left(1\right){\int }_{v_1}^{\theta }\varrho \frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right)d\varrho ]{\left(f_2^{\prime }\left(\theta \right)\right)}^{2}d\theta {)}^{\frac{1}{2}}\hfill \end{array}$$

Proof. 

Applying Theorem 5 for ${ħ}_1\left(\theta \right)={ħ}_2\left(\theta \right)=1$, $\theta \in [v_1,v_2]$, the desired result is obtained. □

Theorem 6.

Let $\mathcal{F}$ be measurable, increasing, positive and monotone function on $(v_1,v_2)$, and having a continuous derivative ${\mathcal{F}}^{\prime }$ on $(v_1,v_2)$. Assume that $f_1:[v_1,v_2]\to \mathbb{R}$ is absolutely continuous function on $[v_1,v_2]$, and $f_2:[v_1,v_2]\to \mathbb{R}$ is non-decreasing on $[v_1,v_2]$. Moreover, suppose that both ${ħ}_1,{ħ}_2:[v_1,v_2]\to {\mathbb{R}}^{+}$ are positive integrable. If $\left(f_1^{\prime }\right)\in L^{\infty }[v_1,v_2]$, then we have

$$\begin{array}{cc}\hfill & |{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{ħ}_1\left(v_2\right){}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_2{f}_1{f}_2\right)\left(v_2\right)+{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{ħ}_2\left(v_2\right){}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_1{f}_1{f}_2\right)\left(v_2\right)\hfill \\ \hfill -& {}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_1{f}_1\right)\left(v_2\right){}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_2{f}_2\right)\left(v_2\right)-{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_2{f}_1\right){\left(v_2\right)}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_1{f}_2\right)\left(v_2\right)|\hfill \\ \hfill \le & \frac{\left|\right|f_1^{\prime }{\left|\right|}_{\infty }}{\varpi \left(\theta \right)}[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{v}_2{ħ}_1\left(v_2\right){\int }_{v_1}^{\theta }\frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right){ħ}_2\left(\varrho \right)d\varrho \hfill \\ \hfill -& {}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{ħ}_1\left(v_2\right){\int }_{v_1}^{\theta }\varrho \frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right){ħ}_2\left(\varrho \right)d\varrho ]{\int }_{v_1}^{v_2}{f}_2^{\prime }\left(\theta \right)d\theta .\hfill \end{array}$$

(15)

Proof. 

Consider the left-hand side of (15), we have

$$\begin{array}{cc}\hfill & |{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{ħ}_1\left(v_2\right){}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_2{f}_1{f}_2\right)\left(v_2\right)+{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{ħ}_2\left(v_2\right){}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_1{f}_1{f}_2\right)\left(v_2\right)\hfill \\ \hfill -& {}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_2{f}_1\right)\left(v_2\right){}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_1{f}_2\right)\left(v_2\right)-{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_1{f}_1\right)\left(v_2\right){}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_2{f}_2\right)\left(v_2\right)|\hfill \\ \hfill \le & \frac{1}{{\varpi }^2\left(\theta \right)}|{\int }_{v_1}^{v_2}{\int }_{v_1}^{v_2}\frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\xi \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\xi \right)}\frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\xi \right){\mathcal{F}}^{\prime }\left(\xi \right)\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right){ħ}_1\left(\xi \right){ħ}_2\left(\varrho \right)\hfill \\ \hfill \times & \left[\left(f_1\left(\xi \right)-f_1\left(\varrho \right)\right)\left(f_2\left(\xi \right)-f_2\left(\varrho \right)\right)\right]d\xi d\varrho |\hfill \\ \hfill \le & \frac{1}{{\varpi }^2\left(\theta \right)}|{\int }_{v_1}^{v_2}{\int }_{v_1}^{v_2}\frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\xi \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\xi \right)}\frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\xi \right){\mathcal{F}}^{\prime }\left(\xi \right)\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right){ħ}_1\left(\xi \right){ħ}_2\left(\varrho \right)\hfill \\ \hfill \times & \left|\frac{\left(f_1\left(\xi \right)-f_1\left(\varrho \right)\right)}{\xi -\varrho }\right|\left|(\xi -\varrho )\left(f_2\left(\xi \right)-f_2\left(\varrho \right)\right)\right|d\xi d\varrho \hfill \\ \hfill \le & \frac{\left|\right|f_1^{\prime }{\left|\right|}_{\infty }}{{\varpi }^2\left(\theta \right)}|{\int }_{v_1}^{v_2}{\int }_{v_1}^{v_2}\frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\xi \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\xi \right)}\frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\xi \right){\mathcal{F}}^{\prime }\left(\xi \right)\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right){ħ}_1\left(\xi \right){ħ}_2\left(\varrho \right)\hfill \\ \hfill \times & \left(\xi -\varrho \right)\left({\int }_{v_1}^{v_2}{f}_2^{\prime }\left(\vartheta \right)d\vartheta \right)d\xi d\varrho \hfill \\ \hfill \le & \frac{\left|\right|f_1^{\prime }{\left|\right|}_{\infty }}{{\varpi }^2\left(\theta \right)}[{\int }_{v_1}^{v_2}\frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\xi \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\xi \right)}\xi {\mathcal{F}}^{\prime }\left(\xi \right)\varpi \left(\xi \right){ħ}_2\left(\xi \right)d\xi {\int }_{v_1}^{\theta }\frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\xi \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}{\mathcal{F}}^{\prime }\left(\varrho \right)\varpi \left(\varrho \right){ħ}_2\left(\varrho \right)d\varrho \hfill \\ \hfill -& {\int }_{v_1}^{v_2}\frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\xi \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\xi \right)}{\mathcal{F}}^{\prime }\left(\xi \right)\varpi \left(\varrho \right){ħ}_2\left(\xi \right)d\xi {\int }_{v_1}^{\theta }\varrho \frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\xi \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}{\mathcal{F}}^{\prime }\left(\varrho \right)\varpi \left(\varrho \right){ħ}_2\left(\varrho \right)d\varrho ]{\int }_{v_1}^{v_2}{f}_2^{\prime }\left(\theta \right)d\theta \hfill \end{array}$$

Hence taking (7) into account, the proof of (15) is completed.

□

Corollary 7.

Let $\mathcal{F}$ be measurable, increasing, positive and monotone function on $(v_1,v_2)$, and having a continuous derivative ${\mathcal{F}}^{\prime }$ on $(v_1,v_2)$. Assume that $f_1:[v_1,v_2]\to \mathbb{R}$ is absolutely continuous function on $[v_1,v_2]$, and $f_2:[v_1,v_2]\to \mathbb{R}$ is non-decreasing on $[v_1,v_2]$. Moreover, suppose that both ${ħ}_1,{ħ}_2:[v_1,v_2]\to {\mathbb{R}}^{+}$ are positive integrable. If $\left(f_1^{\prime }\right)\in L^{\infty }[v_1,v_2]$, then we have

$$\begin{array}{cc}\hfill & |{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left(1\right){}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_2{f}_1{f}_2\right)\left(v_2\right)+{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{ħ}_2\left(v_2\right){}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left(f_1{f}_2\right)\left(v_2\right)\hfill \\ \hfill -& {}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left(f_1\right)\left(v_2\right){}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_2{f}_2\right)\left(v_2\right)-{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_2{f}_1\right){\left(v_2\right)}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left(f_2\right)\left(v_2\right)|\hfill \\ \hfill \le & \frac{\left|\right|f_1^{\prime }{\left|\right|}_{\infty }}{\varpi \left(\theta \right)}[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{v}_2{\int }_{v_1}^{\theta }\frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right){ħ}_2\left(\varrho \right)d\varrho \hfill \\ \hfill -& {}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left(1\right){\int }_{v_1}^{\theta }\varrho \frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right){ħ}_2\left(\varrho \right)d\varrho ]{\int }_{v_1}^{v_2}{f}_2^{\prime }\left(\theta \right)d\theta .\hfill \end{array}$$

Proof. 

Applying Theorem 6 for ${ħ}_1\left(\theta \right)=1$, $\theta \in [v_1,v_2]$, the desired result is obtained. □

Corollary 8.

Let $\mathcal{F}$ be measurable, increasing, positive and monotone function on $(v_1,v_2)$, and having a continuous derivative ${\mathcal{F}}^{\prime }$ on $(v_1,v_2)$. Assume that $f_1:[v_1,v_2]\to \mathbb{R}$ is absolutely continuous function on $[v_1,v_2]$, and $f_2:[v_1,v_2]\to \mathbb{R}$ is non-decreasing on $[v_1,v_2]$. Moreover, suppose that both ${ħ}_1,{ħ}_2:[v_1,v_2]\to {\mathbb{R}}^{+}$ are positive integrable. If $\left(f_1^{\prime }\right)\in L^{\infty }[v_1,v_2]$, then we have

$$\begin{array}{cc}\hfill & |{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{ħ}_1\left(v_2\right){}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left(f_1{f}_2\right)\left(v_2\right)+{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left(1\right){}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left(f_1{f}_2\right)\left(v_2\right)\hfill \\ \hfill -& {}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_1{f}_1\right)\left(v_2\right){}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left(f_2\right)\left(v_2\right)-{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left(f_1\right){\left(v_2\right)}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_1{f}_2\right)\left(v_2\right)|\hfill \\ \hfill \le & \frac{\left|\right|f_1^{\prime }{\left|\right|}_{\infty }}{\varpi \left(\theta \right)}[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{v}_2{ħ}_1\left(v_2\right){\int }_{v_1}^{\theta }\frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right)d\varrho \hfill \\ \hfill -& {}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{ħ}_1\left(v_2\right){\int }_{v_1}^{\theta }\varrho \frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right)d\varrho ]{\int }_{v_1}^{v_2}{f}_2^{\prime }\left(\theta \right)d\theta .\hfill \end{array}$$

Proof. 

Applying Theorem 6 for ${ħ}_2\left(\theta \right)=1$, $\theta \in [v_1,v_2]$, the desired result is obtained. □

Corollary 9.

Let $\mathcal{F}$ be measurable, increasing, positive and monotone function on $(v_1,v_2)$, and having a continuous derivative ${\mathcal{F}}^{\prime }$ on $(v_1,v_2)$. Assume that $f_1:[v_1,v_2]\to \mathbb{R}$ is absolutely continuous function on $[v_1,v_2]$, and $f_2:[v_1,v_2]\to \mathbb{R}$ is non-decreasing on $[v_1,v_2]$. Moreover, suppose that both ${ħ}_1,{ħ}_2:[v_1,v_2]\to {\mathbb{R}}^{+}$ are positive integrable. If $\left(f_1^{\prime }\right)\in L^{\infty }[v_1,v_2]$, then we have

$$\begin{array}{cc}\hfill & |{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left(1\right){}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left(f_1{f}_2\right)\left(v_2\right)-{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left(f_1\right)\left(v_2\right){}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left(f_2\right)\left(v_2\right)|\hfill \\ \hfill \le & \frac{\left|\right|f_1^{\prime }{\left|\right|}_{\infty }}{2\varpi \left(\theta \right)}[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{v}_2{\int }_{v_1}^{\theta }\frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right)d\varrho \hfill \\ \hfill -& {}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left(1\right){\int }_{v_1}^{\theta }\varrho \frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right)d\varrho ]{\int }_{v_1}^{v_2}{f}_2^{\prime }\left(\theta \right)d\theta .\hfill \end{array}$$

Proof. 

Applying Theorem 6 for ${ħ}_1\left(\theta \right)={ħ}_2\left(\theta \right)=1$, $\theta \in [v_1,v_2]$, the desired result is obtained. □

Theorem 7.

Let $\mathcal{F}$ be measurable, increasing, positive and monotone function on $(v_1,v_2)$ and having continuous derivative ${\mathcal{F}}^{\prime }$ on $(v_1,v_2)$. Assume that $f_1,f_2:[v_1,v_2]\to \mathbb{R}$ are absolutely continuous on $[v_1,v_2]$ and $f_2:[v_1,v_2]\to \mathbb{R}$ is nondecreasing on $[v_1,v_2]$. Suppose that ${ħ}_1,{ħ}_2:[v_1,v_2]\to {\mathbb{R}}^{+}$ are positive integrable. If $f_1^{\prime },f_2^{\prime }\in L^{\infty }[v_1,v_2]$, then we have

$$\begin{array}{cc}\hfill & |{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{ħ}_1\left(v_2\right){}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_2{f}_1{f}_2\right)\left(v_2\right)+{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{ħ}_2\left(v_2\right){}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_1{f}_1{f}_2\right)\left(v_2\right)\hfill \\ \hfill -& {}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_2{f}_1\right)\left(v_2\right){}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_1{f}_2\right)\left(x_2\right)-{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_1{f}_1\right)\left(v_2\right){}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_2{f}_2\right)\left(v_2\right)|\hfill \\ \hfill \le & \frac{\left|\right|f_1^{\prime }{\left|\right|}_{\infty }\left|\right|f_2^{\prime }{\left|\right|}_{\infty }}{\varpi \left(\theta \right)}[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{v}_2{ħ}_1\left(v_2\right){\int }_{v_1}^{\theta }\frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right){ħ}_2\left(\varrho \right)d\varrho \hfill \\ \hfill -& 2{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{v}_2{ħ}_1\left(v_2\right){\int }_{v_1}^{\theta }\varrho \frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right){ħ}_2\left(\varrho \right)d\varrho \hfill \\ \hfill +& {}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{ħ}_1\left(v_2\right){\int }_{v_1}^{\theta }{\varrho }^2\frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right){ħ}_2\left(\varrho \right)d\varrho ].\hfill \end{array}$$

(16)

Proof. 

Consider the left-hand side of (16), we have

$$\begin{array}{cc}\hfill & |{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{ħ}_2\left(v_2\right){}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_2{f}_1{f}_2\right)\left(v_2\right)+{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{ħ}_2\left(v_2\right){}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_1{f}_1{f}_2\right)\left(v_2\right)\hfill \\ \hfill -& {}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_2{f}_1\right)\left(v_2\right){}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_1{f}_2\right)\left(x_2\right)-{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_1{f}_1\right)\left(v_2\right){}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_2{f}_2\right)\left(v_2\right)|\hfill \\ \hfill \le & \frac{1}{{\varpi }^2\left(\theta \right)}|{\int }_{v_1}^{v_2}{\int }_{v_1}^{v_2}\frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\xi \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\xi \right)}\frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\hfill \\ \hfill \times & \varpi \left(\xi \right){\mathcal{F}}^{\prime }\left(\xi \right){ħ}_1\left(\xi \right)\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right){ħ}_2\left(\varrho \right)\left[\left(f_1\left(\xi \right)-f_1\left(\varrho \right)\right)\left(f_2\left(\xi \right)-f_2\left(\varrho \right)\right)\right]d\xi d\varrho |\hfill \\ \hfill \le & \frac{1}{{\varpi }^2\left(\theta \right)}{\int }_{v_1}^{v_2}{\int }_{v_1}^{v_2}\frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\xi \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\xi \right)}\frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\xi \right){\mathcal{F}}^{\prime }\left(\xi \right){ħ}_1\left(\xi \right)\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right){ħ}_2\left(\varrho \right)\hfill \\ \hfill \times & \left|\frac{\left(f_1\left(\xi \right)-f_1\left(\varrho \right)\right)}{\xi -\varrho }\right|\left|\frac{\left(f_2\left(\xi \right)-f_2\left(\varrho \right)\right)}{\xi -\varrho }\right|{(\xi -\varrho )}^{2}d\xi d\varrho \hfill \\ \hfill \le & \frac{\left|\right|f_1^{\prime }{\left|\right|}_{\infty }\left|\right|f_2^{\prime }{\left|\right|}_{\infty }}{{\varpi }^2\left(\theta \right)}{\int }_{v_1}^{v_2}{\int }_{v_1}^{\theta }\frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\xi \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\xi \right)}\frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\xi \right){\mathcal{F}}^{\prime }\left(\xi \right){ħ}_1\left(\xi \right)\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right){ħ}_2\left(\varrho \right)\hfill \\ \hfill \times & \left({\xi }^2-2\xi \varrho +{\varrho }^2\right)d\xi d\varrho \hfill \\ \hfill \le & \frac{\left|\right|f_1^{\prime }{\left|\right|}_{\infty }\left|\right|f_2^{\prime }{\left|\right|}_{\infty }}{{\varpi }^2\left(\theta \right)}[{\int }_{v_1}^{v_2}\frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\xi \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\xi \right)}{\xi }^2\varpi \left(\xi \right){\mathcal{F}}^{\prime }\left(\xi \right){ħ}_1\left(\xi \right)d\xi {\int }_{v_1}^{\theta }\frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right){ħ}_2\left(\varrho \right)d\varrho \hfill \\ \hfill -& 2{\int }_{v_1}^{v_2}\frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\xi \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\xi \right)}\xi \varpi \left(\xi \right){\mathcal{F}}^{\prime }\left(\xi \right){ħ}_1\left(\xi \right)d\xi {\int }_{v_1}^{\theta }\varrho \frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right){ħ}_2\left(\varrho \right)d\varrho \hfill \\ \hfill +& {\int }_{v_1}^{v_2}\frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\xi \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\xi \right)}\varpi \left(\xi \right){\mathcal{F}}^{\prime }\left(\xi \right){ħ}_1\left(\xi \right)d\xi {\int }_{v_1}^{\theta }{\varrho }^2\frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right){ħ}_2\left(\varrho \right)d\varrho ].\hfill \end{array}$$

Hence, by using (7), the proof of the theorem is completed. □

Corollary 10.

Let $\mathcal{F}$ be measurable, increasing, positive and monotone function on $(v_1,v_2)$ and having continuous derivative ${\mathcal{F}}^{\prime }$ on $(v_1,v_2)$. Assume that $f_1,f_2:[v_1,v_2]\to \mathbb{R}$ are absolutely continuous on $[v_1,v_2]$ and $f_2:[v_1,v_2]\to \mathbb{R}$ is nondecreasing on $[v_1,v_2]$. Suppose that ${ħ}_1,{ħ}_2:[v_1,v_2]\to {\mathbb{R}}^{+}$ are positive integrable. If $f_1^{\prime },f_2^{\prime }\in L^{\infty }[v_1,v_2]$, then we have

$$\begin{array}{cc}\hfill & |{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left(1\right){}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_2{f}_1{f}_2\right)\left(v_2\right)+{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{ħ}_2\left(v_2\right){}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left(f_1{f}_2\right)\left(v_2\right)\hfill \\ \hfill -& {}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_2{f}_1\right)\left(v_2\right){}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left(f_2\right)\left(x_2\right)-{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left(f_1\right)\left(v_2\right){}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_2{f}_2\right)\left(v_2\right)|\hfill \\ \hfill \le & \frac{\left|\right|f_1^{\prime }{\left|\right|}_{\infty }\left|\right|f_2^{\prime }{\left|\right|}_{\infty }}{\varpi \left(\theta \right)}[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{v}_2{\int }_{v_1}^{\theta }\frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right){ħ}_2\left(\varrho \right)d\varrho \hfill \\ \hfill -& 2{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{v}_2{\int }_{v_1}^{\theta }\varrho \frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right){ħ}_2\left(\varrho \right)d\varrho \hfill \\ \hfill +& {}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left(1\right){\int }_{v_1}^{\theta }{\varrho }^2\frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right){ħ}_2\left(\varrho \right)d\varrho ].\hfill \end{array}$$

Proof. 

Setting ${ħ}_1\left(\theta \right)=1$, $\theta \in [v_1,v_2]$ in Theorem 7, then the desired result is obtained. □

Corollary 11.

Let $\mathcal{F}$ be measurable, increasing, positive and monotone function on $(v_1,v_2)$ and having continuous derivative ${\mathcal{F}}^{\prime }$ on $(v_1,v_2)$. Assume that $f_1,f_2:[v_1,v_2]\to \mathbb{R}$ are absolutely continuous on $[v_1,v_2]$ and $f_2:[v_1,v_2]\to \mathbb{R}$ is nondecreasing on $[v_1,v_2]$. Suppose that ${ħ}_1,{ħ}_2:[v_1,v_2]\to {\mathbb{R}}^{+}$ are positive integrable. If $f_1^{\prime },f_2^{\prime }\in L^{\infty }[v_1,v_2]$, then we have

$$\begin{array}{cc}\hfill & |{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{ħ}_1\left(v_2\right){}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left(f_1{f}_2\right)\left(v_2\right)+{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left(1\right){}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_1{f}_1{f}_2\right)\left(v_2\right)\hfill \\ \hfill -& {}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left(f_1\right)\left(v_2\right){}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_1{f}_2\right)\left(x_2\right)-{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left({ħ}_1{f}_1\right)\left(v_2\right){}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left(f_2\right)\left(v_2\right)|\hfill \\ \hfill \le & \frac{\left|\right|f_1^{\prime }{\left|\right|}_{\infty }\left|\right|f_2^{\prime }{\left|\right|}_{\infty }}{\varpi \left(\theta \right)}[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{v}_2{ħ}_1\left(v_2\right){\int }_{v_1}^{\theta }\frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right)d\varrho \hfill \\ \hfill -& 2{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{v}_2{ħ}_1\left(v_2\right){\int }_{v_1}^{\theta }\varrho \frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right)d\varrho \hfill \\ \hfill +& {}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{ħ}_1\left(v_2\right){\int }_{v_1}^{\theta }{\varrho }^2\frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right)d\varrho ].\hfill \end{array}$$

Proof. 

Setting ${ħ}_2\left(\theta \right)=1$, $\theta \in [v_1,v_2]$ in Theorem 7, then the desired result is proven. □

Corollary 12.

Let $\mathcal{F}$ be measurable, increasing, positive and monotone function on $(v_1,v_2)$ and having continuous derivative ${\mathcal{F}}^{\prime }$ on $(v_1,v_2)$. Assume that $f_1,f_2:[v_1,v_2]\to \mathbb{R}$ are absolutely continuous on $[v_1,v_2]$ and $f_2:[v_1,v_2]\to \mathbb{R}$ is nondecreasing on $[v_1,v_2]$. Suppose that ${ħ}_1,{ħ}_2:[v_1,v_2]\to {\mathbb{R}}^{+}$ are positive integrable. If $f_1^{\prime },f_2^{\prime }\in L^{\infty }[v_1,v_2]$, then we have

$$\begin{array}{cc}\hfill & |{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left(1\right){}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left(f_1{f}_2\right)\left(v_2\right)+{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left(1\right){}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left(f_1{f}_2\right)\left(v_2\right)\hfill \\ \hfill -& {}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left(f_1\right)\left(v_2\right){}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left(f_2\right)\left(x_2\right)-{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left(f_1\right)\left(v_2\right){}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}\left(f_2\right)\left(v_2\right)|\hfill \\ \hfill \le & \frac{\left|\right|f_1^{\prime }{\left|\right|}_{\infty }\left|\right|f_2^{\prime }{\left|\right|}_{\infty }}{\varpi \left(\theta \right)}[{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{v}_2{\int }_{v_1}^{\theta }\frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right)d\varrho \hfill \\ \hfill -& 2{}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{v}_2{\int }_{v_1}^{\theta }\varrho \frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right)d\varrho \hfill \\ \hfill +& {}_{\varpi }^{\mathcal{F}}{\mathcal{I}}_{v_1+}^{\mathsf{\Psi }}{\int }_{v_1}^{\theta }{\varrho }^2\frac{\mathsf{\Psi }\left(\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)\right)}{\mathcal{F}\left(v_2\right)-\mathcal{F}\left(\varrho \right)}\varpi \left(\varrho \right){\mathcal{F}}^{\prime }\left(\varrho \right)d\varrho ].\hfill \end{array}$$

Proof. 

Setting ${ħ}_1\left(\theta \right)={ħ}_2\left(\theta \right)=1$, $\theta \in [v_1,v_2]$ in Theorem 7, then the desired result is obtained. □

Remark 6.

One can easily derive some new inequalities by applying the following conditions.

i. Setting ${ħ}_1\left(\theta \right)={ħ}_2\left(\vartheta \right)$, $\mathcal{F}\left(\theta \right)=\theta$ and $\mathsf{\Psi }\left(\mathcal{F}\right(\theta \left)\right)=\theta$ throughout in the paper.

ii. Setting ${ħ}_1\left(\theta \right)={ħ}_2\left(\theta \right)=1$ and $\mathcal{F}\left(\theta \right)=\theta$ and $\mathsf{\Psi }\left(\mathcal{F}\right(\theta \left)\right)=\theta$ throughout in the paper.

Remark 7.

Throughout in this article, if we put $\varpi \left(\theta \right)=1$ and $\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)\right)=\frac{\mathcal{F}{\left(\theta \right)}^{\kappa }}{\mathsf{\Gamma }\left(\kappa \right)}$, then all the inequalities will reduce to the work of Bezziou et al. [27].

Remark 8.

Throughout in this article, if we consider $\varpi \left(\theta \right)=1$ and $\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)\right)=\frac{\mathcal{F}{\left(\theta \right)}^{\frac{\kappa }{\eta }}}{{\mathsf{\Gamma }}_{\eta }\left(\kappa \right)}$, then all the inequalities will reduce to the work of Rahman et al. [41].

Remark 9.

Taking ${ħ}_1\left(\theta \right)={ħ}_2\left(\theta \right)$, $\varpi \left(\theta \right)=1$, $\mathsf{\Psi }\left(\mathcal{F}\right(\theta \left)\right)=\theta$ and $\mathsf{\Psi }\left(\mathcal{F}\left(\theta \right)\right)=\frac{{\left(\theta \right)}^{\kappa }}{\mathsf{\Gamma }\left(\kappa \right)}$ in Theorems 4–7, the results of Bezziou et al. [42] are restored.

3. Concluding Remarks

In the study of mathematics and related subjects, mathematical inequalities are extremely important. Fractional integral inequalities are now useful in determining the uniqueness of fractional partial differential equation solutions. They also guarantee the boundedness of fractional boundary value problem solutions. These suggestions have promoted the future research in the subject of integral inequalities to investigate the extensions of integral inequalities using fractional calculus operators. In the present investigation, we have proposed some double-weighted generalized fractional integral inequalities by utilizing more generalized class of fractional integrals associated with integrable, measurable, positive and monotone function $\mathcal{F}$ in its kernel. The derived inequalities are more general than the existing inequalities cited therein. All the classical inequalities can be easily restored by applying specific conditions on $\mathcal{F}$ and $\mathsf{\Psi }\left(\theta \right)$ given in Remark 3. Also, we can derive some new weighted type double fractional integral inequalities by applying specific conditions on $\mathcal{F}$ and $\mathsf{\Psi }\left(\theta \right)$ given in Remark 2. In future research, some new other type of inequalities will be derived by employing the proposed operator. The special cases of the obtained result can be found in [24,25,27,41,42].