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On generalized fractional integral inequalities for the monotone weighted Chebyshev functionals

Abstract

In this paper, we establish the generalized Riemann–Liouville (RL) fractional integrals in the sense of another increasing, positive, monotone, and measurable function Ψ. We determine certain new double-weighted type fractional integral inequalities by utilizing the said integrals. We also give some of the new particular inequalities of the main result. Note that we can form various types of new inequalities of fractional integrals by employing conditions on the function Ψ given in the paper. We present some corollaries as particular cases of the main results.

Introduction

The Chebyshev functional is given by (see [7, 10])

$$\begin{aligned} \mathscr {T}(\mathscr {U},\mathscr {V},\mu ) =& \int _{x_{1}}^{x_{2}}\mu ( \tau )\,d\tau \int _{x_{1}}^{x_{2}} \mu (\tau )\mathscr {U}(\tau ) \mathscr {V}(\tau )\,d\tau \\ &{}- \int _{x_{1}}^{x_{2}}\mu (\tau )\mathscr {U}( \tau )\,d \tau \int _{x_{1}}^{x_{2}} \mu (\tau )\mathscr {V}(\tau )\,d \tau , \end{aligned}$$
(1)

where \(\mathscr {U}\) and \(\mathscr {V}\) are integrable functions on \([x_{1},x_{2}]\), and μ is a positive integrable function on \([x_{1},x_{2}]\). Applications of functional (1) are found in probability and statistical problems. Further applications can be found in [6, 16, 36]. In [9, 35] the authors defined the following extended Chebyshev functional:

$$\begin{aligned} \mathscr {T}(\mathscr {U},\mathscr {V}, \mu , \nu ) =& \int _{x_{1}}^{x_{2}} \nu (\tau )\,d\tau \int _{x_{1}}^{x_{2}}\mu (\tau )\mathscr {U}(\tau ) \mathscr {V}(\tau )\,d\tau \\ &{}+ \int _{x_{1}}^{x_{2}}\mu (\tau )\,d\tau \int _{x_{1}}^{x_{2}} \nu (\tau )\mathscr {U}(\tau ) \mathscr {V}(\tau )\,d\tau \\ &{}- \int _{x_{1}}^{x_{2}}\mu (\tau )\mathscr {U}(\tau )\,d \tau \int _{x_{1}}^{x_{2}} \nu (\tau )\mathscr {V}(\tau )\,d \tau \\ &{}- \int _{x_{1}}^{x_{2}}\nu (\tau ) \mathscr {U}(\tau )\,d \tau \int _{x_{1}}^{x_{2}}\mu (\tau )\mathscr {V}( \tau )\,d \tau , \end{aligned}$$
(2)

where \(\mathscr {U}\) and \(\mathscr {V}\) are integrable functions on \([x_{1},x_{2}]\), and μ and ν are positive integrable functions on \([x_{1},x_{2}]\). The functions \(\mathscr {U}\) and \(\mathscr {V}\) are said to be synchronous on \([x_{1},x_{2}]\) if

$$ \bigl(\mathscr {U}(\rho )-\mathscr {U}(\zeta ) \bigr) \bigl( \mathscr {V}(\rho )- \mathscr {V}(\zeta ) \bigr)\geq 0,\quad \rho ,\zeta \in [x_{1},x_{2}]. $$

The functions \(\mathscr {U}\) and \(\mathscr {V}\) are said to be asynchronous on \([x_{1},x_{2}]\) if the inequality reversed, that is,

$$ \bigl(\mathscr {U}(\rho )-\mathscr {U}(\zeta ) \bigr) \bigl( \mathscr {V}(\rho )- \mathscr {V}(\zeta ) \bigr)\leq 0,\quad \rho ,\zeta \in [x_{1},x_{2}]. $$

If the functions \(\mathscr {U}\) and \(\mathscr {V}\) are synchronous on \([r,s]\), then \(\mathscr {T} (\mathscr {U}, \mathscr {V},\mu )\geq 0\) and \(\mathscr {T} (\mathscr {U}, \mathscr {V}, \mu ,\nu )\geq 0\). For further details, the reader may consult Kuang [27] and Mitrinovic [35]. If we consider \(\mu (\vartheta )=\nu (\vartheta )=1\), \(\vartheta \in [x_{1},x_{2}]\), then \(\mathscr {T} (\mathscr {U}, \mathscr {V},\mu )=\frac{1}{2} \mathscr {T} (\mathscr {U}, \mathscr {V},\mu ,\nu )\). In [3, 12, 34, 44], various researchers gave valuable consideration to functionals (1) and (2). Recently, Rahman et al. [57] defined fractional conformable inequalities for the Chebyshev functionals (1) and (2).

Awan et al. [2] presented the following result: If Φ is an absolutely continuous on \([x_{1},x_{2}]\) such that \((\varPhi ^{\prime } )^{2}\in L_{1}[x_{1},x_{2}]\) and μ is a positive integrable function on \([x_{1},x_{2}]\), then the following inequality holds;

$$\begin{aligned} \mathscr {T} (\varPhi ,\varPhi ,\mu ) \leq& \frac{1}{Q^{2}(x_{2})} \int _{x_{1}}^{x_{2}} \biggl[ \int _{x_{1}}^{\tau }\mu (\tau )\,d\tau \int _{x_{1}}^{x_{2}}\tau \mu (\tau )\,d\tau - \int _{x_{1}}^{x_{2}}\mu ( \tau )\,d\tau \int _{x_{1}}^{\tau }\tau \mu (\tau )\,d\tau \biggr] \\ &{}\times\bigl[ \varPhi ^{\prime }(\theta ) \bigr]^{2}\,d\theta , \end{aligned}$$

where \(Q(x_{2})=\int _{x_{1}}^{x_{2}} \mu (\tau )\,d\tau \).

Bezziou et al. [5] presented the following result.

Theorem 1.1

Let\(\varPhi : [x_{1},x_{2}]\rightarrow \mathbb{R}\)be an absolutely continuous function such that\((\varPhi ^{\prime } )^{2}\in L_{1}[x_{1},x_{2}]\), and let\(\mu :[x_{1},x_{2}]\rightarrow \mathbb{R}^{+}\)be an integrable function. Then we have the following inequality for\(\kappa >0\):

$$ \mathscr {J}_{x_{1}}^{\tau }\mu (x_{2}) \mathscr {J}_{x_{1}}^{\tau }\mu \varPhi ^{2}(x_{2})- \bigl(\mathscr {J}_{x_{1}}^{\tau }\mu \varPhi (x_{2}) \bigr)^{2}\leq \int _{x_{1}}^{x_{2}}\varLambda (\theta ) \bigl[\varPhi ^{\prime }(\theta ) \bigr]^{2}\,d\theta $$

with

$$ \varLambda (\theta )=\frac{1}{2} \biggl[\mathscr {J}_{x_{1}}^{\tau }x_{2} \mu (x_{2}) \int _{x_{1}}^{\tau }(x_{2}-\tau )^{\tau -1}\mu (\tau )\,d\tau -\mathscr {J}_{x_{1}}^{\tau } \mu (x_{2}) \int _{x_{1}}^{\tau }\tau (x_{2}- \tau )^{\tau -1}\mu (\tau )\,d\tau \biggr], $$

where\(\mathscr {J}_{x_{1}}^{\tau }\)is the classical RL-fractional integral.

Dahmani and Bounoua [13] established the following result.

Theorem 1.2

Let\(\varPhi : [x_{1},x_{2}]\rightarrow \mathbb{R}\)be an absolutely continuous such that\((\varPhi ^{\prime } )^{2}\in L_{1}[x_{1},x_{2}]\), and let\(\mu :[x_{1},x_{2}]\rightarrow \mathbb{R}^{+}\)be an integrable function. Then for all\(\kappa >0\)and\(\theta \in [x_{1},x_{2}]\), the following inequality holds;

$$\begin{aligned} &\frac{1}{\mathscr {J}_{x_{1}}^{\tau }\mu (\theta )}\mathscr {J}_{x_{1}}^{\tau } \bigl(\mu \varPhi ^{2} \bigr) (\theta )- \biggl[ \frac{1}{\mathscr {J}_{x_{1}}^{\tau }\mu (\theta )} \mathscr {J}_{x_{1}}^{\tau } (\mu \varPhi ) (\theta ) \biggr]^{2} \\ &\quad \leq \frac{1}{ [\mathscr {J}_{x_{1}}^{\tau }\mu (\theta ) ]^{2}} \int _{x_{1}}^{\theta }Q_{\theta }(\tau ) \bigl[\varPhi ^{\prime }(\tau ) \bigr]^{2}\,d\tau \end{aligned}$$

with

$$\begin{aligned} Q_{\theta }(\tau ) =&\frac{1}{\varGamma (\tau )} \biggl[\mathscr {J}_{x_{1}}^{\tau } \bigl(\theta \mu (\theta )\bigr) \int _{x_{1}}^{\rho }\mu (\vartheta ) ( \theta - \vartheta )^{\tau -1}\,d\vartheta \\ &{}-\mathscr {J}_{x_{1}}^{\kappa} \mu (\theta ) \int _{x_{1}}^{\rho }\vartheta \mu (\vartheta ) (\theta - \vartheta )^{\tau -1}\,d\vartheta \biggr], \end{aligned}$$

where\(\mathscr {J}_{x_{1}}^{\tau }\)is the classical Riemann–Liouville fractional integral.

In the last few decades, the researchers investigated different kinds of integral inequalities by considering various integral approaches. In [14] the authors gave weighted Grüss-type inequalities by taking RL-fractional integrals into account. Dahmani [8] proposed some new inequalities in the sense of fractional integrals. Several inequalities for the extended gamma function and confluent hypergeometric k-function are found by Nisar et al. [38]. Nisar et al. [39] used Riemann–Liouville and Hadamard k-fractional derivatives and investigated Gronwall-type inequalities with applications. Rahman et al. [55] studied \((k,\rho )\)-fractional integrals and investigated the corresponding inequalities. Sarikaya and Budak [59] proposed Ostrowski-type inequalities by considering local fractional integrals. Sarikaya et al. [60] proposed the idea of generalized \((k,s)\)-fractional integrals with applications. Set et al. [61] investigated Grüss-type inequalities for the generalized k-fractional integrals. Recently, Jarad et al. [22, 23] proposed the idea of fractional conformable and proportional fractional integral operators. Huang et al. [20] recently presented generalized Hermite–Hadamard-type inequalities for k-fractional conformable integrals. Qi et al. [45] proposed Chebyshev-type inequalities by using generalized k-fractional conformable integrals. Rahman et al. [56] investigated Chebyshev-type inequalities by utilizing fractional conformable integrals. Chebyshev-type inequalities and Minkowski-type inequalities involving generalized conformable integrals can be found in the work of Nisar et al. [42, 43]. Recently, Tassaddiq et al. [63] proposed certain inequalities for the weighted and extended Chebyshev functionals by using fractional conformable integrals. Nisar et al. [40] presented some new classes of inequalities for an n (\(n\in \mathbb{N}\)) family of positive continuous and decreasing functions via generalized conformable fractional integrals. Nisar et al. [41] established generalized fractional integral inequalities via the Marichev–Saigo–Maeda (MSM) fractional integral operators. Rahman et al. [54] recently investigated Grüss-type inequalities for generalized k-fractional conformable integrals. Minkowski’s inequalities, fractional Hadamard proportional integral inequalities, and fractional proportional inequalities for convex functions by employing fractional proportional integrals can be found in [4653]. In addition, various applications of fractional calculus can be found in [1, 17, 18, 2832, 62, 64].

The paper is organized as follows. Some auxiliary results are presented in Sect. 2. In Sect. 3, we present double-weighted fractional integral inequalities for the Chebyshev functionals. In Sect. 4, we retrieve several particular cases of the results. A concluding remark is given in Sect. 5.

Auxiliary results

In this section, we present some well-known definitions and mathematical preliminaries of fractional calculus.

Definition 2.1

([26, 58])

Let \(\mathscr {U}\in L[x_{1},x_{2}]\). Then the classical left- and right-sided RL-fractional integrals of order \(\tau >0\) and \(x_{1}\geq 0\) are respectively defined by

$$ \bigl({}_{x_{1}}\mathscr {J}^{\tau }\mathscr {U} \bigr) (\vartheta )= \frac{1}{\varGamma (\tau )} \int _{a}^{\vartheta }(\vartheta -\varrho )^{ \tau -1}\mathscr {U}(\varrho )\,d\varrho ,\quad x_{1}< \vartheta , $$
(3)

and

$$ \bigl(\mathscr {J}_{x_{2}}^{\tau }\mathscr {U} \bigr) (\vartheta )= \frac{1}{\varGamma (\tau )} \int _{\vartheta }^{x_{2}} (\varrho -\vartheta )^{ \tau -1}\mathscr {U}(\varrho )\,d\varrho , \quad \vartheta < x_{2}, $$
(4)

where Γ is the standard gamma function.

Definition 2.2

([37])

Let \(\mathscr {U}\in L[x_{1},x_{2}]\). Then the generalized left- and right-sided RL κ-fractional integrals of order \(\tau >0\) and \(x_{1}\geq 0\) are respectively defined by

$$ \bigl({}_{x_{1}}\mathscr {J}_{\kappa }^{\tau } \mathscr {U} \bigr) ( \vartheta )=\frac{1}{\kappa \varGamma _{\kappa }(\tau )} \int _{a}^{\vartheta }(\vartheta -\varrho )^{\frac{\tau }{\kappa }-1}\mathscr {U}( \varrho )\,d\varrho , \quad x_{1}< \vartheta , $$
(5)

and

$$ \bigl(\mathscr {J}_{x_{2}, \kappa }^{\tau }\mathscr {U} \bigr) ( \vartheta )=\frac{1}{\kappa \varGamma _{\kappa }(\tau )} \int _{\vartheta }^{x_{2}} (\varrho -\vartheta )^{\frac{\tau }{\kappa }-1}\mathscr {U}(\varrho )\,d\varrho , \quad \vartheta < x_{2}, $$
(6)

where \(\varGamma _{\kappa }\) is the κ-gamma function defined in [15].

Remark 2.1

Applying Definition 2.2 for \(\kappa =1\), we get Definition 2.1.

Definition 2.3

([26])

Let \(\mathscr {U}:[x_{1},x_{2}]\rightarrow \mathbb{R}\) be an integrable function, and let Ψ be an increasing positive function on \((x_{1},x_{2}]\) with continuous derivative \(\varPsi ^{\prime }\) on \((x_{1},x_{2})\). Then the left- and right-sided generalized RL fractional integrals of a function \(\mathscr {U}\) concerning another function Ψ are respectively defined by

$$ \bigl({}_{x_{1}}^{\varPsi }\mathscr {J}^{\tau } \mathscr {U} \bigr) (\rho ) = \frac{1}{\varGamma (\tau )} \int _{x_{1}}^{\vartheta } \bigl(\varPsi ( \vartheta )- \varPsi ( \varrho ) \bigr)^{\frac{\tau }{\kappa }-1}\varPsi ^{\prime }(\varrho )\mathscr {U(\varrho )}\,d\varrho ,\quad x_{1}< \vartheta , $$
(7)

and

$$ \bigl({}^{\varPsi }\mathscr {J}_{x_{2}}^{\tau } \mathscr {U} \bigr) (\rho ) = \frac{1}{\varGamma (\tau )} \int _{\vartheta }^{x_{2}} \bigl(\varPsi ( \varrho )- \varPsi ( \vartheta ) \bigr)^{\frac{\tau }{\kappa }-1}\varPsi ^{\prime }(\varrho )\mathscr {U( \varrho )}\,d\varrho ,\quad \vartheta < x_{2}, $$
(8)

where \(\kappa >0\) and \(\tau \in \mathbb{C}\) with \(\Re (\tau )>0\).

Definition 2.4

([33])

Let \(\mathscr {U}:[x_{1},x_{2}]\rightarrow \mathbb{R}\) be an integrable function, and let Ψ be an increasing positive function on \((x_{1},x_{2}]\) with continuous derivative \(\varPsi ^{\prime }\) on \((x_{1},x_{2})\). Then the left- and right-sided generalized RL κ-fractional integrals of a function \(\mathscr {U}\) concerning another function Ψ are respectively defined by

$$ \bigl({}_{x_{1}}^{\varPsi }\mathscr {J}_{\kappa }^{\tau } \mathscr {U} \bigr) ( \rho ) =\frac{1}{\kappa \varGamma _{\kappa }(\tau )} \int _{x_{1}}^{\vartheta } \bigl(\varPsi (\vartheta )- \varPsi ( \varrho ) \bigr)^{ \frac{\tau }{\kappa }-1}\varPsi ^{\prime }(\varrho )\mathscr {U(\varrho )}\,d\varrho ,\quad x_{1}< \vartheta , $$
(9)

and

$$ \bigl({}^{\varPsi }\mathscr {J}_{x_{2},\kappa }^{\tau } \mathscr {U} \bigr) ( \rho ) =\frac{1}{\kappa \varGamma _{\kappa }(\tau )} \int _{\vartheta }^{x_{2}} \bigl(\varPsi (\varrho )- \varPsi ( \vartheta ) \bigr)^{ \frac{\tau }{\kappa }-1}\varPsi ^{\prime }(\varrho )\mathscr {U( \varrho )}\,d\varrho ,\quad \vartheta < x_{2}, $$
(10)

where \(\kappa >0\) and \(\tau \in \mathbb{C}\) with \(\Re (\tau )>0\).

Remark 2.2

The following particular cases are easily derived:

  1. i.

    Applying Definition 2.4 for \(\varPsi (\vartheta )=\vartheta \), we get Definition 2.2,

  2. ii.

    Applying Definition 2.4 for \(\kappa =1\), we get Definition 2.3,

  3. iii.

    Applying Definition 2.4 for \(\varPsi (\vartheta )=\ln \vartheta \), we get the generalized Hadamard κ-fractional integrals defined in [21],

  4. iv.

    Applying Definition 2.4 for \(\varPsi (\vartheta )=\ln \vartheta \) and \(\kappa =1\) leads to the Hadamard fractional integrals defined in [26],

  5. v.

    Applying Definition 2.4 for \(\varPsi (\vartheta )=\frac{\vartheta ^{\tau }}{\tau }\), \(\tau >0\), and \(\kappa =1\) leads to the Katugampola fractional integrals [24],

  6. vi.

    Applying Definition 2.4 for \(\varPsi (\vartheta )=\frac{\vartheta ^{\alpha +s}}{\alpha +s}\) and \(\kappa =1\) (where \(\alpha \in (0,1]\), \(s\in \mathbb{R}\), and \(\mu +s\neq 0\)) leads to the generalized fractional conformable integrals defined by Khan and Khan [25],

  7. vii.

    Applying Definition 2.4 for \(\varPsi (\vartheta )=\frac{(\vartheta -x_{1})^{\alpha }}{\alpha }\) and \(\varPsi (\vartheta )=\frac{-(x_{2}-\vartheta )^{\alpha }}{\alpha }\), \(\alpha >0\), leads to the \((k,\alpha )\)-fractional conformable integrals defined by Habib et al. [19].

  8. viii.

    Applying Definition 2.4 for \(\varPsi (\vartheta )=\frac{(\vartheta -x_{1})^{\alpha }}{\alpha }\), \(\varPsi (\vartheta )=\frac{-(x_{2}-\vartheta )^{\alpha }}{\alpha }\), \(\alpha >0\), and \(\kappa =1\) leads to the conformable fractional integrals defined by Jarad et al. [23],

  9. ix.

    Applying Definition 2.4 for \(\varPsi (\vartheta )=\vartheta \) and \(\kappa =1\), we get Definition 2.1.

Some double-weighted generalized fractional integral inequalities

In this section, we present some double-weighted generalized fractional integral inequalities. We start by proving the following lemma.

Lemma 3.1

LetΨbe a measurable increasing positive function on\((x_{1},x_{2})\)with continuous derivative\(\varPsi ^{\prime }(\varrho )\)on\([x_{1},x_{2}]\). Let\(\mathscr {V}: [x_{1},x_{2}]\rightarrow \mathbb{R}\)be continuous on\([x_{1},x_{2}]\), and let\(\mu ,\nu : [x_{1},x_{2}]\rightarrow \mathbb{R}^{+}\)be positive integrable. Then for all\(\tau ,\kappa >0\), we have

$$\begin{aligned}& \bigl[{}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }\mu (x_{2}) \bigr] \bigl[{}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau }x_{2}( \nu \mathscr {V}) (x_{2}) \bigr]+ \bigl[{}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }\mathscr {V}(x_{2}) \bigr] \bigl[{}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau }x_{2}( \mu \mathscr {V}) (x_{2}) \bigr] \\& \qquad {}- \bigl[{}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau } \mu (x_{2})x_{2} \mu (x_{2}) \bigr] \bigl[{}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau }( \nu \mathscr {V}) (x_{2}) \bigr]- \bigl[{}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }x_{2}\nu (x_{2}) \bigr] \bigl[{}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }(\mu \mathscr {V}) (x_{2}) \bigr] \\& \quad \leq \frac{1}{\kappa \varGamma _{\kappa }(\tau )} \int _{x_{1}}^{x_{2}} \biggl[{}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }x_{2}\mu (x_{2}) \int _{x_{1}}^{\vartheta } \bigl(\varPsi (x_{2})- \varPsi (\varrho ) \bigr)^{ \frac{\tau }{\kappa }-1}\varPsi ^{\prime }(\varrho )\nu (\varrho )\,d\varrho \\& \qquad {}- {}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau } \mu (x_{2}) \int _{x_{1}}^{ \vartheta }\varrho \bigl(\varPsi (x_{2})-\varPsi (\varrho ) \bigr)^{ \frac{\tau }{\kappa }-1}\varPsi ^{\prime }( \varrho )\nu (\varrho )\,d\varrho \biggr] \bigl(\varPsi ^{\prime }(\vartheta ) \bigr)\,d\vartheta . \end{aligned}$$
(11)

Proof

Suppose that \(\mathscr {U}: [x_{1},x_{2}]\rightarrow \mathbb{R}\) is a continuous function on \([x_{1},x_{2}]\). Then we get

$$\begin{aligned}& \bigl[{}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau } \mu (x_{2}) \bigr] \bigl[{}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }(\nu \mathscr {U} \mathscr {V}) (x_{2}) \bigr]+ \bigl[{}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }\mathscr {V}(x_{2}) \bigr] \bigl[{}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau }( \mu \mathscr {U}\mathscr {V}) (x_{2}) \bigr] \\& \qquad {}- \bigl[{}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau }( \mu \mathscr {U}) (x_{2}) \bigr] \bigl[{}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }(\nu \mathscr {V}) (x_{2}) \bigr]- \bigl[{}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau }( \nu \mathscr {U}) (x_{2}) \bigr] \bigl[{}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }(\mu \mathscr {V}) (x_{2}) \bigr] \\& \quad = \frac{1}{\kappa ^{2}\varGamma _{\kappa }^{2}(\tau )} \int _{x_{1}}^{x_{2}} \int _{x_{1}}^{x_{2}} \bigl(\varPsi (x_{2})- \varPsi (\xi ) \bigr)^{ \frac{\tau }{\kappa }-1}\bigl(\varPsi (x_{2})-\varPsi (\varrho ) \bigr)^{ \frac{\tau }{\kappa }-1} \\& \qquad {}\times \varPsi ^{\prime }(\xi )\mu (\xi )\varPsi ^{\prime }( \varrho )\nu (\varrho ) \bigl[ \bigl(\mathscr {U}(\xi )-\mathscr {U}( \varrho ) \bigr) \bigl(\mathscr {V}( \xi )-\mathscr {V}(\varrho ) \bigr) \bigr]\,d\xi \,d\varrho . \end{aligned}$$

Consequently, it follows that

$$\begin{aligned}& \bigl[{}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }\mu (x_{2}) \bigr] \bigl[{}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau }( \nu \mathscr {U} \mathscr {V}) (x_{2}) \bigr]+ \bigl[{}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }\mathscr {V}(x_{2}) \bigr] \bigl[{}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau }( \mu \mathscr {U}\mathscr {V}) (x_{2}) \bigr] \\& \qquad {}- \bigl[{}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau }( \mu \mathscr {U}) (x_{2}) \bigr] \bigl[{}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }(\nu \mathscr {V}) (x_{2}) \bigr]- \bigl[{}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau }( \nu \mathscr {U}) (x_{2}) \bigr] \bigl[{}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }(\mu \mathscr {V}) (x_{2}) \bigr] \\& \quad = \frac{1}{\kappa ^{2}\varGamma _{\kappa }^{2}(\tau )} \int _{x_{1}}^{x_{2}} \int _{x_{1}}^{x_{2}} \bigl(\varPsi (x_{2})- \varPsi (\xi ) \bigr)^{ \frac{\tau }{\kappa }-1}\bigl(\varPsi (x_{2})-\varPsi (\varrho ) \bigr)^{ \frac{\tau }{\kappa }-1} \\& \qquad {}\times \varPsi ^{\prime }(\xi )\mu (\xi )\varPsi ^{\prime }( \varrho )\nu (\varrho ) \bigl(\mathscr {U}(\xi )-\mathscr {U}(\varrho ) \bigr) \biggl( \int _{\varrho }^{\xi }\mathscr {V}^{\prime }( \vartheta )\,d\vartheta \biggr)\,d\xi \,d\varrho . \end{aligned}$$
(12)

Utilizing the condition \(x_{1}\leq \varrho \leq \vartheta \leq \xi \leq x_{2}\), we conclude that

$$\begin{aligned}& \bigl[{}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }\mu (x_{2}) \bigr] \bigl[{}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau }( \nu \mathscr {U} \mathscr {V}) (x_{2}) \bigr]+ \bigl[{}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }\mathscr {V}(x_{2}) \bigr] \bigl[{}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau }( \mu \mathscr {U}\mathscr {V}) (x_{2}) \bigr] \\& \qquad {}- \bigl[{}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau }( \mu \mathscr {U}) (x_{2}) \bigr] \bigl[{}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }(\nu \mathscr {V}) (x_{2}) \bigr]- \bigl[{}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau }( \nu \mathscr {U}) (x_{2}) \bigr] \bigl[{}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }(\mu \mathscr {V}) (x_{2}) \bigr] \\& \quad = \frac{1}{\kappa ^{2}\varGamma _{\kappa }^{2}(\tau )} \int _{x_{1}}^{x_{2}} \biggl[ \int _{x_{1}}^{\vartheta } \bigl(\varPsi (x_{2})- \varPsi (\varrho ) \bigr)^{\frac{\tau }{\kappa }-1}\varPsi ^{\prime }(\varrho )\nu (\varrho ) \\& \qquad {}\times \int _{x_{1}}^{x_{2}} \bigl(\varPsi (x_{2})- \varPsi (\xi ) \bigr)^{ \frac{\tau }{\kappa }-1} \bigl(\mathscr {U}(\xi )-\mathscr {U}(\varrho ) \bigr) \varPsi ^{\prime }(\xi )\mu (\xi ) \,d\xi \,d\varrho \biggr] \bigl( \mathscr {V}^{\prime }(\vartheta ) \bigr)\,d\vartheta . \end{aligned}$$
(13)

Applying (13) to the particular case \(\mathscr {U}(x)=x\), we can write

$$\begin{aligned}& \bigl[{}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau } \mu (x_{2}) \bigr] \bigl[{}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }x_{2}(\nu \mathscr {V}) (x_{2}) \bigr]+ \bigl[{}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }\mathscr {V}(x_{2}) \bigr] \bigl[{}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau }x_{2}( \mu \mathscr {V}) (x_{2}) \bigr] \\& \qquad {}- \bigl[{}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau }x_{2}( \mu ) (x_{2}) \bigr] \bigl[{}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }(\nu \mathscr {V}) (x_{2}) \bigr]- \bigl[{}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau }x_{2}( \nu ) (x_{2}) \bigr] \bigl[{}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }(\mu \mathscr {V}) (x_{2}) \bigr] \\& \quad = \frac{1}{\kappa ^{2}\varGamma _{\kappa }^{2}(\tau )} \int _{x_{1}}^{x_{2}} \biggl[ \int _{x_{1}}^{\vartheta } \bigl(\varPsi (x_{2})- \varPsi (\varrho ) \bigr)^{\frac{\tau }{\kappa }-1}\varPsi ^{\prime }(\varrho )\nu (\varrho ) \\& \qquad {}\times \int _{x_{1}}^{x_{2}} \bigl(\varPsi (x_{2})- \varPsi (\xi ) \bigr)^{ \frac{\tau }{\kappa }-1} (\xi -\varrho ) \varPsi ^{\prime }(\xi ) \mu (\xi ) \,d\xi \,d\varrho \biggr] \bigl(\mathscr {V}^{\prime }(\vartheta ) \bigr)\,d\vartheta \\& \quad = \frac{1}{\kappa \varGamma _{\kappa }(\tau )} \int _{x_{1}}^{x_{2}} \biggl[ \frac{1}{\kappa \varGamma _{\kappa }(\tau )} \int _{x_{1}}^{x_{2}} \bigl( \varPsi (x_{2})- \varPsi (\xi ) \bigr)^{\frac{\tau }{\kappa }-1} \varPsi ^{\prime }(\xi )\xi \mu (\xi ) \,d \xi \\& \qquad {}\times \int _{x_{1}}^{\vartheta } \bigl(\varPsi (x_{2})- \varPsi (\varrho ) \bigr)^{\frac{\tau }{\kappa }-1}\varPsi ^{\prime }(\varrho )\nu (\varrho )\,d\varrho -\frac{1}{\kappa \varGamma _{\kappa }(\tau )} \\& \qquad {}\times \int _{x_{1}}^{x_{2}} \bigl(\varPsi (x_{2})- \varPsi (\xi ) \bigr)^{ \frac{\tau }{\kappa }-1} \varPsi ^{\prime }(\xi )\mu (\xi ) \,d\xi \times \int _{x_{1}}^{\vartheta } \\& \qquad {}\times \bigl(\varPsi (x_{2})-\varPsi (\varrho ) \bigr)^{ \frac{\tau }{\kappa }-1} \varPsi ^{\prime }(\varrho )\varrho \nu (\varrho )\,d\varrho \biggr] \bigl( \mathscr {V}^{\prime }(\vartheta ) \bigr)\,d\vartheta . \end{aligned}$$

The latter by (9) gives

$$\begin{aligned}& \bigl[{}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau } \mu (x_{2}) \bigr] \bigl[{}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }x_{2}(\nu \mathscr {V}) (x_{2}) \bigr]+ \bigl[{}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }\mathscr {V}(x_{2}) \bigr] \bigl[{}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau }x_{2}( \mu \mathscr {V}) (x_{2}) \bigr] \\& \qquad {}- \bigl[{}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau }x_{2}( \mu ) (x_{2}) \bigr] \bigl[{}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }(\nu \mathscr {V}) (x_{2}) \bigr]- \bigl[{}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau }x_{2}( \nu ) (x_{2}) \bigr] \bigl[{}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }(\mu \mathscr {V}) (x_{2}) \bigr] \\& \quad = \frac{1}{\kappa \varGamma _{\kappa }(\tau )} \int _{x_{1}}^{x_{2}} \biggl[ {}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }x_{2}\mu (x_{2}) \int _{x_{1}}^{ \vartheta } \bigl(\varPsi (x_{2})- \varPsi (\varrho ) \bigr)^{ \frac{\tau }{\kappa }-1}\varPsi ^{\prime }(\varrho )\nu (\varrho )\,d\varrho \\& \qquad {}- {}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau } \mu (x_{2}) \int _{x_{1}}^{ \vartheta }\varrho \bigl(\varPsi (x_{2})-\varPsi (\varrho ) \bigr)^{ \frac{\tau }{\kappa }-1}\varPsi ^{\prime }( \varrho ) \nu (\varrho )\,d\varrho \biggr] \bigl(\mathscr {V}^{\prime }( \vartheta ) \bigr)\,d\vartheta , \end{aligned}$$

which completes the proof. □

Based on Lemma 3.1, we prove the following theorem.

Theorem 3.1

LetΨbe a measurable increasing positive function on\((x_{1},x_{2})\)with continuous derivative\(\varPsi ^{\prime }(\varrho )\)on\([x_{1},x_{2}]\). Let\(\varPhi : [x_{1},x_{2}]\rightarrow \mathbb{R}\)be an absolutely continuous function with\((\varPhi ^{\prime })^{2}\in L_{1}[x_{1},x_{2}]\), and let\(\mu ,\nu : [x_{1},x_{2}]\rightarrow \mathbb{R}^{+}\)be positive integrable functions. Then for all\(\tau ,\kappa >0\), we have

$$\begin{aligned}& \bigl[{}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }\mu (x_{2}) \bigr] \bigl[{}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau } \bigl(\nu \varPhi ^{2}\bigr) (x_{2}) \bigr]+ \bigl[{}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau } \nu (x_{2}) \bigr] \bigl[{}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }\bigl(\mu \varPhi ^{2}\bigr) (x_{2}) \bigr] \\& \qquad {}- 2 \bigl[{}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau }( \mu \varPhi ) (x_{2}) \bigr] \bigl[{}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }(\nu \varPhi ) (x_{2}) \bigr] \\& \quad \leq \frac{1}{\kappa \varGamma _{\kappa }(\tau )} \int _{x_{1}}^{x_{2}} \biggl[{}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }x_{2}\mu (x_{2}) \int _{x_{1}}^{\vartheta } \bigl(\varPsi (x_{2})- \varPsi (\varrho ) \bigr)^{ \frac{\tau }{\kappa }-1}\varPsi ^{\prime }(\varrho )\nu (\varrho )\,d\varrho \\& \qquad {}- {}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau } \mu (x_{2}) \int _{x_{1}}^{ \vartheta }\varrho \bigl(\varPsi (x_{2})-\varPsi (\varrho ) \bigr)^{ \frac{\tau }{\kappa }-1}\varPsi ^{\prime }( \varrho )\nu (\varrho )\,d\varrho \biggr] \bigl(\varPhi ^{\prime }(\vartheta ) \bigr)^{2}\,d\vartheta . \end{aligned}$$
(14)

Proof

By employing definition (9) and Lemma 3.1 we obtain

$$\begin{aligned}& \bigl[{}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau } \mu (x_{2}) \bigr] \bigl[{}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }\bigl(\nu \varPhi ^{2}\bigr) (x_{2}) \bigr]+ \bigl[{}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }\nu (x_{2}) \bigr] \bigl[{}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau } \bigl(\mu \varPhi ^{2}\bigr) (x_{2}) \bigr] \\& \qquad {}- 2 \bigl[{}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau }( \mu \varPhi ) (x_{2}) \bigr] \bigl[{}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }(\nu \varPhi ) (x_{2}) \bigr] \\& \quad = \frac{1}{\kappa ^{2}\varGamma _{\kappa }^{2}(\tau )} \int _{x_{1}}^{x_{2}} \int _{x_{1}}^{x_{2}} \bigl(\varPsi (x_{2})- \varPsi (\xi ) \bigr)^{ \frac{\tau }{\kappa }-1} \\& \qquad {}\times \bigl(\varPsi (x_{2})-\varPsi (\varrho ) \bigr)^{ \frac{\tau }{\kappa }-1} \varPsi ^{\prime }(\xi )\mu (\xi )\varPsi ^{\prime }( \varrho )\nu (\varrho ) \bigl(\varPhi (\xi )-\varPhi (\varrho ) \bigr)^{2}\,d\xi \,d\varrho \\& \quad = \frac{1}{\kappa ^{2}\varGamma _{\kappa }^{2}(\tau )} \int _{x_{1}}^{x_{2}} \int _{x_{1}}^{x_{2}} \bigl(\varPsi (x_{2})- \varPsi (\xi ) \bigr)^{ \frac{\tau }{\kappa }-1} \\& \qquad {}\times \bigl(\varPsi (x_{2})-\varPsi (\varrho ) \bigr)^{ \frac{\tau }{\kappa }-1} \varPsi ^{\prime }(\xi )\mu (\xi )\varPsi ^{\prime }( \varrho )\nu (\varrho ) (\xi -\varrho )^{2} \biggl( \frac{\varPhi (\xi )-\varPhi (\varrho )}{\xi -\varrho } \biggr)^{2}\,d\xi \,d\varrho . \end{aligned}$$

Consequently, it follows that

$$\begin{aligned}& \bigl[{}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau } \mu (x_{2}) \bigr] \bigl[{}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }\bigl(\nu \varPhi ^{2}\bigr) (x_{2}) \bigr]+ \bigl[{}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }\nu (x_{2}) \bigr] \bigl[{}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau } \bigl(\mu \varPhi ^{2}\bigr) (x_{2}) \bigr] \\& \qquad {}- 2 \bigl[{}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau }( \mu \varPhi ) (x_{2}) \bigr] \bigl[{}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }(\nu \varPhi ) (x_{2}) \bigr] \\& \quad \leq \frac{1}{\kappa ^{2}\varGamma _{\kappa }^{2}(\tau )} \int _{x_{1}}^{x_{2}} \int _{x_{1}}^{x_{2}} \bigl(\varPsi (x_{2})- \varPsi (\xi ) \bigr)^{ \frac{\tau }{\kappa }-1} \\& \qquad {}\times \bigl(\varPsi (x_{2})-\varPsi (\varrho ) \bigr)^{ \frac{\tau }{\kappa }-1} \varPsi ^{\prime }(\xi )\mu (\xi )\varPsi ^{\prime }( \varrho )\nu (\varrho ) (\xi -\varrho )^{2} \biggl( \frac{\int _{\varrho }^{\xi }\varPhi ^{\prime }(\vartheta )\,d\vartheta }{\xi -\varrho } \biggr)^{2}\,d\xi \,d\varrho . \end{aligned}$$

By the Cauchy–Schwarz inequality [11] we get

$$\begin{aligned}& \bigl[{}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }\mu (x_{2}) \bigr] \bigl[{}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau } \bigl(\nu \varPhi ^{2}\bigr) (x_{2}) \bigr]+ \bigl[{}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau } \nu (x_{2}) \bigr] \bigl[{}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }\bigl(\mu \varPhi ^{2}\bigr) (x_{2}) \bigr] \\& \qquad {}- 2 \bigl[{}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau }( \mu \varPhi ) (x_{2}) \bigr] \bigl[{}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }(\nu \varPhi ) (x_{2}) \bigr] \\& \quad = \frac{1}{\kappa ^{2}\varGamma _{\kappa }^{2}(\tau )} \int _{x_{1}}^{x_{2}} \int _{x_{1}}^{x_{2}} \bigl(\varPsi (x_{2})- \varPsi (\xi ) \bigr)^{ \frac{\tau }{\kappa }-1} \\& \qquad {}\times \bigl(\varPsi (x_{2})-\varPsi (\varrho ) \bigr)^{ \frac{\tau }{\kappa }-1} \varPsi ^{\prime }(\xi )\mu (\xi )\varPsi ^{\prime }( \varrho )\nu (\varrho ) (\xi -\varrho )^{2} \biggl( \frac{\int _{\varrho }^{\xi }\varPhi ^{\prime }(\vartheta )\,d\vartheta }{\xi -\varrho } \biggr)^{2}\,d\xi \,d\varrho \\& \quad \leq \frac{1}{\kappa ^{2}\varGamma _{\kappa }^{2}(\tau )} \int _{x_{1}}^{x_{2}} \int _{x_{1}}^{x_{2}} \bigl(\varPsi (x_{2})- \varPsi (\xi ) \bigr)^{ \frac{\tau }{\kappa }-1}\bigl(\varPsi (x_{2})-\varPsi (\varrho ) \bigr)^{ \frac{\tau }{\kappa }-1} \\& \qquad {}\times \varPsi ^{\prime }(\xi )\mu (\xi )\varPsi ^{\prime }( \varrho )\nu (\varrho ) (\xi -\varrho )^{2} \biggl( \frac{ (\int _{\varrho }^{\xi }\,d\vartheta )^{\frac{1}{2}} (\int _{\varrho }^{\xi } (\varPhi ^{\prime }(\vartheta ) )^{2}\,d\vartheta )^{\frac{1}{2}}}{\xi -\varrho } \biggr)^{2}\,d\xi \,d\varrho \\& \quad \leq \frac{1}{\kappa ^{2}\varGamma _{\kappa }^{2}(\tau )} \int _{x_{1}}^{x_{2}} \int _{x_{1}}^{x_{2}} \bigl(\varPsi (x_{2})- \varPsi (\xi ) \bigr)^{ \frac{\tau }{\kappa }-1} \\& \qquad {}\times \bigl(\varPsi (x_{2})-\varPsi (\varrho ) \bigr)^{ \frac{\tau }{\kappa }-1} \varPsi ^{\prime }(\xi )\mu (\xi )\varPsi ^{\prime }( \varrho )\nu (\varrho ) (\xi -\varrho )^{2} \biggl( \frac{ (\int _{\varrho }^{\xi } (\varPhi ^{\prime }(\vartheta ) )^{2}\,d\vartheta )}{\xi -\varrho } \biggr)\,d\xi \,d \varrho \\& \quad \leq \frac{1}{\kappa ^{2}\varGamma _{\kappa }^{2}(\tau )} \int _{x_{1}}^{x_{2}} \int _{x_{1}}^{x_{2}} \bigl(\varPsi (x_{2})- \varPsi (\xi ) \bigr)^{ \frac{\tau }{\kappa }-1} \\& \qquad {}\times \bigl(\varPsi (x_{2})-\varPsi (\varrho ) \bigr)^{ \frac{\tau }{\kappa }-1} \varPsi ^{\prime }(\xi )\mu (\xi )\varPsi ^{\prime }( \varrho )\nu (\varrho ) (\xi -\varrho ) \biggl( \int _{\varrho }^{\xi } \bigl(\varPhi ^{\prime }( \vartheta ) \bigr)^{2}\,d\vartheta \biggr)\,d\xi \,d\varrho . \end{aligned}$$
(15)

Hence using (13) and (15), we conclude the proof. □

Corollary 3.1

LetΨbe a measurable increasing positive function on\((x_{1},x_{2})\)with continuous derivative\(\varPsi ^{\prime }(\varrho )\)on\([x_{1},x_{2}]\). Let\(\varPhi : [x_{1},x_{2}]\rightarrow \mathbb{R}\)be an absolutely continuous function with\((\varPhi ^{\prime })^{2}\in L_{1}[x_{1},x_{2}]\), and let\(\nu : [x_{1},x_{2}]\rightarrow \mathbb{R}^{+}\)be positive integrable. Then for all\(\tau ,\kappa >0\), we have

$$\begin{aligned}& \biggl[ \frac{ (\varPsi (x_{2})-\varPsi (x_{1}) )^{\frac{\tau }{\kappa }}}{\varGamma _{\kappa }(\tau +\kappa )} \biggr] \bigl[{}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }\bigl(\nu \varPhi ^{2}\bigr) (x_{2}) \bigr]+ \bigl[{}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }\nu (x_{2}) \bigr] \bigl[{}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau } \bigl(\varPhi ^{2}\bigr) (x_{2}) \bigr] \\& \qquad {}- 2 \bigl[{}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau }( \varPhi ) (x_{2}) \bigr] \bigl[{}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }(\nu \varPhi ) (x_{2}) \bigr] \\& \quad \leq \frac{1}{\kappa \varGamma _{\kappa }(\tau )} \int _{x_{1}}^{x_{2}} \biggl[{}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }x_{2} \int _{x_{1}}^{\vartheta } \bigl(\varPsi (x_{2})- \varPsi (\varrho ) \bigr)^{ \frac{\tau }{\kappa }-1}\varPsi ^{\prime }(\varrho )\nu (\varrho )\,d\varrho \\& \qquad {}- \frac{ (\varPsi (x_{2})-\varPsi (x_{1}) )^{\frac{\tau }{\kappa }}}{\varGamma _{\kappa }(\tau +\kappa )} \int _{x_{1}}^{\vartheta }\varrho \bigl(\varPsi (x_{2})-\varPsi (\varrho ) \bigr)^{\frac{\tau }{\kappa }-1}\varPsi ^{\prime }( \varrho )\nu (\varrho )\,d\varrho \biggr] \bigl(\varPhi ^{\prime }(\vartheta ) \bigr)^{2}\,d\vartheta . \end{aligned}$$

Proof

By considering \(\mu (\vartheta )=1\), \(\vartheta \in [x_{1},x_{2}]\), in Theorem 3.1 we obtain the desired result. □

Corollary 3.2

LetΨbe a measurable increasing positive function on\((x_{1},x_{2})\)with continuous derivative\(\varPsi ^{\prime }(\varrho )\)on\([x_{1},x_{2}]\). Let\(\varPhi : [x_{1},x_{2}]\rightarrow \mathbb{R}\)be an absolutely continuous function with\((\varPhi ^{\prime })^{2}\in L_{1}[x_{1},x_{2}]\), and let\(\mu : [x_{1},x_{2}]\rightarrow \mathbb{R}^{+}\)be positive integrable. Then for all\(\tau ,\kappa >0\), we have

$$\begin{aligned}& \bigl[{}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau } \mu (x_{2}) \bigr] \bigl[{}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }\bigl(\varPhi ^{2}\bigr) (x_{2}) \bigr]+ \biggl[ \frac{ (\varPsi (x_{2})-\varPsi (x_{1}) )^{\frac{\tau }{\kappa }}}{\varGamma _{\kappa }(\tau +\kappa )} \biggr] \bigl[{}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau } \bigl(\varPhi ^{2}\bigr) (x_{2}) \bigr] \\& \qquad {}- 2 \bigl[{}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau }( \mu \varPhi ) (x_{2}) \bigr] \bigl[{}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }(\varPhi ) (x_{2}) \bigr] \\& \quad \leq \frac{1}{\kappa \varGamma _{\kappa }(\tau )} \int _{x_{1}}^{x_{2}} \biggl[{}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }x_{2}\mu (x_{2}) \int _{x_{1}}^{\vartheta } \bigl(\varPsi (x_{2})- \varPsi (\varrho ) \bigr)^{ \frac{\tau }{\kappa }-1}\varPsi ^{\prime }(\varrho )\,d\varrho \\& \qquad {}- {}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau } \mu (x_{2}) \int _{x_{1}}^{ \vartheta }\varrho \bigl(\varPsi (x_{2})-\varPsi (\varrho ) \bigr)^{ \frac{\tau }{\kappa }-1}\varPsi ^{\prime }( \varrho )\,d\varrho \biggr] \bigl( \varPhi ^{\prime }(\vartheta ) \bigr)^{2}\,d\vartheta . \end{aligned}$$

Proof

By considering \(\nu (\vartheta )=1\), \(\vartheta \in [x_{1},x_{2}]\), in Theorem 3.1 we get the desired result. □

Corollary 3.3

LetΨbe a measurable increasing positive function on\((x_{1},x_{2})\)with continuous derivative\(\varPsi ^{\prime }(\varrho )\)on\([x_{1},x_{2}]\). Let\(\varPhi : [x_{1},x_{2}]\rightarrow \mathbb{R}\)be an absolutely continuous function with\((\varPhi ^{\prime })^{2}\in L_{1}[x_{1},x_{2}]\). Then for all\(\tau ,\kappa >0\), we have

$$\begin{aligned}& \biggl[ \frac{ (\varPsi (x_{2})-\varPsi (x_{1}) )^{\frac{\tau }{\kappa }}}{\varGamma _{\kappa }(\tau +\kappa )} \biggr] \bigl[{}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }\bigl(\varPhi ^{2}\bigr) (x_{2}) \bigr] - \bigl[{}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }(\varPhi ) (x_{2}) \bigr]^{2} \\& \quad \leq \frac{1}{2\kappa \varGamma _{\kappa }(\tau )} \int _{x_{1}}^{x_{2}} \biggl[{}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }x_{2} \int _{x_{1}}^{\vartheta } \bigl(\varPsi (x_{2})- \varPsi (\varrho ) \bigr)^{ \frac{\tau }{\kappa }-1}\varPsi ^{\prime }(\varrho )\,d\varrho \\& \qquad {}- \frac{ (\varPsi (x_{2})-\varPsi (x_{1}) )^{\frac{\tau }{\kappa }}}{\varGamma _{\kappa }(\tau +\kappa )} \int _{x_{1}}^{\vartheta }\varrho \bigl(\varPsi (x_{2})-\varPsi (\varrho ) \bigr)^{\frac{\tau }{\kappa }-1}\varPsi ^{\prime }( \varrho )\,d\varrho \biggr] \bigl(\varPhi ^{\prime }(\vartheta ) \bigr)^{2}\,d\vartheta . \end{aligned}$$

Proof

Taking \(\mu (\vartheta )=\nu (\vartheta )=1\), \(\vartheta \in [x_{1},x_{2}]\), in Theorem 3.1, we obtain the desired result. □

Theorem 3.2

LetΨbe a measurable increasing positive function on\((x_{1},x_{2})\)with continuous derivative\(\varPsi ^{\prime }(\varrho )\)on\([x_{1},x_{2}]\). Let\(f_{1}, f_{2}: [x_{1},x_{2}]\rightarrow \mathbb{R}\)be absolutely continuous functions with\((f_{1}^{\prime })^{2}\in L_{1}[x_{1},x_{2}]\)and\((f_{2}^{\prime })^{2}\in L_{1}[x_{1},x_{2}]\), and let\(\mu ,\nu : [x_{1},x_{2}]\rightarrow \mathbb{R}^{+}\)be positive integrable. Then for all\(\tau ,\kappa >0\), we have

$$\begin{aligned}& \bigl\vert {}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }\mu (x_{2}){}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }(\nu f_{1}f_{2}) (x_{2})+ {}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau } \nu (x_{2}){}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau }( \mu f_{1}f_{2}) (x_{2}) \\& \qquad {}- {}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau }( \nu f_{1}) (x_{2}){}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }(\mu f_{2}) (x_{2})- {}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau }( \mu f_{1}) (x_{2}){}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }(\nu f_{2}) (x_{2}) \bigr\vert \\& \quad \leq \frac{1}{\kappa \varGamma _{\kappa }(\tau )} \biggl( \int _{x_{1}}^{x_{2}} \biggl[{}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }x_{2}\mu (x_{2}) \int _{x_{1}}^{\vartheta } \bigl(\varPsi (x_{2})- \varPsi (\varrho ) \bigr)^{ \frac{\tau }{\kappa }-1}\varPsi ^{\prime }(\varrho )\nu (\varrho )\,d\varrho \\& \qquad {}- {}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau } \mu (x_{2}) \int _{x_{1}}^{ \vartheta }\varrho \bigl(\varPsi (x_{2})-\varPsi (\varrho ) \bigr)^{ \frac{\tau }{\kappa }-1}\varPsi ^{\prime }( \varrho )\nu (\varrho )\,d\varrho \biggr] \bigl(f_{1}^{\prime }( \vartheta ) \bigr)^{2}\,d\vartheta \biggr)^{\frac{1}{2}} \\& \qquad {}\times \biggl( \int _{x_{1}}^{x_{2}} \biggl[{}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }x_{2}\mu (x_{2}) \int _{x_{1}}^{\vartheta } \bigl(\varPsi (x_{2})- \varPsi (\varrho ) \bigr)^{\frac{\tau }{\kappa }-1}\varPsi ^{\prime }(\varrho ) \nu ( \varrho )\,d\varrho \\& \qquad {}- {}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau } \mu (x_{2}) \int _{x_{1}}^{ \vartheta }\varrho \bigl(\varPsi (x_{2})-\varPsi (\varrho ) \bigr)^{ \frac{\tau }{\kappa }-1}\varPsi ^{\prime }( \varrho )\nu (\varrho )\,d\varrho \biggr] \bigl(f_{2}^{\prime }( \vartheta ) \bigr)^{2}\,d\vartheta \biggr)^{\frac{1}{2}}. \end{aligned}$$
(16)

Proof

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

$$\begin{aligned}& \bigl\vert {}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau } \mu (x_{2}){}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau }( \nu f_{1}f_{2}) (x_{2})+ {}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }\nu (x_{2}){}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }(\mu f_{1}f_{2}) (x_{2}) \\& \qquad {}- {}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau }( \nu f_{1}) (x_{2}){}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }(\mu f_{2}) (x_{2})- {}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau }( \mu f_{1}) (x_{2}){}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }(\nu f_{2}) (x_{2}) \bigr\vert \\& \quad \leq \frac{1}{\kappa ^{2}\varGamma _{\kappa }^{2}(\tau )} \biggl( \int _{x_{1}}^{x_{2}} \int _{x_{1}}^{x_{2}} \bigl(\varPsi (x_{2})- \varPsi (\xi ) \bigr)^{ \frac{\tau }{\kappa }-1} \bigl(\varPsi (x_{2})-\varPsi (\varrho ) \bigr)^{ \frac{\tau }{\kappa }-1} \varPsi ^{\prime }(\xi )\mu (\xi ) \\& \qquad {}\times \varPsi ^{\prime }(\varrho )\nu (\varrho ) \bigl(f_{1}( \xi )-f_{1}( \varrho ) \bigr)^{2} \,d\xi \,d\varrho \biggr)^{\frac{1}{2}}\times \biggl( \int _{x_{1}}^{x_{2}} \int _{x_{1}}^{x_{2}} \bigl(\varPsi (x_{2})- \varPsi ( \xi ) \bigr)^{\frac{\tau }{\kappa }-1} \\& \qquad {}\times \bigl(\varPsi (x_{2})-\varPsi (\varrho ) \bigr)^{ \frac{\tau }{\kappa }-1} \varPsi ^{\prime }(\xi )\mu (\xi ) \varPsi ^{\prime }( \varrho )\nu (\varrho ) \bigl(f_{2}(\xi )-f_{2}(\varrho ) \bigr)^{2} \,d\xi \,d\varrho \biggr)^{\frac{1}{2}} \\& \quad \leq \frac{1}{\kappa ^{2}\varGamma _{\kappa }^{2}(\tau )} \biggl( \int _{x_{1}}^{x_{2}} \int _{x_{1}}^{x_{2}} \bigl(\varPsi (x_{2})- \varPsi (\xi ) \bigr)^{ \frac{\tau }{\kappa }-1} \bigl(\varPsi (x_{2})-\varPsi (\varrho ) \bigr)^{ \frac{\tau }{\kappa }-1} \varPsi ^{\prime }(\xi )\mu (\xi ) \\& \qquad {}\times \varPsi ^{\prime }(\varrho )\nu (\varrho ) \biggl( \int _{\varrho }^{\xi }f_{1}^{\prime }( \vartheta )\,d\vartheta \biggr)^{2} \,d\xi \,d\varrho \biggr)^{\frac{1}{2}}\times \biggl( \int _{x_{1}}^{x_{2}} \int _{x_{1}}^{x_{2}} \bigl(\varPsi (x_{2})- \varPsi (\xi ) \bigr)^{\frac{\tau }{\kappa }-1} \\& \qquad {}\times \bigl(\varPsi (x_{2})-\varPsi (\varrho ) \bigr)^{ \frac{\tau }{\kappa }-1} \varPsi ^{\prime }(\xi )\mu (\xi ) \varPsi ^{\prime }( \varrho )\nu (\varrho ) \biggl( \int _{\varrho }^{\xi }f_{2}^{\prime }( \vartheta )\,d\vartheta \biggr)^{2} \,d\xi \,d\varrho \biggr)^{\frac{1}{2}}. \end{aligned}$$

Applying the Cauchy–Schwarz inequality [11] to this inequality, we get

$$\begin{aligned}& \bigl\vert {}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau } \mu (x_{2}){}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau }( \nu f_{1}f_{2}) (x_{2})+ {}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }\nu (x_{2}){}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }(\mu f_{1}f_{2}) (x_{2}) \\& \qquad {}- {}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau }( \nu f_{1}) (x_{2}){}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }(\mu f_{2}) (x_{2})- {}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau }( \mu f_{1}) (x_{2}){}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }(\nu f_{2}) (x_{2}) \bigr\vert \\& \quad \leq \frac{1}{\kappa ^{2}\varGamma _{\kappa }^{2}(\tau )} \biggl[ \int _{x_{1}}^{x_{2}} \int _{x_{1}}^{x_{2}} \bigl(\varPsi (x_{2})- \varPsi (\xi ) \bigr)^{ \frac{\tau }{\kappa }-1} \bigl(\varPsi (x_{2})-\varPsi (\varrho ) \bigr)^{ \frac{\tau }{\kappa }-1} \varPsi ^{\prime }(\xi )\mu (\xi ) \\& \qquad {}\times \varPsi ^{\prime }(\varrho )\nu (\varrho ) \biggl( \biggl( \int _{\varrho }^{\xi }\,d\vartheta \biggr)^{\frac{1}{2}} \biggl( \int _{\varrho }^{\xi } \bigl(f_{1}^{\prime }( \vartheta ) \bigr)^{2}\,d\vartheta \biggr)^{\frac{1}{2}} \biggr)^{2} \,d\xi \,d\varrho \biggr]^{\frac{1}{2}} \\& \qquad {}\times \biggl[ \int _{x_{1}}^{x_{2}} \int _{x_{1}}^{x_{2}} \bigl(\varPsi (x_{2})- \varPsi ( \xi ) \bigr)^{\frac{\tau }{\kappa }-1}\bigl(\varPsi (x_{2})-\varPsi (\varrho ) \bigr)^{ \frac{\tau }{\kappa }-1} \\& \qquad {}\times \varPsi ^{\prime }(\xi )\mu (\xi ) \varPsi ^{\prime }( \varrho )\nu (\varrho ) \biggl( \biggl( \int _{\varrho }^{\xi }\,d\vartheta \biggr)^{\frac{1}{2}} \biggl( \int _{\varrho }^{\xi } \bigl(f_{2}^{\prime }( \vartheta ) \bigr)^{2}\,d\vartheta \biggr)^{\frac{1}{2}} \biggr)^{2} \,d\xi \,d\varrho \biggr]^{\frac{1}{2}} \\& \quad \leq \frac{1}{\kappa ^{2}\varGamma _{\kappa }^{2}(\tau )} \biggl[ \int _{x_{1}}^{x_{2}} \int _{x_{1}}^{x_{2}} \bigl(\varPsi (x_{2})- \varPsi (\xi ) \bigr)^{ \frac{\tau }{\kappa }-1} \bigl(\varPsi (x_{2})-\varPsi (\varrho ) \bigr)^{ \frac{\tau }{\kappa }-1} \varPsi ^{\prime }(\xi )\mu (\xi ) \\& \qquad {}\times \varPsi ^{\prime }(\varrho )\nu (\varrho ) (\xi -\varrho ) \biggl( \int _{\varrho }^{\xi } \bigl(f_{1}^{\prime }( \vartheta ) \bigr)^{2}\,d\vartheta \biggr) \,d\xi \,d\varrho \biggr]^{\frac{1}{2}} \times \biggl[ \int _{x_{1}}^{x_{2}} \int _{x_{1}}^{x_{2}} \bigl(\varPsi (x_{2})- \varPsi (\xi ) \bigr)^{\frac{\tau }{\kappa }-1} \\& \qquad {}\times \bigl(\varPsi (x_{2})-\varPsi (\varrho ) \bigr)^{ \frac{\tau }{\kappa }-1} \varPsi ^{\prime }(\xi )\mu (\xi ) \varPsi ^{\prime }( \varrho )\nu (\varrho ) (\xi -\varrho ) \biggl( \int _{\varrho }^{\xi } \bigl(f_{2}^{\prime }( \vartheta ) \bigr)^{2}\,d\vartheta \biggr) \,d\xi \,d\varrho \biggr]^{\frac{1}{2}} \\& \quad \leq \frac{1}{\kappa ^{2}\varGamma _{\kappa }^{2}(\tau )} \biggl[ \int _{x_{1}}^{x_{2}} \biggl( \int _{x_{1}}^{x_{2}} \xi \bigl(\varPsi (x_{2})-\varPsi (\xi ) \bigr)^{ \frac{\tau }{\kappa }-1}\varPsi ^{\prime }(\xi )\mu (\xi )\,d\xi \\& \qquad {}\times \int _{x_{1}}^{ \vartheta } \bigl(\varPsi (x_{2})- \varPsi (\varrho ) \bigr)^{ \frac{\tau }{\kappa }-1}\varPsi ^{\prime }(\varrho )\nu (\varrho )\,d\varrho \biggr) \bigl(f_{1}^{\prime }(\vartheta ) \bigr)^{2} \biggr]^{\frac{1}{2}} \\& \qquad {}- \biggl[ \int _{x_{1}}^{x_{2}} \biggl( \int _{x_{1}}^{x_{2}} \bigl(\varPsi (x_{2})- \varPsi (\xi ) \bigr)^{\frac{\tau }{\kappa }-1}\varPsi ^{\prime }(\xi )\mu ( \xi )\,d\xi \\& \qquad {}\times \int _{x_{1}}^{\varrho }\varrho \bigl(\varPsi (x_{2})- \varPsi (\varrho ) \bigr)^{\frac{\tau }{\kappa }-1}\varPsi ^{\prime }(\varrho ) \nu (\varrho )\,d\varrho \biggr) \bigl(f_{2}^{\prime }( \vartheta ) \bigr)^{2} \biggr]^{\frac{1}{2}}. \end{aligned}$$

In view of (9), we get the desired proof of (16). □

Corollary 3.4

LetΨbe a measurable increasing positive function on\((x_{1},x_{2})\)with continuous derivative\(\varPsi ^{\prime }(\varrho )\)on\([x_{1},x_{2}]\). Let\(f_{1}, f_{2}: [x_{1},x_{2}]\rightarrow \mathbb{R}\)be absolutely continuous functions with\((f_{1}^{\prime })^{2}\in L_{1}[x_{1},x_{2}]\)and\((f_{2}^{\prime })^{2}\in L_{1}[x_{1},x_{2}]\), and let\(\nu : [x_{1},x_{2}]\rightarrow \mathbb{R}^{+}\)be positive integrable. Then for all\(\tau ,\kappa >0\), we have

$$\begin{aligned}& \biggl\vert \frac{ (\varPsi (x_{2})-\varPsi (x_{1}) )^{\frac{\tau }{\kappa }}}{\varGamma _{\kappa }(\tau +\kappa )} {}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }(\nu f_{1}f_{2}) (x_{2})+ {}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau } \nu (x_{2}){}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau }( f_{1}f_{2}) (x_{2}) \\& \qquad {}- {}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau }( \nu f_{1}) (x_{2}){}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }( f_{2}) (x_{2})- {}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau }(f_{1}) (x_{2}){}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau }( \nu f_{2}) (x_{2}) \biggr\vert \\& \quad \leq \frac{1}{\kappa \varGamma _{\kappa }(\tau )} \biggl( \int _{x_{1}}^{x_{2}} \biggl[{}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }x_{2} \int _{x_{1}}^{\vartheta } \bigl(\varPsi (x_{2})- \varPsi (\varrho ) \bigr)^{ \frac{\tau }{\kappa }-1}\varPsi ^{\prime }(\varrho )\nu (\varrho )\,d\varrho \\& \qquad {}- \frac{ (\varPsi (x_{2})-\varPsi (x_{1}) )^{\frac{\tau }{\kappa }}}{\varGamma _{\kappa }(\tau +\kappa )} \int _{x_{1}}^{\vartheta }\varrho \bigl(\varPsi (x_{2})-\varPsi (\varrho ) \bigr)^{\frac{\tau }{\kappa }-1}\varPsi ^{\prime }( \varrho )\nu (\varrho )\,d\varrho \biggr] \bigl(f_{1}^{\prime }( \vartheta ) \bigr)^{2}\,d\vartheta \biggr)^{\frac{1}{2}} \\& \qquad {}\times \biggl( \int _{x_{1}}^{x_{2}} \biggl[{}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }x_{2} \int _{x_{1}}^{\vartheta } \bigl(\varPsi (x_{2})- \varPsi ( \varrho ) \bigr)^{\frac{\tau }{\kappa }-1}\varPsi ^{\prime }(\varrho )\nu ( \varrho )\,d\varrho \\& \qquad {}- \frac{ (\varPsi (x_{2})-\varPsi (x_{1}) )^{\frac{\tau }{\kappa }}}{\varGamma _{\kappa }(\tau +\kappa )} \int _{x_{1}}^{\vartheta }\varrho \bigl(\varPsi (x_{2})-\varPsi (\varrho ) \bigr)^{\frac{\tau }{\kappa }-1}\varPsi ^{\prime }( \varrho )\nu (\varrho )\,d\varrho \biggr] \bigl(f_{2}^{\prime }( \vartheta ) \bigr)^{2}\,d\vartheta \biggr)^{\frac{1}{2}}. \end{aligned}$$

Proof

Applying Theorem 3.2 with \(\mu (\vartheta )=1\), \(\vartheta \in [x_{1},x_{2}]\), we obtain the desired result. □

Corollary 3.5

LetΨbe a measurable increasing positive function on\((x_{1},x_{2})\)with continuous derivative\(\varPsi ^{\prime }(\varrho )\)on\([x_{1},x_{2}]\). Let\(f_{1}, f_{2}: [x_{1},x_{2}]\rightarrow \mathbb{R}\)be absolutely continuous functions with\((f_{1}^{\prime })^{2}\in L_{1}[x_{1},x_{2}]\)and\((f_{2}^{\prime })^{2}\in L_{1}[x_{1},x_{2}]\), and let\(\mu : [x_{1},x_{2}]\rightarrow \mathbb{R}^{+}\)be positive integrable. Then for all\(\tau ,\kappa >0\), we have

$$\begin{aligned}& \biggl\vert {}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau } \mu (x_{2}){}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau }( f_{1}f_{2}) (x_{2})+ \frac{ (\varPsi (x_{2})-\varPsi (x_{1}) )^{\frac{\tau }{\kappa }}}{\varGamma _{\kappa }(\tau +\kappa )} {}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau }( \mu f_{1}f_{2}) (x_{2}) \\& \qquad {}- {}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau }(f_{1}) (x_{2}){}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau }( \mu f_{2}) (x_{2})- {}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }(\mu f_{1}) (x_{2}){}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau }( f_{2}) (x_{2}) \biggr\vert \\& \quad \leq \frac{1}{\kappa \varGamma _{\kappa }(\tau )} \biggl( \int _{x_{1}}^{x_{2}} \biggl[{}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }x_{2}\mu (x_{2}) \int _{x_{1}}^{\vartheta } \bigl(\varPsi (x_{2})- \varPsi (\varrho ) \bigr)^{ \frac{\tau }{\kappa }-1}\varPsi ^{\prime }(\varrho )\,d\varrho \\& \qquad {}- {}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau } \mu (x_{2}) \int _{x_{1}}^{ \vartheta }\varrho \bigl(\varPsi (x_{2})-\varPsi (\varrho ) \bigr)^{ \frac{\tau }{\kappa }-1}\varPsi ^{\prime }( \varrho )\,d\varrho \biggr] \bigl(f_{1}^{\prime }(\vartheta ) \bigr)^{2}\,d\vartheta \biggr)^{\frac{1}{2}} \\& \qquad {}\times \biggl( \int _{x_{1}}^{x_{2}} \biggl[{}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }x_{2}\mu (x_{2}) \int _{x_{1}}^{\vartheta } \bigl(\varPsi (x_{2})- \varPsi (\varrho ) \bigr)^{\frac{\tau }{\kappa }-1}\varPsi ^{\prime }(\varrho )\,d\varrho \\& \qquad {}- {}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau } \mu (x_{2}) \int _{x_{1}}^{ \vartheta }\varrho \bigl(\varPsi (x_{2})-\varPsi (\varrho ) \bigr)^{ \frac{\tau }{\kappa }-1}\varPsi ^{\prime }( \varrho )\,d\varrho \biggr] \bigl(f_{2}^{\prime }(\vartheta ) \bigr)^{2}\,d\vartheta \biggr)^{\frac{1}{2}}. \end{aligned}$$

Proof

Applying Theorem 3.2 with \(\nu (\vartheta )=1\), \(\vartheta \in [x_{1},x_{2}]\), we obtain the desired result. □

Corollary 3.6

LetΨbe a measurable increasing positive function on\((x_{1},x_{2})\)with continuous derivative\(\varPsi ^{\prime }(\varrho )\)on\([x_{1},x_{2}]\). Let\(f_{1}, f_{2}: [x_{1},x_{2}]\rightarrow \mathbb{R}\)be absolutely continuous functions with\((f_{1}^{\prime })^{2}\in L_{1}[x_{1},x_{2}]\)and\((f_{2}^{\prime })^{2}\in L_{1}[x_{1},x_{2}]\), and let\(\mu : [x_{1},x_{2}]\rightarrow \mathbb{R}^{+}\)be positive integrable. Then for all\(\tau ,\kappa >0\), we have

$$\begin{aligned}& \biggl\vert \frac{ (\varPsi (x_{2})-\varPsi (x_{1}) )^{\frac{\tau }{\kappa }}}{\varGamma _{\kappa }(\tau +\kappa )} {}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }( f_{1}f_{2}) (x_{2}) -{}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau }(f_{1}) (x_{2}){}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau }( \mu f_{2}) (x_{2}) \\& \qquad {}- {}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }( f_{1}) (x_{2}){}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau }( f_{2}) (x_{2}) \biggr\vert \\& \quad \leq \frac{1}{2\kappa \varGamma _{\kappa }(\tau )} \biggl( \int _{x_{1}}^{x_{2}} \biggl[{}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }x_{2} \int _{x_{1}}^{\vartheta } \bigl(\varPsi (x_{2})- \varPsi (\varrho ) \bigr)^{ \frac{\tau }{\kappa }-1}\varPsi ^{\prime }(\varrho )\,d\varrho \\& \qquad {}- \frac{ (\varPsi (x_{2})-\varPsi (x_{1}) )^{\frac{\tau }{\kappa }}}{\varGamma _{\kappa }(\tau +\kappa )} \int _{x_{1}}^{\vartheta }\varrho \bigl(\varPsi (x_{2})-\varPsi (\varrho ) \bigr)^{\frac{\tau }{\kappa }-1}\varPsi ^{\prime }( \varrho )\,d\varrho \biggr] \bigl(f_{1}^{\prime }(\vartheta ) \bigr)^{2}\,d\vartheta \biggr)^{\frac{1}{2}} \\& \qquad {}\times \biggl( \int _{x_{1}}^{x_{2}} \biggl[{}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }x_{2} \int _{x_{1}}^{\vartheta } \bigl(\varPsi (x_{2})- \varPsi ( \varrho ) \bigr)^{\frac{\tau }{\kappa }-1}\varPsi ^{\prime }(\varrho )\,d\varrho \\& \qquad {}- \frac{ (\varPsi (x_{2})-\varPsi (x_{1}) )^{\frac{\tau }{\kappa }}}{\varGamma _{\kappa }(\tau +\kappa )} \int _{x_{1}}^{\vartheta }\varrho \bigl(\varPsi (x_{2})-\varPsi (\varrho ) \bigr)^{\frac{\tau }{\kappa }-1}\varPsi ^{\prime }( \varrho )\,d\varrho \biggr] \bigl(f_{2}^{\prime }(\vartheta ) \bigr)^{2}\,d\vartheta \biggr)^{\frac{1}{2}}. \end{aligned}$$

Proof

Applying Theorem 3.2 with \(\mu (\vartheta )=\nu (\vartheta )=1\), \(\vartheta \in [x_{1},x_{2}]\), we obtain the desired result. □

Theorem 3.3

LetΨbe measurable increasing positive function on\((x_{1},x_{2})\)with continuous derivative\(\varPsi ^{\prime }(\varrho )\)on\([x_{1},x_{2}]\). Let\(f_{1}: [x_{1},x_{2}]\rightarrow \mathbb{R}\)be an absolutely continuous function with\((f_{1}^{\prime })\in L^{\infty }[x_{1},x_{2}]\), and let\(f_{2}: [x_{1},x_{2}]\rightarrow \mathbb{R}\)be nondecreasing. Moreover, let\(\mu ,\nu : [x_{1},x_{2}]\rightarrow \mathbb{R}^{+}\)be positive integrable, Then for all\(\tau ,\kappa >0\), we have

$$\begin{aligned}& \bigl\vert {}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }\mu (x_{2}){}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }(\nu f_{1}f_{2}) (x_{2})+ {}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau } \nu (x_{2}){}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau }( \mu f_{1}f_{2}) (x_{2}) \\& \qquad {}- {}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau }( \nu f_{1}) (x_{2}){}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }(\mu f_{2}) (x_{2})- {}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau }( \mu f_{1}) (x_{2}){}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }(\nu f_{2}) (x_{2}) \bigr\vert \\& \quad \leq \frac{ \Vert f_{1}^{\prime } \Vert _{\infty }}{\kappa \varGamma _{\kappa }(\tau )} \biggl[{}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }x_{2}\mu (x_{2}) \int _{x_{1}}^{\vartheta } \bigl(\varPsi (x_{2})- \varPsi (\varrho ) \bigr)^{ \frac{\tau }{\kappa }-1}\varPsi ^{\prime }(\varrho )\nu (\varrho )\,d\varrho \\& \qquad {}- {}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau } \mu (x_{2}) \int _{x_{1}}^{ \vartheta }\varrho \bigl(\varPsi (x_{2})-\varPsi (\varrho ) \bigr)^{ \frac{\tau }{\kappa }-1}\varPsi ^{\prime }( \varrho )\nu (\varrho )\,d\varrho \biggr] \int _{x_{1}}^{x_{2}}f_{2}^{\prime }( \vartheta )\,d\vartheta . \end{aligned}$$
(17)

Proof

Considering the left-hand side of (17), we have

$$\begin{aligned}& \bigl\vert {}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau } \mu (x_{2}){}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau }( \nu f_{1}f_{2}) (x_{2})+ {}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }\nu (x_{2}){}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }(\mu f_{1}f_{2}) (x_{2}) \\& \qquad {}- {}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau }( \nu f_{1}) (x_{2}){}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }(\mu f_{2}) (x_{2})- {}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau }( \mu f_{1}) (x_{2}){}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }(\nu f_{2}) (x_{2}) \bigr\vert \\& \quad \leq \frac{1}{\kappa ^{2}\varGamma _{\kappa }^{2}(\tau )} \biggl\vert \int _{x_{1}}^{x_{2}} \int _{x_{1}}^{x_{2}} \bigl(\varPsi (x_{2})- \varPsi (\xi ) \bigr)^{ \frac{\tau }{\kappa }-1} \bigl(\varPsi (x_{2})-\varPsi (\varrho ) \bigr)^{ \frac{\tau }{\kappa }-1}\varPsi ^{\prime }(\xi )\mu (\xi )\varPsi ^{\prime }( \varrho )\nu (\varrho ) \\& \qquad {}\times \bigl[ \bigl(f_{1}(\xi )-f_{1}(\varrho ) \bigr) \bigl(f_{2}( \xi )-f_{2}(\varrho ) \bigr) \bigr]\,d \xi \,d\varrho \biggr\vert \\& \quad \leq \frac{1}{\kappa ^{2}\varGamma _{\kappa }^{2}(\tau )} \int _{x_{1}}^{x_{2}} \int _{x_{1}}^{x_{2}} \bigl(\varPsi (x_{2})- \varPsi (\xi ) \bigr)^{ \frac{\tau }{\kappa }-1} \bigl(\varPsi (x_{2})-\varPsi (\varrho ) \bigr)^{ \frac{\tau }{\kappa }-1}\varPsi ^{\prime }(\xi )\mu (\xi )\varPsi ^{\prime }( \varrho )\nu (\varrho ) \\& \qquad {}\times \biggl\vert \frac{ (f_{1}(\xi )-f_{1}(\varrho ) )}{\xi -\varrho } \biggr\vert {} \bigl\vert (\xi -\varrho ) \bigl(f_{2}(\xi )-f_{2}(\varrho ) \bigr) \bigr\vert \,d\xi \,d\varrho \\& \leq \frac{ \Vert f_{1}^{\prime } \Vert _{\infty }}{\kappa ^{2}\varGamma _{\kappa }^{2}(\tau )} \int _{x_{1}}^{x_{2}} \int _{x_{1}}^{\vartheta } \bigl(\varPsi (x_{2})- \varPsi (\xi ) \bigr)^{\frac{\tau }{\kappa }-1} \bigl(\varPsi (x_{2})-\varPsi ( \varrho ) \bigr)^{\frac{\tau }{\kappa }-1}\varPsi ^{\prime }(\xi )\mu (\xi ) \varPsi ^{\prime }(\varrho )\nu (\varrho ) \\& \qquad {}\times (\xi -\varrho ) \biggl( \int _{x_{1}}^{x_{2}} f_{2}^{\prime }( \vartheta )\,d\vartheta \biggr)\,d\xi \,d\varrho \\& \quad \leq \frac{ \Vert f_{1}^{\prime } \Vert _{\infty }}{\kappa ^{2}\varGamma _{\kappa }^{2}(\tau )} \biggl[ \int _{x_{1}}^{x_{2}} \bigl(\varPsi (x_{2})- \varPsi (\xi ) \bigr)^{ \frac{\tau }{\kappa }-1}\xi \varPsi ^{\prime }(\xi )\mu (\xi )\,d\xi \\& \qquad {}\times\int _{x_{1}}^{ \vartheta } \bigl(\varPsi (x_{2})- \varPsi (\varrho ) \bigr)^{ \frac{\tau }{\kappa }-1} \varPsi ^{\prime }(\varrho )\nu ( \varrho )\,d\varrho - \int _{x_{1}}^{x_{2}} \bigl(\varPsi (x_{2})- \varPsi (\xi ) \bigr)^{ \frac{\tau }{\kappa }-1}\varPsi ^{\prime }(\xi )\mu (\xi )\,d\xi \\& \qquad {}\times\int _{x_{1}}^{ \vartheta }\varrho \bigl(\varPsi (x_{2})-\varPsi (\varrho ) \bigr)^{ \frac{\tau }{\kappa }-1} \varPsi ^{\prime }(\varrho )\nu (\varrho )\,d\varrho \biggr] \int _{x_{1}}^{x_{2}} f_{2}^{\prime }( \vartheta )\,d\vartheta. \end{aligned}$$

Hence taking (9) into account, we complete the proof of (17). □

Corollary 3.7

LetΨbe a measurable increasing positive function on\((x_{1},x_{2})\)with continuous derivative\(\varPsi ^{\prime }(\varrho )\)on\([x_{1},x_{2}]\). Let\(f_{1}: [x_{1},x_{2}]\rightarrow \mathbb{R}\)be an absolutely continuous function with\((f_{1}^{\prime })\in L^{\infty }[x_{1},x_{2}]\), and let\(f_{2}: [x_{1},x_{2}]\rightarrow \mathbb{R}\)be a nondecreasing function. Suppose that\(\nu : [x_{1},x_{2}]\rightarrow \mathbb{R}^{+}\)is positive integrable. Then for all\(\tau ,\kappa >0\), we have

$$\begin{aligned}& \biggl\vert \frac{ (\varPsi (x_{2})-\varPsi (x_{1}) )^{\frac{\tau }{\kappa }}}{\varGamma _{\kappa }(\tau +\kappa )} {}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }(\nu f_{1}f_{2}) (x_{2})+ {}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau } \nu (x_{2}){}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau }( f_{1}f_{2}) (x_{2}) \\& \qquad {}- {}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau }( \nu f_{1}) (x_{2}){}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }( f_{2}) (x_{2})- {}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau }( f_{1}) (x_{2}){}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }(\nu f_{2}) (x_{2}) \biggr\vert \\& \quad \leq \frac{ \Vert f_{1}^{\prime } \Vert _{\infty }}{\kappa \varGamma _{\kappa }(\tau )} \biggl[{}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }x_{2} \int _{x_{1}}^{\vartheta } \bigl(\varPsi (x_{2})- \varPsi (\varrho ) \bigr)^{ \frac{\tau }{\kappa }-1}\varPsi ^{\prime }(\varrho )\nu (\varrho )\,d\varrho \\& \qquad {}- \frac{ (\varPsi (x_{2})-\varPsi (x_{1}) )^{\frac{\tau }{\kappa }}}{\varGamma _{\kappa }(\tau +\kappa )} \int _{x_{1}}^{\vartheta }\varrho \bigl(\varPsi (x_{2})-\varPsi (\varrho ) \bigr)^{\frac{\tau }{\kappa }-1}\varPsi ^{\prime }( \varrho )\nu (\varrho )\,d\varrho \biggr] \int _{x_{1}}^{x_{2}}f_{2}^{\prime }( \vartheta )\,d\vartheta . \end{aligned}$$

Proof

Applying Theorem 3.3 with \(\mu (\vartheta )=1\), \(\vartheta \in [x_{1},x_{2}]\), we obtain the desired result. □

Corollary 3.8

LetΨbe a measurable increasing positive function on\((x_{1},x_{2})\)with continuous derivative\(\varPsi ^{\prime }(\varrho )\)on\([x_{1},x_{2}]\). Let\(f_{1}: [x_{1},x_{2}]\rightarrow \mathbb{R}\)be an absolutely continuous function with\((f_{1}^{\prime })\in L^{\infty }[x_{1},x_{2}]\), and let\(f_{2}: [x_{1},x_{2}]\rightarrow \mathbb{R}\)be nondecreasing. Suppose that\(\mu : [x_{1},x_{2}]\rightarrow \mathbb{R}^{+}\)is positive integrable. Then for all\(\tau ,\kappa >0\), we have

$$\begin{aligned}& \biggl\vert {}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau } \mu (x_{2}){}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau }( \nu f_{1}f_{2}) (x_{2})+ {} \frac{ (\varPsi (x_{2})-\varPsi (x_{1}) )^{\frac{\tau }{\kappa }}}{\varGamma _{\kappa }(\tau +\kappa )} {}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau }( \mu f_{1}f_{2}) (x_{2}) \\& \qquad {}- {}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau }( f_{1}) (x_{2}){}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }(\mu f_{2}) (x_{2})- {}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau }( \mu f_{1}) (x_{2}){}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }( f_{2}) (x_{2}) \biggr\vert \\& \quad \leq \frac{ \Vert f_{1}^{\prime } \Vert _{\infty }}{\kappa \varGamma _{\kappa }(\tau )} \biggl[{}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }x_{2}\mu (x_{2}) \int _{x_{1}}^{\vartheta } \bigl(\varPsi (x_{2})- \varPsi (\varrho ) \bigr)^{ \frac{\tau }{\kappa }-1}\varPsi ^{\prime }(\varrho )\,d\varrho \\& \qquad {}- {}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau } \mu (x_{2}) \int _{x_{1}}^{ \vartheta }\varrho \bigl(\varPsi (x_{2})-\varPsi (\varrho ) \bigr)^{ \frac{\tau }{\kappa }-1}\varPsi ^{\prime }( \varrho )\,d\varrho \biggr] \int _{x_{1}}^{x_{2}}f_{2}^{\prime }( \vartheta )\,d\vartheta . \end{aligned}$$

Proof

Applying Theorem 3.3 with \(\nu (\vartheta )=1\), \(\vartheta \in [x_{1},x_{2}]\), we obtain the desired result. □

Corollary 3.9

LetΨbe a measurable increasing positive function on\((x_{1},x_{2})\)with continuous derivative\(\varPsi ^{\prime }(\varrho )\)on\([x_{1},x_{2}]\). Let\(f_{1}: [x_{1},x_{2}]\rightarrow \mathbb{R}\)be an absolutely continuous function with\((f_{1}^{\prime })\in L^{\infty }[x_{1},x_{2}]\), and let\(f_{2}: [x_{1},x_{2}]\rightarrow \mathbb{R}\)be nondecreasing. Then for all\(\tau ,\kappa >0\), we have

$$\begin{aligned}& \biggl\vert \frac{ (\varPsi (x_{2})-\varPsi (x_{1}) )^{\frac{\tau }{\kappa }}}{\varGamma _{\kappa }(\tau +\kappa )} {}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }(\nu f_{1}f_{2}) (x_{2}) -{}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau }( f_{1}) (x_{2}){}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }( f_{2}) (x_{2}) \biggr\vert \\& \quad \leq \frac{ \Vert f_{1}^{\prime } \Vert _{\infty }}{2\kappa \varGamma _{\kappa }(\tau )} \biggl[{}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }x_{2} \int _{x_{1}}^{\vartheta } \bigl(\varPsi (x_{2})- \varPsi (\varrho ) \bigr)^{ \frac{\tau }{\kappa }-1}\varPsi ^{\prime }(\varrho )\nu (\varrho )\,d\varrho \\& \qquad {}- \frac{ (\varPsi (x_{2})-\varPsi (x_{1}) )^{\frac{\tau }{\kappa }}}{\varGamma _{\kappa }(\tau +\kappa )} \int _{x_{1}}^{\vartheta }\varrho \bigl(\varPsi (x_{2})-\varPsi (\varrho ) \bigr)^{\frac{\tau }{\kappa }-1}\varPsi ^{\prime }( \varrho )\nu (\varrho )\,d\varrho \biggr] \int _{x_{1}}^{x_{2}}f_{2}^{\prime }( \vartheta )\,d\vartheta . \end{aligned}$$

Proof

Applying Theorem 3.3 with \(\mu (\vartheta )=\nu (\vartheta )=1\), \(\vartheta \in [x_{1},x_{2}]\), we obtain the desired result. □

Theorem 3.4

LetΨbe a measurable increasing positive function on\((x_{1},x_{2})\)with continuous derivative\(\varPsi ^{\prime }(\varrho )\)on\([x_{1},x_{2}]\). Let\(f_{1}, f_{2}: [x_{1},x_{2}]\rightarrow \mathbb{R}\)be absolutely continuous functions, and let\(f_{2}: [x_{1},x_{2}]\rightarrow \mathbb{R}\)be nondecreasing. Suppose that\(\mu ,\nu : [x_{1},x_{2}]\rightarrow \mathbb{R}^{+}\)are positive integrable. If\(f_{1}^{\prime }, f_{2}^{\prime }\in L^{\infty }[x_{1},x_{2}]\), then for all\(\tau ,\kappa >0\), we have

$$\begin{aligned}& \bigl\vert {}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }\mu (x_{2}){}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }(\nu f_{1}f_{2}) (x_{2})+ {}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau } \nu (x_{2}){}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau }( \mu f_{1}f_{2}) (x_{2}) \\& \qquad {}- {}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau }( \nu f_{1}) (x_{2}){}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }(\mu f_{2}) (x_{2})- {}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau }( \mu f_{1}) (x_{2}){}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }(\nu f_{2}) (x_{2}) \bigr\vert \\& \quad \leq \frac{ \Vert f_{1}^{\prime } \Vert _{\infty } \Vert f_{2}^{\prime } \Vert _{\infty }}{\kappa \varGamma _{\kappa }(\tau )} \biggl[{}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }x_{2}^{2}\mu (x_{2}) \int _{x_{1}}^{\vartheta } \bigl(\varPsi (x_{2})- \varPsi (\varrho ) \bigr)^{ \frac{\tau }{\kappa }-1}\varPsi ^{\prime }(\varrho )\nu (\varrho )\,d\varrho \\& \qquad {}- 2{}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau }x_{2} \mu (x_{2}) \int _{x_{1}}^{ \vartheta }\varrho \bigl(\varPsi (x_{2})-\varPsi (\varrho ) \bigr)^{ \frac{\tau }{\kappa }-1}\varPsi ^{\prime }( \varrho )\nu (\varrho )\,d\varrho \\& \qquad {}+ {}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau } \mu (x_{2}) \int _{x_{1}}^{ \vartheta }\varrho ^{2} \bigl( \varPsi (x_{2})-\varPsi (\varrho ) \bigr)^{ \frac{\tau }{\kappa }-1}\varPsi ^{\prime }(\varrho )\nu (\varrho )\,d\varrho \biggr]. \end{aligned}$$
(18)

Proof

Considering the left-hand side of (18), we have

$$\begin{aligned}& \bigl\vert {}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau } \mu (x_{2}){}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau }( \nu f_{1}f_{2}) (x_{2})+ {}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }\nu (x_{2}){}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }(\mu f_{1}f_{2}) (x_{2}) \\& \qquad {}- {}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau }( \nu f_{1}) (x_{2}){}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }(\mu f_{2}) (x_{2})- {}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau }( \mu f_{1}) (x_{2}){}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }(\nu f_{2}) (x_{2}) \bigr\vert \\& \quad \leq \frac{1}{\kappa ^{2}\varGamma _{\kappa }^{2}(\tau )} \biggl\vert \int _{x_{1}}^{x_{2}} \int _{x_{1}}^{x_{2}} \bigl(\varPsi (x_{2})- \varPsi (\xi ) \bigr)^{ \frac{\tau }{\kappa }-1} \bigl(\varPsi (x_{2})-\varPsi (\varrho ) \bigr)^{ \frac{\tau }{\kappa }-1}\varPsi ^{\prime }(\xi )\mu (\xi )\varPsi ^{\prime }( \varrho )\nu (\varrho ) \\& \qquad {}\times \bigl[ \bigl(f_{1}(\xi )-f_{1}(\varrho ) \bigr) \bigl(f_{2}( \xi )-f_{2}(\varrho ) \bigr) \bigr]\,d \xi \,d\varrho \biggr\vert \\& \quad \leq \frac{1}{\kappa ^{2}\varGamma _{\kappa }^{2}(\tau )} \int _{x_{1}}^{x_{2}} \int _{x_{1}}^{x_{2}} \bigl(\varPsi (x_{2})- \varPsi (\xi ) \bigr)^{ \frac{\tau }{\kappa }-1} \bigl(\varPsi (x_{2})-\varPsi (\varrho ) \bigr)^{ \frac{\tau }{\kappa }-1}\varPsi ^{\prime }(\xi )\mu (\xi )\varPsi ^{\prime }( \varrho )\nu (\varrho ) \\& \qquad {}\times \biggl\vert \frac{ (f_{1}(\xi )-f_{1}(\varrho ) )}{\xi -\varrho } \biggr\vert {} \biggl\vert \frac{ (f_{2}(\xi )-f_{2}(\varrho ) )}{\xi -\varrho } \biggr\vert ( \xi -\varrho )^{2}\,d\xi \,d \varrho \\& \quad \leq \frac{ \Vert f_{1}^{\prime } \Vert _{\infty } \Vert f_{2}^{\prime } \Vert _{\infty }}{\kappa ^{2}\varGamma _{\kappa }^{2}(\tau )} \int _{x_{1}}^{x_{2}} \int _{x_{1}}^{\vartheta } \bigl(\varPsi (x_{2})- \varPsi (\xi ) \bigr)^{\frac{\tau }{\kappa }-1} \bigl(\varPsi (x_{2})-\varPsi ( \varrho ) \bigr)^{\frac{\tau }{\kappa }-1}\varPsi ^{\prime }(\xi )\mu (\xi ) \varPsi ^{\prime }(\varrho )\nu (\varrho ) \\& \qquad {}\times \bigl(\xi ^{2} -2\xi \varrho +\varrho ^{2} \bigr)\,d\xi \,d\varrho \\& \quad \leq \frac{ \Vert f_{1}^{\prime } \Vert _{\infty } \Vert f_{2}^{\prime } \Vert _{\infty }}{\kappa ^{2}\varGamma _{\kappa }^{2}(\tau )} \biggl[ \int _{x_{1}}^{x_{2}} \bigl(\varPsi (x_{2})- \varPsi (\xi ) \bigr)^{ \frac{\tau }{\kappa }-1}\xi ^{2}\varPsi ^{\prime }(\xi ) \mu (\xi )\,d\xi \\& \qquad {}\times\int _{x_{1}}^{ \vartheta } \bigl(\varPsi (x_{2})- \varPsi (\varrho ) \bigr)^{ \frac{\tau }{\kappa }-1} \varPsi ^{\prime }(\varrho )\nu ( \varrho )\,d\varrho \\& \qquad {}- 2 \int _{x_{1}}^{x_{2}} \bigl(\varPsi (x_{2})- \varPsi (\xi ) \bigr)^{ \frac{\tau }{\kappa }-1}\xi \varPsi ^{\prime }(\xi )\mu (\xi )\,d\xi \int _{x_{1}}^{ \vartheta }\varrho \bigl(\varPsi (x_{2})-\varPsi (\varrho ) \bigr)^{ \frac{\tau }{\kappa }-1} \varPsi ^{\prime }(\varrho )\nu (\varrho )\,d\varrho \\& \qquad {}+ \int _{x_{1}}^{x_{2}} \bigl(\varPsi (x_{2})- \varPsi (\xi ) \bigr)^{ \frac{\tau }{\kappa }-1}\varPsi ^{\prime }(\xi )\mu (\xi )\,d\xi \int _{x_{1}}^{ \vartheta }\varrho ^{2} \bigl( \varPsi (x_{2})-\varPsi (\varrho ) \bigr)^{ \frac{\tau }{\kappa }-1} \varPsi ^{\prime }(\varrho )\nu (\varrho )\,d\varrho \biggr]. \end{aligned}$$

Hence by (9) we complete the proof. □

Corollary 3.10

LetΨbe a measurable increasing positive function on\((x_{1},x_{2})\)with continuous derivative\(\varPsi ^{\prime }(\varrho )\)on\([x_{1},x_{2}]\). Let\(f_{1}, f_{2}: [x_{1},x_{2}]\rightarrow \mathbb{R}\)be absolutely continuous functions, and let\(f_{2}: [x_{1},x_{2}]\rightarrow \mathbb{R}\)be nondecreasing. Suppose that\(\nu : [x_{1},x_{2}]\rightarrow \mathbb{R}^{+}\)is positive integrable. If\(f_{1}^{\prime }, f_{2}^{\prime }\in L^{\infty }[x_{1},x_{2}]\), then for all\(\tau ,\kappa >0\), we have

$$\begin{aligned}& \biggl\vert \frac{ (\varPsi (x_{2})-\varPsi (x_{1}) )^{\frac{\tau }{\kappa }}}{\varGamma _{\kappa }(\tau +\kappa )} {}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }(\nu f_{1}f_{2}) (x_{2})+ {}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau } \nu (x_{2}){}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau }( f_{1}f_{2}) (x_{2}) \\& \qquad {}- {}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau }( \nu f_{1}) (x_{2}){}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }( f_{2}) (x_{2})- {}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau }( f_{1}) (x_{2}){}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }(\nu f_{2}) (x_{2}) \biggr\vert \\& \quad \leq \frac{ \Vert f_{1}^{\prime } \Vert _{\infty } \Vert f_{2}^{\prime } \Vert _{\infty }}{\kappa \varGamma _{\kappa }(\tau )} \biggl[{}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }x_{2}^{2} \int _{x_{1}}^{\vartheta } \bigl(\varPsi (x_{2})- \varPsi (\varrho ) \bigr)^{ \frac{\tau }{\kappa }-1}\varPsi ^{\prime }(\varrho )\nu (\varrho )\,d\varrho \\& \qquad {}- 2{}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau }x_{2} \int _{x_{1}}^{ \vartheta }\varrho \bigl(\varPsi (x_{2})-\varPsi (\varrho ) \bigr)^{ \frac{\tau }{\kappa }-1}\varPsi ^{\prime }( \varrho )\nu (\varrho )\,d\varrho \\& \qquad {}+ \frac{ (\varPsi (x_{2})-\varPsi (x_{1}) )^{\frac{\tau }{\kappa }}}{\varGamma _{\kappa }(\tau +\kappa )} \int _{x_{1}}^{\vartheta }\varrho ^{2} \bigl( \varPsi (x_{2})-\varPsi ( \varrho ) \bigr)^{\frac{\tau }{\kappa }-1}\varPsi ^{\prime }(\varrho )\nu ( \varrho )\,d\varrho \biggr]. \end{aligned}$$

Proof

Setting \(\mu (\vartheta )=1\), \(\vartheta \in [x_{1}, x_{2}]\), in Theorem 3.4, we obtain the desired result. □

Corollary 3.11

LetΨbe a measurable increasing positive function on\((x_{1},x_{2})\)with continuous derivative\(\varPsi ^{\prime }(\varrho )\)on\([x_{1},x_{2}]\). Let\(f_{1}, f_{2}: [x_{1},x_{2}]\rightarrow \mathbb{R}\)be absolutely continuous functions, and let\(f_{2}: [x_{1},x_{2}]\rightarrow \mathbb{R}\)be nondecreasing. Suppose that\(\mu : [x_{1},x_{2}]\rightarrow \mathbb{R}^{+}\)is positive integrable. If\(f_{1}^{\prime }, f_{2}^{\prime }\in L^{\infty }[x_{1},x_{2}]\), then for all\(\tau ,\kappa >0\), we have

$$\begin{aligned}& \biggl\vert {}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau } \mu (x_{2}){}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau }( f_{1}f_{2}) (x_{2})+ {} \frac{ (\varPsi (x_{2})-\varPsi (x_{1}) )^{\frac{\tau }{\kappa }}}{\varGamma _{\kappa }(\tau +\kappa )} {}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau }( \mu f_{1}f_{2}) (x_{2}) \\& \qquad {}- {}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau }( f_{1}) (x_{2}){}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }(\mu f_{2}) (x_{2})- {}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau }( \mu f_{1}) (x_{2}){}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }( f_{2}) (x_{2}) \biggr\vert \\& \quad \leq \frac{ \Vert f_{1}^{\prime } \Vert _{\infty } \Vert f_{2}^{\prime } \Vert _{\infty }}{\kappa \varGamma _{\kappa }(\tau )} \biggl[{}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }x_{2}^{2}\mu (x_{2}) \int _{x_{1}}^{\vartheta } \bigl(\varPsi (x_{2})- \varPsi (\varrho ) \bigr)^{ \frac{\tau }{\kappa }-1}\varPsi ^{\prime }(\varrho )\,d\varrho \\& \qquad {}- 2{}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau }x_{2} \mu (x_{2}) \int _{x_{1}}^{ \vartheta }\varrho \bigl(\varPsi (x_{2})-\varPsi (\varrho ) \bigr)^{ \frac{\tau }{\kappa }-1}\varPsi ^{\prime }( \varrho )\,d\varrho \\& \qquad {}+ {}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau } \mu (x_{2}) \int _{x_{1}}^{ \vartheta }\varrho ^{2} \bigl( \varPsi (x_{2})-\varPsi (\varrho ) \bigr)^{ \frac{\tau }{\kappa }-1}\varPsi ^{\prime }(\varrho )\,d\varrho \biggr]. \end{aligned}$$

Proof

Setting \(\nu (\vartheta )=1\), \(\vartheta \in [x_{1}, x_{2}]\), in Theorem 3.4, we obtain the desired result. □

Corollary 3.12

LetΨbe a measurable increasing positive function on\((x_{1},x_{2})\)with continuous derivative\(\varPsi ^{\prime }(\varrho )\)on\([x_{1},x_{2}]\). Let\(f_{1}, f_{2}: [x_{1},x_{2}]\rightarrow \mathbb{R}\)be absolutely continuous functions, and let\(f_{2}: [x_{1},x_{2}]\rightarrow \mathbb{R}\)be nondecreasing. If\(f_{1}^{\prime }, f_{2}^{\prime }\in L^{\infty }[x_{1},x_{2}]\), then for all\(\tau ,\kappa >0\), we have

$$\begin{aligned}& \biggl\vert \frac{ (\varPsi (x_{2})-\varPsi (x_{1}) )^{\frac{\tau }{\kappa }}}{\varGamma _{\kappa }(\tau +\kappa )} {}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }( f_{1}f_{2}) (x_{2}) -{}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau }( f_{1}) (x_{2}){}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }( f_{2}) (x_{2}) \biggr\vert \\& \quad \leq \frac{ \Vert f_{1}^{\prime } \Vert _{\infty } \Vert f_{2}^{\prime } \Vert _{\infty }}{2\kappa \varGamma _{\kappa }(\tau )} \biggl[{}_{x_{1}}^{\varPsi } \mathscr {I}_{\kappa }^{\tau }x_{2}^{2} \int _{x_{1}}^{\vartheta } \bigl(\varPsi (x_{2})- \varPsi (\varrho ) \bigr)^{ \frac{\tau }{\kappa }-1}\varPsi ^{\prime }(\varrho )\,d\varrho \\& \qquad {}- 2{}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau }x_{2} \int _{x_{1}}^{ \vartheta }\varrho \bigl(\varPsi (x_{2})-\varPsi (\varrho ) \bigr)^{ \frac{\tau }{\kappa }-1}\varPsi ^{\prime }( \varrho )\,d\varrho \\& \qquad {}+{}_{x_{1}}^{\varPsi }\mathscr {I}_{\kappa }^{\tau } \mu (x_{2}) \int _{x_{1}}^{\vartheta } \varrho ^{2} \bigl( \varPsi (x_{2})-\varPsi (\varrho ) \bigr)^{ \frac{\tau }{\kappa }-1}\varPsi ^{\prime }(\varrho )\,d\varrho \biggr]. \end{aligned}$$

Proof

Setting \(\mu (\vartheta )=\nu (\vartheta )=1\), \(\vartheta \in [x_{1}, x_{2}]\), in Theorem 3.4, we obtain the desired result. □

Particular cases

Here we present some inequalities in terms of the Riemann–Liouville κ-fractional integrals, which are the particular cases of the main results.

Theorem 4.1

Suppose that\(\varPhi : [x_{1},x_{2}]\rightarrow \mathbb{R}\)is absolutely continuous on\([x_{1},x_{2}]\)with\((\varPhi ^{\prime })^{2}\in L_{1}[x_{1},x_{2}]\)and that\(\mu ,\nu : [x_{1},x_{2}]\rightarrow \mathbb{R}^{+}\)are positive integrable. Then for all\(\tau ,\kappa >0\), we have

$$\begin{aligned}& \bigl[{}_{x_{1}}\mathscr {I}_{\kappa }^{\tau }\mu (x_{2}) \bigr] \bigl[ {}_{x_{1}}\mathscr {I}_{\kappa }^{\tau } \bigl(\nu \varPhi ^{2}\bigr) (x_{2}) \bigr]+ \bigl[{}_{x_{1}}\mathscr {I}_{\kappa }^{\tau }\nu (x_{2}) \bigr] \bigl[ {}_{x_{1}}\mathscr {I}_{\kappa }^{\tau } \bigl(\mu \varPhi ^{2}\bigr) (x_{2}) \bigr] \\& \qquad {}- 2 \bigl[{}_{x_{1}}\mathscr {I}_{\kappa }^{\tau }(\mu \varPhi ) (x_{2}) \bigr] \bigl[{}_{x_{1}}\mathscr {I}_{\kappa }^{\tau }( \nu \varPhi ) (x_{2}) \bigr] \\& \quad \leq \frac{1}{\kappa \varGamma _{\kappa }(\tau )} \int _{x_{1}}^{x_{2}} \biggl[{}_{x_{1}} \mathscr {I}_{\kappa }^{\tau }x_{2}\mu (x_{2}) \int _{x_{1}}^{\vartheta } (x_{2}-\varrho )^{\frac{\tau }{\kappa }-1}\nu ( \varrho )\,d\varrho \\& \qquad {}- {}_{x_{1}}\mathscr {I}_{\kappa }^{\tau }\mu (x_{2}) \int _{x_{1}}^{ \vartheta }\varrho (x_{2}-\varrho )^{\frac{\tau }{\kappa }-1} \nu (\varrho )\,d\varrho \biggr] \bigl(\varPhi ^{\prime }( \vartheta ) \bigr)^{2}\,d\vartheta . \end{aligned}$$

Proof

Applying Theorem 3.1 with \(\varPsi (\vartheta )=\vartheta \) gives the proof of the theorem. □

Theorem 4.2

Suppose that\(f_{1}, f_{2}: [x_{1},x_{2}]\rightarrow \mathbb{R}\)are absolutely continuous functions with\((f_{1}^{\prime })^{2}\in L_{1}[x_{1},x_{2}]\)and\((f_{2}^{\prime })^{2}\in L_{1}[x_{1},x_{2}]\)and that\(\mu ,\nu : [x_{1},x_{2}]\rightarrow \mathbb{R}^{+}\)are positive integrable. Then for all\(\tau ,\kappa >0\), we have

$$\begin{aligned}& \bigl\vert {}_{x_{1}}\mathscr {I}_{\kappa }^{\tau } \mu (x_{2}){}_{x_{1}} \mathscr {I}_{\kappa }^{\tau }( \nu f_{1}f_{2}) (x_{2})+ {}_{x_{1}} \mathscr {I}_{\kappa }^{\tau }\nu (x_{2}){}_{x_{1}} \mathscr {I}_{\kappa }^{ \tau }(\mu f_{1}f_{2}) (x_{2}) \\& \qquad {}- {}_{x_{1}}\mathscr {I}_{\kappa }^{\tau }(\nu f_{1}) (x_{2}){}_{x_{1}} \mathscr {I}_{\kappa }^{\tau }( \mu f_{2}) (x_{2})- {}_{x_{1}} \mathscr {I}_{\kappa }^{\tau }(\mu f_{1}) (x_{2}){}_{x_{1}}\mathscr {I}_{\kappa }^{\tau }( \nu f_{2}) (x_{2}) \bigr\vert \\& \quad \leq \frac{1}{\kappa \varGamma _{\kappa }(\tau )} \biggl( \int _{x_{1}}^{x_{2}} \biggl[{}_{x_{1}} \mathscr {I}_{\kappa }^{\tau }x_{2}\mu (x_{2}) \int _{x_{1}}^{\vartheta } (x_{2}-\varrho )^{\frac{\tau }{\kappa }-1}\nu ( \varrho )\,d\varrho \\& \qquad {}- {}_{x_{1}}\mathscr {I}_{\kappa }^{\tau }\mu (x_{2}) \int _{x_{1}}^{ \vartheta }\varrho (x_{2}-\varrho )^{\frac{\tau }{\kappa }-1} \nu (\varrho )\,d\varrho \biggr] \bigl(f_{1}^{\prime }( \vartheta ) \bigr)^{2}\,d\vartheta \biggr)^{\frac{1}{2}} \\& \qquad {}\times \biggl( \int _{x_{1}}^{x_{2}} \biggl[{}_{x_{1}} \mathscr {I}_{\kappa }^{\tau }x_{2}\mu (x_{2}) \int _{x_{1}}^{\vartheta } (x_{2}- \varrho )^{\frac{\tau }{\kappa }-1}\nu (\varrho )\,d\varrho \\& \qquad {}- {}_{x_{1}}\mathscr {I}_{\kappa }^{\tau }\mu (x_{2}) \int _{x_{1}}^{ \vartheta }\varrho (x_{2}-\varrho )^{\frac{\tau }{\kappa }-1} \nu (\varrho )\,d\varrho \biggr] \bigl(f_{2}^{\prime }( \vartheta ) \bigr)^{2}\,d\vartheta \biggr)^{\frac{1}{2}}. \end{aligned}$$
(19)

Proof

Applying Theorem 3.2 with \(\varPsi (\vartheta )=\vartheta \) gives the proof of the theorem. □

Similarly, we can get several new inequalities in terms of the Riemann–Liouville κ-fractional integrals for \(\varPsi (\vartheta )=\vartheta \) in Theorems 3.33.4. Also, employing Corollaries 3.13.12 for \(\varPsi (\vartheta )=\vartheta \) results in various new inequalities.

Remark 4.1

We can also establish other types of new inequalities by taking the following assumptions:

  1. i.

    Setting \(\mu (\vartheta )=\nu (\vartheta )\) and \(\varPsi (\vartheta )=\vartheta \) throughout the paper.

  2. ii.

    Setting \(\mu (\vartheta )=\nu (\vartheta )=1\) and \(\varPsi (\vartheta )=\vartheta \) throughout the paper.

Remark 4.2

If we take \(\kappa =1\), then all established results reduce to the work of Bezziou et al. [5].

Remark 4.3

Setting \(\mu (\vartheta )=\nu (\vartheta )\), \(\kappa =1\), and \(\varPsi (\vartheta )=\vartheta \) in Theorems 3.13.4 restores the results of Bezziou et al. [4].

Concluding remarks

In this present paper, we derived some double-weighted generalized fractional integral inequalities by employing the generalized Riemann–Liouville κ-fractional integrals containing another function Ψ in the kernels, where Ψ is integrable, measurable, positive, and monotone. We can quickly form many new fractional integral inequalities for different fractional definitions by considering Remark 2.2.

References

  1. 1.

    Alshabanat, A., Jleli, M., Kumar, S., Samet, B.: Generalization of Caputo–Fabrizio fractional derivative and applications to electrical circuits. Front. Phys. 8, 64 (2020)

    Google Scholar 

  2. 2.

    Awan, K.M., Pecaric, J., Rehman, A.: Steffensen’s generalization of Chebyshev inequality. J. Math. Inequal. 9(1), 155–163 (2015)

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Belarbi, S., Dahmani, Z.: On some new fractional integral inequalities. J. Inequal. Pure Appl. Math. 10(3), 1–12 (2009)

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Bezziou, M., Dahmani, Z., Khameli, A.: Some weighted inequalities of Chebyshev type via RL-approach. Mathematica 60(83), 12–20 (2018)

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Bezziou, M., Dahmani, Z., Khameli, A.: On some double-weighted fractional integral inequalities. Sarajevo J. Math. 15(28), 23–36 (2019)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Cerone, P., Dragomir, S.S.: A refinement of the Gruss inequality and applications. Tamkang J. Math. 38(1), 37–49 (2007)

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Chebyshev, P.L.: Sur les expressions approximatives des intégrales définies par les autres prises entre les mêmes limites. Proc. Math. Soc. Charkov. 2, 93–98 (1882)

    Google Scholar 

  8. 8.

    Dahmani, Z.: New inequalities in fractional integrals. Int. J. Nonlinear Sci. 9, 493–497 (2010)

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Dahmani, Z.: New inequalities in fractional integrals. Int. J. Nonlinear Sci. 9(4), 493–497 (2010)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Dahmani, Z.: The Riemann–Liouville operator to generate some new inequalities. Int. J. Nonlinear Sci. 12, 452–455 (2011)

    MathSciNet  MATH  Google Scholar 

  11. 11.

    Dahmani, Z.: About some integral inequalities using Riemann–Liouville integrals. Gen. Math. 20(4), 63–69 (2012)

    Google Scholar 

  12. 12.

    Dahmani, Z., Benzidane, A.: New inequalities using Q-fractional theory. Bull. Math. Anal. Appl. 4(1), 190–196 (2012)

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Dahmani, Z., Bounoua, M.D.: Further results on Chebyshev and Steffensen inequalities. Kyungpook Math. J. 58, 55–66 (2018)

    MathSciNet  MATH  Google Scholar 

  14. 14.

    Dahmani, Z., Tabharit, L.: On weighted Gruss type inequalities via fractional integration. J. Adv. Res. Pure Math. 2, 31–38 (2010)

    MathSciNet  Google Scholar 

  15. 15.

    Diaz, R., Pariglan, E.: On hypergeometric functions and Pochhammer k-symbol. Divulg. Mat. 15(2), 179–192 (2007)

    MathSciNet  MATH  Google Scholar 

  16. 16.

    Dragomir, S.S.: A generalization of Gruss’s inequality in inner product spaces and applications. J. Math. Anal. Appl. 237(1), 74–82 (1999)

    MathSciNet  MATH  Google Scholar 

  17. 17.

    Ghanbari, B., Kumar, S., Kumar, R.: A study of behaviour for immune and tumor cells in immunogenetic tumour model with non-singular fractional derivative. Chaos Solitons Fractals 133, 109619 (2020)

    MathSciNet  Google Scholar 

  18. 18.

    Goufo, E.F., Kumar, S., Mugisha, S.B.: Similarities in a fifth-order evolution equation with and with no singular kernel. Chaos Solitons Fractals 130, 109467 (2020)

    MathSciNet  Google Scholar 

  19. 19.

    Habib, S., Mubeen, S., Naeem, M.N.: Chebyshev type integral inequalities for generalized k-fractional conformable integrals. J. Inequal. Spec. Funct. 9(4), 53–65 (2018)

    MathSciNet  Google Scholar 

  20. 20.

    Huang, C.J., Rahman, G., Nisar, K.S., Ghaffar, A., Qi, F.: Some inequalities of Hermite–Hadamard type for k-fractional conformable integrals. Aust. J. Math. Anal. Appl. 16(1), 1–9 (2019)

    MathSciNet  MATH  Google Scholar 

  21. 21.

    Iqbal, S., Mubeen, S., Tomar, M.: On Hadamard k-fractional integrals. J. Fract. Calc. Appl. 9, 255–267 (2018)

    MathSciNet  MATH  Google Scholar 

  22. 22.

    Jarad, F., Abdeljawad, T., Alzabut, J.: Generalized fractional derivatives generated by a class of local proportional derivatives. Eur. Phys. J. 226, 3457–3471 (2017). https://doi.org/10.1140/epjst/e2018-00021-7

    Article  Google Scholar 

  23. 23.

    Jarad, F., Ugurlu, E., Abdeljawad, T., Baleanu, D.: On a new class of fractional operators. Adv. Differ. Equ. 2017, 247 (2017)

    MathSciNet  MATH  Google Scholar 

  24. 24.

    Katugampola, U.N.: Approach to a generalized fractional integral. Appl. Math. Comput. 218, 860–865 (2011)

    MathSciNet  MATH  Google Scholar 

  25. 25.

    Khan, T.U., Khan, M.A.: Generalized conformable fractional integral operators. J. Comput. Appl. Math. 346, 378–389 (2018). https://doi.org/10.1016/j.cam.2018.07.018

    Article  MATH  Google Scholar 

  26. 26.

    Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Application of Fractional Differential Equations. Elsevier, Amsterdam (2006)

    Google Scholar 

  27. 27.

    Kuang, J.C.: Applied Inequalities. Shandong Sciences and Technologie Press, Jinan (2004)

    Google Scholar 

  28. 28.

    Kumar, S.: A new fractional modeling arising in engineering sciences and its analytical approximate solution. Alex. Eng. J. 52(4), 813–819 (2013)

    Google Scholar 

  29. 29.

    Kumar, S., Ahmadian, A., Kumar, R., Kumar, D., Singh, J., Baleanu, D., Salimi, M.: An efficient numerical method for fractional SIR epidemic model of infectious disease by using Bernstein wavelets. Mathematics 8(4), 558 (2020)

    Google Scholar 

  30. 30.

    Kumar, S., Ghosh, S., Samet, B., Goufo, E.F.: An analysis for heat equations arises in diffusion process using new Yang–Abdel–Aty–Cattani fractional operator. Math. Methods Appl. Sci. 43, 6062–6080 (2020)

    Google Scholar 

  31. 31.

    Kumar, S., Kumar, R., Agarwal, R.P., Samet, B.: A study of fractional Lotka–Volterra population model using Haar wavelet and Adams–Bashforth–Moulton methods. Math. Methods Appl. Sci. 43(8), 5564–5578 (2020)

    Google Scholar 

  32. 32.

    Kumar, S., Nisar, K.S., Kumar, R., Cattani, C., Samet, B.: A new Rabotnov fractional-exponential function based fractional derivative for diffusion equation under external force. Math. Methods Appl. Sci. 43(7), 4460–4471 (2020). https://doi.org/10.1002/mma.6208

    Article  Google Scholar 

  33. 33.

    Kwun, Y.C., Farid, G., Nazeer, W., Ullah, S., Kang, S.M.: Generalized Riemann–Liouville k-fractional integrals associated with Ostrowski type inequalities and error bounds of Hadamard inequalities. IEEE Access 6, 64946–64953 (2018)

    Google Scholar 

  34. 34.

    McD Mercer, A.: An improvement of the Gruss inequality. J. Inequal. Pure Appl. Math. 10(4), Art. 93 (2005)

    MathSciNet  MATH  Google Scholar 

  35. 35.

    Mitrinovic, D.S.: Analytic Inequalities. Springer, Berlin (1970)

    Google Scholar 

  36. 36.

    Mitrinovic, D.S., Pecaric, J.E., Fink, A.M.: Classical and New Inequalities in Analysis. Kluwer Academic, Dordrecht (1993)

    Google Scholar 

  37. 37.

    Mubeen, S., Habibullah, G.M.: k-Fractional integrals and application. Int. J. Contemp. Math. Sci. 7, 89–94 (2012)

    MathSciNet  MATH  Google Scholar 

  38. 38.

    Nisar, K.S., Qi, F., Rahman, G., Mubeen, S., Arshad, M.: Some inequalities involving the extended gamma function and the Kummer confluent hypergeometric k-function. J. Inequal. Appl. 2018, 135 (2018)

    MathSciNet  Google Scholar 

  39. 39.

    Nisar, K.S., Rahman, G., Choi, J., Mubeen, S., Arshad, M.: Certain Gronwall type inequalities associated with Riemann–Liouville k- and Hadamard k-fractional derivatives and their applications. East Asian Math. J. 34(3), 249–263 (2018)

    MATH  Google Scholar 

  40. 40.

    Nisar, K.S., Rahman, G., Khan, A.: Some new inequalities for generalized fractional conformable integral operators. Adv. Differ. Equ. 2019, 427 (2019). https://doi.org/10.1186/s13662-019-2362-3

    MathSciNet  Article  Google Scholar 

  41. 41.

    Nisar, K.S., Rahman, G., Khan, A., Tassaddiq, A., Abouzaid, M.S.: Certain generalized fractional integral inequalities. AIMS Math. 5(2), 1588–1602 (2020). https://doi.org/10.3934/math.2020108

    Article  Google Scholar 

  42. 42.

    Nisar, K.S., Rahman, G., Mehrez, K.: Chebyshev type inequalities via generalized fractional conformable integrals. J. Inequal. Appl. 2019, 245 (2019). https://doi.org/10.1186/s13660-019-2197-1

    MathSciNet  Article  Google Scholar 

  43. 43.

    Nisar, K.S., Tassadiq, A., Rahman, G., Khan, A.: Some inequalities via fractional conformable integral operators. J. Inequal. Appl. 2019, 217 (2019). https://doi.org/10.1186/s13660-019-2170-z

    MathSciNet  Article  Google Scholar 

  44. 44.

    Ostrowski, A.M.: On an integral inequality. Aequ. Math. 4, 358–373 (1970)

    MATH  Google Scholar 

  45. 45.

    Qi, F., Rahman, G., Hussain, S.M., Du, W.S., Nisar, K.S.: Some inequalities of Čebyšev type for conformable k-fractional integral operators. Symmetry 10, 614 (2018). https://doi.org/10.3390/sym10110614

    Article  MATH  Google Scholar 

  46. 46.

    Rahman, G., Abdeljawad, T., Jarad, F., Khan, A., Nisar, K.S.: Certain inequalities via generalized proportional Hadamard fractional integral operators. Adv. Differ. Equ. 2019, 454 (2019). https://doi.org/10.1186/s13662-019-2381-0

    MathSciNet  Article  Google Scholar 

  47. 47.

    Rahman, G., Abdeljawad, T., Jarad, F., Nisar, K.S.: Bounds of generalized proportional fractional integrals in general form via convex functions and their applications. Mathematics 8, 113 (2020). https://doi.org/10.3390/math8010113

    Article  Google Scholar 

  48. 48.

    Rahman, G., Abdeljawad, T., Jarad, F., Nisar, K.S.: Bounds of generalized proportional fractional integrals in general form via convex functions and their applications. Mathematics 8, 113 (2020). https://doi.org/10.3390/math8010113

    Article  Google Scholar 

  49. 49.

    Rahman, G., Abdeljawad, T., Khan, A., Nisar, K.S.: Some fractional proportional integral inequalities. J. Inequal. Appl. 2019, 244 (2019). https://doi.org/10.1186/s13660-019-2199-z

    MathSciNet  Article  Google Scholar 

  50. 50.

    Rahman, G., Khan, A., Abdeljawad, T., Nisar, K.S.: The Minkowski inequalities via generalized proportional fractional integral operators. Adv. Differ. Equ. 2019, 287 (2019). https://doi.org/10.1186/s13662-019-2229-7

    MathSciNet  Article  Google Scholar 

  51. 51.

    Rahman, G., Nisar, K.S., Abdeljawad, T.: Certain Hadamard proportional fractional integral inequalities. Mathematics 8, 504 (2020). https://doi.org/10.3390/math8040504

    Article  Google Scholar 

  52. 52.

    Rahman, G., Nisar, K.S., Abdeljawad, T., Ullah, S.: Certain fractional proportional integral inequalities via convex functions. Mathematics 8, 222 (2020). https://doi.org/10.3390/math8020222

    Article  Google Scholar 

  53. 53.

    Rahman, G., Nisar, K.S., Abdeljawad, T., Ullah, S.: Certain fractional proportional integral inequalities via convex functions. Mathematics 8, 222 (2020). https://doi.org/10.3390/math8020222

    Article  Google Scholar 

  54. 54.

    Rahman, G., Nisar, K.S., Ghaffar, A., Qi, F.: Some inequalities of the Grüss type for conformable k-fractional integral operators. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 114, 9 (2020). https://doi.org/10.1007/s13398-019-00731-3

    Article  MATH  Google Scholar 

  55. 55.

    Rahman, G., Nisar, K.S., Mubeen, S., Choi, J.: Certain inequalities involving the \((k,\rho )\)-fractional integral operator. Far East J. Math. Sci.: FJMS 103(11), 1879–1888 (2018)

    Google Scholar 

  56. 56.

    Rahman, G., Nisar, K.S., Qi, F.: Some new inequalities of the Grüss type for conformable fractional integrals. AIMS Math. 3(4), 575–583 (2018)

    MATH  Google Scholar 

  57. 57.

    Rahman, G., Ullah, Z., Khan, A., Set, E., Nisar, K.S.: Certain Chebyshev type inequalities involving fractional conformable integral operators. Mathematics 7, 364 (2019). https://doi.org/10.3390/math7040364

    Article  Google Scholar 

  58. 58.

    Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives, Theory and Applications. Gordon & Breach, Yverdon (1993)

    Google Scholar 

  59. 59.

    Sarikaya, M.Z., Budak, H.: Generalized Ostrowski type inequalities for local fractional integrals. Proc. Am. Math. Soc. 145(4), 1527–1538 (2017)

    MathSciNet  MATH  Google Scholar 

  60. 60.

    Sarikaya, M.Z., Dahmani, Z., Kiris, M.E., Ahmad, F.: \((k, s)\)-Riemann–Liouville fractional integral and applications. Hacet. J. Math. Stat. 45(1), 77–89 (2016)

    MathSciNet  MATH  Google Scholar 

  61. 61.

    Set, E., Tomar, M., Sarikaya, M.Z.: On generalized Grüss type inequalities for k-fractional integrals. Appl. Math. Comput. 269, 29–34 (2015)

    MathSciNet  MATH  Google Scholar 

  62. 62.

    Sharma, B., Kumar, S., Cattani, C., Baleanu, D.: Nonlinear dynamics of Cattaneo–Christov heat flux model for third-grade power-law fluid. J. Comput. Nonlinear Dyn. 15(1), 011009 (2020). https://doi.org/10.1115/1.4045406

    Article  Google Scholar 

  63. 63.

    Tassaddiq, A., Rahman, G., Nisar, K.S., Samraiz, M.: Certain fractional conformable inequalities for the weighted and the extended Chebyshev functionals. Adv. Differ. Equ. 2020, 96 (2020). https://doi.org/10.1186/s13662-020-2543-0

    MathSciNet  Article  Google Scholar 

  64. 64.

    Veeresha, P., Prakasha, D.G., Kumar, S.: A fractional model for propagation of classical optical solitons by using nonsingular derivative. Math. Methods Appl. Sci., 1–15 (2020). https://doi.org/10.1002/mma.6335

    Article  Google Scholar 

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The author T. Abdeljawad would like to thank Prince Sultan University for funding this work through research group Nonlinear Analysis Methods in Applied Mathematics (NAMAM) group number RG-DES-2017-01-17.

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Rahman, G., Nisar, K.S., Ghanbari, B. et al. On generalized fractional integral inequalities for the monotone weighted Chebyshev functionals. Adv Differ Equ 2020, 368 (2020). https://doi.org/10.1186/s13662-020-02830-7

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Keywords

  • Fractional integrals
  • The generalized fractional integrals
  • Fractional integral inequalities
  • The Chebyshev functional