1 Introduction

The integral inequalities play a major role in the field of differential equations and applied mathematics. Applications of integral inequalities are found in applied sciences, such as statistical problems, transform theory, numerical quadrature, and probability. In the last few years, many researchers have established various types of integral inequalities by employing different approaches. The interested readers are suggested to see [4, 5, 8, 14, 15, 17, 18]. In [28, 39–41] the researchers established different kinds of integral inequalities by employing various types of fractional integrals.

In the last few years, the field of fractional calculus has been extensively studied due to wide applications in diverse domains. Several different kinds of fractional integral and derivative operators have been investigated. We refer the readers to [1–3, 6, 7, 11, 21, 24]. In [20, 22] the authors introduced the idea of fractional conformable integral operators. Jarad et al. [19] introduced the idea of proportional fractional integral operators. Recently, the researchers have established certainly remarkable inequalities, properties, and applications of the fractional conformable integrals and generalized proportional integrals [27, 29, 33–37, 42]. Rahman et al. [37] recently established bounds of proportional fractional integrals for convex functions and their applications.

We consider the following extended Chebyshev functional:

$$\begin{aligned} \mathfrak{T} (\mathcal{U}, \mathcal{V},\mu ,\nu ) =& \int _{r}^{s} \mu (\vartheta )\,d\vartheta \int _{r}^{s} \nu (\vartheta )\mathcal{U}( \vartheta )\mathcal{V}(\vartheta )\,d\vartheta \\ &{}+ \int _{r}^{s} \nu ( \vartheta )\,d\vartheta \int _{r}^{s} \mu (\vartheta )\mathcal{U}( \vartheta )\mathcal{V}(\vartheta )\,d\vartheta \\ &{}- \biggl( \int _{r}^{s} \nu (\vartheta )\mathcal{U}( \vartheta )\,d\vartheta \biggr) \biggl( \int _{r}^{s} \mu (\vartheta )\mathcal{V}( \vartheta )\,d\vartheta \biggr) \\ &{}- \biggl( \int _{r}^{s} \mu (\vartheta ) \mathcal{U}( \vartheta )\,d\vartheta \biggr) \biggl( \int _{r}^{s} \nu ( \vartheta )\mathcal{V}( \vartheta )\,d\vartheta \biggr), \end{aligned}$$
(1)

where the functions \(\mathcal{U}\) and \(\mathcal{V}\) are integrable on \([r,s]\), and the functions μ and ν are positive and integrable on \([r,s]\). The functions \(\mathcal{U}\) and \(\mathcal{V}\) are said to be synchronous on \([r,s]\) if

$$ \bigl(\mathcal{U}(\rho )-\mathcal{U}(\zeta ) \bigr) \bigl( \mathcal{V}(\rho )- \mathcal{V}(\zeta ) \bigr)\geq 0,\quad \rho ,\zeta \in [r,s]. $$

The functions \(\mathcal{U}\) and \(\mathcal{V}\) are said to be asynchronous on \([r,s]\) if the inequality is reversed, that is,

$$ \bigl(\mathcal{U}(\rho )-\mathcal{U}(\zeta ) \bigr) \bigl( \mathcal{V}(\rho )- \mathcal{V}(\zeta ) \bigr)\leq 0,\quad \rho ,\zeta \in [r,s]. $$

If the functions \(\mathcal{U}\) and \(\mathcal{V}\) are synchronous on \([r,s]\), then \(\mathfrak{T} (\mathcal{U}, \mathcal{V},\mu ,\nu )\geq 0\). For more detail, see Kuang [23] and Mitrinovic [26]. The Chebyshev functional (1) leads to the Chebyshev inequality [12] if \(\mu (\vartheta )=\nu (\vartheta )=1\), \(\vartheta \in [r,s]\). Ostrowski [30] obtained the following generalization of the Chebyshev inequality: Let \(\mathcal{U}\) and \(\mathcal{V}\) be differentiable and synchronous functions on \([r,s]\) such that \(|\mathcal{U}^{\prime }(\vartheta )|\geq m\) and \(|\mathcal{V}^{\prime }(\vartheta )|\geq k\) for \(\vartheta \in [r,s]\), and let μ be a positive integrable function on \([r,s]\). Then

$$ \mathfrak{T} (\mathcal{U}, \mathcal{V},\mu )=\mathfrak{T} (\mathcal{U}, \mathcal{V},\mu ,\nu )\geq mk\mathfrak{T} (\vartheta -a, \vartheta -a;\mu )\geq 0. $$

If the functions \(\mathcal{U}\) and \(\mathcal{V}\) are asynchronous on \([r,s]\), then

$$ \mathfrak{T} (\mathcal{U}, \mathcal{V},\mu )=\mathfrak{T} (\mathcal{U}, \mathcal{V},\mu ,\nu )\leq mk\mathfrak{T} (\vartheta -a, b-\vartheta ;\mu )\leq 0. $$

If \(\mathcal{U}\) and \(\mathcal{V}\) are differentiable functions on \([r,s]\) such that \(|\mathcal{U}^{\prime }(\vartheta )|\geq M\) and \(| \mathcal{V}^{\prime }(\vartheta )|\geq K\) for \(\vartheta \in [r,s]\) and μ is a positive integrable function on \([r,s]\), then

$$ \bigl\vert \mathfrak{T} (\mathcal{U}, \mathcal{V},\mu ) \bigr\vert \leq MK \mathfrak{T} (\vartheta -a, \vartheta -a;\mu )\leq 0. $$

The researchers studied the functional \(\mathfrak{T} (\mathcal{U}, \mathcal{V},\mu )\) and established several extensions and generalizations, which can be found in [9, 10, 16, 25].

2 Preliminaries

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

Definition 2.1

([31, 38])

The Riemann–Liouville (left and right)-sided fractional integrals of order \(\kappa >0\) are respectively given by

$$ \bigl({}_{r}\mathfrak{J}^{\kappa }\mathcal{U} \bigr) (x)= \frac{1}{\Gamma (\kappa )} \int _{r}^{\vartheta }(\vartheta -\xi )^{ \kappa -1} \mathcal{U}(\xi )\,d\xi ,\quad r< \vartheta , $$
(2)

and

$$ \bigl(\mathfrak{J}_{b}^{\kappa }\mathcal{U} \bigr) (\vartheta )= \frac{1}{\Gamma (\kappa )} \int _{\vartheta }^{s}(\xi -\vartheta )^{ \kappa -1} \mathcal{U}(\xi )\,d\xi , \quad \vartheta < s, $$
(3)

where \(\Gamma (\kappa )\) is the classic gamma function.

Definition 2.2

([19])

The left-sided PFI (proportional fractional integral) is defined by

$$ \bigl({}_{r}\mathfrak{J}^{\kappa ,\omega }\mathcal{U} \bigr) ( \vartheta )=\frac{1}{\omega ^{\kappa }\Gamma (\kappa )} \int _{r}^{\vartheta }\exp \biggl[\frac{\omega -1}{\omega }( \vartheta -\xi )\biggr] ( \vartheta -\xi )^{\kappa -1}\mathcal{U}(\xi )\,d\xi ,\quad r< \vartheta , $$
(4)

where \(\kappa >0\) is the order of PFI, and \(\omega \in (0,1]\) is the proportionality index.

Definition 2.3

The right-sided PFI (proportional fractional integral) is defined by

$$ \bigl(\mathfrak{J}_{b}^{\kappa ,\omega }\mathcal{U} \bigr) (x)= \frac{1}{\omega ^{\kappa }\Gamma (\kappa )} \int _{\vartheta }^{s}\exp \biggl[ \frac{\omega -1}{\omega }( \xi -\vartheta )\biggr](\xi -\vartheta )^{\kappa -1} \mathcal{U}(\xi )\,d\xi ,\quad \vartheta < s. $$
(5)

Remark 2.1

Setting \(\omega =1\) in (4) and (5), we obtain the Riemann–Liouville integrals (2) and (3), respectively.

In this paper, we consider the following one-sided PFI-operator.

Definition 2.4

The one-sided PFI is defined by

$$ \bigl(\mathfrak{J}_{0}^{\kappa ,\omega }\mathcal{U} \bigr) ( \vartheta )= \bigl(\mathfrak{J}^{\kappa ,\omega }\mathcal{U} \bigr) ( \vartheta )=\frac{1}{\omega ^{\kappa }\Gamma (\kappa )} \int _{0}^{\vartheta }\exp \biggl[\frac{\omega -1}{\omega }( \vartheta -\xi )\biggr] ( \vartheta -\xi )^{\kappa -1}\mathcal{U}(\xi )\,d\xi , $$
(6)

where \(\kappa >0\) is the order of PFI, and \(\omega \in (0,1]\) is the proportionality index.

Definition 2.5

The left-sided Hadamard fractional integral of order \(\kappa >0\) is defined by

$$ \bigl({}_{r}\mathcal{H}^{\kappa } \mathcal{U} \bigr) (\vartheta )= \frac{1}{\Gamma (\kappa )} \int _{r}^{\vartheta }(\ln \vartheta -\ln t)^{ \kappa -1}\frac{\mathcal{U}(\xi )}{\xi }\,d\xi , \quad r< \vartheta . $$
(7)

Definition 2.6

The right-sided Hadamard fractional integral of order \(\kappa >0\) is defined by

$$ \bigl(\mathcal{H}_{b}^{\kappa } \mathcal{U} \bigr) (\vartheta )= \frac{1}{\Gamma (\kappa )} \int _{\vartheta }^{s}(\ln \xi -\ln \vartheta )^{\kappa -1}\frac{\mathcal{U}(\xi )}{\xi }\,d\xi ,\quad \vartheta < s. $$
(8)

Definition 2.7

The one-sided Hadamard fractional integral of order \(\kappa >0\) is defined by

$$ \bigl(\mathcal{H}_{1,\vartheta }^{\kappa } \mathcal{U} \bigr) ( \vartheta )=\frac{1}{\Gamma (\kappa )} \int _{1}^{\vartheta }(\ln \vartheta -\ln \xi )^{\kappa -1}\frac{\mathcal{U}(\xi )}{\xi }\,d\xi ,\quad \vartheta >1. $$
(9)

Rahman et al. [32] recently presented the following generalized Hadamard proportional fractional integrals.

Definition 2.8

The left-sided Hadamard proportional fractional integral is defined by

$$\begin{aligned}& \bigl({}_{r}\mathcal{H}^{\kappa ,\omega } \mathcal{U} \bigr) ( \vartheta ) \\& \quad =\frac{1}{\omega ^{\kappa }\Gamma (\kappa )} \int _{r}^{\vartheta }\exp \biggl[\frac{\omega -1}{\omega }( \ln \vartheta -\ln \xi )\biggr]( \ln \vartheta -\ln \xi )^{\kappa -1} \frac{\mathcal{U}(\xi )}{\xi }\,d\xi ,\quad r< \vartheta . \end{aligned}$$
(10)

Definition 2.9

The right-sided Hadamard proportional fractional integral is defined by

$$\begin{aligned}& \bigl(\mathcal{H}_{b}^{\kappa ,\omega } \mathcal{U} \bigr) ( \vartheta ) \\& \quad =\frac{1}{\omega ^{\kappa }\Gamma (\kappa )} \int _{\vartheta }^{s} \exp \biggl[\frac{\omega -1}{\omega }( \ln \xi -\ln \vartheta )\biggr](\ln \xi - \ln \vartheta )^{\kappa -1} \frac{\mathcal{U}(\xi )}{\xi }\,d\xi , \quad \vartheta < s. \end{aligned}$$
(11)

Definition 2.10

The one-sided Hadamard proportional fractional integral is defined by

$$\begin{aligned}& \bigl(\mathcal{H}_{1,\vartheta }^{\kappa ,\omega } \mathcal{U} \bigr) ( \vartheta ) \\& \quad =\frac{1}{\omega ^{\kappa }\Gamma (\kappa )} \int _{1}^{\vartheta }\exp \biggl[\frac{\omega -1}{\omega }( \ln \vartheta -\ln \xi )\biggr]( \ln \vartheta -\ln \xi )^{\kappa -1} \frac{\mathcal{U}(\xi )}{\xi }\,d\xi ,\quad \vartheta >1, \end{aligned}$$
(12)

where \(\kappa >0\) is the order, and \(\omega \in (0,1]\) is the proportionality index.

Remark 2.2

Setting \(\omega =1\), (10)–(12) lead to integrals (7)–(9), respectively.

The following results can be easily proved.

Lemma 2.1

$$\begin{aligned}& \biggl(\mathcal{H}_{1,\vartheta }^{\kappa ,\omega }\exp \biggl[ \frac{\omega -1}{\omega }(\ln \vartheta )\biggr](\ln \vartheta )^{\lambda -1} \biggr) (\vartheta ) \\& \quad = \frac{\Gamma (\lambda )}{\omega ^{\kappa }\Gamma (\kappa +\lambda )} \exp \biggl[\frac{\omega -1}{\omega }(\ln \vartheta )\biggr](\ln \vartheta )^{ \kappa +\lambda -1}, \end{aligned}$$
(13)

and

$$ \bigl(\mathcal{H}_{1,\vartheta }^{\kappa ,\omega } \bigr) \bigl( \mathcal{H}_{1,\vartheta }^{\lambda ,\omega } \bigr)\mathcal{U}( \vartheta )= \bigl(\mathcal{H}_{1,\vartheta }^{\kappa +\lambda ,\omega } \bigr)\mathcal{U}(\vartheta ) $$
(14)

(the semigroup property).

Remark 2.3

Setting \(\omega =1\), (13) reduces to (see [38])

$$ \bigl(\mathcal{H}_{1,\vartheta }^{\kappa }(\ln \vartheta )^{\lambda -1} \bigr) (\vartheta )= \frac{\Gamma (\lambda )}{\Gamma (\kappa +\lambda )}(\ln \vartheta )^{ \kappa +\lambda -1}. $$
(15)

The paper is organized as follows. In Sect. 3, we present two integral inequalities for the extended Chebyshev functional. The first result concerns one-parameter PFI, and the second one deals with two-parameter PFI. In Sect. 4, we establish integral inequalities for the extended Chebyshev functional by employing the Hadamard proportional fractional integral.

3 Certain fractional proportional integral inequalities

In this section, we present proportional fractional integral inequalities for the extended Chebyshev functional (1) by utilizing the PFI (6). To establish our main result, we first prove the following lemma.

Lemma 3.1

Let \(\mathcal{U}\) and \(\mathcal{V}\) be synchronous functions on \([0,\infty )\), and let \(v,w:[0,\infty )\rightarrow [0,\infty )\). Then for all \(\vartheta >0\), \(\kappa >0\), and \(\omega \in (0,1]\), we have the following inequality for the PFI-operator (6):

$$\begin{aligned} &\mathfrak{J}^{\kappa ,\omega } (v ) (\vartheta ) \mathfrak{J}^{\kappa ,\omega } (w\mathcal{U}\mathcal{V} ) ( \vartheta )+ \mathfrak{J}^{\kappa ,\omega } (w ) (\vartheta ) \mathfrak{J}^{\kappa ,\omega } (v \mathcal{U}\mathcal{V} ) ( \vartheta ) \\ &\quad \geq \mathfrak{J}^{\kappa ,\omega } (v\mathcal{U} ) ( \vartheta ) \mathfrak{J}^{\kappa ,\omega } (w\mathcal{V} ) ( \vartheta )+\mathfrak{J}^{\kappa ,\omega } (w\mathcal{U} ) ( \vartheta ) \mathfrak{J}^{\kappa ,\omega } (v\mathcal{V} ) ( \vartheta ). \end{aligned}$$
(16)

Proof

Since the functions \(\mathcal{U}\) and \(\mathcal{V}\) are synchronous on \([0,\infty )\), for all \(\rho \geq 0\) and \(\zeta \geq 0\), we have

$$ \bigl(\mathcal{U}(\rho )-\mathcal{U}(\zeta ) \bigr) \bigl( \mathcal{V}(\rho )- \mathcal{V}(\zeta ) \bigr)\geq 0. $$

Therefore

$$ \mathcal{U}(\rho )\mathcal{V}(\rho )+\mathcal{U}(\zeta ) \mathcal{V}( \zeta )\geq \mathcal{U}(\rho )\mathcal{V}(\zeta )+\mathcal{U}(\zeta ) \mathcal{V}(\rho ). $$
(17)

Multiplying (17) by \(\frac{1}{\omega ^{\kappa }\Gamma (\kappa )}e^{\frac{\omega -1}{\omega }( \vartheta -\rho )}(\vartheta -\rho )^{\kappa -1}v(\rho )\), \(\rho \in (0, \vartheta )\), and integrating the obtained inequality with respect to ρ over \((0,\vartheta )\), we get

$$\begin{aligned}& \frac{1}{\omega ^{\kappa }\Gamma (\kappa )} \int _{0}^{\vartheta }e^{ \frac{\omega -1}{\omega }(\vartheta -\rho )}(\vartheta -\rho )^{ \kappa -1}v(\rho )\mathcal{U}(\rho )\mathcal{V}(\rho )\,d\rho \\& \qquad {}+ \mathcal{U}( \zeta )\mathcal{V}(\zeta ) \frac{1}{\omega ^{\kappa }\Gamma (\kappa )} \int _{0}^{\vartheta }e^{ \frac{\omega -1}{\omega }(\vartheta -\rho )}(\vartheta -\rho )^{ \kappa -1}v(\rho ) \,d\rho \\& \quad \geq \mathcal{V}(\zeta )\frac{1}{\omega ^{\kappa }\Gamma (\kappa )} \int _{0}^{\vartheta }e^{\frac{\omega -1}{\omega }(\vartheta -\rho )}( \vartheta -\rho )^{\kappa -1}v(\rho )\mathcal{U}(\rho )\,d\rho \\& \qquad {}+ \mathcal{U}(\zeta ) \frac{1}{\omega ^{\kappa }\Gamma (\kappa )} \int _{0}^{\vartheta }e^{\frac{\omega -1}{\omega }(\vartheta -\rho )}(\vartheta - \rho )^{\kappa -1}v(\rho ). \mathcal{V}(\rho )\,d\rho . \end{aligned}$$

In view of (6), we get

$$\begin{aligned} &\mathfrak{J}^{\kappa ,\omega } (v\mathcal{U}\mathcal{V} ) ( \vartheta )+ \mathcal{U}(\zeta )\mathcal{V}(\zeta )\mathfrak{J}^{ \kappa ,\omega } (v ) (\vartheta ) \\ &\quad \geq \mathcal{V}(\zeta )\mathfrak{J}^{\kappa ,\omega } (v \mathcal{U} ) ( \vartheta )+\mathcal{U}(\zeta )\mathfrak{J}^{ \kappa ,\omega } (v\mathcal{V} ) ( \vartheta ). \end{aligned}$$
(18)

Now multiplying (18) by \(\frac{1}{\omega ^{\kappa }\Gamma (\kappa )}e^{\frac{\omega -1}{\omega }( \vartheta -\zeta )}(\vartheta -\zeta )^{\kappa -1}w(\zeta )\), \(\zeta \in (0,\vartheta )\), and integrating the obtained inequality with respect to ζ over \((0,\vartheta )\), we obtain

$$\begin{aligned}& \mathfrak{J}^{\kappa ,\omega } (v\mathcal{U}\mathcal{V} ) ( \vartheta )\frac{1}{\omega ^{\kappa }\Gamma (\kappa )} \int _{0}^{\vartheta }e^{\frac{\omega -1}{\omega }(\vartheta -\zeta )}(\vartheta - \zeta )^{\kappa -1}w(\zeta )\,d\zeta \\& \qquad {}+ \mathfrak{J}^{\kappa ,\omega } (v ) (\vartheta ) \frac{1}{\omega ^{\kappa }\Gamma (\kappa )} \int _{0}^{\vartheta }e^{ \frac{\omega -1}{\omega }(\vartheta -\zeta )}(\vartheta -\zeta )^{ \kappa -1}w(\zeta ) \mathcal{U}(\zeta )\mathcal{V}(\zeta )\,d\zeta \\& \quad \geq \mathfrak{J}^{\kappa ,\omega } (v\mathcal{U} ) ( \vartheta ) \frac{1}{\omega ^{\kappa }\Gamma (\kappa )} \int _{0}^{\vartheta }e^{\frac{\omega -1}{\omega }(\vartheta -\zeta )}(\vartheta - \zeta )^{\kappa -1}w(\zeta )\mathcal{V}(\zeta )\,d\zeta \\& \qquad {}+ \mathfrak{J}^{\kappa ,\omega } (v\mathcal{V} ) ( \vartheta ) \frac{1}{\omega ^{\kappa }\Gamma (\kappa )} \int _{0}^{\vartheta }e^{\frac{\omega -1}{\omega }(\vartheta -\zeta )}(\vartheta - \zeta )^{\kappa -1}w(\zeta )\mathcal{U}(\zeta )\,d\zeta , \end{aligned}$$
(19)

which in view of (6) gives the desired inequality (16). □

Theorem 3.1

Let \(\mathcal{U}\) and \(\mathcal{V}\) be synchronous functions on \([0,\infty )\), and let \(r, p, q:[0,\infty )\rightarrow [0,\infty )\). Then for all \(\vartheta >0\), \(\kappa >0m\) and \(\omega \in (0,1]\), we have the following inequality for the PFI-operator (6):

$$\begin{aligned}& 2\mathfrak{J}^{\kappa ,\omega } (r ) (\vartheta ) \bigl[ \mathfrak{J}^{\kappa ,\omega }p(\vartheta )\mathfrak{J}^{\kappa , \omega } (q \mathcal{U}\mathcal{V} ) (\vartheta )+ \mathfrak{J}^{\kappa ,\omega } (q ) ( \vartheta ) \mathfrak{J}^{\kappa ,\omega } (p\mathcal{U}\mathcal{V} ) ( \vartheta ) \bigr] \\& \qquad {}+ 2\mathfrak{J}^{\kappa ,\omega } (p ) (\vartheta ) \mathfrak{J}^{\kappa ,\omega } (q ) (\vartheta ) (r \mathcal{U}\mathcal{V} ) (\vartheta ) \\& \quad \geq \mathfrak{J}^{\kappa ,\omega } (r ) (\vartheta ) \bigl[ \mathfrak{J}^{\kappa ,\omega } (p\mathcal{U} ) ( \vartheta )\mathfrak{J}^{\kappa ,\omega } (q\mathcal{V} ) ( \vartheta )+ \mathfrak{J}^{\kappa ,\omega } (q\mathcal{U} ) ( \vartheta )\mathfrak{J}^{\kappa ,\omega } (p\mathcal{V} ) ( \vartheta ) \bigr] \\& \qquad {}+ \mathfrak{J}^{\kappa ,\omega } (p ) (\vartheta ) \bigl[ \mathfrak{J}^{\kappa ,\omega } (r\mathcal{U} ) (\vartheta ) \mathfrak{J}^{\kappa ,\omega } (q\mathcal{V} ) ( \vartheta )+ \mathfrak{J}^{\kappa ,\omega } (q\mathcal{U} ) (\vartheta ) \mathfrak{J}^{\kappa ,\omega } (r\mathcal{V} ) (\vartheta ) \bigr] \\& \qquad {}+ \mathfrak{J}^{\kappa ,\omega } (q ) (\vartheta ) \bigl[ \mathfrak{J}^{\kappa ,\omega } (r\mathcal{U} ) (\vartheta ) \mathfrak{J}^{\kappa ,\omega } (p\mathcal{V} ) ( \vartheta )+ \mathfrak{J}^{\kappa ,\omega } (p\mathcal{U} ) (\vartheta ) \mathfrak{J}^{\kappa ,\omega } (r\mathcal{V} ) (\vartheta ) \bigr] . \end{aligned}$$
(20)

Proof

Taking \(v=p\) and \(w=q\) in Lemma 3.1, we obtain

$$\begin{aligned} &\mathfrak{J}^{\kappa ,\omega } (p ) (\vartheta ) \mathfrak{J}^{\kappa ,\omega } (q\mathcal{U}\mathcal{V} ) ( \vartheta )+ \mathfrak{J}^{\kappa ,\omega } (q ) (\vartheta ) \mathfrak{J}^{\kappa ,\omega } (p \mathcal{U}\mathcal{V} ) ( \vartheta ) \\ &\quad \geq \mathfrak{J}^{\kappa ,\omega } (p\mathcal{U} ) ( \vartheta ) \mathfrak{J}^{\kappa ,\omega } (q\mathcal{V} ) ( \vartheta )+\mathfrak{J}^{\kappa ,\omega } (q\mathcal{U} ) ( \vartheta ) \mathfrak{J}^{\kappa ,\omega } (p\mathcal{V} ) ( \vartheta ). \end{aligned}$$
(21)

Multiplying both sides of (21) by \(\mathfrak{J}^{\kappa ,\omega } (r )(\vartheta )\), we get

$$\begin{aligned} &\mathfrak{J}^{\kappa ,\omega } (r ) (\vartheta ) \bigl[ \mathfrak{J}^{\kappa ,\omega } (p ) (\vartheta )\mathfrak{J}^{ \kappa ,\omega } (q \mathcal{U}\mathcal{V} ) (\vartheta )+ \mathfrak{J}^{\kappa ,\omega } (q ) ( \vartheta )\mathfrak{J}^{ \kappa ,\omega } (p\mathcal{U}\mathcal{V} ) (\vartheta ) \bigr] \\ &\quad \geq \mathfrak{J}^{\kappa ,\omega } (r ) (\vartheta ) \bigl[ \mathfrak{J}^{\kappa ,\omega } (p\mathcal{U} ) ( \vartheta )\mathfrak{J}^{\kappa ,\omega } (q\mathcal{V} ) ( \vartheta )+\mathfrak{J}^{\kappa ,\omega } (q\mathcal{U} ) ( \vartheta ) \mathfrak{J}^{\kappa ,\omega } (p\mathcal{V} ) ( \vartheta ) \bigr]. \end{aligned}$$
(22)

Now taking \(v=r\) and \(w=q\) in Lemma 3.1, we have

$$\begin{aligned} &\mathfrak{J}^{\kappa ,\omega } (r ) (\vartheta ) \mathfrak{J}^{\kappa ,\omega } (q\mathcal{U}\mathcal{V} ) ( \vartheta )+ \mathfrak{J}^{\kappa ,\omega } (q ) (\vartheta ) \mathfrak{J}^{\kappa ,\omega } (r \mathcal{U}\mathcal{V} ) ( \vartheta ) \\ &\quad \geq \mathfrak{J}^{\kappa ,\omega } (r\mathcal{U} ) ( \vartheta ) \mathfrak{J}^{\kappa ,\omega } (q\mathcal{V} ) ( \vartheta )+\mathfrak{J}^{\kappa ,\omega } (q\mathcal{U} ) ( \vartheta ) \mathfrak{J}^{\kappa ,\omega } (r\mathcal{V} ) ( \vartheta ). \end{aligned}$$
(23)

Multiplying both sides of (23) by \(\mathfrak{J}^{\kappa ,\omega } (p )(\vartheta )\), we get

$$\begin{aligned} &\mathfrak{J}^{\kappa ,\omega } (p ) (\vartheta ) \bigl[ \mathfrak{J}^{\kappa ,\omega } (r ) (\vartheta )\mathfrak{J}^{ \kappa ,\omega } (q \mathcal{U}\mathcal{V} ) (\vartheta )+ \mathfrak{J}^{\kappa ,\omega } (q ) ( \vartheta )\mathfrak{J}^{ \kappa ,\omega } (r\mathcal{U}\mathcal{V} ) (\vartheta ) \bigr] \\ &\quad \geq \mathfrak{J}^{\kappa ,\omega } (p ) (\vartheta ) \bigl[ \mathfrak{J}^{\kappa ,\omega } (r\mathcal{U} ) ( \vartheta )\mathfrak{J}^{\kappa ,\omega } (q\mathcal{V} ) ( \vartheta )+ \mathfrak{J}^{\kappa ,\omega } (q\mathcal{U} ) ( \vartheta ) \mathfrak{J}^{\kappa ,\omega } (r\mathcal{V} ) ( \vartheta ) \bigr]. \end{aligned}$$
(24)

Similarly, taking \(v=r\) and \(w=p\) in Lemma 3.1 and then multiplying both sides of the resultant inequality by \(\mathfrak{J}^{\kappa ,\omega } (q )(\vartheta )\), we obtain

$$\begin{aligned} &\mathfrak{J}^{\kappa ,\omega } (q ) (\vartheta ) \bigl[ \mathfrak{J}^{\kappa ,\omega } (r ) (\vartheta )\mathfrak{J}^{ \kappa ,\omega } (p \mathcal{U}\mathcal{V} ) (\vartheta )+ \mathfrak{J}^{\kappa ,\omega } (p ) ( \vartheta )\mathfrak{J}^{ \kappa ,\omega } (r\mathcal{U}\mathcal{V} ) (\vartheta ) \bigr] \\ &\quad \geq \mathfrak{J}^{\kappa ,\omega } (q ) (\vartheta ) \bigl[ \mathfrak{J}^{\kappa ,\omega } (r\mathcal{U} ) ( \vartheta )\mathfrak{J}^{\kappa ,\omega } (p\mathcal{V} ) ( \vartheta )+ \mathfrak{J}^{\kappa ,\omega } (p\mathcal{U} ) ( \vartheta ) \mathfrak{J}^{\kappa ,\omega } (r\mathcal{V} ) ( \vartheta ) \bigr]. \end{aligned}$$
(25)

Hence by adding (22), (24), and (25) we get the desired statement (20). □

Remark 3.1

Setting \(\omega =1\) in Theorem 3.1, we get Theorem 2 proved by Dahmani [14].

Lemma 3.2

Let \(\mathcal{U}\) and \(\mathcal{V}\) be synchronous functions on \([0,\infty )\), and let \(v,w:[0,\infty )\rightarrow [0,\infty )\). Then for all \(\vartheta >0\), \(\kappa , \eta >0\), and \(\omega \in (0,1]\), we have the following inequality for the PFI-operator (6):

$$\begin{aligned} &\mathfrak{J}^{\kappa ,\omega } (v ) (\vartheta ) \mathfrak{J}^{\eta ,\omega } (w\mathcal{U}\mathcal{V} ) ( \vartheta )+ \mathfrak{J}^{\eta ,\omega } (w ) (\vartheta ) \mathfrak{J}^{\kappa ,\omega } (v \mathcal{U}\mathcal{V} ) ( \vartheta ) \\ &\quad \geq \mathfrak{J}^{\kappa ,\omega } (v\mathcal{U} ) ( \vartheta ) \mathfrak{J}^{\eta ,\omega } (w\mathcal{V} ) ( \vartheta )+\mathfrak{J}^{\eta ,\omega } (w\mathcal{U} ) ( \vartheta ) \mathfrak{J}^{\kappa ,\omega } (v\mathcal{V} ) ( \vartheta ). \end{aligned}$$
(26)

Proof

Multiplying (19) by \(\frac{1}{\omega ^{\eta }\Gamma (\eta )}e^{\frac{\omega -1}{\omega }( \vartheta -\zeta )}(\vartheta -\zeta )^{\eta -1}w(\zeta ), \zeta \in (0, \vartheta )\) and integrating the obtained inequality with respect to ζ over \((0,\vartheta )\), we obtain

$$\begin{aligned}& \mathfrak{J}^{\kappa ,\omega } (v\mathcal{U}\mathcal{V} ) ( \vartheta ) \frac{1}{\omega ^{\eta }\Gamma (\eta )} \int _{0}^{\vartheta }e^{ \frac{\omega -1}{\omega }(\vartheta -\zeta )}(\vartheta -\zeta )^{ \eta -1}w(\zeta )\,d\zeta \\& \qquad {}+ \mathfrak{J}^{\kappa ,\omega } (v ) (\vartheta ) \frac{1}{\omega ^{\eta }\Gamma (\eta )} \int _{0}^{\vartheta }e^{ \frac{\omega -1}{\omega }(\vartheta -\zeta )}(\vartheta -\zeta )^{ \eta -1}w(\zeta )\mathcal{U}(\zeta )\mathcal{V}(\zeta )\,d\zeta \\& \quad \geq \mathfrak{J}^{\kappa ,\omega } (v\mathcal{U} ) ( \vartheta ) \frac{1}{\omega ^{\eta }\Gamma (\eta )} \int _{0}^{\vartheta }e^{ \frac{\omega -1}{\omega }(\vartheta -\zeta )}(\vartheta -\zeta )^{ \eta -1}w(\zeta )\mathcal{V}(\zeta )\,d\zeta \\& \qquad {}+ \mathfrak{J}^{\kappa ,\omega } (v\mathcal{V} ) ( \vartheta ) \frac{1}{\omega ^{\eta }\Gamma (\eta )} \int _{0}^{\vartheta }e^{ \frac{\omega -1}{\omega }(\vartheta -\zeta )}(\vartheta -\zeta )^{ \eta -1}w(\zeta )\mathcal{U}(\zeta )\,d\zeta , \end{aligned}$$

which by (6) gives

$$\begin{aligned} &\mathfrak{J}^{\kappa ,\omega } (v ) (\vartheta ) \mathfrak{J}^{\eta ,\omega } (w \mathcal{U}\mathcal{V} ) ( \vartheta )+\mathfrak{J}^{\eta ,\omega } (w ) ( \vartheta ) \mathfrak{J}^{\kappa ,\omega } (v\mathcal{U}\mathcal{V} ) ( \vartheta ) \\ &\quad \geq \mathfrak{J}^{\kappa ,\omega } (v\mathcal{U} ) ( \vartheta ) \mathfrak{J}^{\eta ,\omega } (w\mathcal{V} ) ( \vartheta )+\mathfrak{J}^{\eta ,\omega } (w\mathcal{U} ) ( \vartheta ) \mathfrak{J}^{\kappa ,\omega } (v\mathcal{V} ) ( \vartheta ). \end{aligned}$$

This completes the proof of Lemma 3.2. □

Remark 3.2

Setting \(\kappa =\eta \) in Lemma 3.2, we get Lemma 3.1.

Theorem 3.2

Let \(\mathcal{U}\) and \(\mathcal{V}\) be synchronous functions on \([0,\infty )\), and let \(r, p, q:[0,\infty )\rightarrow [0,\infty )\). Then for all \(\vartheta >0\), \(\kappa >0m\) and \(\omega \in (0,1]\), we have the following inequality for the PFI-operator (6):

$$\begin{aligned}& \mathfrak{J}^{\kappa ,\omega } (r ) (\vartheta ) \bigl[ \mathfrak{J}^{\kappa ,\omega }p(\vartheta )\mathfrak{J}^{\eta ,\omega } (q \mathcal{U}\mathcal{V} ) (\vartheta )+2\mathfrak{J}^{ \kappa ,\omega } (p ) ( \vartheta )\mathfrak{J}^{\eta , \omega } (q\mathcal{U}\mathcal{V} ) (\vartheta ) \\& \qquad {}+ \mathfrak{J}^{\eta ,\omega } (q ) (\vartheta ) \mathfrak{J}^{ \kappa ,\omega } (p \mathcal{U}\mathcal{V} ) (\vartheta ) \bigr] \\& \qquad {}+ \bigl[\mathfrak{J}^{\kappa ,\omega } (p ) (\vartheta ) \mathfrak{J}^{\eta ,\omega } (q ) (\vartheta )+ \mathfrak{J}^{ \eta ,\omega } (p ) (\vartheta ) \mathfrak{J}^{\kappa ,\omega } (q ) (\vartheta ) \bigr] \mathfrak{J}^{\kappa ,\omega } (r\mathcal{U}\mathcal{V} ) (\vartheta ) \\& \quad \geq \mathfrak{J}^{\kappa ,\omega } (r ) (\vartheta ) \bigl[ \mathfrak{J}^{\kappa ,\omega } (p\mathcal{U} ) ( \vartheta )\mathfrak{J}^{\eta ,\omega } (q\mathcal{V} ) ( \vartheta )+ \mathfrak{J}^{\eta ,\omega } (q\mathcal{U} ) ( \vartheta )\mathfrak{J}^{\kappa ,\omega } (p\mathcal{V} ) ( \vartheta ) \bigr] \\& \qquad {}+ \mathfrak{J}^{\kappa ,\omega } (p ) (\vartheta ) \bigl[ \mathfrak{J}^{\kappa ,\omega } (r\mathcal{U} ) (\vartheta ) \mathfrak{J}^{\eta ,\omega } (q\mathcal{V} ) ( \vartheta )+ \mathfrak{J}^{\eta ,\omega } (q\mathcal{U} ) (\vartheta ) \mathfrak{J}^{\kappa ,\omega } (r\mathcal{V} ) (\vartheta ) \bigr] \\& \qquad {}+ \mathfrak{J}^{\kappa ,\omega } (q ) (\vartheta ) \bigl[ \mathfrak{J}^{\kappa ,\omega } (r\mathcal{U} ) (\vartheta ) \mathfrak{J}^{\eta ,\omega } (p\mathcal{V} ) ( \vartheta )+ \mathfrak{J}^{\eta ,\omega } (p\mathcal{U} ) (\vartheta ) \mathfrak{J}^{\kappa ,\omega } (r\mathcal{V} ) (\vartheta ) \bigr] . \end{aligned}$$
(27)

Proof

Taking \(v=p\) and \(w=q\) in Lemma 3.2, we obtain

$$\begin{aligned} &\mathfrak{J}^{\kappa ,\omega } (p ) (\vartheta ) \mathfrak{J}^{\eta ,\omega } (q\mathcal{U}\mathcal{V} ) ( \vartheta )+ \mathfrak{J}^{\eta ,\omega } (q ) (\vartheta ) \mathfrak{J}^{\kappa ,\omega } (p \mathcal{U}\mathcal{V} ) ( \vartheta ) \\ &\quad \geq \mathfrak{J}^{\kappa ,\omega } (p\mathcal{U} ) ( \vartheta ) \mathfrak{J}^{\eta ,\omega } (q\mathcal{V} ) ( \vartheta )+\mathfrak{J}^{\eta ,\omega } (q\mathcal{U} ) ( \vartheta ) \mathfrak{J}^{\kappa ,\omega } (p\mathcal{V} ) ( \vartheta ). \end{aligned}$$
(28)

Multiplying (28) by \(\mathfrak{J}^{\kappa ,\omega } (r )(\vartheta )\), we get

$$\begin{aligned} &\mathfrak{J}^{\kappa ,\omega } (r ) (\vartheta ) \bigl[ \mathfrak{J}^{\kappa ,\omega } (p ) (\vartheta )\mathfrak{J}^{ \eta ,\omega } (q \mathcal{U}\mathcal{V} ) (\vartheta )+ \mathfrak{J}^{\eta ,\omega } (q ) ( \vartheta )\mathfrak{J}^{ \kappa ,\omega } (p\mathcal{U}\mathcal{V} ) (\vartheta ) \bigr] \\ &\quad \geq \mathfrak{J}^{\kappa ,\omega } (r ) (\vartheta ) \bigl[ \mathfrak{J}^{\kappa ,\omega } (p\mathcal{U} ) ( \vartheta )\mathfrak{J}^{\eta ,\omega } (q\mathcal{V} ) ( \vartheta )+\mathfrak{J}^{\eta ,\omega } (q\mathcal{U} ) ( \vartheta ) \mathfrak{J}^{\kappa ,\omega } (p\mathcal{V} ) ( \vartheta ) \bigr]. \end{aligned}$$
(29)

Setting \(v=r\) and \(w=q\) in Lemma 3.2, we obtain

$$\begin{aligned} &\mathfrak{J}^{\kappa ,\omega } (r ) (\vartheta ) \mathfrak{J}^{\eta ,\omega } (q\mathcal{U}\mathcal{V} ) ( \vartheta )+ \mathfrak{J}^{\eta ,\omega } (q ) (\vartheta ) \mathfrak{J}^{\kappa ,\omega } (r \mathcal{U}\mathcal{V} ) ( \vartheta ) \\ &\quad \geq \mathfrak{J}^{\kappa ,\omega } (r\mathcal{U} ) ( \vartheta ) \mathfrak{J}^{\eta ,\omega } (q\mathcal{V} ) ( \vartheta )+\mathfrak{J}^{\eta ,\omega } (q\mathcal{U} ) ( \vartheta ) \mathfrak{J}^{\kappa ,\omega } (r\mathcal{V} ) ( \vartheta ). \end{aligned}$$
(30)

Multiplying (30) by \(\mathfrak{J}^{\kappa ,\omega } (p )(\vartheta )\), we get

$$\begin{aligned} &\mathfrak{J}^{\kappa ,\omega } (p ) (\vartheta ) \bigl[ \mathfrak{J}^{\kappa ,\omega } (r ) (\vartheta )\mathfrak{J}^{ \eta ,\omega } (q \mathcal{U}\mathcal{V} ) (\vartheta )+ \mathfrak{J}^{\eta ,\omega } (q ) ( \vartheta )\mathfrak{J}^{ \kappa ,\omega } (r\mathcal{U}\mathcal{V} ) (\vartheta ) \bigr] \\ &\quad \geq \mathfrak{J}^{\kappa ,\omega } (p ) (\vartheta ) \bigl[ \mathfrak{J}^{\kappa ,\omega } (r\mathcal{U} ) ( \vartheta )\mathfrak{J}^{\eta ,\omega } (q\mathcal{V} ) ( \vartheta )+\mathfrak{J}^{\eta ,\omega } (q\mathcal{U} ) ( \vartheta ) \mathfrak{J}^{\kappa ,\omega } (r\mathcal{V} ) ( \vartheta ) \bigr]. \end{aligned}$$
(31)

Similarly, setting \(v=r\) and \(w=p\) in Lemma 3.2 and multiplying both sides of the resultant inequality by \(\mathfrak{J}^{\kappa ,\omega } (q )(\vartheta )\), we obtain

$$\begin{aligned} &\mathfrak{J}^{\kappa ,\omega } (q ) (\vartheta ) \bigl[ \mathfrak{J}^{\kappa ,\omega } (r ) (\vartheta )\mathfrak{J}^{ \eta ,\omega } (p \mathcal{U}\mathcal{V} ) (\vartheta )+ \mathfrak{J}^{\eta ,\omega } (p ) ( \vartheta )\mathfrak{J}^{ \kappa ,\omega } (r\mathcal{U}\mathcal{V} ) (\vartheta ) \bigr] \\ &\quad \geq \mathfrak{J}^{\kappa ,\omega } (q ) (\vartheta ) \bigl[ \mathfrak{J}^{\kappa ,\omega } (r\mathcal{U} ) ( \vartheta )\mathfrak{J}^{\eta ,\omega } (p\mathcal{V} ) ( \vartheta )+ \mathfrak{J}^{\eta ,\omega } (p\mathcal{U} ) ( \vartheta ) \mathfrak{J}^{\kappa ,\omega } (r\mathcal{V} ) ( \vartheta ) \bigr]. \end{aligned}$$
(32)

Hence we obtain statement (27) by adding inequalities (29), (31), and (32). □

Remark 3.3

Setting \(\kappa =\eta \) in Theorem 3.2, we get Theorem 3.1.

Remark 3.4

Setting \(\omega =1\) in Theorem 3.2, we get Theorem 4 proved by Dahmani [14].

Remark 3.5

Inequalities (20) and (27) will be reversed in the following cases:

  1. (i)

    The functions \(\mathcal{U}\) and \(\mathcal{V}\) are asynchronous on \([0,\infty )\).

  2. (ii)

    The functions r, p, and q are negative on \([0,\infty )\).

  3. (iii)

    Two of the functions r, p, and q are positive, and the third one is negative on \([0,\infty )\).

Remark 3.6

The Chebyshev inequality [12] on \([0,x]\) can be obtained for any \(\vartheta \in [0,\infty )\) if we set \(\kappa =\eta =\omega =1\) and \(p(\vartheta )=q(\vartheta )=r(\vartheta )=1\) in Theorem 3.2.

4 Inequalities via Hadamard proportional fractional integral

In this section, we present some inequalities for the extended Chebyshev functional (1) by employing the Hadamard proportional fractional integral (12).

Lemma 4.1

Let \(\mathcal{U}\) and \(\mathcal{V}\) be synchronous functions on \([0,\infty )\), and let \(v,w:[0,\infty )\rightarrow [0,\infty )\). Then for all \(\vartheta >1\), \(\kappa >0\), and \(\omega \in (0,1]\), we have the following inequality for (12):

$$\begin{aligned} &\mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (v ) ( \vartheta )\mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (w \mathcal{U}\mathcal{V} ) ( \vartheta )+\mathcal{H}_{1,\vartheta }^{ \kappa ,\omega } (w ) (\vartheta ) \mathcal{H}_{1,\vartheta }^{ \kappa ,\omega } (v\mathcal{U}\mathcal{V} ) ( \vartheta ) \\ &\quad \geq \mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (v\mathcal{U} ) (\vartheta ) \mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (w \mathcal{V} ) (\vartheta )+ \mathcal{H}_{1,\vartheta }^{\kappa , \omega } (w\mathcal{U} ) (\vartheta ) \mathcal{H}_{1, \vartheta }^{\kappa ,\omega } (v\mathcal{V} ) (\vartheta ). \end{aligned}$$
(33)

Proof

Since the functions \(\mathcal{U}\) and \(\mathcal{V}\) are synchronous on \([0,\infty )\), for all \(\rho \geq 0\), \(\zeta \geq 0\), we have

$$ \bigl(\mathcal{U}(\rho )-\mathcal{U}(\zeta ) \bigr) \bigl( \mathcal{V}(\rho )- \mathcal{V}(\zeta ) \bigr)\geq 0. $$

Therefore

$$ \mathcal{U}(\rho )\mathcal{V}(\rho )+\mathcal{U}(\zeta ) \mathcal{V}( \zeta )\geq \mathcal{U}(\rho )\mathcal{V}(\zeta )+\mathcal{U}(\zeta ) \mathcal{V}(\rho ). $$
(34)

Multiplying (34) by \(\frac{1}{\omega ^{\kappa }\Gamma (\kappa )}e^{\frac{\omega -1}{\omega }( \ln \vartheta -\ln \rho )}(\ln \vartheta -\ln \rho )^{\kappa -1} \frac{v(\rho )}{\rho }, \rho \in (1,\vartheta )\), and integrating the obtained inequality with respect to ρ over \((1,\vartheta )\), we get

$$\begin{aligned}& \frac{1}{\omega ^{\kappa }\Gamma (\kappa )} \int _{1}^{\vartheta }e^{ \frac{\omega -1}{\omega }(\ln \vartheta -\ln \rho )}(\ln \vartheta - \ln \rho )^{\kappa -1}v(\rho )\mathcal{U}(\rho )\mathcal{V}(\rho ) \frac{d\rho }{\rho } \\& \qquad {}+ \mathcal{U}(\zeta )\mathcal{V}(\zeta ) \frac{1}{\omega ^{\kappa }\Gamma (\kappa )} \int _{1}^{\vartheta }e^{ \frac{\omega -1}{\omega }(\ln \vartheta -\ln \rho )}(\ln \vartheta - \ln \rho )^{\kappa -1}v(\rho ) \frac{d\rho }{\rho } \\& \quad \geq \mathcal{V}(\zeta )\frac{1}{\omega ^{\kappa }\Gamma (\kappa )} \int _{1}^{\vartheta }e^{\frac{\omega -1}{\omega }(\ln \vartheta -\ln \rho )}(\ln \vartheta - \ln \rho )^{\kappa -1}v(\rho )\mathcal{U}( \rho )\frac{d\rho }{\rho } \\& \qquad {}+ \mathcal{U}(\zeta )\frac{1}{\omega ^{\kappa }\Gamma (\kappa )} \int _{1}^{\vartheta }e^{\frac{\omega -1}{\omega }(\ln \vartheta -\ln \rho )}(\ln \vartheta - \ln \rho )^{\kappa -1}v(\rho ) \mathcal{V}(\rho ) \frac{d\rho }{\rho }. \end{aligned}$$

In view of (12), we get

$$\begin{aligned} &\mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (v\mathcal{U} \mathcal{V} ) (\vartheta )+ \mathcal{U}(\zeta )\mathcal{V}( \zeta ) \mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (v ) ( \vartheta ) \\ &\quad \geq \mathcal{V}(\zeta )\mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (v \mathcal{U} ) (\vartheta )+\mathcal{U}(\zeta ) \mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (v\mathcal{V} ) ( \vartheta ). \end{aligned}$$
(35)

Now multiplying (35) by \(\frac{1}{\omega ^{\kappa }\Gamma (\kappa )}e^{\frac{\omega -1}{\omega }( \ln \vartheta -\ln \zeta )}(\ln \vartheta -\ln \zeta )^{\kappa -1} \frac{w(\zeta )}{\zeta }, \zeta \in (1,\vartheta )\), and integrating the obtained inequality with respect to ζ over \((1,\vartheta )\), we obtain

$$\begin{aligned}& \mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (v\mathcal{U} \mathcal{V} ) ( \vartheta ) \frac{1}{\omega ^{\kappa }\Gamma (\kappa )} \int _{1}^{\vartheta }e^{ \frac{\omega -1}{\omega }(\ln \vartheta -\ln \zeta )}(\ln \vartheta - \ln \zeta )^{\kappa -1}w(\zeta )\frac{d\zeta }{\zeta } \\& \qquad {}+ \mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (v ) ( \vartheta ) \frac{1}{\omega ^{\kappa }\Gamma (\kappa )} \int _{1}^{\vartheta }e^{\frac{\omega -1}{\omega }(\ln \vartheta -\ln \zeta )}( \ln \vartheta - \ln \zeta )^{\kappa -1}w(\zeta ) \mathcal{U}(\zeta ) \mathcal{V}(\zeta ) \frac{d\zeta }{\zeta } \\& \quad \geq \mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (v\mathcal{U} ) (\vartheta ) \frac{1}{\omega ^{\kappa }\Gamma (\kappa )} \int _{1}^{\vartheta }e^{\frac{\omega -1}{\omega }(\ln \vartheta -\ln \zeta )}( \ln \vartheta - \ln \zeta )^{\kappa -1}w(\zeta )\mathcal{V}(\zeta ) \frac{d\zeta }{\zeta } \\& \qquad {}+ \mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (v\mathcal{V} ) (\vartheta ) \frac{1}{\omega ^{\kappa }\Gamma (\kappa )} \int _{1}^{\vartheta }e^{\frac{\omega -1}{\omega }(\ln \vartheta -\ln \zeta )}( \ln \vartheta - \ln \zeta )^{\kappa -1}w(\zeta )\mathcal{U}(\zeta ) \frac{d\zeta }{\zeta }, \end{aligned}$$

which in view of (6) gives the desired inequality (16). □

Theorem 4.1

Let \(\mathcal{U}\) and \(\mathcal{V}\) be synchronous functions on \([0,\infty )\), and let \(r, p, q:[0,\infty )\rightarrow [0,\infty )\). Then for all \(\vartheta >1\), \(\kappa >0\), and \(\omega \in (0,1]\), we have the following inequality for (12):

$$\begin{aligned}& 2\mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (r ) ( \vartheta ) \bigl[\mathcal{H}_{1,\vartheta }^{\kappa ,\omega }p( \vartheta ) \mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (q \mathcal{U}\mathcal{V} ) ( \vartheta )+ \mathcal{H}_{1,\vartheta }^{ \kappa ,\omega } (q ) (\vartheta ) \mathcal{H}_{1,\vartheta }^{ \kappa ,\omega } (p\mathcal{U}\mathcal{V} ) ( \vartheta ) \bigr] \\& \qquad {}+ 2\mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (p ) ( \vartheta ) \mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (q ) ( \vartheta ) (r\mathcal{U} \mathcal{V} ) (\vartheta ) \\& \quad \geq \mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (r ) ( \vartheta ) \bigl[ \mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (p \mathcal{U} ) (\vartheta ) \mathcal{H}_{1,\vartheta }^{\kappa , \omega } (q\mathcal{V} ) (\vartheta )+ \mathcal{H}_{1, \vartheta }^{\kappa ,\omega } (q\mathcal{U} ) (\vartheta ) \mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (p\mathcal{V} ) ( \vartheta ) \bigr] \\& \qquad {}+ \mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (p ) ( \vartheta ) \bigl[ \mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (r \mathcal{U} ) (\vartheta ) \mathcal{H}_{1,\vartheta }^{\kappa , \omega } (q\mathcal{V} ) (\vartheta )+ \mathcal{H}_{1, \vartheta }^{\kappa ,\omega } (q\mathcal{U} ) (\vartheta ) \mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (r\mathcal{V} ) ( \vartheta ) \bigr] \\& \qquad {}+ \mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (q ) ( \vartheta ) \bigl[ \mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (r \mathcal{U} ) (\vartheta ) \mathcal{H}_{1,\vartheta }^{\kappa , \omega } (p\mathcal{V} ) (\vartheta )+ \mathcal{H}_{1, \vartheta }^{\kappa ,\omega } (p\mathcal{U} ) (\vartheta ) \mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (r\mathcal{V} ) ( \vartheta ) \bigr] . \end{aligned}$$
(36)

Proof

Taking \(v=p\) and \(w=q\) in Lemma 4.1, we obtain

$$\begin{aligned} &\mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (p ) ( \vartheta )\mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (q \mathcal{U}\mathcal{V} ) ( \vartheta )+\mathcal{H}_{1,\vartheta }^{ \kappa ,\omega } (q ) (\vartheta ) \mathcal{H}_{1,\vartheta }^{ \kappa ,\omega } (p\mathcal{U}\mathcal{V} ) ( \vartheta ) \\ &\quad \geq \mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (p\mathcal{U} ) (\vartheta ) \mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (q \mathcal{V} ) (\vartheta )+ \mathcal{H}_{1,\vartheta }^{\kappa , \omega } (q\mathcal{U} ) (\vartheta ) \mathcal{H}_{1, \vartheta }^{\kappa ,\omega } (p\mathcal{V} ) (\vartheta ). \end{aligned}$$
(37)

Multiplying (37) by \(\mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (r )( \vartheta )\), we get

$$\begin{aligned} &\mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (r ) ( \vartheta ) \bigl[\mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (p ) (\vartheta ) \mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (q \mathcal{U}\mathcal{V} ) ( \vartheta )+\mathcal{H}_{1,\vartheta }^{ \kappa ,\omega } (q ) (\vartheta ) \mathcal{H}_{1,\vartheta }^{ \kappa ,\omega } (p\mathcal{U}\mathcal{V} ) ( \vartheta ) \bigr] \\ &\quad \geq \mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (r ) ( \vartheta ) \bigl[ \mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (p \mathcal{U} ) (\vartheta ) \mathcal{H}_{1,\vartheta }^{\kappa , \omega } (q\mathcal{V} ) (\vartheta )+ \mathcal{H}_{1, \vartheta }^{\kappa ,\omega } (q\mathcal{U} ) (\vartheta ) \mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (p\mathcal{V} ) ( \vartheta ) \bigr]. \end{aligned}$$
(38)

Now taking \(v=r\) and \(w=q\) in Lemma 4.1, we have

$$\begin{aligned} &\mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (r ) ( \vartheta )\mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (q \mathcal{U}\mathcal{V} ) ( \vartheta )+\mathcal{H}_{1,\vartheta }^{ \kappa ,\omega } (q ) (\vartheta ) \mathcal{H}_{1,\vartheta }^{ \kappa ,\omega } (r\mathcal{U}\mathcal{V} ) ( \vartheta ) \\ &\quad \geq \mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (r\mathcal{U} ) (\vartheta ) \mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (q \mathcal{V} ) (\vartheta )+ \mathcal{H}_{1,\vartheta }^{\kappa , \omega } (q\mathcal{U} ) (\vartheta ) \mathcal{H}_{1, \vartheta }^{\kappa ,\omega } (r\mathcal{V} ) (\vartheta ). \end{aligned}$$
(39)

Multiplying (39) by \(\mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (p )( \vartheta )\), we get

$$\begin{aligned} &\mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (p ) ( \vartheta ) \bigl[\mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (r ) (\vartheta ) \mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (q \mathcal{U}\mathcal{V} ) ( \vartheta )+\mathcal{H}_{1,\vartheta }^{ \kappa ,\omega } (q ) (\vartheta ) \mathcal{H}_{1,\vartheta }^{ \kappa ,\omega } (r\mathcal{U}\mathcal{V} ) ( \vartheta ) \bigr] \\ &\quad \geq \mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (p ) ( \vartheta ) \bigl[ \mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (r \mathcal{U} ) (\vartheta ) \mathcal{H}_{1,\vartheta }^{\kappa , \omega } (q\mathcal{V} ) (\vartheta )+ \mathcal{H}_{1, \vartheta }^{\kappa ,\omega } (q\mathcal{U} ) (\vartheta ) \mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (r\mathcal{V} ) ( \vartheta ) \bigr]. \end{aligned}$$
(40)

Similarly, taking \(v=r\) and \(w=p\) in Lemma 4.1 and then multiplying both sides of the resultant inequality by \(\mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (q )( \vartheta )\), we obtain

$$\begin{aligned} &\mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (q ) ( \vartheta ) \bigl[\mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (r ) (\vartheta ) \mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (p \mathcal{U}\mathcal{V} ) ( \vartheta )+ \mathcal{H}_{1,\vartheta }^{ \kappa ,\omega } (p ) (\vartheta ) \mathcal{H}_{1,\vartheta }^{ \kappa ,\omega } (r\mathcal{U}\mathcal{V} ) ( \vartheta ) \bigr] \\ &\quad \geq \mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (q ) ( \vartheta ) \bigl[ \mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (r \mathcal{U} ) (\vartheta ) \mathcal{H}_{1,\vartheta }^{\kappa , \omega } (p\mathcal{V} ) (\vartheta )+ \mathcal{H}_{1, \vartheta }^{\kappa ,\omega } (p\mathcal{U} ) (\vartheta ) \mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (r\mathcal{V} ) ( \vartheta ) \bigr]. \end{aligned}$$
(41)

Hence by adding (38), (40), and (41) we get the desired statement (36). □

Remark 4.1

Setting \(\omega =1\) in Theorem 4.1, we get Theorem 3.2 proved by Chinchane and Pachpatte [13].

Lemma 4.2

Let \(\mathcal{U}\) and \(\mathcal{V}\) be synchronous functions on \([0,\infty )\), and let \(v,w:[0,\infty )\rightarrow [0,\infty )\). Then for all \(\vartheta >1\), \(\kappa , \eta >0\), and \(\omega \in (0,1]\), we have the following inequality for (12):

$$\begin{aligned} &\mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (v ) ( \vartheta )\mathcal{H}_{1,\vartheta }^{\eta ,\omega } (w \mathcal{U}\mathcal{V} ) ( \vartheta )+\mathcal{H}_{1,\vartheta }^{ \eta ,\omega } (w ) (\vartheta ) \mathcal{H}_{1,\vartheta }^{ \kappa ,\omega } (v\mathcal{U}\mathcal{V} ) ( \vartheta ) \\ &\quad \geq \mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (v\mathcal{U} ) (\vartheta ) \mathcal{H}_{1,\vartheta }^{\eta ,\omega } (w \mathcal{V} ) (\vartheta )+ \mathcal{H}_{1,\vartheta }^{\eta , \omega } (w\mathcal{U} ) (\vartheta ) \mathcal{H}_{1, \vartheta }^{\kappa ,\omega } (v\mathcal{V} ) (\vartheta ). \end{aligned}$$
(42)

Proof

Multiplying (35) by \(\frac{1}{\omega ^{\eta }\Gamma (\eta )}e^{\frac{\omega -1}{\omega }( \ln \vartheta -\ln \zeta )}(\ln \vartheta -\ln \zeta )^{\eta -1} \frac{w(\zeta )}{\zeta }, \zeta \in (1,\vartheta )\), and integrating the resultant inequality with respect to ζ over \((1,\vartheta )\), we obtain

$$\begin{aligned}& \mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (v\mathcal{U} \mathcal{V} ) ( \vartheta )\frac{1}{\omega ^{\eta }\Gamma (\eta )} \int _{1}^{\vartheta }e^{\frac{\omega -1}{\omega }(\ln \vartheta -\ln \zeta )}(\ln \vartheta - \ln \zeta )^{\eta -1}w(\zeta ) \frac{d\zeta }{\zeta } \\& \qquad {}+ \mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (v ) ( \vartheta ) \frac{1}{\omega ^{\eta }\Gamma (\eta )} \int _{1}^{\vartheta }e^{ \frac{\omega -1}{\omega }(\ln \vartheta -\ln \zeta )}(\ln \vartheta - \ln \zeta )^{\eta -1}w(\zeta )\mathcal{U}(\zeta )\mathcal{V}(\zeta ) \frac{d\zeta }{\zeta } \\& \quad \geq \mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (v\mathcal{U} ) (\vartheta ) \frac{1}{\omega ^{\eta }\Gamma (\eta )} \int _{1}^{\vartheta }e^{\frac{\omega -1}{\omega }(\ln \vartheta -\ln \zeta )}( \ln \vartheta - \ln \zeta )^{\eta -1}w(\zeta )\mathcal{V}(\zeta ) \frac{d\zeta }{\zeta } \\& \qquad {}+ \mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (v\mathcal{V} ) (\vartheta ) \frac{1}{\omega ^{\eta }\Gamma (\eta )} \int _{1}^{\vartheta }e^{\frac{\omega -1}{\omega }(\ln \vartheta -\ln \zeta )}( \ln \vartheta - \ln \zeta )^{\eta -1}w(\zeta )\mathcal{U}(\zeta ) \frac{d\zeta }{\zeta }, \end{aligned}$$

which by (12) gives

$$\begin{aligned}& \mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (v ) ( \vartheta ) \mathcal{H}_{1,\vartheta }^{\eta ,\omega } (w \mathcal{U}\mathcal{V} ) ( \vartheta )+\mathcal{H}_{1,\vartheta }^{ \eta ,\omega } (w ) (\vartheta ) \mathcal{H}_{1,\vartheta }^{ \kappa ,\omega } (v\mathcal{U}\mathcal{V} ) ( \vartheta ) \\& \quad \geq \mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (v\mathcal{U} ) (\vartheta ) \mathcal{H}_{1,\vartheta }^{\eta ,\omega } (w \mathcal{V} ) (\vartheta )+ \mathcal{H}_{1,\vartheta }^{\eta , \omega } (w\mathcal{U} ) (\vartheta ) \mathcal{H}_{1, \vartheta }^{\kappa ,\omega } (v\mathcal{V} ) (\vartheta ). \end{aligned}$$

This completes the proof of Lemma 4.2. □

Remark 4.2

Setting \(\kappa =\eta \) in Lemma 4.2, we get Lemma 4.1.

Theorem 4.2

Let \(\mathcal{U}\) and \(\mathcal{V}\) be synchronous functions on \([0,\infty )\), and let \(r, p, q:[0,\infty )\rightarrow [0,\infty )\). Then for all \(\vartheta >1\), \(\kappa >0\), and \(\omega \in (0,1]\), we have the following inequality for (12):

$$\begin{aligned}& \mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (r ) ( \vartheta ) \bigl[\mathcal{H}_{1,\vartheta }^{\kappa ,\omega }p( \vartheta ) \mathcal{H}_{1,\vartheta }^{\eta ,\omega } (q \mathcal{U}\mathcal{V} ) ( \vartheta )+2\mathcal{H}_{1,\vartheta }^{ \kappa ,\omega } (p ) (\vartheta ) \mathcal{H}_{1,\vartheta }^{ \eta ,\omega } (q\mathcal{U}\mathcal{V} ) ( \vartheta ) \\& \qquad {}+ \mathcal{H}_{1,\vartheta }^{\eta ,\omega } (q ) (\vartheta ) \mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (p\mathcal{U} \mathcal{V} ) ( \vartheta ) \bigr] \\& \qquad {}+ \bigl[\mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (p ) ( \vartheta ) \mathcal{H}_{1,\vartheta }^{\eta ,\omega } (q ) ( \vartheta )+ \mathcal{H}_{1,\vartheta }^{\eta ,\omega } (p ) ( \vartheta ) \mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (q ) ( \vartheta ) \bigr] \mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (r \mathcal{U}\mathcal{V} ) ( \vartheta ) \\& \quad \geq \mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (r ) ( \vartheta ) \bigl[ \mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (p \mathcal{U} ) (\vartheta ) \mathcal{H}_{1,\vartheta }^{\eta , \omega } (q\mathcal{V} ) (\vartheta )+ \mathcal{H}_{1, \vartheta }^{\eta ,\omega } (q\mathcal{U} ) (\vartheta ) \mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (p\mathcal{V} ) ( \vartheta ) \bigr] \\& \qquad {}+ \mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (p ) ( \vartheta ) \bigl[ \mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (r \mathcal{U} ) (\vartheta ) \mathcal{H}_{1,\vartheta }^{\eta , \omega } (q\mathcal{V} ) (\vartheta )+ \mathcal{H}_{1, \vartheta }^{\eta ,\omega } (q\mathcal{U} ) (\vartheta ) \mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (r\mathcal{V} ) ( \vartheta ) \bigr] \\& \qquad {}+ \mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (q ) ( \vartheta ) \bigl[ \mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (r \mathcal{U} ) (\vartheta ) \mathcal{H}_{1,\vartheta }^{\eta , \omega } (p\mathcal{V} ) (\vartheta )+ \mathcal{H}_{1, \vartheta }^{\eta ,\omega } (p\mathcal{U} ) (\vartheta ) \mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (r\mathcal{V} ) ( \vartheta ) \bigr] . \end{aligned}$$
(43)

Proof

Taking \(v=p\) and \(w=q\) in Lemma 4.2, we obtain

$$\begin{aligned} &\mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (p ) ( \vartheta )\mathcal{H}_{1,\vartheta }^{\eta ,\omega } (q \mathcal{U}\mathcal{V} ) ( \vartheta )+\mathcal{H}_{1,\vartheta }^{ \eta ,\omega } (q ) (\vartheta ) \mathcal{H}_{1,\vartheta }^{ \kappa ,\omega } (p\mathcal{U}\mathcal{V} ) ( \vartheta ) \\ &\quad \geq \mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (p\mathcal{U} ) (\vartheta ) \mathcal{H}_{1,\vartheta }^{\eta ,\omega } (q \mathcal{V} ) (\vartheta )+ \mathcal{H}_{1,\vartheta }^{\eta , \omega } (q\mathcal{U} ) (\vartheta ) \mathcal{H}_{1, \vartheta }^{\kappa ,\omega } (p\mathcal{V} ) (\vartheta ). \end{aligned}$$
(44)

Multiplying (44) by \(\mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (r )( \vartheta )\), we get

$$\begin{aligned} &\mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (r ) ( \vartheta ) \bigl[\mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (p ) (\vartheta ) \mathcal{H}_{1,\vartheta }^{\eta ,\omega } (q \mathcal{U}\mathcal{V} ) ( \vartheta )+\mathcal{H}_{1,\vartheta }^{ \eta ,\omega } (q ) (\vartheta ) \mathcal{H}_{1,\vartheta }^{ \kappa ,\omega } (p\mathcal{U}\mathcal{V} ) ( \vartheta ) \bigr] \\ &\quad \geq \mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (r ) ( \vartheta ) \bigl[ \mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (p \mathcal{U} ) (\vartheta ) \mathcal{H}_{1,\vartheta }^{\eta , \omega } (q\mathcal{V} ) (\vartheta )+ \mathcal{H}_{1, \vartheta }^{\eta ,\omega } (q\mathcal{U} ) (\vartheta ) \mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (p\mathcal{V} ) ( \vartheta ) \bigr]. \end{aligned}$$
(45)

Setting \(v=r\) and \(w=q\) in Lemma 4.2, we obtain

$$\begin{aligned} &\mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (r ) ( \vartheta )\mathcal{H}_{1,\vartheta }^{\eta ,\omega } (q \mathcal{U}\mathcal{V} ) ( \vartheta )+\mathcal{H}_{1,\vartheta }^{ \eta ,\omega } (q ) (\vartheta ) \mathcal{H}_{1,\vartheta }^{ \kappa ,\omega } (r\mathcal{U}\mathcal{V} ) ( \vartheta ) \\ &\quad \geq \mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (r\mathcal{U} ) (\vartheta ) \mathcal{H}_{1,\vartheta }^{\eta ,\omega } (q \mathcal{V} ) (\vartheta )+ \mathcal{H}_{1,\vartheta }^{\eta , \omega } (q\mathcal{U} ) (\vartheta ) \mathcal{H}_{1, \vartheta }^{\kappa ,\omega } (r\mathcal{V} ) (\vartheta ). \end{aligned}$$
(46)

Multiplying (46) by \(\mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (p )( \vartheta )\), we get

$$\begin{aligned} &\mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (p ) ( \vartheta ) \bigl[\mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (r ) (\vartheta ) \mathcal{H}_{1,\vartheta }^{\eta ,\omega } (q \mathcal{U}\mathcal{V} ) ( \vartheta )+\mathcal{H}_{1,\vartheta }^{ \eta ,\omega } (q ) (\vartheta ) \mathcal{H}_{1,\vartheta }^{ \kappa ,\omega } (r\mathcal{U}\mathcal{V} ) ( \vartheta ) \bigr] \\ &\quad \geq \mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (p ) ( \vartheta ) \bigl[ \mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (r \mathcal{U} ) (\vartheta ) \mathcal{H}_{1,\vartheta }^{\eta , \omega } (q\mathcal{V} ) (\vartheta )+ \mathcal{H}_{1, \vartheta }^{\eta ,\omega } (q\mathcal{U} ) (\vartheta ) \mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (r\mathcal{V} ) ( \vartheta ) \bigr]. \end{aligned}$$
(47)

Similarly, setting \(v=r\) and \(w=p\) in Lemma 4.2 and multiplying both sides of the resultant inequality by \(\mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (q )( \vartheta )\), we obtain

$$\begin{aligned} &\mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (q ) ( \vartheta ) \bigl[\mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (r ) (\vartheta ) \mathcal{H}_{1,\vartheta }^{\eta ,\omega } (p \mathcal{U}\mathcal{V} ) ( \vartheta )+ \mathcal{H}_{1,\vartheta }^{ \eta ,\omega } (p ) (\vartheta ) \mathcal{H}_{1,\vartheta }^{ \kappa ,\omega } (r\mathcal{U}\mathcal{V} ) ( \vartheta ) \bigr] \\ &\quad \geq \mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (q ) ( \vartheta ) \bigl[ \mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (r \mathcal{U} ) (\vartheta ) \mathcal{H}_{1,\vartheta }^{\eta , \omega } (p\mathcal{V} ) (\vartheta )+ \mathcal{H}_{1, \vartheta }^{\eta ,\omega } (p\mathcal{U} ) (\vartheta ) \mathcal{H}_{1,\vartheta }^{\kappa ,\omega } (r\mathcal{V} ) ( \vartheta ) \bigr]. \end{aligned}$$
(48)

Hence we obtain the desired statement (43) by adding inequalities (45), (47), and (48). □

Remark 4.3

Setting \(\kappa =\eta \) in Theorem 4.2, we get Theorem 4.1.

Remark 4.4

Setting \(\omega =1\) in Theorem 3.2, we get Theorem 3.4 proved by Chinchane and Pachpatte [13]

Remark 4.5

Inequalities (36) and (43) will be reversed in the following cases:

  1. (i)

    The functions \(\mathcal{U}\) and \(\mathcal{V}\) are asynchronous on \([0,\infty )\).

  2. (ii)

    The functions r, p, and q are negative on \([0,\infty )\).

  3. (iii)

    Two of the functions r, p, and q are positive, and the third one is negative on \([0,\infty )\).

5 Concluding remarks

In the last few decades, fractional calculus has been extensively studied due to its wide applications in diverse fields cited in the literature. Based on that notion, the idea of generalized proportional fractional integral operators concerning the exponential function in their kernels was recently introduced by Jarad et al. [19]. Later on, the Hadamard proportional fractional integrals were introduced by Rahman et al. [32], who established certain inequalities for convex functions by employing the Hadamard proportional fractional integrals. Recently, many researchers established integral inequalities by employing generalized proportional fractional integral operators cited in the literature. In this paper, we established integral inequalities for the extended Chebyshev functional by utilizing the generalized proportional fractional and Hadamard proportional fractional integrals. Particular cases of our results can be found in the works of Dahmani [14] and Chinchane and Pachpatte [13].