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On Grüss inequalities within generalized K-fractional integrals

Abstract

In this paper, we introduce the generalized \(\mathcal{K}\)-fractional integral in the frame of a new parameter \(\mathcal{K}>0\). This paper offers some new important inequalities of Grüss type using the generalized \(\mathcal{K}\)-fractional integral and associated integral inequalities. Our results with this new integral operator have the abilities to be implemented for the evaluation of many mathematical problems related to the real world applications.

Introduction

There are numerous problems wherein fractional derivatives (non-integer order derivatives and integrals) attain a valuable position [19]. It must be emphasized that fractional derivatives are furnished in many techniques, especially, characterizing three distinct approaches, which we are able to mention in an effort to grow the work in certainly one of them. Every classical fractional operator is typically described in terms of a particular significance. There are many well-recognized definitions of fractional operators, we can point out the Riemann-Liouville, Caputo, Grunwald–Letnikov, and Hadamard operators [10], whose formulations include integrals with singular kernels and which may be used to check the issues involving the reminiscence effect [11]. Some special new formulas for the fractional operators can be found in the literature [12]. These new formulas are different from the classical formulas in numerous components. As an example, classical fractional derivatives are described in such a manner that in the limit wherein the order of the derivative is an integer, one recovers the classical derivatives in the sense of Newton and Leibniz. There have additionally been currently proposed new fractional operators [13] with a corresponding integral whose kernel may be a non-singular mapping, as an instance, a Mittag-Leffler function. Additionally, in such instances, integer-order derivatives are rediscovered by supposing suitable limits for the values of their parameters.

On the other hand, there are numerous approaches to acquiring a generalization of classical fractional integrals. Many authors introduced new fractional operators generated from general classical local derivatives (see [1417] and the references therein). Other authors introduced a parameter and enunciated a generalization for fractional integrals on a selected space. These are called generalized \(\mathcal{K}\)-fractional integrals. For such operators, we refer to [4, 1820] and the works cited in them.

It is well known that inequalities have potential applications in the technology, scientific studies, and analysis [2132] and numerous mathematical problems such as approximation theory, statistical analysis, and human social sciences [3359]. In perspective on the more extensive applications, such variants have acquired large interest. Presently, authors have provided the unique version of such inequalities, which may be beneficial in the investigation of diverse forms of integrodifferential and difference equations. Those variants are an extensive instrument to take a gander at the classes of differential and integral equations.

The most celebrated Grüss inequality can be described as follows.

Theorem 1.1

(see [60])

Let\(\mathcal{R}\)be a set of real numbers, \(m,\mathcal{M}, n,\mathcal{N}\in\mathcal{R}\)and\(\mathcal{P},\mathcal{S}:[\sigma_{1},\sigma_{2}]\rightarrow\mathcal{R}\)be two positive functions such that\(m\leq\mathcal{P}(\mathfrak{d}_{1})\leq\mathcal{M} \)and\(n\leq\mathcal{S}(\mathfrak{d}_{1})\leq\mathcal{N}\)for all\(\mathfrak{d}_{1}\in[\sigma_{1},\sigma_{2}]\). Then

$$ \begin{aligned}[b] &\biggl\vert \frac{1}{\sigma_{2}-\sigma_{1}} \int_{\sigma_{1}}^{\sigma_{2}} \mathcal{P}(\mathfrak{d}_{1}) \mathcal{S}(\mathfrak{d}_{1})\,d\mathfrak{d}_{1}- \frac{1}{(\sigma_{2}-\sigma_{1})^{2}} \int_{\sigma_{1}}^{\sigma_{2}} \mathcal{P}(\mathfrak{d}_{1}) \,d\mathfrak{d}_{1} \int_{\sigma _{1}}^{\sigma_{2}}\mathcal{S}(\mathfrak{d}_{1}) \,d\mathfrak{d}_{1} \biggr\vert \\ &\quad\leq\frac{1}{4}(\mathcal{M}-m) (\mathcal{N}-n),\end{aligned} $$
(1.1)

where the constant\(1/4\)cannot be improved.

Grüss inequality (1.1) connects the integral of the product of two functions with the product of their integrals. It is extensively identified that continuous and discrete cases of Grüss type variants play a considerable job in examining the qualitative conduct of differential and integral equations. Inspired by Grüss inequality (1.1), we intend to show modified versions of (1.1) by using generalized \(\mathcal{K}\)-fractional integrals. For the reason that such variants are supposed to be vital, the exploration has continued to develop the investigations for such kinds of variants. Our findings and their utilities appeared in a variety of academic papers (see [6164]). Amongst such sorts of inequalities, the Grüss inequality is one of the most fascinating inequalities.

We are influenced to take a look at this inequality for the generalized \(\mathcal{K}\)-fractional integral. As a consequence, we obtain numerous fractional integral inequalities, such inequalities are worthwhile in the fields of fractional differential equations. Also we apply these fractional inequalities to find new versions for the generalized Riemann–Liouville fractional integral. In this sequel, we present some preliminary results in order to prove our main results later.

Definition 1.2

Let \(\mathcal{P}\in L_{1}([\sigma_{1},\sigma_{2}])\) (the Lebesgue space). Then the left- and right-sided generalized Riemann–Liouville fractional integrals of order \(\rho>0\) are defined by

$$\begin{aligned} \mathfrak{J}_{\sigma_{1}^{+}}^{\varrho}\mathcal{P}(\varrho)= \frac {1}{\varGamma(\rho)} \int_{\sigma_{1}}^{\varrho}(\varrho-\zeta)^{\rho-1} \mathcal{P}(\zeta )\,d\zeta,\quad \varrho>\sigma_{1}, \end{aligned}$$

and

$$\begin{aligned} \mathfrak{J}_{\sigma_{2}^{-}}^{\varrho}\mathcal{P}(\varrho)= \frac {1}{\varGamma(\rho)} \int_{\varrho}^{\sigma_{2}}(\zeta-\lambda)^{\rho-1} \mathcal{P}(\zeta )\,d\zeta,\quad \varrho< \sigma_{2}, \end{aligned}$$

where \(\varGamma(\cdot)\) is the gamma function [65, 66].

Now, we give the definition of more general fractional integral, which is mainly due to Mubeen and Habibullah [18].

Definition 1.3

(see [18])

Let \(\mathcal{P}\in L_{1}([\sigma_{1},\sigma_{2}])\) (the Lebsegue space). Then the left-sided and right-sided \(\mathcal{K}\)-fractional integrals of order \(\rho, \mathcal{K}>0\) are defined by

$$\begin{aligned} \mathfrak{J}_{\sigma_{1}^{+}}^{\rho,\mathcal{K}}\mathcal{P}(\varrho )= \frac{1}{\mathcal{K}\varGamma_{\mathcal{K}}(\rho)} \int_{\sigma_{1}}^{\varrho}(\varrho-\zeta)^{\frac{\rho}{\mathcal {K}}-1} \mathcal{P}(\zeta)\,d\zeta \quad(\varrho>\sigma_{1}) \end{aligned}$$

and

$$\begin{aligned} \mathfrak{J}_{\sigma_{2}^{-}}^{\rho,\mathcal{K}}\mathcal{P}(\varrho )= \frac{1}{\mathcal{K}\varGamma_{\mathcal{K}}(\rho)} \int_{\varrho}^{\sigma_{2}}(\zeta-\varrho)^{\frac{\rho}{\mathcal {K}}-1} \mathcal{P}(\zeta)\,d\zeta \quad(\varrho< \sigma_{2}). \end{aligned}$$

Further, we demonstrate the concept of generalized Riemann–Liouville fractional integral as follows.

Definition 1.4

(see [11, 67])

Let \(\sigma_{1},\sigma_{2}\in(-\infty, +\infty)\) such that \(\sigma_{1}<\sigma_{2}\) and \(\varPsi(\zeta)\) be an increasing and positive monotone function on \((\sigma_{1},\sigma_{2}]\). Then the left-sided and right-sided generalized Riemann–Liouville fractional integrals of a function \(\mathcal{P}\) with respect to another function Ψ of order \(\rho>0\) are defined by

$$\begin{aligned}& \mathfrak{J}_{\sigma_{1}^{+}}^{\rho,\varPsi}\mathcal{P}(\varrho )= \frac{1}{\varGamma(\varrho)} \int_{\sigma_{1}}^{\varrho}\varPsi ^{\prime}(\zeta ) \bigl(\varPsi(\varrho)-\varPsi(\zeta) \bigr)^{\rho-1} \mathcal{P}(\zeta ) \,d\zeta, \\& \mathfrak{J}_{\sigma_{2}^{-}}^{\rho,\varPsi}\mathcal{P}(\varrho)= \frac{1}{\varGamma(\rho)} \int_{\varrho}^{\sigma_{2}}\varPsi^{\prime }(\zeta) \bigl(\varPsi(\zeta)-\varPsi(\varrho) \bigr)^{\rho-1} \mathcal{P}(\zeta)\,d \zeta. \end{aligned}$$
(1.2)

We define the more general form of generalized \(\mathcal{K}\)-fractional integral as follows.

Definition 1.5

Let \(\sigma_{1},\sigma_{2}\in(-\infty, +\infty)\) such that \(\sigma_{1}<\sigma_{2}\) and \(\varPsi(\zeta)\) be an increasing and positive monotone function on \((\sigma_{1},\sigma_{2}]\). Then the left-sided and right-sided generalized \(\mathcal{K}\)-fractional integrals of a function \(\mathcal{P}\) with respect to another function Ψ of order \(\rho, \mathcal{K}>0\) are defined by

$$\begin{aligned}& \mathfrak{J}_{\varPsi,\sigma_{1}^{+}}^{\rho,\mathcal{K}}\mathcal {P}(\varrho)= \frac{1}{ \mathcal{K}\varGamma_{\mathcal{K}}(\rho)} \int_{\sigma_{1}}^{\varrho }\varPsi^{\prime}(\zeta) \bigl(\varPsi(\varrho)-\varPsi(\zeta) \bigr)^{\frac{\rho}{\mathcal{K}}-1} \mathcal{P}(\zeta) \,d\zeta, \\& \mathfrak{J}_{\varPsi,\sigma_{2}^{-}}^{\rho,\mathcal{K}}\mathcal {P}(\varrho)= \frac{1}{\mathcal{K}\varGamma_{\mathcal{K}}(\rho)} \int_{\varrho }^{\sigma_{2}}\varPsi^{\prime}(\zeta) \bigl(\varPsi(\zeta)-\varPsi(\varrho ) \bigr)^{\frac{\rho}{\mathcal{K}}-1} \mathcal{P}(\zeta) \,d\zeta, \end{aligned}$$
(1.3)

where \(\varGamma_{\mathcal{K}}\) is the \(\mathcal{K}\)-gamma function.

Remark 1.6

From Definition 1.5 we clearly see that Definition 1.2, Definition 1.3, and Definition 1.4 can be obtained if we take \(\mathcal{K}=1\), \(\varPsi(\varrho)=\varrho\), and \(\varPsi(\varrho)=\varrho\) and \(\mathcal{K}=1\), respectively.

Grüss type inequality and its useful consequences are investigated by Kacar et al. [68]. Motivated by [68], we provide new and novel results using generalized \(\mathcal{K}\)-fractional integral related to (1.1). Consequently, the effects furnished in this research paper are an extra generalization.

Main results

Theorem 2.1

Let\(\mathcal{K}, \rho, \delta>0\), \(\mathcal{P}\)be a positive function on\([0,\infty)\)andΨbe an increasing and positive function on\([0,\infty)\)such that\(\varPsi^{\prime}(x)\)is continuous on\([0,\infty)\)with\(\varPsi(0)=0\). Suppose that there exist integrable functions\(\varTheta_{1}\), \(\varTheta_{2}\)on\([0,\infty)\)such that

$$\begin{aligned} \varTheta_{1}(\varrho)\leq\mathcal{P}(\varrho)\leq \varTheta_{2}(\varrho) \end{aligned}$$
(2.1)

for all\(\varrho\in[0,\infty)\). Then we have

$$\begin{aligned}& \mathfrak{J}_{\varPsi,0^{+}}^{\delta,\mathcal{K}}\varTheta _{1}( \varrho)\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\mathcal {P}(\varrho) + \mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\varTheta_{2}(\varrho ) \mathfrak{J}_{\varPsi,0^{+}}^{\delta,\mathcal{K}}\mathcal{P}(\varrho ) \\& \quad\geq \mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\varTheta_{2}( \varrho )\mathfrak{J}_{\varPsi,0^{+}}^{\delta,\mathcal{K}}\varTheta_{1}( \varrho)+ \mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\mathcal{P}(\varrho ) \mathfrak{J}_{\varPsi,0^{+}}^{\delta,\mathcal{K}}\mathcal{P}(\varrho). \end{aligned}$$
(2.2)

Proof

Using (2.1), for all \(\mathfrak{d}_{1}\geq0\), \(\mathfrak{d_{2}}\geq0\), we have

$$\begin{aligned}& \bigl(\varTheta_{2}(\mathfrak{d}_{1})-\mathcal{P}( \mathfrak{d}_{1}) \bigr) \bigl(\mathcal{P}(\mathfrak{d_{2}})- \varTheta_{1}(\mathfrak{d_{2}}) \bigr)\geq0, \\& \varTheta_{2}(\mathfrak{d}_{1})\mathcal{P}( \mathfrak{d_{2}})+\varTheta _{1}(\mathfrak{d_{2}}) \mathcal{P}(\mathfrak{d}_{1})\geq\varTheta _{1}( \mathfrak{d_{2}}) \varTheta_{2}(\mathfrak{d}_{1})+ \mathcal{P}(\mathfrak{d}_{1})\mathcal {P}(\mathfrak{d_{2}}). \end{aligned}$$
(2.3)

If we multiply both sides of (2.3) by \(\frac{ (\varPsi(\varrho)-\varPsi(\mathfrak{d}_{1}) )^{\frac{\rho }{\mathcal{K}}-1}\varPsi^{\prime}(\mathfrak{d}_{1})}{\mathcal {K}\varGamma_{\mathcal{K}}(\rho)}\) and integrate with respect to \(\mathfrak{d}_{1}\) on \((0,\varrho)\), we obtain

$$\begin{aligned}& \mathcal{P}(\mathfrak{d_{2}})\frac{1}{\mathcal{K}\varGamma_{\mathcal{K}}(\rho)} \int_{0}^{\varrho} \bigl(\varPsi(\varrho)-\varPsi( \mathfrak {d}_{1}) \bigr) ^{\frac{\rho}{\mathcal{K}}-1}\varPsi^{\prime}( \mathfrak {d}_{1})\varTheta_{2}(\mathfrak{d}_{1}) \,d\mathfrak{d}_{1} \\& \qquad{} +\varTheta_{1}(\mathfrak{d_{2}})\frac{1}{\mathcal{K}\varGamma _{\mathcal{K}}(\rho)} \int_{0}^{\varrho} \bigl(\varPsi(\varrho)-\varPsi ( \mathfrak{d}_{1}) \bigr)^{\frac{\rho}{\mathcal{K}}-1} \varPsi^{\prime}( \mathfrak{d}_{1})\mathcal{P}(\mathfrak{d}_{1})\, d \mathfrak{d}_{1} \\& \quad\geq\varTheta_{1}(\mathfrak{d_{2}}) \frac{1}{\mathcal{K}\varGamma _{\mathcal{K}}(\rho)} \int_{0}^{\varrho} \bigl(\varPsi(\varrho)-\varPsi ( \mathfrak{d}_{1} ) \bigr)^{\frac{\rho}{\mathcal{K}}-1}\varPsi^{\prime}( \mathfrak {d}_{1})\varTheta_{2}(\mathfrak{d}_{1}) \,d\mathfrak{d}_{1} \\& \quad\quad{}+\mathcal{P}(\mathfrak{d_{2}})\frac{1}{\mathcal{K}\varGamma _{\mathcal{K}}(\rho)} \int_{0}^{\varrho} \bigl(\varPsi(\varrho)-\varPsi ( \mathfrak{d}_{1}) \bigr)^{\frac{\rho}{\mathcal{K}}-1}\varPsi^{\prime}( \mathfrak {d}_{1})\mathcal{P}(\mathfrak{d}_{1})\,d \mathfrak{d}_{1}, \end{aligned}$$

which can be written as

$$\begin{aligned}& \mathcal{P}(\mathfrak{d_{2}})\mathfrak{J}_{\varPsi,0^{+}}^{\rho ,\mathcal{K}} \varTheta_{2}(\varrho)+\varTheta_{1}( \mathfrak{d_{2}}) \mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}} \mathcal{P}(\varrho) \\& \quad\geq\varTheta_{1}(\mathfrak{d_{2}}) \mathfrak{J}_{\varPsi,0^{+}}^{\rho ,\mathcal{K}}\varTheta_{2}(\varrho)+ \mathcal{P}(\mathfrak{d_{2}}) \mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}} \mathcal{P}(\varrho). \end{aligned}$$
(2.4)

Multiplying both sides of (2.4) by \(\frac{ (\varPsi(\varrho)-\varPsi(\mathfrak{d_{2}}) )^{\frac {\delta}{\mathcal{K}}-1}\varPsi^{\prime}(\mathfrak{d_{2}})}{\mathcal {K}\varGamma_{\mathcal{K}}(\delta)}\) and integrating with respect to \(\mathfrak{d_{2}}\) on \((0,\varrho)\), we get

$$\begin{aligned}& \mathfrak{J}_{\varPsi,0^{+}}^{\delta,\mathcal{K}}\varTheta _{1}(\varrho)\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\mathcal {P}(\varrho) +\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\varTheta_{2}( \varrho )\mathfrak{J}_{\varPsi,0^{+}}^{\delta,\mathcal{K}}\mathcal{P}(\varrho ) \\& \quad\geq \mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\varTheta_{2}( \varrho )\mathfrak{J}_{\varPsi,0^{+}}^{\delta,\mathcal{K}}\varTheta_{1}( \varrho)+ \mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\mathcal{P}(\varrho ) \mathfrak{J}_{\varPsi,0^{+}}^{\delta,\mathcal{K}}\mathcal{P}(\varrho). \end{aligned}$$
(2.5)

This completes the proof of Theorem 2.1. □

Corollary 2.2

Let\(\varPsi(\varrho)=\varrho\). Then Theorem 2.1leads to the inequality for\(\mathcal{K}\)-fractional integral as follows:

$$\begin{aligned}& \begin{gathered} \mathfrak{J}^{\delta,\mathcal{K}}\varTheta_{1}(\varrho)\mathfrak {J}^{\rho,\mathcal{K}}\mathcal{P}(\varrho) +\mathfrak{J}^{\rho,\mathcal{K}} \varTheta_{2}(\varrho)\mathfrak {J}^{\delta,\mathcal{K}}\mathcal{P}( \varrho)\\ \quad\geq \mathfrak{J}^{\rho,\mathcal{K}}\varTheta_{2}(\varrho) \mathfrak {J}^{\delta,\mathcal{K}}\varTheta_{1}(\varrho)+ \mathfrak{J}^{\rho,\mathcal{K}}\mathcal{P}(\varrho)\mathfrak{J}^{\delta ,\mathcal{K}} \mathcal{P}(\varrho).\end{gathered} \end{aligned}$$

Corollary 2.3

Let\(m,\mathcal{M}\in\mathcal{R}\)with\(m<\mathcal{M}\in\mathcal{R}\), \(\mathcal{K}, \rho,\delta>0\)and\(\mathcal{P}\)be a positive function on\([0,\infty)\)such that\(m\leq\mathcal{P}(\varrho)\leq\mathcal{M}\). Then one has

$$\begin{aligned}& m\frac{\varPsi^{\frac{\delta}{\mathcal{K}}}(\varrho)}{\varGamma _{\mathcal{K}}(\delta+\mathcal{K})}\mathfrak{J}_{\varPsi,0^{+}}^{\rho ,\mathcal{K}} \mathcal{P}(\varrho) +\mathcal{M}\frac{\varPsi^{\frac{\rho}{\mathcal{K}}}(\varrho )}{\varGamma_{\mathcal{K}}(\rho+\mathcal{K})}\mathfrak{J}_{\varPsi ,0^{+}}^{\delta,\mathcal{K}} \mathcal{P}(\varrho) \\& \quad\geq m\mathcal{M}\frac{\varPsi^{\frac{\delta}{\mathcal{K}}}(\varrho )}{\varGamma_{\mathcal{K}}(\delta+\mathcal{K})}\frac{\varPsi^{\frac{\rho }{\mathcal{K}}}(\varrho)}{\varGamma_{\mathcal{K}}(\rho+\mathcal{K})}+ \mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\mathcal{P}(\varrho ) \mathfrak{J}_{\varPsi,0^{+}}^{\delta,\mathcal{K}}\mathcal{P}(\varrho). \end{aligned}$$

Remark 2.4

Theorem 2.1, Corollary 2.2, and Corollary 2.3 lead to the following conclusions:

  1. (1)

    If \(\mathcal{K}=1\) in Theorem 2.1, then we get Theorem 2.11 in [68].

  2. (2)

    If \(\mathcal{K}=1\) and \(\varPsi(\varrho)=\varrho\) in Theorem 2.1, then we get Theorem 2 in [69].

  3. (3)

    If \(\mathcal{K}=1\) in Corollary 2.2, then we get Corollary 2.14 in [68].

  4. (4)

    If \(\mathcal{K}=1\) and \(\varPsi(\varrho)=\varrho\) in Corollary 2.3, then we obtain Corollary 3 in [69].

Theorem 2.5

Let\(\mathcal{K}>0\), \(\rho,\delta>0\), \(\mathcal{P}\)and\(\mathcal{S}\)be two positive functions on\([0,\infty)\), andΨbe an increasing and positive function on\([0,\infty)\)such that\(\varPsi(0)=0\)and\(\varPsi^{\prime}\)is continuous on\([0,\infty)\). Suppose that (2.1) holds and there exist integrable functions\(\varphi_{1}\)and\(\varphi_{2}\)on\([0,\infty)\)such that

$$\begin{aligned} \varphi_{1}(\varrho)\leq\mathcal{S}(\varrho)\leq \varphi_{2}(\varrho) \end{aligned}$$
(2.6)

for all\(\varrho\in[0,\infty)\). Then we have four inequalities as follows:

$$\begin{aligned}& \textup{(a)}\quad\mathfrak{J}_{\varPsi,0^{+}}^{\delta,\mathcal{K}}\varphi _{1}(\varrho)\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\mathcal {P}(\varrho) +\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\varTheta_{2}( \varrho )\mathfrak{J}_{\varPsi,0^{+}}^{\delta,\mathcal{K}}\mathcal{S}(\varrho) \\& \phantom{\textup{(a)}\quad}\quad\geq \mathfrak{J}_{\varPsi,0^{+}}^{\delta,\mathcal{K}}\varphi_{2}( \varrho )\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}} \varTheta_{2}( \varrho)+\mathfrak{J}_{\varPsi,0^{+}}^{\rho}\mathcal {P}(\varrho) \mathfrak{J}_{\varPsi,0^{+}}^{\delta,\mathcal{K}}\mathcal {S}(\varrho), \\& \textup{(b)}\quad\mathfrak{J}_{\varPsi,0^{+}}^{\delta,\mathcal{K}}\varTheta _{1}(\varrho)\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\mathcal {S}(\varrho) +\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\varphi_{2}( \varrho )\mathfrak{J}_{\varPsi,0^{+}}^{\delta,\mathcal{K}}\mathcal{P}(\varrho) \\& \phantom{\textup{(b)}\quad}\quad\geq \mathfrak{J}_{\varPsi,0^{+}}^{\delta,\mathcal{K}}\varTheta_{1}( \varrho )\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}} \varphi_{2}( \varrho)+\mathfrak{J}_{\varPsi,0^{+}}^{\delta,\mathcal {K}}\mathcal{P}(\varrho) \mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal {K}}\mathcal{S}(\varrho), \\& \textup{(c)}\quad\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\varTheta _{2}(\varrho)\mathfrak{J}_{\varPsi,0^{+}}^{\delta,\mathcal{K}}\varphi _{2}(\varrho) +\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}} \mathcal{P}(\varrho )\mathfrak{J}_{\varPsi,0^{+}}^{\delta,\mathcal{K}}\mathcal{S}( \varrho) \\& \phantom{\textup{(c)}\quad}\quad\geq \mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\varTheta_{2}( \varrho )\mathfrak{J}_{\varPsi,0^{+}}^{\delta,\mathcal{K}} \mathcal{S}(\varrho)+ \mathfrak{J}_{\varPsi,0^{+}}^{\delta,\mathcal {K}}\varphi_{2}(\varrho) \mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal {K}}\mathcal{P}(\varrho), \\& \textup{(d)}\quad\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\varTheta _{1}(\varrho)\mathfrak{J}_{\varPsi,0^{+}}^{\delta,\mathcal{K}}\varphi _{1}(\varrho) +\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}} \mathcal{P}(\varrho )\mathfrak{J}_{\varPsi,0^{+}}^{\delta,\mathcal{K}}\mathcal{S}( \varrho) \\& \phantom{\textup{(d)}\quad}\quad\geq \mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\varTheta_{1}( \varrho )\mathfrak{J}_{\varPsi,0^{+}}^{\delta,\mathcal{K}} \mathcal{S}(\varrho)+ \mathfrak{J}_{\varPsi,0^{+}}^{\delta,\mathcal {K}}\varphi_{1}(\varrho) \mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\mathcal{P}(\varrho). \end{aligned}$$
(2.7)

Proof

For any \(\varrho\in[0,\infty)\), it follows from (2.1) and (2.6) that

$$\begin{aligned} \bigl(\varTheta_{2}(\mathfrak{d}_{1})-\mathcal{P}( \mathfrak{d}_{1}) \bigr) \bigl(\mathcal{S}(\mathfrak{d_{2}})- \varphi_{1}(\mathfrak{d_{2}}) \bigr)\geq0 \end{aligned}$$

and

$$\begin{aligned} \varTheta_{2}(\mathfrak{d}_{1})\mathcal{S}( \mathfrak{d_{2}})+\varphi _{1}(\mathfrak{d_{2}}) \mathcal{P}(\mathfrak{d}_{1})\geq \varphi_{1}( \mathfrak{d_{2}})\varTheta_{2}(\mathfrak{d}_{1})+ \mathcal {P}(\mathfrak{d}_{1})\mathcal{S}(\mathfrak{d_{2}}). \end{aligned}$$
(2.8)

Taking product on both sides of (2.8) by \(\frac{ (\varPsi(\varrho)-\varPsi(\mathfrak{d}_{1}) )^{\frac{\rho }{\mathcal{K}}-1}\varPsi^{\prime}(\mathfrak{d}_{1})}{\mathcal {K}\varGamma_{\mathcal{K}}(\rho)}\) and integrating the resulting identity for the variable \(\mathfrak{d}_{1}\) on \((0,\varrho)\) give

$$\begin{aligned}& \mathcal{S}(\mathfrak{d_{2}})\frac{1}{K\varGamma_{k}(\rho)} \int _{0}^{\varrho} \bigl(\varPsi(\varrho)-\varPsi( \mathfrak{d}_{1}) \bigr)^{\frac{\rho}{\mathcal{K}}-1} \varPsi^{\prime}( \mathfrak{d}_{1})\varTheta_{2}(\mathfrak{d}_{1}) \, d\mathfrak{d}_{1} \\& \quad\quad{}+\varphi_{1}(\mathfrak{d_{2}}) \frac{1}{\mathcal{K}\varGamma_{\mathcal{K}}(\rho)} \int_{0}^{\varrho} \bigl(\varPsi(\varrho)-\varPsi( \mathfrak{d}_{1}) \bigr)^{\frac{\rho }{\mathcal{K}}-1}\varPsi^{\prime}( \mathfrak{d}_{1})\mathcal{P}(\mathfrak {d}_{1})\,d \mathfrak{d}_{1} \\& \quad\geq \varphi_{1}(\mathfrak{d_{2}}) \frac{1}{\mathcal{K}\varGamma_{\mathcal{K}}(\rho)} \int_{0}^{\varrho } \bigl(\varPsi(\varrho)-\varPsi( \mathfrak{d}_{1}) \bigr)^{\frac{\rho }{\mathcal{K}}-1} \varPsi^{\prime}( \mathfrak{d}_{1})\varTheta_{2}(\mathfrak{d}_{1}) \, d\mathfrak{d}_{1} \\& \quad\quad{}+\mathcal{S}(\mathfrak{d_{2}}) \frac{1}{\mathcal{K}\varGamma_{\mathcal{K}}(\rho)} \int_{0}^{\varrho } \bigl(\varPsi(\varrho)-\varPsi( \mathfrak{d}_{1}) \bigr)^{\frac{\rho }{\mathcal{K}}-1} \varPsi^{\prime}( \mathfrak{d}_{1})\mathcal{P}(\mathfrak{d}_{1})\, d \mathfrak{d}_{1}, \end{aligned}$$

which can be rewritten as

$$\begin{aligned}& \mathcal{S}(\mathfrak{d_{2}})\mathfrak{J}_{\varPsi,0^{+}}^{\rho ,\mathcal{K}} \varTheta_{2}(\varrho) +\varphi_{1}(\mathfrak{d_{2}}) \mathfrak{J}_{\varPsi,0^{+}}^{\rho ,\mathcal{K}} \mathcal{P}(\varrho) \\& \quad\geq\varphi_{1}(\mathfrak{d_{2}})\mathfrak{J}_{\varPsi,0^{+}}^{\rho ,\mathcal{K}} \varTheta_{2}(\varrho) +\mathcal{S}(\mathfrak{d_{2}}) \mathfrak{J}_{\varPsi,0^{+}}^{\rho ,\mathcal{K}}\mathcal{P}(\varrho). \end{aligned}$$
(2.9)

Again, taking product on both sides of (2.9) by \(\frac{ (\varPsi(\varrho)-\varPsi(\mathfrak{d_{2}}) )^{\frac {\delta}{\mathcal{K}}-1}\varPsi^{\prime}(\mathfrak{d_{2}})}{\mathcal {K}\varGamma_{\mathcal{K}}(\delta)}\) and integrating the resulting identity for the variable \(\mathfrak{d_{2}}\) on \((0,\varrho)\), we obtain

$$\begin{aligned}& \mathfrak{J}_{\varPsi,0^{+}}^{\delta,\mathcal{K}}\varphi_{1}(\varrho )\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\mathcal{P}(\varrho) + \mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\varTheta_{2}(\varrho ) \mathfrak{J}_{\varPsi,0^{+}}^{\delta,\mathcal{K}}\mathcal{S}(\varrho) \\& \quad \geq \mathfrak{J}_{\varPsi,0^{+}}^{\delta,\mathcal{K}}\varphi_{1}( \varrho )\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\varTheta_{2}( \varrho )+\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\mathcal{P}(\varrho) \mathfrak{J}_{\varPsi,0^{+}}^{\delta,\mathcal{K}}\mathcal{S}(\varrho). \end{aligned}$$

This proves part (a). To prove parts (b)–(d), we consider the subsequent inequalities:

$$\begin{aligned}& \text{(b)}\quad \bigl(\varphi_{2}(\mathfrak{d}_{1})- \mathcal{S}(\mathfrak{d}_{1}) \bigr) \bigl(\mathcal{P}( \mathfrak{d_{2}})-\varTheta_{1}(\mathfrak{d_{2}}) \bigr)\geq0, \\& \text{(c)}\quad \bigl(\varTheta_{2}(\mathfrak{d}_{1})- \mathcal{P}(\mathfrak{d}_{1}) \bigr) \bigl(\mathcal{S}( \mathfrak{d_{2}})-\varphi_{2}(\mathfrak{d_{2}}) \bigr)\leq0, \\& \text{(d)}\quad \bigl(\varTheta_{1}(\mathfrak{d}_{1})- \mathcal{P}(\mathfrak{d}_{1}) \bigr) \bigl(\mathcal{S}( \mathfrak{d_{2}})-\varphi_{1}(\mathfrak{d_{2}}) \bigr)\leq0. \end{aligned}$$

We use similar arguments as those in the proof of part (a) to get the rest of the inequalities. □

The following inequalities are special cases of Theorem 2.5.

Corollary 2.6

Let\(\mathcal{K}>0\), \(\rho, \delta>0\), and\(\mathcal{P}\)and\(\mathcal{S}\)be two positive functions on\([0,\infty)\). Suppose that there exist real constantsm, \(\mathcal{M}\), n, \(\mathcal{N}\)such that

$$ m\leq\mathcal{P}(\varrho)\leq\mathcal{M},\qquad n\leq\mathcal{S}(\varrho)\leq \mathcal{N} $$

for all\(\varrho\in[0,\infty)\). Then we obtain

$$\begin{aligned}& \textup{(i)}\quad n\frac{\varPsi^{\frac{\delta}{\mathcal{K}}}(\varrho)}{\varGamma _{\mathcal{K}}(\delta+\mathcal{K})}\mathfrak{J}_{\varPsi,0^{+}}^{\rho ,\mathcal{K}} \mathcal{P}(\varrho)+ \mathcal{M}\frac{\varPsi^{\frac{\rho}{\mathcal{K}}}(\varrho)}{\varGamma _{\mathcal{K}}(\rho+\mathcal{K})}\mathfrak{J}_{\varPsi,0^{+}}^{\delta ,\mathcal{K}} \mathcal{S}(\varrho) \\& \phantom{\textup{(i)}\quad}\quad\geq n\mathcal{M}\frac{\varPsi^{\frac{\delta}{\mathcal{K}}}(\varrho )}{\varGamma_{\mathcal{K}}(\delta+\mathcal{K})}\frac{\varPsi^{\frac{\rho }{\mathcal{K}}}(\varrho)}{ \varGamma_{\mathcal{K}}(\rho+\mathcal{K})}+ \mathfrak{J}_{\varPsi ,0^{+}}^{\rho,\mathcal{K}}\mathcal{P}(\varrho) \mathfrak{J}_{\varPsi ,0^{+}}^{\delta,\mathcal{K}}\mathcal{S}(\varrho), \\& \textup{(ii)}\quad m\frac{\varPsi^{\frac{\delta}{\mathcal{K}}}(\varrho)}{\varGamma _{\mathcal{K}}(\delta+\mathcal{K})}\mathfrak{J}_{\varPsi,0^{+}}^{\rho ,\mathcal{K}} \mathcal{S}(\varrho)+ \mathcal{N}\frac{\varPsi^{\frac{\rho}{\mathcal{K}}}(\varrho)}{\varGamma _{\mathcal{K}}(\rho+\mathcal{K})}\mathfrak{J}_{\varPsi,0^{+}}^{\delta ,\mathcal{K}} \mathcal{P}(\varrho) \\& \phantom{\textup{(ii)}\quad}\quad\geq m\mathcal{N}\frac{\varPsi{\frac{\delta}{\mathcal{K}}}(\varrho )}{\varGamma_{\mathcal{K}}(\delta+\mathcal{K})}\frac{\varPsi^{\frac{\rho }{\mathcal{K}}}(\varrho)}{ \varGamma_{\mathcal{K}}(\rho+\mathcal{K})}+ \mathfrak{J}_{\varPsi ,0^{+}}^{\delta,\mathcal{K}}\mathcal{P}(\varrho) \mathfrak{J}_{\varPsi ,0^{+}}^{\rho,\mathcal{K}}\mathcal{S}(\varrho), \\& \textup{(iii)}\quad \mathcal{N}\mathcal{M}\frac{\varPsi^{\frac{\delta}{\mathcal {K}}}(\varrho)}{\varGamma_{\mathcal{K}}(\delta+\mathcal{K})}\frac{\varPsi ^{\frac{\rho}{\mathcal{K}}}(\varrho)}{ \varGamma_{\mathcal{K}}(\rho+\mathcal{K})}+ \mathfrak{J}_{\varPsi ,0^{+}}^{\rho,\mathcal{K}}\mathcal{P}(\varrho) \mathfrak{J}_{\varPsi,0^{+}}^{\delta,\mathcal{K}}\mathcal{S}(\varrho ) \\& \phantom{\textup{(iii)}\quad}\quad\geq \mathcal{M}\frac{\varPsi^{\frac{\rho}{\mathcal{K}}}(\varrho)}{\varGamma (\varGamma_{\mathcal{K}}(\rho+\mathcal{K})} \mathfrak{J}_{\varPsi,0^{+}}^{\delta,\mathcal{K}} \mathcal{S}(\varrho)+ \mathcal{N}\frac{\varPsi{\frac{\delta}{\mathcal{K}}}(\varrho )}{\varGamma_{\mathcal{K}}(\delta+\mathcal{K})}\mathfrak{J}_{\varPsi ,0^{+}}^{\rho,\mathcal{K}} \mathcal{P}(\varrho) , \\& \textup{(iv)}\quad nm\frac{\varPsi{\frac{\delta}{\mathcal{K}}}(\varrho)}{\varGamma _{\mathcal{K}}(\delta+\mathcal{K})} \frac{\varPsi^{\frac{\rho}{\mathcal{K}}}(\varrho)}{\varGamma_{\mathcal{K}}(\rho+\mathcal{K})} +\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}} \mathcal{P}(\varrho) \mathfrak{J}_{\varPsi,0^{+}}^{\delta,\mathcal{K}} \mathcal{S}( \varrho) \\& \phantom{\textup{(iv)}\quad}\quad\geq m\frac{\varPsi^{\frac{\rho}{\mathcal{K}}}(\varrho)}{\varGamma_{\mathcal{K}}(\rho+\mathcal{K})} \mathfrak{J}_{\varPsi,0^{+}}^{\delta,\mathcal{K}} \mathcal{S}(\varrho)+ n\frac{\varPsi^{\frac{\delta}{\mathcal{K}}}(\varrho)}{\varGamma _{\mathcal{K}}(\delta+\mathcal{K})}\mathfrak{J}_{\varPsi,0^{+}}^{\rho ,\mathcal{K}} \mathcal{P}(\varrho). \end{aligned}$$
(2.10)

Corollary 2.7

Let\(\mathcal{K}>0\), \(\rho, \delta>0\), \(\mathcal{P},\mathcal{S}\in L_{1}[0,\infty]\)and\(\varPsi(\varrho)=\varrho\). Suppose that there exist real constantsm, \(\mathcal{M}\), n, \(\mathcal{N}\)such that

$$ m\leq\mathcal{P}(\varrho)\leq\mathcal{M},\qquad n\leq\mathcal{S}(\varrho) \leq \mathcal{N} $$

for all\(\varrho\in[0,\infty)\), then one has

$$\begin{aligned}& \textup{(a)}\quad\mathfrak{J}^{\delta,\mathcal{K}}\varphi_{1}(\varrho ) \mathfrak{J}^{\rho,\mathcal{K}}\mathcal{P}(\varrho) +\mathfrak{J}^{\rho,\mathcal{K}} \varTheta_{2}(\varrho)\mathfrak {J}^{\delta,\mathcal{K}}\mathcal{S}( \varrho) \\& \phantom{\textup{(a)}\quad}\quad\geq \mathfrak{J}^{\delta,\mathcal{K}}\varphi_{1}(\varrho) \mathfrak{J}^{\rho ,\mathcal{K}}\varTheta_{2}(\varrho) + \mathfrak{J}^{\rho,\mathcal{K}}\mathcal{P}(\varrho) \mathfrak{J}^{\delta,\mathcal{K}} \mathcal{S}(\varrho), \\& \textup{(b)}\quad\mathfrak{J}^{\delta,\mathcal{K}}\varTheta_{1}(\varrho ) \mathfrak{J}^{\rho,\mathcal{K}}\mathcal{S}(\varrho) +\mathfrak{J}^{\rho,\mathcal{K}} \varphi_{2}(\varrho)\mathfrak {J}^{\delta,\mathcal{K}}\mathcal{P}(\varrho) \\& \phantom{\textup{(a)}\quad}\quad\geq \mathfrak{J}^{\delta,\mathcal{K}}\varTheta_{1}(\varrho) \mathfrak {J}^{\rho,\mathcal{K}}\varphi_{2}(\varrho) + \mathfrak{J}^{\delta,\mathcal{K}}\mathcal{P}(\varrho) \mathfrak{J}^{\rho,\mathcal{K}} \mathcal{S}(\varrho), \\& \textup{(c)}\quad\mathfrak{J}^{\rho,\mathcal{K}}\varTheta_{2}(\varrho) \mathfrak {J}^{\delta,\mathcal{K}}\varphi_{2}(\varrho) + \mathfrak{J}^{\rho,\mathcal{K}}\mathcal{P}(\varrho)\mathfrak {J}^{\delta,\mathcal{K}} \mathcal{S}(\varrho) \\& \phantom{\textup{(a)}\quad}\quad\geq \mathfrak{J}^{\rho,\mathcal{K}}\varTheta_{2}(\varrho) \mathfrak {J}^{\delta,\mathcal{K}} \mathcal{S}(\varrho)+\mathfrak{J}^{\delta,\mathcal{K}} \varphi _{2}(\varrho) \mathfrak{J}^{\rho,\mathcal{K}}\mathcal{P}( \varrho), \\& \textup{(d)}\quad\mathfrak{J}^{\rho,\mathcal{K}}\varTheta_{1}(\varrho) \mathfrak {J}^{\delta,\mathcal{K}}\varphi_{1}(\varrho) + \mathfrak{J}^{\rho,\mathcal{K}}\mathcal{P}(\varrho)\mathfrak {J}^{\delta,\mathcal{K}} \mathcal{S}(\varrho) \\& \phantom{\textup{(a)}\quad}\quad\geq \mathfrak{J}^{\rho,\mathcal{K}}\varTheta_{1}(\varrho) \mathfrak {J}^{\delta,\mathcal{K}} \mathcal{S}(\varrho)+\mathfrak{J}^{\delta,\mathcal{K}} \varphi _{1}(\varrho) \mathfrak{J}^{\rho,\mathcal{K}}\mathcal{P}( \varrho). \end{aligned}$$

Remark 2.8

From Theorem 2.5, and Corollaries 2.6 and 2.7 we get four conclusions as follows:

  1. (1)

    If \(\mathcal{K}=1\), then Theorem 2.5 leads to Theorem 2.15 in [68].

  2. (2)

    If \(\mathcal{K}=1\) and \(\varPsi(\varrho)=\varrho\), then Theorem 2.5 gives Theorem 5 in [69].

  3. (3)

    If \(\mathcal{K}=1\), then Corollary 2.6 becomes Corollary 2.16 in [68].

  4. (4)

    If \(\mathcal{K}=1\) and \(\varPsi(\varrho)=\varrho\), then Corollary 6 in [69] can be derived from Corollary 2.7.

Lemma 2.9

Let\(\mathcal{K}>0\), \(\rho,\delta>0\), \(\mathcal{P}\)be a positive function on\([0,\infty)\), \(\varTheta_{1}\), \(\varTheta_{2}\)be two integrable functions on\([0,\infty)\), andΨbe an increasing and positive function on\([0,\infty)\)such that\({\varPsi(0)=0}\)and\(\varPsi^{\prime}\)is continuous on\([0,\infty)\). If (2.1) holds, then

$$\begin{aligned}& \begin{aligned}[b] &\frac{\varPsi^{\frac{\rho}{\mathcal{K}}}(\varrho)}{\varGamma _{\mathcal{K}}(\rho+\mathcal{K})}\mathfrak{J}_{\varPsi,0^{+}}^{\rho ,\mathcal{K}} \mathcal{P}^{2}(\varrho)- \bigl(\mathfrak{J}_{\varPsi,0^{+}}^{\rho ,\mathcal{K}} \mathcal{P}(\varrho) \bigr)^{2} \\ &\quad= \bigl(\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal {K}}\varTheta_{2}( \varrho)-\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal {K}}\mathcal{P}(\varrho) \bigr) \bigl(\mathfrak{J}_{\varPsi,0^{+}}^{\rho ,\mathcal{K}}\mathcal{P}(\varrho) - \mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\varTheta_{1}(\varrho ) \bigr) \\ &\quad\quad -\frac{\varPsi^{\frac{\rho}{\mathcal{K}}}(\varrho)}{\varGamma_{\mathcal {K}}(\rho+\mathcal{K})} \bigl(\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal {K}} \varTheta_{2}(\varrho) -\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}} \mathcal{P}(\varrho ) \bigr) \bigl(\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}} \mathcal {P}(\varrho) -\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}} \varTheta_{1}(\varrho ) \bigr) \\ &\quad\quad+ \frac{\varPsi^{\frac{\rho}{\mathcal{K}}}(\varrho)}{\varGamma_{\mathcal {K}}(\rho+\mathcal{K})}\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal {K}} \bigl(\varTheta_{1}(\varrho)\mathcal{P}(\varrho) \bigr) - \mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\varTheta_{1}(\varrho ) \mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\mathcal{P}(\varrho ) \\ &\quad\quad +\frac{\varPsi^{\frac{\rho}{\mathcal{K}}}(\varrho)}{\varGamma_{\mathcal {K}}(\rho+\mathcal{K})}\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal {K}} \bigl(\varTheta_{2}(\varrho)\mathcal{P}(\varrho) \bigr) - \mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\varTheta_{2}(\varrho ) \mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\mathcal{P}(\varrho ) \\ &\quad\quad+ \mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}} \bigl(\varTheta _{1}(\varrho) \bigr)\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}} \bigl(\varTheta_{2}(\varrho) \bigr)-\frac{\varPsi^{\frac{\rho}{\mathcal {K}}}(\varrho)}{ \varGamma_{\mathcal{K}}(\rho+\mathcal{K})} \mathfrak{J}_{\varPsi ,0^{+}}^{\rho,\mathcal{K}} \bigl(\varTheta_{1}( \varrho)\varTheta_{2}(\varrho) \bigr).\end{aligned} \end{aligned}$$
(2.11)

Proof

Since \(\mathfrak{d}_{1}, \mathfrak{d_{2}}>0\), we have

$$\begin{aligned} & \bigl(\varTheta_{2}(\mathfrak{d_{2}})-\mathcal{P}( \mathfrak {d_{2}}) \bigr) \bigl(\mathcal{P}(\mathfrak{d}_{1})- \varTheta_{1}(\mathfrak {d}_{1}) \bigr)+ \bigl( \varTheta_{2}(\mathfrak{d}_{1})-\mathcal{P}( \mathfrak{d}_{1}) \bigr) \bigl(\mathcal{P}(\mathfrak{d_{2}})- \varTheta_{1}(\mathfrak{d_{2}}) \bigr) \\ &\qquad{}- \bigl(\varTheta_{2}(\mathfrak{d}_{1})-\mathcal{P}( \mathfrak{d}_{1}) \bigr) \bigl(\mathcal{P}(\mathfrak{d}_{1})- \varTheta_{1}(\mathfrak{d}_{1}) \bigr)- \bigl( \varTheta_{2}(\mathfrak{d_{2}})-\mathcal{P}( \mathfrak{d_{2}}) \bigr) \bigl(\mathcal{P}(\mathfrak{d_{2}})- \varTheta_{1}(\mathfrak{d_{2}}) \bigr) \\ &\quad= \mathcal{P}^{2}(\mathfrak{d}_{1})+ \mathcal{P}^{2}(\mathfrak {d_{2}})-2\mathcal{P}( \mathfrak{d}_{1})\mathcal{P}(\mathfrak {d_{2}})+ \varTheta_{2}(\mathfrak{d_{2}}) \mathcal{P}( \mathfrak{d}_{1})+\varTheta_{1}(\mathfrak{d}_{1}) \mathcal{P}(\mathfrak{d_{2}}) \\ &\qquad{} -\varTheta_{1}(\mathfrak{d}_{1}) \varTheta_{2}(\mathfrak {d_{2}})+\varTheta_{2}( \mathfrak{d}_{1})\mathcal{P}(\mathfrak {d_{2}})+ \varTheta_{1}(\mathfrak{d_{2}}) \mathcal{P}( \mathfrak{d}_{1})-\varTheta_{1}(\mathfrak{d_{2}}) \varTheta _{2}(\mathfrak{d}_{1}) \\ &\qquad {}-\varTheta_{2}(\mathfrak{d}_{1})\mathcal{P}( \mathfrak{d}_{1})+\varTheta _{1}(\mathfrak{d}_{1}) \varTheta_{2}(\mathfrak{d}_{1})-\varTheta _{1}(\mathfrak{d}_{1}) \mathcal{P}( \mathfrak{d}_{1})-\varTheta_{2}(\mathfrak{d_{2}}) \mathcal {P}(\mathfrak{d_{2}}) \\ &\qquad {}+\varTheta_{1}(\mathfrak{d_{2}}) \varTheta_{2}(\mathfrak {d_{2}})-\varTheta_{1}( \mathfrak{d_{2}})\mathcal{P}(\mathfrak{d_{2}}). \end{aligned}$$
(2.12)

Taking product on both sides of (2.12) by \(\frac{1}{\mathcal{K}\varGamma_{\mathcal{K}}(\rho)} (\varPsi(\varrho )-\varPsi(\mathfrak{d}_{1}) )^{\frac{\rho}{\mathcal{K}}-1}\varPsi ^{\prime}(\mathfrak{d}_{1})\) and integrating the resulting identity for the variable \(\mathfrak{d}_{1}\) on \((0,\varrho)\), we get

$$\begin{aligned} & \bigl(\varTheta_{2}(\mathfrak{d_{2}})-\mathcal{P}( \mathfrak {d_{2}}) \bigr) \bigl(\mathfrak{J}_{\varPsi,0^{+}}^{\rho, \mathcal{K}} \mathcal{P}(\varrho)-\mathfrak{J}_{\varPsi,0^{+}}^{\rho ,\mathcal{K}} \varTheta_{1}(\varrho) \bigr) \\ &\qquad{}+ \bigl(\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}} \varTheta _{2}(\varrho)-\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}} \mathcal {P}(\varrho) \bigr) \bigl(\mathcal{P}(\mathfrak{d_{2}})- \varTheta _{1}(\mathfrak{d_{2}}) \bigr) \\ &\qquad{}-\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}} \bigl( \bigl( \varTheta_{2}(\varrho)-\mathcal{P}(\varrho) \bigr) \bigl(\mathcal {P}( \varrho)-\varTheta_{1}(\varrho) \bigr) \bigr) \\ &\qquad{}- \bigl( \varTheta_{2}(\mathfrak{d_{2}})-\mathcal{P}( \mathfrak{d_{2}}) \bigr) \bigl(\mathcal{P}(\mathfrak{d_{2}})- \varTheta_{1}(\mathfrak{d_{2}}) \bigr) \frac{\varPsi^{\frac{\rho}{\mathcal{K}}}(\varrho)}{ \varGamma_{\mathcal{K}}(\rho+\mathcal{K})} \\ &\quad= \mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\mathcal{P}^{2}( \varrho )+\frac{\varPsi^{\frac{\rho}{\mathcal{K}}}(\varrho)}{\varGamma_{\mathcal {K}}(\varrho+\mathcal{K})}\mathcal{P}^{2}(\mathfrak{d_{2}}) \\ &\qquad{}-2 \mathcal {P}(\mathfrak{d_{2}})\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal {K}} \mathcal{P}(\varrho) +\varTheta_{2}(\mathfrak{d_{2}}) \mathfrak{J}_{\varPsi,0^{+}}^{\rho ,\mathcal{K}}\mathcal{P}(\varrho)+\mathcal{P}( \mathfrak{d_{2}})\mathfrak {J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}} \varTheta_{1}(\varrho) \\ &\qquad{} -\varTheta_{2}(\mathfrak{d_{2}}) \mathfrak{J}_{\varPsi,0^{+}}^{\rho ,\mathcal{K}}\varTheta_{1}(\varrho)+ \mathcal{P}(\mathfrak {d_{2}})\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}} \varTheta _{2}(\varrho) \\ &\qquad{}+\varTheta_{1}( \mathfrak{d_{2}}) \mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}} \mathcal{P}(\varrho )-\varTheta_{1}(\mathfrak{d_{2}}) \mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\varTheta_{2}(\varrho ) \\ &\quad\quad{} -\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}} \bigl(\varTheta _{2}(\varrho)\mathcal{P}(\varrho) \bigr)+\mathfrak{J}_{\varPsi ,0^{+}}^{\rho,\mathcal{K}} \bigl(\varTheta_{1}(\varrho)\varTheta _{2}(\varrho) \bigr) -\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}} \bigl(\varTheta _{1}(\varrho)\mathcal{P}(\varrho) \bigr) \\ &\qquad{}-\varTheta_{2}( \mathfrak {d_{2}})\mathcal{P}(\mathfrak{d_{2}}) \frac{\varPsi^{\frac{\rho}{\mathcal{K}}}(\varrho)}{\varGamma_{\mathcal {K}}(\rho+\mathcal{K})} +\varTheta_{1}(\mathfrak{d_{2}}) \varTheta_{2}(\mathfrak{d_{2}})\frac {\varPsi^{\frac{\rho}{\mathcal{K}}}(\varrho)}{ \varGamma_{\mathcal{K}}(\rho+\mathcal{K})} \\ &\qquad{}- \varTheta_{1}(\mathfrak {d_{2}})\mathcal{P}( \mathfrak{d_{2}})\frac{\varPsi^{\frac{\rho}{\mathcal {K}}}(\varrho)}{\varGamma_{\mathcal{K}}(\rho+\mathcal{K})}. \end{aligned}$$
(2.13)

Taking product on both sides of (2.13) by \(\frac{1}{\mathcal{K}\varGamma_{\mathcal{K}}(\rho)} (\varPsi(\varrho )-\varPsi(\mathfrak{d_{2}}) )^{\frac{\rho}{\mathcal{K}}-1}\varPsi ^{\prime}(\mathfrak{d_{2}})\) and integrating the resulting identity for the variable \(\mathfrak{d_{2}}\) on \((0,\varrho)\), we get

$$\begin{aligned} & \bigl(\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\varTheta _{2}(\varrho)-\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\mathcal {P}(\varrho) \bigr) \bigl(\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}} \mathcal{P}(\varrho )-\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}} \varTheta_{1}(\varrho ) \bigr) \\ &\qquad{}+ \bigl(\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}} \varTheta _{2}(\varrho)-\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}} \mathcal {P}(\varrho) \bigr) \bigl(\mathfrak{J}_{\varPsi,0^{+}}^{\rho ,\mathcal{K}} \mathcal{P}( \varrho)-\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}} \varTheta_{1}( \varrho) \bigr) \\ &\qquad{}-\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}} \bigl(\varTheta _{2}(\varrho)-\mathcal{P}(\varrho) \bigr) \bigl(\mathcal{P}(\varrho )-\varTheta_{1}(\varrho) \bigr) \frac{\varPsi^{\frac{\rho}{\mathcal{K}}}}{\varGamma_{\mathcal {K}}(\varrho+\mathcal{K})} \\ &\qquad{}- \bigl(\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}(\varrho)}\varTheta _{2}(\varrho)-\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\mathcal {P}(\varrho) \bigr) \bigl( \mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}} \mathcal{P}(\varrho )-\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}} \varTheta_{1}(\varrho ) \bigr)\frac{\varPsi^{\frac{\rho}{\mathcal{K}}(\varrho)}}{\varGamma _{\mathcal{K}}(\rho+\mathcal{K})} \\ &\quad= \mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\mathcal{P}^{2}( \varrho )\frac{\varPsi^{\frac{\rho}{\mathcal{K}}}(\varrho)}{\varGamma_{\mathcal {K}}(\rho+\mathcal{K})}+ \frac{\varPsi^{\frac{\rho}{\mathcal{K}}}(\varrho)}{\varGamma_{\mathcal {K}}(\rho+\mathcal{K})}\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal {K}} \mathcal{P}^{2}(\mathfrak{d_{2}})- 2\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}} \mathcal{P}(\varrho )\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\mathcal{P}( \varrho) \\ &\qquad{}+\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal {K}}\varTheta_{2}( \varrho)\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal {K}}\mathcal{P}(\varrho)+ \mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\mathcal{P}(\varrho ) \mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\varTheta_{1}(\varrho) - \mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\varTheta_{2}(\varrho ) \mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\varTheta_{1}(\varrho) \\ &\qquad{}+\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal {K}}\mathcal{P}(\varrho) \mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal {K}}\varTheta_{2}(\varrho)+ \mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\varTheta_{1}(\varrho ) \mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\mathcal{P}(\varrho )- \mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\varTheta_{1}(\varrho) \mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\varTheta_{2}(\varrho) \\ &\qquad{}-\frac{\varPsi^{\frac{\rho}{\mathcal{K}}}(\varrho )}{\varGamma_{\mathcal{K}}(\rho+\mathcal{K})}\mathfrak{J}_{\varPsi ,0^{+}}^{\rho,\mathcal{K}} \bigl( \varTheta_{2}(\varrho)\mathcal{P}(\varrho ) \bigr)+ \frac{\varPsi^{\frac{\rho}{\mathcal{K}}}(\varrho)}{\varGamma_{\mathcal {K}}(\rho+\mathcal{K})}\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal {K}} \bigl( \varTheta_{1}(\varrho)\varTheta_{2}(\varrho) \bigr) \\ &\qquad{}-\frac{\varPsi^{\frac{\rho}{\mathcal{K}}}(\varrho )}{\varGamma_{\mathcal{K}}(\rho+\mathcal{K})}\mathfrak{J}^{\rho,\mathcal {K}}_{\varPsi} \bigl(\varTheta_{1}(\varrho)\mathcal{P}(\varrho) \bigr) - \frac{\varPsi^{\frac{\rho}{\mathcal{K}}}(\varrho)}{\varGamma_{\mathcal {K}}(\rho+\mathcal{K})} \mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}} \bigl( \varTheta _{2}(\varrho)\mathcal{P}(\varrho) \bigr) \\ &\qquad{} +\frac{\varPsi^{\frac{\rho}{\mathcal{K}}}(\varrho)}{\varGamma_{\mathcal {K}}(\rho+\mathcal{K})}\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal {K}} \bigl(\varTheta_{1}(\varrho)\varTheta_{2}(\varrho) \bigr)-\frac{\varPsi ^{\frac{\rho}{\mathcal{K}}}(\varrho)}{ \varGamma_{\mathcal{K}}(\rho+\mathcal{K})}\mathfrak{J}_{\varPsi ,0^{+}}^{\rho,\mathcal{K}} \bigl( \varTheta_{1}(\varrho)\mathcal{P}(\varrho) \bigr). \end{aligned}$$
(2.14)

This is the proof of Lemma 2.9. □

Corollary 2.10

Let\(\varPsi(\varrho)=\varrho\)in Lemma 2.9, then we have the\(\mathcal{K}\)-fractional integral inequality

$$\begin{aligned} &\frac{\varrho^{\frac{\rho}{\mathcal{K}}}}{\varGamma_{\mathcal {K}}(\rho+\mathcal{K})} \mathfrak{J}^{\rho,\mathcal{K}}\mathcal{P}^{2}( \varrho)- \bigl(\mathfrak {J}^{\rho,\mathcal{K}}\mathcal{P}(\varrho) \bigr)^{2} \\ &\quad= \bigl(\mathfrak{J}^{\rho,\mathcal{K}}\varTheta_{2}(\varrho )- \mathfrak{J}^{\rho,\mathcal{K}}\mathcal{P}(\varrho) \bigr) \bigl(\mathfrak {J}^{\rho,\mathcal{K}}\mathcal{P}(\varrho) -\mathfrak{J}^{\rho,\mathcal{K}} \varTheta_{1}(\varrho) \bigr) \\ &\qquad{} -\frac{\varrho^{\frac{\rho}{\mathcal{K}}}}{\varGamma_{\mathcal{K}}(\rho +\mathcal{K})} \bigl(\mathfrak{J}^{\rho,\mathcal{K}} \varTheta_{2}(\varrho) -\mathfrak{J}^{\rho,\mathcal{K}}\mathcal{P}( \varrho) \bigr) \bigl(\mathfrak {J}^{\rho,\mathcal{K}}\mathcal{P}(\varrho) - \mathfrak{J}^{\rho,\mathcal{K}}\varTheta_{1}(\varrho) \bigr) \\ &\qquad{}+ \frac{\varrho^{\frac{\rho}{\mathcal{K}}}}{\varGamma_{\mathcal{K}}(\rho +\mathcal{K})} (\mathfrak{J}^{\rho,\mathcal{K}} \bigl(\varTheta _{1}(\varrho)\mathcal{P}(\varrho) \bigr) -\mathfrak{J}^{\rho,\mathcal{K}} \varTheta_{1}(\varrho)\mathfrak {J}^{\rho,\mathcal{K}}\mathcal{P}( \varrho) \\ &\qquad {}+ \frac{\varrho^{\frac{\rho}{\mathcal{K}}}}{\varGamma_{\mathcal{K}}(\rho +\mathcal{K})}\mathfrak{J}^{\rho,\mathcal{K}} \bigl( \varTheta_{2}(\varrho )\mathcal{P}(\varrho) \bigr) - \mathfrak{J}^{\rho,\mathcal{K}}\varTheta_{2}(\varrho)\mathfrak {J}^{\rho,\mathcal{K}}\mathcal{P}(\varrho) \\ &\qquad{}+ \mathfrak{J}^{\rho,\mathcal{K}} \bigl(\varTheta_{1}(\varrho) \bigr)\mathfrak{J}^{\rho,\mathcal{K}} \bigl(\varTheta_{2}(\varrho) \bigr)-\frac {\varrho^{\frac{\rho}{\mathcal{K}}}}{\varGamma_{\mathcal{K}}(\rho+\mathcal {K})}\mathfrak{J}^{\rho,\mathcal{K}} \bigl( \varTheta_{1}(\varrho)\varTheta_{2}(\varrho) \bigr). \end{aligned}$$
(2.15)

Corollary 2.11

Let\(m<\mathcal{M}\), \(\mathcal{K}, \rho,\delta>0\), \(\mathcal{P}\)be a positive function on\([0,\infty)\)such that\(m\leq\mathcal{P}(\varrho)\leq\mathcal{M}\), andΨbe a positive and increasing function on\([0,\infty)\)such that\(\varPsi(0)=0\)and\(\varPsi^{\prime}\)is continuous on\([0,\infty)\). Then

$$\begin{aligned} \begin{aligned}[b] &\frac{\varPsi^{\frac{\rho}{\mathcal{K}}}(\varrho)}{\varGamma _{\mathcal{K}}(\rho+\mathcal{K})}\mathfrak{J}_{\varPsi,0^{+}}^{\rho ,\mathcal{K}} \mathcal{P}^{2}(\varrho)- \bigl(\mathfrak{J}_{\varPsi,0^{+}}^{\rho ,\mathcal{K}} \mathcal{P}(\varrho) \bigr)^{2} \\ &\quad= \biggl(\mathcal{M}\frac{\varPsi^{\frac{\rho}{\mathcal {K}}}(\varrho)}{\varGamma_{\mathcal{K}}(\rho+\mathcal{K})}-\mathfrak {J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}} \mathcal{P}(\varrho) \biggr) \biggl( \mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}} \mathcal{P}(\varrho )-m\frac{\varPsi^{\frac{\rho}{\mathcal{K}}}(\varrho)}{\varGamma _{\mathcal{K}}(\rho+\mathcal{K})} \biggr) \\ &\quad\quad-\frac{\varPsi^{\frac{\rho}{\mathcal {K}}}(\varrho)}{\varGamma_{\mathcal{K}}(\rho+\mathcal{K})} \mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}} \bigl(\mathcal {M}-\mathcal{P}(\varrho) \bigr) \bigl(\mathcal{P}(\varrho)-m\bigr) ).\end{aligned} \end{aligned}$$
(2.16)

Remark 2.12

From Lemma 2.9 and Corollary 2.11 we get three conclusions as follows.

  1. (1)

    If \(\mathcal{K}=1\), then Lemma 2.9 leads to Lemma 2.19 in [68].

  2. (2)

    If \(\mathcal{K}=1\) and \(\varPsi(\varrho)=\varrho\), then Lemma 2.9 becomes Lemma 7 in [69].

  3. (3)

    If \(\mathcal{K}=1\) and \(\varPsi(\varrho)=\varrho\), then Corollary 2.11 leads to Corollary 8 in [69].

Theorem 2.13

Let\(\mathcal{K}, \varrho>0\), \(\mathcal{P}\), \(\mathcal{S}\), \(\varTheta_{1}\), \(\varTheta_{2}\), \(\varphi_{1}\)and\(\varphi_{2}\)be six integrable functions defined on\([0,\infty)\), andΨbe an increasing and positive function on\([0,\infty)\)such that\(\varPsi^{\prime}(x)\)is continuous on\([0,\infty)\)and\(\varPsi(0)=0\). If conditions (2.1) and (2.6) are satisfied, then one has

$$\begin{aligned}[b] &\biggl\vert \frac{\varPsi^{{\frac{\rho}{\mathcal{K}}}}}{\varGamma _{\mathcal{K}}(\rho+\mathcal{K})}\mathfrak{J}_{\varPsi,0^{+}}^{\rho ,\mathcal{K}} \bigl(\mathcal{P}(\varrho)\mathcal{S}(\varrho) \bigr) -\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}} \mathcal{P}(\varrho )\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\mathcal{S}( \varrho) \biggr\vert \\ &\quad\leq\sqrt{\mathfrak{T}(\mathcal{P},\varTheta_{1},\varTheta _{2})\mathfrak{T}(\mathcal{S},\varphi_{1}, \varphi_{2})},\end{aligned} $$
(2.17)

where

$$\begin{aligned} \mathfrak{T}(\mathcal{P},\varTheta_{1},\varTheta_{2}) =& \bigl(\mathfrak {J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\varTheta_{2}( \varrho)-\mathfrak {J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\mathcal{P}(\varrho) \bigr) \bigl(\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\mathcal{P}(\varrho )- \mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\varTheta_{1}(\varrho ) \bigr) \\ &{}+ \frac{\varPsi^{{\frac{\rho}{\mathcal{K}}}}(\varrho)}{\varGamma_{\mathcal{K}}(\rho+\mathcal{K})}\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal {K}} \bigl(\varTheta_{1}(\varrho)\mathcal{P}(\varrho) \bigr) - \mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\varTheta_{1}(\varrho ) \mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\mathcal{P}(\varrho) \\ &{}+\frac{\varPsi^{{\frac{\rho}{\mathcal{K}}}}(\varrho )}{\varGamma_{\mathcal{K}}(\rho+\mathcal{K})}\mathfrak{J}_{\varPsi ,0^{+}}^{\rho,\mathcal{K}} \varTheta_{2}(\varrho)\mathcal{P}(\varrho) \\ &{} -\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}} \varTheta_{2}(\varrho )\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}} \mathcal{P}(\varrho)+ \mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}} \varTheta_{1}(\varrho )\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}} \varTheta_{2}(\varrho ) \\ &{}-\frac{\varPsi^{{\frac{\rho}{\mathcal{K}}}}(\varrho)}{ \varGamma_{\mathcal{K}}(\rho+\mathcal{K})}\mathfrak{J}_{\varPsi ,0^{+}}^{\rho,\mathcal{K}} \bigl( \varTheta_{1}(\varrho)\varTheta _{2}(\varrho) \bigr) \end{aligned}$$

and

$$\begin{aligned} \mathfrak{T}(\mathcal{S},\varphi_{1},\varphi_{2}) =& \bigl(\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\varphi _{2}( \varrho)-\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\mathcal {S}(\varrho) \bigr) \bigl(\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\mathcal{S}(\varrho )- \mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\varphi_{1}(\varrho ) \bigr) \\ &{}+ \frac{\varPsi^{{\frac{\rho}{\mathcal{K}}}}(\varrho)}{\varGamma_{\mathcal{K}}(\rho+\mathcal{K})}\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal {K}} \bigl(\varphi_{1}(\varrho)\mathcal{S}(\varrho) \bigr) - \mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\varphi_{1}(\varrho ) \mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\mathcal{S}(\varrho) \\ &{}+\frac{\varPsi^{{\frac{\rho}{\mathcal{K}}}}(\varrho )}{\varGamma_{\mathcal{K}}(\rho+\mathcal{K})} \mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}} \bigl(\varphi_{2}(\varrho )\mathcal{S}(\varrho) \bigr) \\ &{}- \mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\varphi_{2}(\varrho ) \mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\mathcal{S}(\varrho)+ \mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\varphi_{1}(\varrho ) \mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\varphi_{2}(\varrho ) \\ &{} -\frac{\varPsi^{{\frac{\rho}{\mathcal{K}}}}(\varrho)}{ \varGamma_{\mathcal{K}}(\rho+\mathcal{K})}\mathfrak{J}_{\varPsi ,0^{+}}^{\rho,\mathcal{K}} \bigl(\varphi_{1}(\varrho)\varphi_{2}(\varrho ) \bigr). \end{aligned}$$

Proof

Let \(\varrho>0\), \(\mathfrak{d}_{1},\mathfrak{d_{2}}\in(0,\varrho)\), \(\mathcal{P}\) and \(\mathcal{S}\) be two positive functions defined on \([0,\infty)\) such that (2.1) and (2.6) are satisfied, and \(\mathfrak{T}(\mathfrak{d}_{1},\mathfrak{d_{2}})\) be defined by

$$\begin{aligned} \mathfrak{T}(\mathfrak{d}_{1},\mathfrak{d_{2}})= \bigl( \mathcal {P}(\mathfrak{d}_{1})-\mathcal{P}(\mathfrak{d_{2}}) \bigr) \bigl(\mathcal{S}(\mathfrak{d}_{1})-\mathcal{S}( \mathfrak{d_{2}}) \bigr). \end{aligned}$$
(2.18)

Taking product on both sides of (2.18) by \(\frac{ (\varPsi(\varrho)-\varPsi(\mathfrak{d}_{1}) )^{\frac{\rho }{\mathcal{K}}-1}\varPsi^{\prime} (\mathfrak{d}_{1}) (\varPsi(\varrho)-\varPsi(\mathfrak{d_{2}}) )^{\frac{\rho}{\mathcal{K}}-1}\varPsi^{\prime}(\mathfrak{d_{2}})}{ 2 (\mathcal{K}\varGamma_{\mathcal{K}}(\rho) )^{2}}\) and integrating the resulting identity for the variable \(\mathfrak{d}_{1}\) and \(\mathfrak{d_{2}}\) from 0 to ϱ, we obtain

$$\begin{aligned} &\frac{1}{2 (\mathcal{K}\varGamma_{\mathcal{K}}(\rho) )^{2}} \int_{0}^{\varrho} \int_{0}^{\varrho} \bigl(\varPsi(\varrho)-\varPsi( \mathfrak{d}_{1}) \bigr)^{{\frac{\rho }{\mathcal{K}}}-1} \bigl(\varPsi(\varrho)- \varPsi(\mathfrak{d_{2}}) \bigr) ^{{\frac{\rho}{\mathcal{K}}}-1}\mathfrak{T}( \mathfrak{d}_{1},\mathfrak{d_{2}}) \\ &\qquad{}\times\varPsi^{\prime}( \mathfrak{d}_{1})\varPsi^{\prime}(\mathfrak{d_{2}}) \, d\mathfrak{d}_{1}\,d\mathfrak{d_{2}} \\ &\quad=\frac{\psi^{{\frac{\rho}{\mathcal{K}}}}(\varrho )}{\varGamma_{\mathcal{K}}(\rho+\mathcal{K})}\mathfrak{J}_{\varPsi ,0^{+}}^{\rho,\mathcal{K}} \bigl( \mathcal{P}(\varrho)\mathcal{S}(\varrho ) \bigr) -\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}} \bigl(\mathcal{P}(\varrho )\bigr)\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}} \bigl(\mathcal{S}(\varrho)\bigr). \end{aligned}$$
(2.19)

Applying the Cauchy–Schwarz inequality to (2.19), we have

$$\begin{aligned} & \biggl(\frac{1}{2 (\mathcal{K}\varGamma_{\mathcal{K}}(\rho) )^{2}} \int_{0}^{\varrho} \int_{0}^{\varrho} \bigl(\varPsi(\varrho)-\varPsi( \mathfrak{d}_{1}) \bigr)^{{\frac{\rho }{\mathcal{K}}}-1} \bigl(\varPsi(\varrho)- \varPsi(\mathfrak{d_{2}}) \bigr)^{{\frac{\rho}{\mathcal{K}}}-1} \bigl(\mathcal{P}( \mathfrak{d}_{1}) -\mathcal{P}(\mathfrak{d_{2}}) \bigr) \\ &\qquad{}\times \bigl(\mathcal{S}(\mathfrak{d}_{1})-\mathcal{S}( \mathfrak{d_{2}}) \bigr)\varPsi ^{\prime}( \mathfrak{d}_{1})\varPsi^{\prime}(\mathfrak{d_{2}}) \, d\mathfrak{d}_{1}\,d\mathfrak{d_{2}} \biggr)^{2} \\ &\quad\leq\frac{1}{2 (\mathcal{K}\varGamma_{\mathcal {K}}(\rho) )^{2}} \int_{0}^{\varrho} \int_{0}^{\varrho} \bigl(\varPsi(\varrho)-\varPsi( \mathfrak{d}_{1}) \bigr)^{{\frac{\rho }{\mathcal{K}}}-1} \bigl(\varPsi(\varrho)- \varPsi(\mathfrak{d_{2}}) \bigr)^{{\frac{\rho}{\mathcal{K}}}-1} \bigl(\mathcal{P}( \mathfrak{d}_{1})-\mathcal{P}(\mathfrak{d_{2}}) \bigr)^{2} \\ &\qquad{}\times\varPsi^{\prime}(\mathfrak{d}_{1})\varPsi ^{\prime}(\mathfrak{d_{2}})\,d\mathfrak{d}_{1} \,d\mathfrak{d_{2}} \\ &\qquad{}\times\frac{1}{2 (\mathcal{K}\varGamma_{\mathcal {K}}(\rho) )^{2}} \int_{0}^{\varrho} \int_{0}^{\varrho} \bigl(\varPsi(\varrho)-\varPsi( \mathfrak{d}_{1}) \bigr)^{{\frac{\rho }{\mathcal{K}}}-1} \bigl(\varPsi(\varrho)- \varPsi(\mathfrak{d_{2}}) \bigr)^{{\frac{\rho}{\mathcal{K}}}-1} \bigl(\mathcal{S}( \mathfrak{d}_{1})-\mathcal{S}(\mathfrak{d_{2}}) \bigr)^{2} \\ &\qquad{}\times\varPsi^{\prime}(\mathfrak{d}_{1}) \varPsi^{\prime}(\mathfrak{d_{2}})\,d\mathfrak{d}_{1} \,d\mathfrak{d_{2}}. \end{aligned}$$
(2.20)

From (2.19) and (2.20) we get

$$\begin{aligned} & \biggl(\frac{\psi^{{\frac{\rho}{\mathcal{K}}}}(\varrho)}{\varGamma _{\mathcal{K}}(\rho+\mathcal{K})}\mathfrak{J}_{\varPsi,0^{+}}^{\rho ,\mathcal{K}} \bigl(\mathcal{P}(\varrho)\mathcal{S}(\varrho) \bigr) -\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}} \bigl(\mathcal{P}(\varrho )\bigr)\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}} \bigl(\mathcal{S}(\varrho )\bigr) \biggr)^{2} \\ &\quad\leq \biggl(\frac{\varPsi^{{\frac{\rho}{\mathcal {K}}}}(\varrho)}{\varGamma_{\mathcal{K}}(\rho+\mathcal{K})} \mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}} \mathcal{P}^{2}(\varrho )- \bigl(\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}} \mathcal {P}(\varrho) \bigr)^{2} \biggr) \\ &\qquad{}\times \biggl(\frac{\varPsi^{{\frac{\rho}{\mathcal{K}}}}(\varrho)}{\varGamma _{\mathcal{K}}(\rho+\mathcal{K})} \mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}} \mathcal{S}^{2}(\varrho )- \bigl(\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}} \mathcal {S}(\varrho) \bigr)^{2} \biggr). \end{aligned}$$
(2.21)

Since \((\varTheta_{2}(\varrho)-\mathcal{P}(\varrho) ) (\mathcal {P}(\varrho)-\varTheta_{1}(\varrho) )\geq0\) and \((\varphi_{2}(\varrho)-\mathcal{S}(\varrho) ) (\mathcal {S}(\varrho)-\varphi_{1}(\varrho) )\geq0\) for \(\varrho\in[0,\infty)\), we have

$$\begin{aligned}& \frac{\varPsi^{{\frac{\rho}{\mathcal{K}}}}(\varrho)}{\varGamma _{\mathcal{K}}(\rho+\mathcal{K})} \mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}} \bigl( \varTheta _{2}(\varrho)-\mathcal{P}(\varrho) \bigr) \bigl(\mathcal{P}( \varrho )-\varTheta_{1}(\varrho) \bigr)\geq0, \end{aligned}$$
(2.22)
$$\begin{aligned}& \frac{\varPsi^{{\frac{\rho}{\mathcal{K}}}}(\varrho)}{\varGamma _{\mathcal{K}}(\rho+\mathcal{K})}\mathfrak{J}_{\varPsi,0^{+}}^{\rho ,\mathcal{K}} \bigl( \varphi_{2}(\varrho)-\mathcal{S}(\varrho) \bigr) \bigl(\mathcal {S}( \varrho)-\varphi_{1}(\varrho) \bigr)\geq0. \end{aligned}$$
(2.23)

Thus from Lemma 2.9 we obtain

$$\begin{aligned}& \biggl(\frac{\varPsi^{{\frac{\rho}{\mathcal{K}}}}(\varrho)}{\varGamma _{\mathcal{K}}(\rho+\mathcal{K})} \mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}} \mathcal{P}^{2}(\varrho )- \bigl(\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}} \mathcal {P}(\varrho) \bigr)^{2} \biggr) \\& \quad\leq \bigl(\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal {K}}\varTheta_{2}( \varrho)-\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal {K}}\mathcal{P}(\varrho) \bigr) \bigl(\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\mathcal{P}(\varrho )- \mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\varTheta_{1}(\varrho ) \bigr) \\& \qquad{}+ \frac{\varPsi^{{\frac{\rho}{\mathcal{K}}}}(\varrho)}{\varGamma_{\mathcal{K}}(\rho+\mathcal{K})}\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal {K}} \bigl( \varTheta_{1}(\varrho)\mathcal{P}(\varrho) \bigr) \\& \qquad{}-\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal {K}}\varTheta_{1}( \varrho)\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal {K}}\mathcal{P}(\varrho) + \frac{\varPsi^{{\frac{\rho}{\mathcal{K}}}}(\varrho)}{\varGamma _{\mathcal{K}}(\rho+\mathcal{K})}\mathfrak{J}_{\varPsi,0^{+}}^{\rho ,\mathcal{K}} \varTheta_{2}(\varrho)\mathcal{P}(\varrho) -\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}} \varTheta_{2}(\varrho )\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}} \mathcal{P}(\varrho ) \\& \qquad{}+ \mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\varTheta_{1}( \varrho )\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\varTheta_{2}( \varrho )-\frac{\varPsi^{{\frac{\rho}{\mathcal{K}}}}(\varrho)}{ \varGamma_{\mathcal{K}}(\rho+\mathcal{K})}\mathfrak{J}_{\varPsi ,0^{+}}^{\rho,\mathcal{K}} \bigl(\varTheta_{1}(\varrho)\varTheta _{2}(\varrho) \bigr) \\& \quad=\mathfrak{T}(\mathcal{P},\varTheta_{1},\varTheta_{2}), \end{aligned}$$
(2.24)
$$\begin{aligned}& \biggl(\frac{\varPsi^{{\frac{\rho}{\mathcal{K}}}}(\varrho)}{\varGamma _{\mathcal{K}}(\rho+\mathcal{K})} \mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}} \mathcal{S}^{2}(\varrho )- \bigl(\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}} \mathcal {S}(\varrho) \bigr)^{2} \biggr) \\& \quad\leq \bigl(\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal {K}}\varphi_{2}( \varrho)-\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal {K}}\mathcal{S}(\varrho) \bigr) \bigl(\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\mathcal{S}(\varrho )- \mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\varphi_{1}(\varrho ) \bigr) \\& \qquad{}+ \frac{\varPsi^{{\frac{\rho}{\mathcal{K}}}}(\varrho)}{\varGamma_{\mathcal{K}}(\rho+\mathcal{K})}\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal {K}} \bigl( \varphi_{1}(\varrho)\mathcal{S}(\varrho) \bigr) \\& \quad\quad{}-\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal {K}}\varphi_{1}( \varrho)\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal {K}}\mathcal{S}(\varrho) + \frac{\varPsi^{{\frac{\rho}{\mathcal{K}}}}(\varrho)}{\varGamma _{\mathcal{K}}(\rho+\mathcal{K})} \mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}} \bigl( \varphi_{2}(\varrho )\mathcal{S}(\varrho) \bigr) - \mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\varphi_{2}(\varrho ) \mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\mathcal{S}(\varrho ) \\& \quad\quad{}+ \mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\varphi_{1}( \varrho )\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\varphi_{2}( \varrho )-\frac{\varPsi^{{\frac{\rho}{\mathcal{K}}}}(\varrho)}{ \varGamma_{\mathcal{K}}(\rho+\mathcal{K})}\mathfrak{J}_{\varPsi ,0^{+}}^{\rho,\mathcal{K}} \bigl(\varphi_{1}(\varrho)\varphi_{2}(\varrho ) \bigr) \\& \quad=\mathfrak{T}(\mathcal{S},\varphi_{1},\varphi_{2}). \end{aligned}$$
(2.25)

Therefore, inequality (2.17) follows from (2.20), (2.24), and (2.25). □

Corollary 2.14

Let\(m, M, n, N\in\mathcal{R}\), \(\mathfrak{T}(\mathcal{P},\varTheta_{1},\varTheta_{2})= \mathfrak{T}(\mathcal{P},m,M)\)and\(\mathfrak{T}(\mathcal{S},\varphi_{1},\varphi_{2})= \mathfrak{T}(\mathcal{S},n,N)\). Then inequality (2.17) reduces to

$$\begin{aligned}& \biggl\vert \frac{\varPsi^{\frac{\rho}{\mathcal{K}}}(\varrho)}{\varGamma _{\mathcal{K}}(\rho+\mathcal{K})}\mathfrak{J}_{\varPsi,0^{+}}^{\rho ,\mathcal{K}} \bigl(\mathcal{P}(\varrho)\mathcal{S}(\varrho) \bigr) -\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}} \mathcal{P}(\varrho )\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\mathcal{S}( \varrho) \biggr\vert \\& \quad\leq \biggl(\frac{\varPsi^{\frac{\rho}{\mathcal{K}}}(\varrho )}{2\varGamma_{\mathcal{K}}(\rho+\mathcal{K})} \biggr)^{2}(M-m) (N-n). \end{aligned}$$

Corollary 2.15

Let\(m, \mathcal{M}, n, \mathcal{N}\in\mathcal{R}\), \(\varPsi(\varrho)=\varrho\), \(\mathfrak{T}(\mathcal{P},\varTheta_{1},\varTheta_{2})= \mathfrak{T}(\mathcal{P},m,\mathcal{M})\)and\(\mathfrak{T}(\mathcal{S},\varphi_{1},\varphi_{2})= \mathfrak{T}(\mathcal{S},n,\mathcal{N})\). Then inequality (2.17) leads to

$$\begin{aligned}& \biggl\vert \frac{\varrho^{\frac{\rho}{\mathcal{K}}}}{\varGamma_{\mathcal{K}}(\rho+\mathcal{K})}\mathfrak{J}^{\rho,\mathcal{K}} \bigl(\mathcal {P}(\varrho)\mathcal{S}(\varrho) \bigr) -\mathfrak{J}^{\rho,\mathcal{K}}\mathcal{P}( \varrho)\mathfrak{J}^{\rho ,\mathcal{K}}\mathcal{S}(\varrho) \biggr\vert \\& \quad\leq \biggl(\frac{\varrho^{\frac{\rho}{\mathcal{K}}}}{\varGamma _{\mathcal{K}}(\rho+\mathcal{K})} \biggr)^{2}(\mathcal{M}-m) ( \mathcal{N}-n). \end{aligned}$$

Remark 2.16

Theorem 2.13 and Corollary 2.14 lead to four conclusions as follows:

  1. (1)

    If \(\mathcal{K}=1\), then Theorem 2.13 leads to Theorem 2.23 in [68].

  2. (2)

    If \(\mathcal{K}=1\) and \(\varPsi(\varrho)=\varrho\), then Theorem 2.13 gives Theorem 9 in [69].

  3. (3)

    If \(\mathcal{K}=1\), then Corollary 2.14 leads to Corollary 2.26 in [68].

  4. (4)

    If \(\mathcal{K}=1\) and \(\varPsi(\varrho)=\varrho\), then Corollary 2.14 gives Remark 10 in [69].

Example 2.17

Let \(\mathcal{K},\rho,\delta>0\), \(\mathcal{P}\) and \(\mathcal{S}\) be two positive functions defined on \([0,\infty)\), and Ψ be an increasing and positive function on \([0,\infty)\) such that \(\varPsi^{\prime}\) is continuous on \([0,\infty)\) with \(\varPsi(0)=0\). Then one has

$$\begin{aligned}& \textup{(a)}\quad q\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\mathcal{P}^{p}( \varrho) +p\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\mathcal {S}^{q}(\varrho) \geq pq\frac{\varGamma_{\mathcal{K}}(\rho+\mathcal{K})}{\varPsi^{\frac{\rho }{\mathcal{K}}} (\varrho)} \bigl[ \mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\mathcal {S}(\varrho) \mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\mathcal {P}(\varrho) \bigr], \\& \textup{(b)}\quad q\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\mathcal{P}^{p}( \varrho )\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\mathcal{S}^{p}( \varrho) +p\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\mathcal {P}^{q}(\varrho)\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}} \mathcal {S}^{q}(\varrho) \geq pq \bigl(\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}} \mathcal {P}(\varrho)\mathcal{S}(\varrho) \bigr)^{2}, \\& \textup{(c)}\quad q\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\mathcal{P}^{p}( \varrho )\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\mathcal{S}^{q}( \varrho) +p\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\mathcal {P}^{q}(\varrho)\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}} \mathcal {S}^{p}(\varrho)\\& \phantom{\textup{(c)}\quad} \quad\geq pq\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}} \mathcal {P}(\varrho)\mathcal{S}^{p-1}(\varrho)\mathfrak{J}_{\varPsi,0^{+}}^{\rho ,\mathcal{K}} \mathcal{P}\mathcal{S}^{q-1}(\varrho), \\& \textup{(d)}\quad q\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\mathcal{P}^{p}( \varrho )\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\mathcal{S}^{q}( \varrho) +p\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\mathcal {P}^{p}(\varrho)\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}} \mathcal {S}^{q}(\varrho)\\& \phantom{\textup{(d)}\quad} \quad\geq pq\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}} \mathcal {P}^{p-1}(\varrho)\mathcal{S}^{q-1}(\varrho) \mathfrak{J}_{\varPsi ,0^{+}}^{\rho,\mathcal{K}}\mathcal{P}\mathcal{S}(\varrho), \\& \textup{(e)}\quad q\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\mathcal{P}^{p}( \varrho )\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\mathcal{S}^{2}( \varrho) +p\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\mathcal {P}^{2}(\varrho)\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\mathcal {S}^{q}(\varrho) \\& \phantom{\textup{(e)}\quad}\quad\geq pq\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}} \mathcal {P}(\varrho)\mathcal{S}(\varrho)\mathfrak{J}_{\varPsi,0^{+}}^{\rho ,\mathcal{K}} \mathcal{P}^{\frac{2}{q}}\mathcal{S}^{\frac{2}{p}}(\varrho ), \\& \textup{(f)}\quad q\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\mathcal{P}^{2}( \varrho )\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\mathcal{S}^{q}( \varrho) +p\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\mathcal {P}^{p}(\varrho)\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}} \mathcal {S}^{2}(\varrho) \\& \phantom{\textup{(f)}\quad}\quad\geq pq\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}} \mathcal{P}^{\frac {2}{p}}(\varrho)\mathcal{S} ^{\frac{2}{q}}(\varrho) \mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal {K}}\mathcal{P}^{p-1} \mathcal{S}^{q-1}(\varrho), \\& \textup{(g)}\quad q\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\mathcal{P}^{2} \mathcal {S}^{q}(\varrho) +p\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}} \mathcal {P}^{2}(\varrho)\mathcal{S}^{p}(\varrho) \\& \phantom{\textup{(g)}\quad}\quad\geq pq \frac{\varGamma_{\mathcal{K}}(\rho+\mathcal{K})}{ \varPsi^{\frac{\rho}{\mathcal{K}}}(\varrho)}\mathfrak{J}_{\varPsi ,0^{+}}^{\rho,\mathcal{K}} \mathcal{P}^{\frac{2}{p}}(\varrho)\mathcal{S} ^{q-1}(\varrho) \mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\mathcal {P}^{\frac{2}{q}}( \varrho)\mathcal{S}^{p-1}(\varrho). \end{aligned}$$

Proof

According to the well-known Young inequality [70]

$$\begin{aligned} \frac{1}{p}a^{p}+\frac{1}{q}b^{q} \geq ab \quad \biggl(a,b\geq0,p, q>1,\frac{1}{p}+\frac{1}{q}=1 \biggr), \end{aligned}$$
(2.26)

and putting \(a=\mathcal{P}(\mathfrak{d}_{1})\) and \(b=\mathcal{S}(\mathfrak{d_{2}})\), we have

$$\begin{aligned} \frac{1}{p}\mathcal{P}^{p}(\mathfrak{d}_{1})+ \frac{1}{q}\mathcal {S}^{q}(\mathfrak{d_{2}}) \geq \mathcal{P}(\mathfrak{d}_{1})\mathcal{S}(\mathfrak{d_{2}}) \end{aligned}$$
(2.27)

for all \(\mathcal{P}(\mathfrak{d}_{1}),\mathcal{S}(\mathfrak{d_{2}})\geq0\).

Taking product on both sides of (2.27) by \(\frac{1}{\mathcal{K}\varGamma_{\mathcal{K}}(\rho)} (\varPsi(\varrho )-\varPsi(\mathfrak{d}_{1}) )^{\frac{\rho}{\mathcal{K}}-1}\varPsi ^{\prime}(\mathfrak{d}_{1})\) (which is positive due to \(\varrho\in(0,\mathfrak{d}_{1})\)) and integrating the resulting identity for the variable \(\mathfrak{d}_{1}\) from 0 to ϱ, we get

$$\begin{aligned} &\frac{1}{p}\frac{1}{\mathcal{K}\varGamma_{\mathcal{K}}(\rho)} \int _{0}^{\varrho} \bigl(\varPsi(\varrho)-\varPsi( \mathfrak{d}_{1}) \bigr)^{\frac{\rho}{\mathcal{K}}-1}\varPsi^{\prime}( \mathfrak{d}_{1})\mathcal {P}^{p}(\mathfrak{d}_{1}) \,d\mathfrak{d}_{1} \\ &\qquad{}+\frac {1}{q}\mathcal{S}^{q}(\mathfrak{d_{2}}) \frac{1}{\mathcal{K}\varGamma_{\mathcal{K}}(\rho)} \int_{0}^{\varrho } \bigl(\varPsi(\varrho)-\varPsi( \mathfrak{d}_{1}) \bigr)^{\frac{\rho }{\mathcal{K}}-1}\varPsi^{\prime}( \mathfrak{d}_{1}) \,d\mathfrak{d}_{1} \\ &\quad\geq\mathcal{S}(\mathfrak{d_{2}}) \frac{1}{\mathcal{K}\varGamma_{\mathcal{K}}(\rho)} \int_{0}^{\varrho } \bigl(\varPsi(\varrho)-\varPsi( \mathfrak{d}_{1}) \bigr)^{\frac{\rho }{\mathcal{K}}-1} \varPsi^{\prime}( \mathfrak{d}_{1})\mathcal{P}(\mathfrak{d}_{1})\, d \mathfrak{d}_{1} \end{aligned}$$
(2.28)

and

$$\begin{aligned} \frac{1}{p}\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\mathcal {P}^{p}(\varrho)+\frac{1}{q}\mathcal{S}^{q}( \mathfrak{d_{2}})\frac {\varPsi^{\frac{\rho}{\mathcal{K}}}(\varrho)}{ \varGamma_{\mathcal{K}}(\rho+\mathcal{K})} \geq \mathcal{S}( \mathfrak{d_{2}})\mathfrak{J}_{\varPsi,0^{+}}^{\rho ,\mathcal{K}} \mathcal{P}(\varrho). \end{aligned}$$
(2.29)

Again, taking product on both sides of (2.29) by \(\frac{1}{\mathcal{K}\varGamma_{\mathcal{K}}(\rho)} (\varPsi(\varrho )-\varPsi(\mathfrak{d_{2}}) )^{\frac{\rho}{\mathcal{K}}-1}\varPsi ^{\prime}(\mathfrak{d_{2}})\) and integrating the resulting identity for the variable \(\mathfrak{d_{2}}\) from 0 to ϱ, we get

$$\begin{aligned} \frac{1}{p}\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\mathcal {P}^{p}(\varrho)\frac{\varPsi^{\frac{\rho}{\mathcal{K}}}(\varrho )}{\varGamma_{\mathcal{K}}(\rho+\mathcal{K})} +\frac{1}{q} \mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\mathcal {S}^{q}( \varrho)\frac{\varPsi^{\frac{\rho}{\mathcal{K}}}(\varrho )}{\varGamma_{\mathcal{K}}(\rho+\mathcal{K})} \geq\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}} \mathcal{S}(\varrho )\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\mathcal{P}( \varrho) \end{aligned}$$
(2.30)

and

$$\begin{aligned} \frac{1}{p}\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\mathcal {P}^{p}(\varrho) +\frac{1}{q}\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}} \mathcal {S}^{q}(\varrho) \geq \frac{\varGamma_{\mathcal{K}}(\rho+\mathcal{K})}{\varPsi^{\frac{\rho }{\mathcal{K}}}(\varrho)} \bigl[ \mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\mathcal{S}(\varrho) \mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\mathcal{P}(\varrho) \bigr], \end{aligned}$$
(2.31)

which implies part (a). The rest of inequalities can be shown in a similar manner by using the subsequent desire of parameters in the Young inequality:

$$\begin{aligned}& \text{(b)}\quad a=\mathcal{P}(\mathfrak{d}_{1})\mathcal{S}(\mathfrak {d_{2}}), \quad\quad b=\mathcal{P}(\mathfrak{d_{2}}) \mathcal {S}(\mathfrak{d}_{1}). \\& \text{(c)} \quad a=\frac{\mathcal{P}(\mathfrak{d}_{1})}{\mathcal {S}(\mathfrak{d}_{1})}, \quad\quad b=\frac{\mathcal{P}(\mathfrak {d_{2}})}{\mathcal{S}(\mathfrak{d_{2}})},\quad \mathcal{S}(\mathfrak {d}_{1})\mathcal{S}(\mathfrak{d_{2}}) \neq0. \\& \text{(d)} \quad a=\frac{\mathcal{P}(\mathfrak{d_{2}})}{\mathcal {P}(\mathfrak{d}_{1})}, \quad\quad b=\frac{\mathcal{S}(\mathfrak {d_{2}})}{\mathcal{S}(\mathfrak{d}_{1})},\quad \mathcal{P}(\mathfrak {d}_{1})\mathcal{S}(\mathfrak{d_{2}}) \neq0. \\& \text{(e)} \quad a=\mathcal{P}(\mathfrak{d}_{1})\mathcal{S}^{\frac {2}{p}}( \mathfrak{d_{2}}), \qquad b=\mathcal{P}^{\frac{2}{q}}(\mathfrak {d_{2}})\mathcal{S}(\mathfrak{d}_{1}). \\& \text{(f)} \quad a=\frac{\mathcal{P}^{\frac{2}{p}}(\mathfrak {d}_{1})}{\mathcal{P}(\mathfrak{d_{2}})}, \quad\quad b=\frac{\mathcal {S}^{\frac{2}{q}}(\mathfrak{d}_{1})}{\mathcal{S}(\mathfrak{d_{2}})},\quad \mathcal{P}(\mathfrak{d_{2}})\mathcal{S}(\mathfrak{d_{2}}) \neq0. \\& \text{(g)} \quad a=\frac{\mathcal{P}^{\frac{2}{p}}(\mathfrak{d}_{1})}{\mathcal {S}(\mathfrak{d_{2}})}, \quad\quad b=\frac{\mathcal{P}^{\frac{2}{q}}(\mathfrak{d_{2}})}{\mathcal {S}(\mathfrak{d}_{1})},\quad \mathcal{S}(\mathfrak{d}_{1})\mathcal {S}(\mathfrak{d_{2}}) \neq0. \end{aligned}$$

Repeating the foregoing argument, we can obtain parts (b)–(g). □

Example 2.18

Let \(\mathcal{K}, \rho, \delta>0\), \(\mathcal{P}\) and \(\mathcal{S}\) be two positive functions defined on \([0,\infty)\), Ψ be an increasing and positive function on \([0,\infty)\) such that \(\varPsi^{\prime}\) is continuous on \([0,\infty)\) and \(\varPsi(0)=0\), and m and \(\mathcal{M}\) be defined by

$$\begin{aligned} m=\min_{0\leq\mathfrak{d}_{1}\leq \varrho}\frac{\mathcal{P}(\mathfrak{d}_{1})}{\mathcal{S}(\mathfrak {d}_{1})}, \quad\quad \mathcal{M}= \max_{0\leq\mathfrak{d}_{1}\leq \varrho}\frac{\mathcal{P}(\mathfrak{d}_{1})}{\mathcal{S}(\mathfrak{d}_{1})}, \end{aligned}$$
(2.32)

respectively. Then we have

$$\begin{aligned}& \textup{(a)}\quad0\leq \mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\mathcal{P}^{2}( \varrho )\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\mathcal{S}^{2}( \varrho) \leq\frac{(m+\mathcal{M})^{2}}{4m\mathcal{M}} \bigl(\mathfrak{J}_{\varPsi ,0^{+}}^{\rho,\mathcal{K}} \mathcal{P}\mathcal{S}(\varrho) \bigr)^{2}, \\& \textup{(b)}\quad0\leq \sqrt{\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\mathcal {P}^{2}(\varrho)\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\mathcal {S}^{2}(\varrho)} - \bigl(\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}} \mathcal {P}(\varrho)\mathcal{S}(\varrho) \bigr) \\& \phantom{\textup{(b)}\quad0}\leq\frac{(\sqrt{\mathcal{M}}-\sqrt{m})^{2}}{2\sqrt{m\mathcal {M}}} \mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\mathcal{P}(\varrho )\mathcal{S}( \varrho). \end{aligned}$$

Proof

It follows from (2.32) and

$$\begin{aligned} \biggl(\frac{\mathcal{P}(\mathfrak{d}_{1})}{\mathcal{S}(\mathfrak {d}_{1})}-m \biggr) \biggl(\mathcal{M}-\frac{\mathcal{P}(\mathfrak{d}_{1})}{\mathcal {S}(\mathfrak{d}_{1})} \biggr) \mathcal{S}^{2}(\mathfrak{d}_{1})\geq0 \quad(0 \leq\varrho\leq \mathfrak{d}_{1}) \end{aligned}$$

that

$$\begin{aligned} \mathcal{P}^{2}(\mathfrak{d}_{1})+m\mathcal{M} \mathcal{S}^{2}(\mathfrak {d}_{1})\leq(m+\mathcal{M}) \mathcal{P}(\mathfrak{d}_{1})\mathcal {S}(\mathfrak{d}_{1}). \end{aligned}$$
(2.33)

Multiplying (2.33) by \(\frac{1}{\mathcal{K}\varGamma_{\mathcal{K}}(\rho)} (\varPsi(\varrho )-\varPsi(\mathfrak{d}_{1}) )^{\frac{\rho}{\mathcal{K}}-1}\varPsi ^{\prime}(\mathfrak{d}_{1})\) and integrating with respect to \(\mathfrak{d}_{1}\) over \((0,\varrho)\), we get

$$\begin{aligned} &\frac{1}{\mathcal{K}\varGamma_{\mathcal{K}}(\rho)} \int_{0}^{\varrho } \bigl(\varPsi(\varrho)-\varPsi( \mathfrak{d}_{1}) \bigr)^{\frac{\rho }{\mathcal{K}}-1} \varPsi^{\prime}( \mathfrak{d}_{1})\mathcal{P}^{2}(\mathfrak{d}_{1}) \, d\mathfrak{d}_{1}\\ &\qquad{}+ m\mathcal{M}\frac{1}{\mathcal{K}\varGamma_{\mathcal{K}}(\rho)} \int _{0}^{\varrho} \bigl(\varPsi(\varrho)-\varPsi( \mathfrak{d}_{1}) \bigr)^{\frac{\rho}{\mathcal{K}}-1} \varPsi^{\prime}( \mathfrak{d}_{1})\mathcal{S}^{2}(\mathfrak{d}_{1}) \, d\mathfrak{d}_{1} \\ &\quad\leq\frac{m+\mathcal{M}}{\mathcal{K}\varGamma_{\mathcal{K}}(\rho )} \int_{0}^{\varrho} \bigl(\varPsi(\varrho)-\varPsi( \mathfrak{d}_{1}) \bigr)^{\frac{\rho}{\mathcal{K}}-1} \varPsi^{\prime}( \mathfrak{d}_{1})\mathcal{P}(\mathfrak{d}_{1})\mathcal {S}(\mathfrak{d}_{1})\,d\mathfrak{d}_{1}, \end{aligned}$$

which implies that

$$\begin{aligned} \mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\mathcal{P}^{2}(\varrho )+m\mathcal{M}\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\mathcal {S}^{2}(\varrho)\leq(m+\mathcal{M}) \mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}} \mathcal{P}(\varrho )\mathcal{S}(\varrho). \end{aligned}$$
(2.34)

Alternately, it follows from \(m,\mathcal{M}>0\) and

$$\begin{aligned} \bigl(\sqrt{\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\mathcal {P}^{2}(\varrho)}-\sqrt{m\mathcal{M} \mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}} \mathcal{S}^{2}(\varrho )} \bigr)^{2}\geq0 \end{aligned}$$

that

$$\begin{aligned} 2\sqrt{\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\mathcal {P}^{2}(\varrho)}\sqrt{m\mathcal{M}\mathfrak{J}_{\varPsi,0^{+}}^{\rho ,\mathcal{K}} \mathcal{S}^{2}(\varrho)} \leq \mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}} \mathcal{P}^{2}(\varrho )+m\mathcal{M}\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}} \mathcal {S}^{2}(\varrho). \end{aligned}$$
(2.35)

Therefore,

$$\begin{aligned} 4m\mathcal{M}\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\mathcal {P}^{2}(\varrho)\mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\mathcal {S}^{2}(\varrho)\leq(m+\mathcal{M})^{2} \bigl( \mathfrak{J}_{\varPsi,0^{+}}^{\rho,\mathcal{K}}\mathcal{P}(\varrho )\mathcal{S}( \varrho) \bigr)^{2} \end{aligned}$$

follows from (2.34) and (2.35), and part (a) is proved. By using a few transformations to part (a), we can obtain part (b). □

Conclusion

This article begins with a succinct review of fractional integrals in the frame of a new fractional integral operator. We characterize the definition of generalized \(\mathcal{K}\)-fractional integral operators. We modify the Grüss type inequality by employing generalized \(\mathcal{K}\)-fractional integrals; specifically, the variant involving fractional integrals in the generalized Riemann–Liouville and \(\mathcal{K}\)-fractional integrals frame is provided. The associated significant variants including summed up generalized \(\mathcal{K}\)-fractional integrals are also outlined. Numerous consequences can be generalized for the utility of these recently presented fractional integral operators.

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Rashid, S., Jarad, F., Noor, M.A. et al. On Grüss inequalities within generalized K-fractional integrals. Adv Differ Equ 2020, 203 (2020). https://doi.org/10.1186/s13662-020-02644-7

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MSC

  • 26E60
  • 26D15
  • 26D10
  • 26A33

Keywords

  • Grüss inequality
  • Generalized \(\mathcal{K}\)-fractional integral
  • Integral inequality