On some generalized Raina-type fractional-order integral operators and related Chebyshev inequalities

1.   Introduction

Fractional calculus emerged as a generalization of conventional calculus. The classical differential and integral operators proposed by Leibniz more than three centuries ago have been extended to the fractional scenario in various different forms. In the way, various applications have been proposed to within mathematics, like Cauchy problems with Caputo Hadamard fractional derivatives ^[1], the synchronization for a class of fractional-order hyperchaotic systems ^[2] the inequality estimates for the boundedness of multilinear singular and fractional integral operators ^[3], the analysis of unstructured-mesh Galerkin finite element method for the two-dimensional multi-term time-space fractional Bloch-Torrey equations on irregular convex domains ^[4] and the design of numerically efficient and conservative model for a Riesz space-fractional Klein-Gordon-Zakharov system ^[5], among various other interesting problems ^[6].

It is worth pointing out that each fractional operator is fully characterized by its own special kernel, and they can be used in a range of particular problems. In this respect, the analysis on the uniqueness of solutions for fractional ordinary and partial differential equations may be performed by employing fractional integral inequalities, which obviously depend on the type of operator. The literature provides accounts of many applications potential applications, like the periodic orbit analysis for the delayed Filippov systems ^[7] the bifurcation of limit cycles at infinity in piecewise polynomial systems ^[8] and the number and stability of limit cycles for planar piecewise linear systems of node-saddle type ^[9]. Indeed, integral inequalities play a major role in the field of differential equations and applied mathematics. More well-known applications of integral inequalities are found in applied sciences, such as statistical problems, transform theory, numerical quadrature, and probability ^[10,11].

In the last few years, many researchers have established various types of integral inequalities by employing different approaches. For example, there are reports on some new inequalities for $(k, s)$-fractional integrals ^[12], certain weighted integral inequalities involving the fractional hypergeometric operators ^[13] and some generalized integral inequalities for Hadamard fractional integrals ^[14]. Moreover, those results provide direct applications to other areas of mathematics, such as conformable fractional integral inequalities ^[15], including those of the Hermite-Hadamard type ^[16]. Moreover, there are reports in the literature on integral inequalities for a fractional operator of a function with respect to another function with nonsingular kernel ^[17] and Hermite-Hadamard-type inequalities for Riemann-Liouville fractional integrals ^[18].

The so-called Chebyshev inequality is one important type of inequality which is related to the synchronous functions ^[19]. This inequality has been extensively studied in the literature. Indeed, these inequalities appear in the study of conformable $\kappa$-fractional integral operators ^[20] in the investigation of inequalities of the Grüss-type for conformable fractional integrals ^[21,22], in Simpson's type integral inequalities for $\kappa$-fractional integrals ^[23], in generalized fractional conformable integrals ^[24], in the study of some fractional integral inequalities for the Katugampola integral operator ^[25], in the theory of Riemann-Liouville fractional operators ^[26], and in the context of generalized conformable fractional operators ^[27,28]. Here, it is worth pointing out that the classical Chebyshev inequality reads as follows ^[19]:

Here, ${\psi_{1}}$ and ${\psi_{2}}$ are supposed to be synchronous and integrable functions on the interval $[d_{1}, d_{2}] \subseteq \mathbb {R}$. Recall that two functions are synchronous on $[d_{1}, d_{2}]$ if

It is worthwhile mentioning that Chebyshev's inequality (1.1) has been extended for functions whose derivatives belong to $L _p$ spaces ^[29,30] and a variant of Chebyshev's inequality was applied to obtain some inequalities for expectation and variance for cumulative distribution functions ^[31]. It is also important to mention that a new class of fractional derivatives and integrals (fractional conformable derivatives and integrals) were recently proposed in the literature ^[32,33]. Later on, the fractional conformable integral operators were introduce ^[34], and various generalizations of the classical inequalities ^[35], including the Hadamard, Hermite-Hadamard and Opial inequalities ^[36], Grüss inequality ^[37], and the Ostrowski and Chebyshev inequalities ^[20]. Recently, some new fractional extensions of Chebyshev's inequality were derived using the generalized Katugampola integrals through the Polya-Szegö inequality ^[38].

In the present manuscript, we will present new Raina-type fractional integral operators, and we will devote our investigation to derive extensions of Chebyshev's inequality which consider the presents of those operators. As one of the most important results in this work, we will prove a version of Chebyshev's inequality for more than two functions. In the way, some Chebyshev-type inequalities will be proved for functions whose derivatives are bounded from below or from above. In addition, an estimate of the Chebyshev functional are proved using the new Raina-type fractional integral operators. Before closing this work we will establish similar inequalities for the Prabhakar-Salim fractional integrals. Overall, the present work proposes more general results than those in various existing reports available in the literature. To show this fact, we will notice that some known results from our general Chebyshev inequalities are obtained as particular cases.

2.   Preliminaries

The present section is devoted to recall various definitions and results from the literature on fractional calculus, and to derive new properties which will be required in the sequel. Most of the standard definitions will be taken from ^[39,40].

Definition 2.1. Let $\imath > 0$ be a real number and suppose that $\psi : [d _1, d _2] \rightarrow \mathbb {R}$ is a function such that $\psi\in L_{1}\left[d_{1}, d_{2}\right]$. Then the standard left and right Riemann-Liouville fractional integral operators of order $\imath$ are defined, respectively, by

In turn, if we agree that $n = \lfloor\imath\rfloor+1$, then the left and right Riemann-Liouville fractional derivatives are given by

Definition 2.2 (Raina ^[41]). Let $\sigma = (\sigma (i)) _{i = 1} ^\infty$ be a bounded real sequence. The modified Mittag-Leffler function with parameters $\varrho, \rho > 0$ and $\sigma$ at ${\bf k} \in \mathbb {C}$ is given by

The following definition introduces the Raina fractional operators. It is important to notice that the those operators are obtained from (2.1) and (2.2) by replacing the Mittag-Leffler kernel with the modified Mittag-Leffler function with parameters $\varrho, \rho > 0$ and $\sigma$. Meantime, the Raina fractional model is proposed in ^[41] as one important models of fractional calculus and this is defined by integral similar to but with a modified Mittag-Leffler function in the kernel. This is defined by the following definition.

Definition 2.3 (Agarwal ^[42]). Suppose that $\varrho, \rho > 0$, and let $w\in \mathbb {R}$. For any function $\psi \in L_{1}[d_{1}, d_{2}]$, the left and right Raina fractional integral operators of $\psi$ are defined by the following integral transforms, respectively:

Various known particular scenarios are obtained depending on specific expressions of $\rho$, $\varrho$ and $\sigma$. For example, if $\rho = 1$, $\varrho = 0$ and $\sigma(i) = ((\imath)_{i}(\beta)_{i}/(\gamma)_{i})$ for $i\in \mathbb {N} \cup \{ 0 \}$ in the formula (2.5), then we get the classical hypergeometric function, which is given by

Here, $\imath$, $\beta$ and $\gamma$ are arbitrary complex or real parameters with the property that $\gamma \neq 0, -1, -2, \ldots$, and $(d_{1})_{i}$ is the Pochhammer symbol defined by

On the other hand, if $\sigma (i) = (1, 1, 1, \ldots)$ with $\rho = \imath > 0$ and $\varrho = 1$ in (2.5), then we get the standard Mittag-Leffler function, which is given by

Extensions of the Mittag-Leffler function are obtained from (2.5) when $\sigma(i) = \frac{(\eta)_{i}}{i!}$, $\rho = \imath$ and $\varrho = \beta,$ see ^[43], and when $\sigma(i) = \frac{(\gamma)_{qi}}{(\delta)_{pi}}$, $\rho = \imath$, and $\varrho = \beta,$ see ^[44].

Let $\sigma = (\sigma (i)) _{i = 1} ^\infty$ be a bounded real sequence and $w\in \mathbb {R}$, then the following lemma holds.

Lemma 2.4. The integral operators $\mathcal{J}_{\rho, \varrho, d_{1}+; w}^{\sigma}\psi$ and $\mathcal{J}_{\rho, \varrho, d_{2}-; w}^{\sigma }\psi$ are bounded on $L_{1}[d_{1}, d_{2}]$, if $\mathfrak{M}: = \mathcal{F}_{\rho, \varrho+1}^{|\sigma|}\left[|w|(d_{2}-d_{1})^{\rho}\right] < \infty$. In fact, for $\psi \in L_{1}\left[d_{1}, d_{2}\right]$, we have

where $\left\Vert \cdot \right\Vert _{p}$ is the usual norm in $L _p$.

Let us define $\mathfrak{G}: = \mathcal{F}_{\rho, \varrho+1}^{\sigma}\left[w(d_{2}-d_{1})^{\rho}\right] < \infty.$ Taking $\sigma(i)\geq0,$ where $i \in \mathbb {N} \cup \{ 0 \}$ and $w\geq0,$ in above lemma, we get the following result:

Raina's fractional operators are important due to their level of generality. More precisely, by specifying the coefficient $\sigma(i)$, we can obtain many useful fractional integral operators. For example, if $\varrho = \imath$, $\sigma(0) = 1$, $\sigma(i) = 0$ for each $i \in \mathbb {N}$ and $w = 0$ in Definition 2.3, then we readily retrieve the left and right Riemann-Liouville fractional integrals. On the other hand, if $\sigma(i) = \frac{(\eta)_{i}}{i!}$, $\rho = \imath$, $\varrho = \beta$ and $d_{1} = 0$ in Definition 2.3, we obtain the well-known Prabhakar fractional integral in ^[43], which is given by

Finally, when $\sigma(i) = \frac{(\gamma)_{qi}}{(\delta)_{pi}}$, $\rho = \imath$ and $\varrho = \beta$, we obtain the well-known Salim fractional integral ^[44]:

where $p, q > 0$ and $q < \imath+p$.

The new Raina fractional integral operators are introduced next.

Definition 2.5. Let $d _1$ and $d _2$ be real numbers satisfying $0\leq d_{1} < d_{2}$, and let $\psi \in L_{1} [d_{1}, d_{2}]$ be any function. The generalized left and right Raina fractional integral operator on $\psi(\ell)$ associated to the parameters $\varrho$, $\rho > 0$, $w\in \mathbb {R}$, $\vartheta \in (0, 1]$, $\varpi\geq 0$, and $\sigma$ any bounded arbitrary sequence of real (or complex) numbers, are respectively defined by the following integral transforms:

and

It is obvious from Definition 2.5 that $^{\vartheta}_{\varpi}\mathcal{J}_{\rho, \varrho, d_{1}+; w}^{\sigma }\psi$ and $^{\vartheta}_{\varpi}\mathcal{J}_{\rho, \varrho, d_{2}-; w}^{\sigma }\psi$ are linear operators, and we claim that they are bounded integral operators on $L_{1}[d_{1}, d_{2}]$, if

Indeed, for $\psi \in L_{1}\left[d_{1}, d_{2}\right]$, we readily obtain that

The relevance of the present study lies in the fact that generalized forms of existing inequalities will be provided ^[45]. To that end, it is important to notice that different expressions of the sequence $\sigma$. As example, if $\rho = 1$, $w = 1$, $\sigma(0) = 1$, and $\sigma(i) = 0$ for each $i \in \mathbb {N}$, we readily obtain the left and right generalized fractional conformable integral operators ^[24,27], which are respectively defined by

If, in addition, $\varpi = 0$, then we readily obtain the Katugampola conformable left and right fractional integral operators, which are respectively defined by

With the parametric choice $\vartheta = 1$, it is easy to verify that left and right Riemann-Liouville fractional integrals are obtained from (2.15) and (2.16), respectively. Similarly, if $\vartheta = 1$, $\varpi = 0$, $\sigma (i) = \frac{(\eta)_{i}}{i!}$, $\rho = \imath$ and $\varrho = \beta$, we readily obtain the Prabhakar fractional integral (2.13). Also, if we let $\sigma (i) = \frac{(\gamma)_{qi}}{(\delta)_{pi}}$, $\rho = \imath$ and $\varrho = \beta$, then we readily retrieve the Salim fractional integral (2.14).

3.   Main results

For the remainder of this work, we suppose in general that $\{\sigma(n)\}_{n\in \mathbb{N}_{0}}$ is a sequence of non-negative real numbers and $w\geq0.$

Theorem 3.1. Let $\varrho, \rho > 0$, $\vartheta \in (0, 1]$ and $\varpi\geq 0$, and suppose that ${\psi_{1}}$ and ${\psi_{2}}$ are two synchronous functions defined on $[0, \infty)$. For all $\ell > 0$, we have

Proof. Since the functions ${\psi_{1}}$ and ${\psi_{2}}$ are synchronous on $[0, \infty)$, then for all $r, s \geq 0$, it follows that $({\psi_{1}}(r)-{\psi_{1}}(s))({\psi_{2}}(r)-{\psi_{2}}(s)) \geq 0$. As a consequence,

Multiplying now both sides of (3.2) by $\left(\frac{\ell^{\vartheta + \varpi }-r^{\vartheta + \varpi }}{\vartheta + \varpi } \right)^{\varrho -1} r^{\vartheta + \varpi -1} \mathcal{F}_{\rho, \varrho}^{\sigma }\left[w(\ell^{\vartheta + \varpi}-r^{\vartheta + \varpi })^{\rho }\right]$ and letting $r \in (0, \ell)$, we may obtain readily that

Integrating over $r\in (0, \ell)$ yields

From here, it is easy to check that

Multiplying this last inequality by $\left(\frac{\ell^{\vartheta + \varpi }-s^{\vartheta + \varpi}}{\vartheta + \varpi}\right)^{\varrho-1}s^{\vartheta + \varpi - 1}\mathcal{F}_{\rho, \varrho}^{\sigma}\left[w(\ell^{\vartheta + \varpi}-s^{\vartheta + \varpi})^{\rho}\right]$ on both sides, letting $s\in (0, \ell)$. Integrate then over all $s\in (0, \ell)$, we obtain that

Rewriting this expression and simplifying, we readily reach the conclusion.

Corollary 3.2. Let $\varrho, \rho > 0$, $\vartheta \in (0, 1]$ and $\varpi\geq 0$, and suppose that ${\psi_{1}}$ and ${\psi_{2}}$ are two synchronous functions defined on $[d _1, \infty)$ where $d _1 > 0$. For all $\ell > d _1$, we have

Proof. The proof is similar to that of the previous result.

Some particular consequences can be derived from Theorem 3.1 for the right Raina conformable fractional integral operator. For example, if $\varrho = \imath$, $\sigma (0) = 1$, $\sigma(i) = 0$ for each $i \in \mathbb {N}$, $w = 1$, $\varpi = 0$ and $\vartheta = 1$ in Theorem 3.1, we obtain that

which is an inequality investigated in [46,Theorem 3.1]. If, in addition, $\varrho = \imath$, $\sigma (0) = 1$ and $w = 0$, it follows that

On the other hand, $\varrho = \imath$, $\sigma (0) = 1$, $\sigma(i) = 0$ for each $i \in \mathbb {N}$ and $w = 1$ yield

which was established in [24,Theorem 2.1]. Finally, if we let $\varpi = 0$, then we retrieve the Chebyshev inequality for the Katugampola fractional integral, which reads

In this point of our discussion, it is useful to note that the Chebyshev inequality for the Katugampola fractional integral operator reported in ^[25] is wrong. Actually, the error results from the integral calculation of the term ${}^{\vartheta}I_{0}^{\imath}(1)$. In fact, the correct expression for this term results from

As we pointed out in our work using suitable parameter values of the Raina conformable fractional integral operator and the conclusion of Theorem 3.1, we can express that result in terms of the Prabhakar and Salim fractional integrals. More concretely, if $\vartheta = 1$, $\varpi = 0$, $\sigma (i) = \frac{(\eta)_{i}}{i!}$, $\rho = \imath$ and $\varrho = \beta$ in Theorem 3.1, we derive the following inequality for the Prabhakar fractional integral:

On the other hand, if we agree that $\vartheta = 1$, $\varpi = 0$, $\sigma(i) = \frac{(\gamma)_{qi}}{(\delta)_{pi}}$, $\rho = \imath$ and $\varrho = \beta$ in Theorem 3.1, we get the corresponding inequality for the Salim fractional integral, which reads

Moreover, taking $\varpi = 0$ and $\vartheta = 1$ in Theorem 3.1, then we obtain [28,Theorem 2] for the classical Raina fractional integral.

Theorem 3.3. Let $\varrho, \rho > 0$, $\vartheta \in (0, 1]$ and $\varpi \geq 0$, and suppose that $\{ \psi_{i} \}_{i = 1}^{n}$ are increasing positive functions defined on $[0, \infty)$. Then, for any $\ell > 0$, we have

Proof. The proof will use mathematical induction. In the case that $n = 1$, is it easy to check that ${} ^{\vartheta}_{\varpi}\mathcal{J}_{\rho, \varrho, 0+;w}^{\sigma }\left(\psi_{1} \right)(\ell) \geq {} ^{\vartheta}_{\varpi}\mathcal{J}_{\rho, \varrho, 0+;w}^{\sigma }\left(\psi_{1}\right)(\ell)$, for all $\ell > 0$. Now, let $n \geq 2$ and suppose that the conclusion is satisfied for $n - 1$ in place of $n$. Since $\{\psi_{i} \}_{i = 1}^{n}$ are all increasing positive functions, then $\prod_{i = 1}^{n}\psi_{i}$ is also an increasing function. We can apply now Theorem 3.1 with ${\psi_{1}}^{\star} = \prod_{i = 1}^{n-1}\psi_{i}$ and ${\psi_{2}}^{\star} = \psi_{n}$ along with the induction hypothesis to obtain

The conclusion of this result follows now from mathematical induction.

One may readily check that the inequality

results from Theorem 3.3 when $\varrho = \imath, \sigma (0) = 1, w = 1, \vartheta = 1$ and $\varpi = 0$. This inequality was obtained in [46,Theorem 3.3]. On the other hand, if $\varrho = \imath$, $\sigma (0) = 1$ and $w = 1$, then we obtain [24,Theorem 2.3]. Meanwhile, if $\varpi = 0$, then we reach [25,Theorem 2.3] for conformable fractional integrals. In similar fashion, we can deduce the following inequalities related to Prabhakar and Salim fractional integrals, respectively:

and

Also, [24,Theorem 2.3] and [25,Theorem 2.3] can be obtained from (3.15). Moreover, if we let $\varpi = 0$ and $\vartheta = 1$ (which concern the classical Raina fractional integral), then Theorem 3.3 reduces to [28,Theorem 4].

Theorem 3.4. Let $\varrho, \rho > 0$, $\vartheta \in (0, 1]$ and $\varpi \geq 0$. Assume that ${\psi_{1}}$ and ${\psi_{2}}$ are two functions such that ${\psi_{1}}$ is increasing and ${\psi_{2}}$ is differentiable. If $m: = \inf_{\ell\geq 0}{\psi_{2}}'(\ell)$ exists in the real numbers and $\phi(\ell) = \ell^{\vartheta+\varpi}$, then

holds true in the following cases:

(1) $\vartheta+\varpi = 1,$ and for all $\ell > 0;$

(2) $\vartheta+\varpi > 1,$ and for all $\ell\in[0, \ell_{0}];$

(3) $\vartheta+\varpi < 1,$ and for all $\ell\geq\ell_{0},$

where $\ell_{0} = (\vartheta+\varpi)^{\frac{1}{1-(\vartheta+\varpi)}}.$

Proof. Define the function $h(\ell):= {\psi_{2}}(\ell)-m\phi(\ell)$, where $\phi(\ell) = \ell^{\vartheta+\varpi}$. One can easily verify that $h$ is increasing and differentiable function according to the above cases. Moreover, notice that

Using then Theorem 3.1, it is possible to check that

This completes the proof of this result.

The conclusion of the following results is reached using the same arguments as in Theorem 3.4, and letting $h(\ell):= {\psi_{2}}(\ell)-M\phi(\ell)$.

Corollary 3.5. Let $\varrho, \rho > 0$, $\vartheta \in (0, 1]$ and $\varpi \geq 0$. Suppose that ${\psi_{1}}, {\psi_{2}}$ are two functions such that ${\psi_{1}}$ is {decreasing} and ${\psi_{2}}$ is differentiable. If $M: = \sup_{\ell \geq 0}{\psi_{2}}'(\ell)$ exists in the real numbers and if $\phi$ is defined as in Theorem 3.4, then

is satisfied for the following cases:

(1) $\vartheta+\varpi = 1,$ and for all $\ell > 0;$

(2) $\vartheta+\varpi > 1,$ and for all $\ell\geq\ell_{0};$

(3) $\vartheta+\varpi < 1,$ and for all $\ell\in[0, \ell_{0}],$

where $\ell_{0} = (\vartheta+\varpi)^{\frac{1}{1-(\vartheta+\varpi)}}.$

Corollary 3.6. Let $\varrho, \rho > 0$, $\vartheta \in (0, 1]$ and $\varpi \geq 0$. Let ${\psi_{1}}$ and ${\psi_{2}}$ be functions such that ${\psi_{1}}$ is increasing, and ${\psi_{1}}$ and ${\psi_{2}}$ are differentiable. If $m_{1} : = \inf_{\ell \geq 0}{\psi_{1}}'(\ell)$ and $m_{2}: = \inf_{\ell \geq 0}{\psi_{2}}'(\ell)$ exist in the real numbers and $\phi$ is defined as in Theorem 3.4, then

is satisfied for the following cases:

(1) $\vartheta+\varpi = 1,$ and for all $\ell > 0;$

(2) $\vartheta+\varpi > 1,$ and for all $\ell\in[0, \ell_{0}];$

(3) $\vartheta+\varpi < 1,$ and for all $\ell\geq\ell_{0},$

where $\ell_{0} = (\vartheta+\varpi)^{\frac{1}{1-(\vartheta+\varpi)}}.$

Proof. The proof is similar to that of Theorem 3.4, setting $h_{1}(\ell) = {\psi_{2}}(\ell)-m_{1}\phi(\ell)$ and $h_{2}(\ell) = {\psi_{2}}(\ell)-m_{2}\phi(\ell)$.

Corollary 3.7. Let $\varrho, \rho > 0$, $\vartheta \in (0, 1]$ and $\varpi \geq 0$, and let ${\psi_{1}}, {\psi_{2}}$ be such that ${\psi_{1}}$ is increasing, and ${\psi_{1}}$ and ${\psi_{2}}$ are differentiable. Let $M_{1}: = \sup_{\ell \geq 0}{\psi_{1}}'(\ell)$ and $M_{2}: = \sup_{\ell \geq 0}{\psi_{2}}'(\ell)$ be real numbers, and let $\phi$ be defined as in Theorem 3.4. Then

is satisfied for the following cases:

(1)$\vartheta+\varpi = 1,$ and for all $\ell > 0;$

(2) $\vartheta+\varpi > 1,$ and for all $\ell\in[0, \ell_{0}];$

(3)$\vartheta+\varpi < 1,$ and for all $\ell\geq\ell_{0},$

where $\ell_{0} = (\vartheta+\varpi)^{\frac{1}{1-(\vartheta+\varpi)}}.$

4.   Some consequences

Beforehand, we would like to point out that various well-known inequalities result as particular cases of Theorem 3.3 and Corollaries 3.5–3.7. More concretely, Chebyshev inequalities for Riemann-Liouville, Prabhakar, Salim and other conformable fractional integral operators are obtained from that result.

More precisely, notice that if $\varrho = \imath$, $\sigma(0) = 1$, $\sigma(i) = 0$ for each $i \in \mathbb {N}$, $w = 1$, $\vartheta = 1$ and $\varpi = 0$, then we reach the Riemann-Liouville fractional integral inequality

(it is worth to point out here that Theorem 3.4 in reference ^[46] is wrong due to an erroneous application of Theorem 3.1 in the same reference). In the case that $\varrho = \imath$, $\sigma(0) = 1$, $\sigma(i) = 0$ for all $i \in \mathbb {N}$, we readily obtain

which was derived in [24,Theorem 2.4]. In turn, if $\varrho = \imath$, $\sigma(0) = 1$, $\sigma(i) = 0$ for each $i \in \mathbb {N}$ and $\varpi = 0$, then we have the following inequality for the Katugampola fractional integral operator:

Finally, if we let $\vartheta = 1$ and $\varpi = 0$ in Theorem 3.4, then we obtain [28,Theorem 5]:

As pointed out previously, it is easy to derive versions of the inequality (3.20) for the Prabhakar and Salim fractional integrals, which read as

and

Similarly, appropriate choices of the parameters in the conformable Raina fractional integral we can reduce the conclusions of Corollaries 3.5–3.7 to some known inequalities.

Theorem 4.1. Let $\varrho, \rho > 0$, $w \in \mathbb {R}$, $\vartheta \in (0, 1]$ and $\varpi \geq 0$. Suppose that $h$ is a positive function on $[0, \infty)$, and that ${\psi_{1}}$ and ${\psi_{2}}$ are differentiable functions on $[0, \infty)$. If $\psi_{1}' \in L_{r}[0, \infty), \psi_{2}' \in L_{s}[0, \infty)$ with $r > 1$ and $\frac {1} {r} + \frac {1} {s} = 1$, then

Proof. Let $h$, ${\psi_{1}}$ and ${\psi_{2}}$ be as in the hypotheses of this theorem. For each $\tau, \nu \in (0, {\bf k})$ and ${\bf k} > 0$ we define the function $H(\tau, \nu):= ({\psi_{1}}(\tau)-{\psi_{1}}(\nu))({\psi_{2}}(\tau)-{\psi_{2}}(\nu))$. Multiplying this identity by $\left(\frac{{\bf k}^{\vartheta + \varpi} -\tau^{\vartheta + \varpi}}{\vartheta + \varpi}\right)^{\varrho -1} \tau^{\vartheta + \varpi -1}\mathcal{F}_{\rho, \varrho}^{\sigma}[w({\bf k}^{\vartheta + \varpi}-\tau^{\vartheta + \varpi})^{\rho}]h(\tau)$ with $\tau \in (0, {\bf k})$, and integrating then over $(0, {\bf k})$, we obtain

Multiplying (4.8) by $\left(\frac{{\bf k}^{\vartheta + \varpi} -\nu^{\vartheta + \varpi}}{\vartheta + \varpi}\right)^{\varrho -1} \nu^{\vartheta + \varpi -1}\mathcal{F}_{\rho, \varrho}^{\sigma}[w({\bf k}^{\vartheta + \varpi}-\nu^{\vartheta+ \varpi})^{\rho}]h(\nu)$ with $\nu \in (0, {\bf k})$, and integrating then over $(0, {\bf k})$ yields

On the other hand, rewriting $H$ as a double integral and using Hölder's inequality for double integrals we obtain

Suppose that $0 < \left|\tau-\nu\right| < {\bf k}$ is satisfied. Using (4.10) in (4.9), applying Hölder's inequality and using that $\psi_{1}' \in L_{r}[0, \infty)$, $\psi_{2}' \in L_{s}[0, \infty)$ and $\frac {1} {r} + \frac {1} {s} = 1$, it follows that

which completes the proof.

Corollary 4.2. Let $\varrho, \rho > 0$, $w \in \mathbb {R}$, $\vartheta \in (0, 1]$ and $\varpi \geq 0$, and suppose that ${\psi_{1}}$ and ${\psi_{2}}$ are two differentiable functions on $[0, \infty)$. If $\psi_{1}'\in L_{r}[0, \infty)$, $\psi_{2}'\in L_{s}[0, \infty)$, $r > 1$ and $\frac {1} {r} + \frac {1} {s} = 1$, then the following inequality is satisfied:

Proof. The proof is reached by applying Theorem 4.1 with $h \equiv 1$.

Before closing this section, it is worth pointing out that some well-known particular inequalities are readily obtained for specific values of the parameters in Theorem 4.1. For example, if $\varrho = \imath$, $\sigma(0) = 1$, $\sigma(i) = 0$ for each $i \in \mathbb {N}$, $w = 0$, $\vartheta = 1$ and $\varpi = 0$, then we obtain the following inequality for the Riemann-Liouville fractional integral operator:

This inequality was derived in [26,Theorem 3.1]. If, in addition, $h\equiv1$, then we reach

which was obtained in [26,Corollary 3.3]. On the other hand, letting $\varrho = \imath$, $\sigma(0) = 1$, $\sigma(i) = 0$ for each $i \in \mathbb {N}$ and $w = 1$ yields

and if $\varpi = 0$, we obtain an inequality corresponding to the Katugampola fractional integral. Meanwhile, if $\sigma(i) = \frac{(\eta)_{i}}{i!}$, $\rho = \imath$, $\varrho = \beta$, $\vartheta = 1$ and $\varpi = 0$ in Theorem 4.1, we obtain the inequality associated to the Prabhakar fractional integral operator. Finally, when $\sigma(i) = \frac{(\gamma)_{qi}}{(\delta)_{pi}}$, $\rho = \imath$, $\varrho = \beta$, $\vartheta = 1$, and $\varpi = 0$ in Theorem 4.1, then we readily obtain the an inequality in terms of the Salim fractional integral.

5.   Conclusions

In this work, we proposed a generalization of the Raina fractional integral operators, and established a generalized form the Chebyshev inequality. Various generalizations of this inequality were obtained, considering various scenarios. More precisely, we established generalizations of the Chebyshev inequality for the product of an arbitrary finite number of functions, and for pairs of functions whose derivatives are bounded from below or from above. In addition, an estimate for the Chebyshev functional was established using the generalized Raina fractional integral operators. Some particular forms of the Chebyshev inequality for the Prabhakar and Salim fractional integral operators were retrieved from our main results. In that sense, the fractional integral operators presented in this work yield more general results.

Acknowledgments

This work was supported by the Taif University Researchers Supporting Project (No. TURSP-2020/155), Taif University, Taif, Saudi Arabia.

The corresponding author (J. E. Macias-Diaz) was funded by the National Council for Science and Technology of Mexico (CONACYT) through grant A1-S-45928.

Conflict of interest

The authors declare no potential conflicts of interest in this work.