Some new Chebyshev type inequalities utilizing generalized fractional integral operators

1.   Introduction

In recent years, there has been an increasing interest in the fractional calculus. One of the significant motivations for such deep interest in the subject is its capability to model a number of natural phenomena, see, for example, the papers ^[9,12]. On the other hand Chebyshev inequality has broad practicability in statistical problems, numerical quadrature, probability and transform theory, and the bounding of special functions. Its basic appeal develops out of a desire to approximate, for instance, information in the form of a particular measure of the product of functions in terms of the products of the individual function measures. It is, also, of great interest in differential and difference equations ^[7,13].

The essential destination of the present study is to prove a Chebyshev type inequality for the generalized fractional integral operators. After some preliminaries and summarization of some previous known results in Section 2, Section 3 deals with general Chebyshev type inequalities for generalized fractional integral operators. Finally, some concluding remarks are given in Section 4.

2.   Preliminaries

In this section we recall some basic definitions and previous results which will be used in what follows.

In 1882, Chebyshev ^[3] proved the following inequality:

Theorem 2.1. Let $f$ and $g$ be two integrable functions in $[0, 1]$. If both functions are simultaneously increasing or decreasing for the same values of $x$ in $[0, 1]$, then

If one function is increasing and the other is decreasing for the same values of $x$ in $[0, 1]$, then (2.1) reverses. In the last years, many papers were devoted to the generalization of the inequalities (2.1), we can mention the works ^{[1,2,4,5,6,8,11,15,16,17,18,19]}.

2.1.   Generalized fractional integral operators

In ^[14], Raina defined the following results connected with the general class of fractional integral operators.

where the coefficients $\sigma \left(k\right)$ $\left(k\in \mathbb{N}_{0} = \mathbb{N\cup }\left \{ 0\right \} \right)$ is a bounded sequence of positive real numbers and $\mathcal{R}$ is the set of real numbers. With the help of (2.2), in ^[14], Raina defined the following fractional integral operators, as follows:

The importance of these operators stems indeed from their generality. Many useful fractional integral operators can be obtained by specializing the coefficient $\sigma \left(k\right)$. Here, we just point out that the classical Riemann-Liouville fractional integrals $I_{a^{+}}^{\alpha }$ of order $\alpha$ defined by (see, ^[10])

follow easily by setting

in (2.3).

In ^[19], Usta et. al gave the following Chebychev type inequalities for the generalized fractional integral operators:

Theorem 2.2. Let $f$ and $g$ be two synchronous functions on $\left[0, \infty \right)$, that is they are having the same sense of variation on $\left[0, \infty \right)$. Then for all $t, \rho, \lambda > 0$ and $w\in \mathbb{R}$, we have

where the coefficients $\sigma \left(k\right)$ $\left(k\in \mathbb{N}_{0} = \mathbb{N\cup }\left\{ 0\right\} \right)$ is a bounded sequence of positive real numbers.

Theorem 2.3. Let $f$ and $g$ be two synchronous functions on $\left[0, \infty \right)$, that is they are having the same sense of variation on $\left[0, \infty \right)$. Then for all $t > 0$ and $\rho _{1}, \rho _{2}, \lambda _{1}, \lambda _{2} > 0$ and $w_{1}, w_{2}\in \mathbb{R}$, we have

where the coefficients $\sigma _{1}\left(k\right), \sigma _{2}\left(k\right)$ $\left(k\in \mathbb{N}_{0} = \mathbb{N\cup }\left\{ 0\right\} \right)$ are bounded sequences of positive real numbers.

For more details, one may consult ^[19].

3.   Main findings & Cumulative results

In this section, we will present some fractional integral inequalities for functions defined on the positive real line with the help of fractional integral operators given above. The inequalities to be given in this section are a generalization of the former inequalities.

Theorem 3.1. Let $f$ and $g$ be two synchronous functions on $\left[0, \infty \right)$, that is they are having the same sense of variation on $\left[0, \infty \right)$ and $h\geq 0$. Then for all $t > 0$ and $\rho_1, \rho_2, \lambda_1, \lambda_2 > 0$ and $w_1, w_2\in \mathbb{R}$, we have

where the coefficients $\sigma_1 \left(k\right), \sigma_2 \left(k\right)$ $\left(k\in \mathbb{N}_{0} = \mathbb{N\cup }\left\{ 0\right\} \right)$ are bounded sequences of positive real numbers.

Proof. Let $f$, $g$ and $h$ be three functions satisfying the conditions of Theorem 3.1. Then, we have

Therefore

Now, by multiplying both sides of (3.1) by $\left(t-\eta \right) ^{\lambda _{1}-1}\left(t-\xi \right) ^{\lambda _{2}-1}\mathcal{F}_{\rho _{1}, \lambda _{1}}^{\sigma _{1}}\left[\omega _{1}\left(t-\eta \right) ^{\rho _{1}}\right] \mathcal{F}_{\rho _{2}, \lambda _{2}}^{\sigma _{2}}\left[\omega _{2}\left(t-\xi \right) ^{\rho _{2}}\right]$, we obtain:

Finally, if we double integrate (3.2) with respect to $\eta$ and $\xi$ over $(0, t)\times (0, t)$, we get the desired result.

Corollary 3.2. Choosing $\lambda _{1} = \lambda _{2} = \lambda,$ $\sigma _{1} = \sigma _{2} = \sigma,$ $\rho _{1} = \rho _{2} = \rho$ and $w_{1} = w_{2} = w$ in Theorem 3.1, we obtain the following inequality

where the coefficients $\sigma \left(k\right)$ $\left(k\in \mathbb{N}_{0} = \mathbb{N\cup }\left\{ 0\right\} \right)$ is a bounded sequence of positive real numbers.

Remark 3.3. If we choose $h = 1$ in Theorem 3.1, Theorem 3.1 reduce to Theorem 2.3 which was proved by Usta et al. in ^[19].

Theorem 3.4. Let $f,$ $g$ and $h$ be three monotonic functions defined on $\left[0, \infty \right)$, satisfying the following

for all $\eta, \xi \in \left[0, t\right],$ then for all $t > 0$ and $\rho_1, \rho_2, \lambda_1, \lambda_2 > 0$ and $w_1, w_2\in \mathbb{R}$, we have

where the coefficients $\sigma_1 \left(k\right), \sigma_2 \left(k\right)$ $\left(k\in \mathbb{N}_{0} = \mathbb{N\cup }\left\{ 0\right\} \right)$ are bounded sequences of positive real numbers.

Proof. The proof is similar to previous theorem.

Theorem 3.5. Let $f,$ $g$ be two functions on $[0, \infty).$ Then for all $t > 0$ and $\rho_1, \rho_2, \lambda_1, \lambda_2 > 0$ and $w_1, w_2\in \mathbb{R}$, we have

where the coefficients $\sigma_1 \left(k\right), \sigma_2 \left(k\right)$ $\left(k\in \mathbb{N}_{0} = \mathbb{N\cup }\left\{ 0\right\} \right)$ are bounded sequences of positive real numbers.

Proof. As

we have

Now, by multiplying both sides of (3.3) by $\left(t-\eta \right) ^{\lambda _{1}-1}\left(t-\xi \right) ^{\lambda _{2}-1}\mathcal{F}_{\rho _{1}, \lambda _{1}}^{\sigma _{1}}\left[\omega _{1}\left(t-\eta \right) ^{\rho _{1}}\right] \mathcal{F}_{\rho _{2}, \lambda _{2}}^{\sigma _{2}}\left[\omega _{2}\left(t-\xi \right) ^{\rho _{2}}\right]$, we get

Finally by double integration (3.4) over $(0, t)\times (0, t)$, we get the desired result.

Corollary 3.6. Choosing $\lambda _{1} = \lambda _{2} = \lambda,$ $\sigma _{1} = \sigma _{2} = \sigma,$ $\rho _{1} = \rho _{2} = \rho$ and $w_{1} = w_{2} = w$ in Theorem 3.5, we obtain the following inequality

In particular

where the coefficients $\sigma \left(k\right)$ $\left(k\in \mathbb{N}_{0} = \mathbb{N\cup }\left\{ 0\right\} \right)$ is a bounded sequence of positive real numbers.

Theorem 3.7. Let $f,$ $g$ be two functions on $[0, \infty).$ Then for all $t > 0$ and $\rho_1, \rho_2, \lambda_1, \lambda_2 > 0$ and $w_1, w_2\in \mathbb{R}$, we have

where the coefficents $\sigma_1 \left(k\right), \sigma_2 \left(k\right)$ $\left(k\in \mathbb{N}_{0} = \mathbb{N\cup }\left\{ 0\right\} \right)$ are bounded sequences of positive real numbers.

Proof. As

we have

Then by multiplying both sides of (3.5) by $\left(t-\eta \right) ^{\lambda _{1}-1}\left(t-\xi \right) ^{\lambda _{2}-1}\mathcal{F}_{\rho _{1}, \lambda _{1}}^{\sigma _{1}}\left[\omega _{1}\left(t-\eta \right) ^{\rho _{1}}\right] \mathcal{F}_{\rho _{2}, \lambda _{2}}^{\sigma _{2}}\left[\omega _{2}\left(t-\xi \right) ^{\rho _{2}}\right]$ and following the similar steps in Theorem 3.5, we get the desired result.

Corollary 3.8. Choosing $\lambda _{1} = \lambda _{2} = \lambda,$ $\sigma _{1} = \sigma _{2} = \sigma,$ $\rho _{1} = \rho _{2} = \rho$ and $w_{1} = w_{2} = w$ in Theorem 3.7, we obtain the following inequality

where the coefficients $\sigma \left(k\right)$ $\left(k\in \mathbb{N}_{0} = \mathbb{N\cup }\left\{ 0\right\} \right)$ is a bounded sequence of positive real numbers.

Lemma 3.9. Let $f: \mathbb{R} \rightarrow \mathbb{R},$ and define

then for all $t, \rho > 0,$ $\lambda > 1$ and $w\in \mathbb{R}$, we have

where the coefficients $\sigma \left(k\right)$ $\left(k\in \mathbb{N}_{0} = \mathbb{N\cup }\left\{ 0\right\} \right)$ is a bounded sequence of positive real numbers.

Proof.

Theorem 3.10. Let $f$ and $g$ be two functions on $[0, \infty),$ Then for all $t, \rho > 0,$ $\lambda _{1}, \lambda _{2} > 1$ and $w_{1}, w_{2}\in \mathbb{R}$, we have

where the coefficients $\sigma_1 \left(k\right), \sigma_2 \left(k\right)$ $\left(k\in \mathbb{N}_{0} = \mathbb{N\cup }\left\{ 0\right\} \right)$ are bounded sequences of positive real numbers.

Proof. From Lemma 3.9 and inequality (2.7), we have

which completes the proof.

Corollary 3.11. Choosing $\lambda _{1} = \lambda _{2} = \lambda,$ $\sigma _{1} = \sigma _{2} = \sigma,$ $\rho _{1} = \rho _{2} = \rho$ and $w_{1} = w_{2} = w$ in Theorem 3.10, we obtain the following inequality

where the coefficients $\sigma \left(k\right)$ $\left(k\in \mathbb{N}_{0} = \mathbb{N\cup }\left\{ 0\right\} \right)$ is a bounded sequence of positive real numbers.

Theorem 3.12. Let $f$ and $g$ be two differentiable function on $[0, \infty).$ Then for all $t, \rho > 0,$ $\lambda _{1}, \lambda _{2} > 0$ and $w_{1}, w_{2}\in \mathbb{R}$, we have

where

and the coefficients $\sigma _{1}\left(k\right), \sigma _{2}\left(k\right)$ $\left(k\in \mathbb{N}_{0} = \mathbb{N\cup }\left\{ 0\right\} \right)$ are bounded sequences of positive real numbers.

Proof. With basic calculation, we have

Then taking modulus of the above equality, we find that

which completes the proof.

Corollary 3.13. Choosing $\lambda _{1} = \lambda _{2} = \lambda,$ $\sigma _{1} = \sigma _{2} = \sigma,$ $\rho _{1} = \rho _{2} = \rho$ and $w_{1} = w_{2} = w$ in Theorem 3.12, we obtain the following inequality

where the coefficients $\sigma \left(k\right)$ $\left(k\in \mathbb{N}_{0} = \mathbb{N\cup }\left\{ 0\right\} \right)$ is a bounded sequence of positive real numbers.

Remark 3.14. If we choose $\lambda _{1} = \alpha,$ $\lambda _{2} = \beta, \ \sigma _{1}(0) = \sigma _{2}(0) = 1, \ $and $w_{1} = w_{2} = 0$, in Theorem 3.1, Theorem 3.4, Theorem 3.5, Theorem 3.7, Theorem 3.10 and Theorem 3.12, then the inequalities reduces to Theorem 2.1, Theorem 2.2, Theorem 2.3, Theorem 2.4, Theorem 2.6 and Theorem 2.7 proved by Sulaiman in ^[18], respectively.

Theorem 3.15. Let $f$ and $g$ be two synchronous functions on $\left[0, \infty \right)$, that is they are having the same sense of variation on $\left[0, \infty \right),$ and let $v_{1},$ $v_{2}:\left[0, \infty \right) \rightarrow \left[0, \infty \right).$ Then for all $t, \rho > 0,$ $\lambda _{1}, \lambda _{2} > 0$ and $w_{1}, w_{2}\in \mathbb{R}$, we have

where the coefficients $\sigma_1 \left(k\right), \sigma_2 \left(k\right)$ $\left(k\in \mathbb{N}_{0} = \mathbb{N\cup }\left\{ 0\right\} \right)$ are bounded sequences of positive real numbers.

Proof. As $f$ and $g$ be two synchronous functions on $\left[0, \infty \right),$ then for all $\eta, \xi \geq 0$ we have

Therefore

Mutlipying both sides of (3.7) by $\left(t-\eta \right) ^{\lambda _{1}-1}\mathcal{F}_{\rho _{1}, \lambda _{1}}^{\sigma _{1}}\left[\omega _{1}\left(t-\eta \right) ^{\rho _{1}}\right] v_{1}(\eta),$ $\eta \in \left(0, t\right),$ we find that

Integrating (3.7) with repect to $\eta$ over $\left(0, t\right),$ we get

Now, similarly, by multiplying both sides of (3.9) by $\left(t-\xi \right) ^{\lambda _{2}-1}\mathcal{F}_{\rho _{2}, \lambda _{2}}^{\sigma _{2}} \left[\omega _{2}\left(t-\xi \right) ^{\rho _{2}}\right] v_{2}(\xi)$, $\xi \in \left(0, t\right)$ and integrating with respect to $\xi$ over $\left(0, t\right),$ we get the desired result.

Corollary 3.16. Choosing $\lambda _{1} = \lambda _{2} = \lambda,$ $\sigma _{1} = \sigma _{2} = \sigma,$ $\rho _{1} = \rho _{2} = \rho$ and $w_{1} = w_{2} = w$ in Theorem 3.15, we obtain the following inequality

where the coefficients $\sigma \left(k\right)$ $\left(k\in \mathbb{N}_{0} = \mathbb{N\cup }\left\{ 0\right\} \right)$ is a bounded sequence of positive real numbers.

Remark 3.17. If we choose $v_{1} = v_{2} = 1$ in Theorem 3.15, the inequality (3.14) reduce to inequality (2.7).

Theorem 3.18. Let $f$ and $g$ be two synchronous functions on $\left[0, \infty \right)$, that is they are having the same sense of variation on $\left[0, \infty \right),$ and let $p, q, r:\left[0, \infty \right) \rightarrow \left[0, \infty \right).$ Then for all $t, \rho > 0,$ $\lambda _{1}, \lambda _{2} > 0$ and $w_{1}, w_{2}\in \mathbb{R}$, we have

where the coefficients $\sigma_1 \left(k\right), \sigma_2 \left(k\right)$ $\left(k\in \mathbb{N}_{0} = \mathbb{N\cup }\left\{ 0\right\} \right)$ are bounded sequences of positive real numbers.

Proof. If we choose $v_{1} = p$ and $v_{2} = q$ in Theorem 3.15, we can write:

Multiplying both sides of (3.10) by $\mathcal{J}_{\rho _{1}, \lambda _{1}, 0+;\omega _{1}}^{\sigma _{1}}\left(r\right) (t),$ we get

If we choose $v_{1} = r$ and $v_{2} = q$ in Theorem 3.15 and multiplying by $\mathcal{J}_{\rho _{1}, \lambda _{1}, 0+;\omega _{1}}^{\sigma _{1}}\left(p\right) (t),$ then we find that

Similarly, if we choose $v_{1} = p$ and $v_{2} = r$ in Theorem 3.15 and multiplying by $\mathcal{J}_{\rho _{1}, \lambda _{1}, 0+;\omega _{1}}^{\sigma _{1}}\left(q\right) (t),$ then we find that

Then by adding the inequalities of (3.11)-(3.13), the desired inequality has been obtained.

Corollary 3.19. Choosing $\lambda _{1} = \lambda _{2} = \lambda,$ $\sigma _{1} = \sigma _{2} = \sigma,$ $\rho _{1} = \rho _{2} = \rho$ and $w_{1} = w_{2} = w$ in Theorem 3.18, we obtain the following inequality

where the coefficients $\sigma \left(k\right)$ $\left(k\in \mathbb{N}_{0} = \mathbb{N\cup }\left\{ 0\right\} \right)$ is a bounded sequence of positive real numbers.

Remark 3.20. If we choose $\lambda _{1} = \alpha,$ $\lambda _{2} = \beta, \ \sigma _{1}(0) = \sigma _{2}(0) = 1, \ $and $w_{1} = w_{2} = 0$, in Theorem 3.15 and Theorem 3.18 then the inequalities reduces to Lemma 3 and Theorem 4 proved by Dahmani in ^[5], respectively.

Theorem 3.21. Let $v_{1}$ and $v_{2}$ be two positive functions on $[0, \infty)$ and let $f$ and $g$ be two differentiable functions on $\left[0, \infty \right)$.If $f^{\prime }\in L_{p}\left([0, \infty)\right),$ $g^{\prime }\in L_{q}\left([0, \infty)\right),$ $p > 1,$ $\frac{1}{p}+\frac{1}{q} = 1,$ then for all $t > 0$ and $\rho _{1}, \rho _{2}, \lambda _{1}, \lambda _{2} > 0$ and $w_{1}, w_{2}\in \mathbb{R}$, we have

where

and the coefficients $\sigma_1 \left(k\right), \sigma_2 \left(k\right)$ $\left(k\in \mathbb{N}_{0} = \mathbb{N\cup }\left\{ 0\right\} \right)$ are bounded sequences of positive real numbers.

Proof. Following the similar steps of proof of Theorem 8, we can write

where

Using the well-known Hölder inequality for double integral, we find that

Substituting (3.15) into (3.14), we have

Appying again Hölder inequality to the right hand side of (3.16), we find that

Now using the fact that $\left \vert \eta -\xi \right \vert \leq t$ and $\frac{1}{p}+\frac{1}{q} = 1,$ we have

Thus the proof is completed.

Corollary 3.22. If we choose $\lambda _{1} = \lambda _{2} = \lambda,$ $\sigma _{1} = \sigma _{2} = \sigma,$ $\rho _{1} = \rho _{2} = \rho,$ $w_{1} = w_{2} = w$ and $v_{1} = v_{2} = v$ in Theorem 3.21, we have the following inequality

where the coefficients $\sigma \left(k\right)$ $\left(k\in \mathbb{N}_{0} = \mathbb{N\cup }\left\{ 0\right\} \right)$ is a bounded sequence of positive real numbers.

Remark 3.23. In particular, putting $\lambda _{1} = \lambda _{2} = \alpha$, $\sigma _{1}(0) = \sigma _{2}(0) = 1$, and $w_{1} = w_{2} = 0,$ then Corollary 3.22 reduce to Theorem 3.1 proved by Dahmani et. al in ^[6].

Corollary 3.24. If we choose $v_{1} = v_{2} = v$ in Theorem 3.21, we have the following inequality

where the coefficients $\sigma_1 \left(k\right), \sigma_2 \left(k\right)$ $\left(k\in \mathbb{N}_{0} = \mathbb{N\cup }\left\{ 0\right\} \right)$ are bounded sequences of positive real numbers.

Remark 3.25. In particular, putting $\lambda _{1} = \alpha$, $\lambda _{2} = \beta,$ $\sigma _{1}(0) = \sigma _{2}(0) = 1$, and $w_{1} = w_{2} = 0,$ then Corollary 3.24 reduce to Theorem 3.2 proved by Dahmani et. al in ^[6].

Corollary 3.26. If we choose $v_{1} = v_{2} = 1$ in Theorem 3.21, we have the following inequality

where the coefficients $\sigma_1 \left(k\right), \sigma_2 \left(k\right)$ $\left(k\in \mathbb{N}_{0} = \mathbb{N\cup }\left\{ 0\right\} \right)$ are bounded sequences of positive real numbers.

Remark 3.27. In particular, putting $\lambda _{1} = \alpha$, $\lambda _{2} = \beta,$ $\sigma _{1}(0) = \sigma _{2}(0) = 1$, and $w_{1} = w_{2} = 0,$ then Corollary 3.26 reduce to Corollary 3.4 given by Dahmani et. al in ^[6].

4.   Concluding remarks

We have introduced a general version of Chebyshev type integral inequality for the generalised fractional integral operators based on two synchronous functions. The established results are generalization of some existing Chebychev type integral inequalities in the previous published studies. For further investigations we propose to consider the Chebyshev type inequalities for other fractional integral operators.

Conflict of interest

All authors declare no conflicts of interest in this paper.