Chebyshev type inequalities involving extended generalized fractional integral operators

Erhan Set; M. Emin Özdemir; Sevdenur Demirbaş; Erhan Set; M. Emin Özdemir; Sevdenur Demirbaş

doi:10.3934/math.2020232

AIMS Mathematics

2020, Volume 5, Issue 4: 3573-3583. doi: 10.3934/math.2020232

Previous Article Next Article

Research article

Chebyshev type inequalities involving extended generalized fractional integral operators

1.
Department of Mathematics, Faculty of Science and Arts, Ordu University, Ordu, Turkey
2.
Department of Mathematics Education, Education Faculty, Bursa Uludağ University, Görükle Campus, Bursa, Turkey

Received: 26 November 2019 Accepted: 27 March 2020 Published: 13 April 2020
MSC : 26A33, 26D10, 33B20

In this paper, mainly by using the extended generalized fractional integral operator that involve a further extension of Mittag-Leffler function in the kernel, we obtain several fractional Chebyshev type integral inequalities. So, results of Dahmani et al. from ^[4] are generalized. Also, it is point out that new results are obtained for different fractional integral operators with the help of special selection of parameters.

Keywords:

Chebyshev inequality,
fractional integral operators

Citation: Erhan Set, M. Emin Özdemir, Sevdenur Demirbaş. Chebyshev type inequalities involving extended generalized fractional integral operators[J]. AIMS Mathematics, 2020, 5(4): 3573-3583. doi: 10.3934/math.2020232

Related Papers:

[1]	Miguel Vivas-Cortez, Pshtiwan O. Mohammed, Y. S. Hamed, Artion Kashuri, Jorge E. Hernández, Jorge E. Macías-Díaz . On some generalized Raina-type fractional-order integral operators and related Chebyshev inequalities. AIMS Mathematics, 2022, 7(6): 10256-10275. doi: 10.3934/math.2022571
[2]	Hari M. Srivastava, Artion Kashuri, Pshtiwan Othman Mohammed, Abdullah M. Alsharif, Juan L. G. Guirao . New Chebyshev type inequalities via a general family of fractional integral operators with a modified Mittag-Leffler kernel. AIMS Mathematics, 2021, 6(10): 11167-11186. doi: 10.3934/math.2021648
[3]	Erhan Deniz, Ahmet Ocak Akdemir, Ebru Yüksel . New extensions of Chebyshev-Pólya-Szegö type inequalities via conformable integrals. AIMS Mathematics, 2020, 5(2): 956-965. doi: 10.3934/math.2020066
[4]	Fuat Usta, Hüseyin Budak, Mehmet Zeki Sarıkaya . Some new Chebyshev type inequalities utilizing generalized fractional integral operators. AIMS Mathematics, 2020, 5(2): 1147-1161. doi: 10.3934/math.2020079
[5]	Shahid Mubeen, Rana Safdar Ali, Iqra Nayab, Gauhar Rahman, Kottakkaran Sooppy Nisar, Dumitru Baleanu . Some generalized fractional integral inequalities with nonsingular function as a kernel. AIMS Mathematics, 2021, 6(4): 3352-3377. doi: 10.3934/math.2021201
[6]	Shuang-Shuang Zhou, Saima Rashid, Erhan Set, Abdulaziz Garba Ahmad, Y. S. Hamed . On more general inequalities for weighted generalized proportional Hadamard fractional integral operator with applications. AIMS Mathematics, 2021, 6(9): 9154-9176. doi: 10.3934/math.2021532
[7]	Shuang-Shuang Zhou, Saima Rashid, Asia Rauf, Fahd Jarad, Y. S. Hamed, Khadijah M. Abualnaja . Efficient computations for weighted generalized proportional fractional operators with respect to a monotone function. AIMS Mathematics, 2021, 6(8): 8001-8029. doi: 10.3934/math.2021465
[8]	Abd-Allah Hyder, Hüseyin Budak, Mohamed A. Barakat . Milne-Type inequalities via expanded fractional operators: A comparative study with different types of functions. AIMS Mathematics, 2024, 9(5): 11228-11246. doi: 10.3934/math.2024551
[9]	Muhammad Amer Latif, Humaira Kalsoom, Zareen A. Khan . Hermite-Hadamard-Fejér type fractional inequalities relating to a convex harmonic function and a positive symmetric increasing function. AIMS Mathematics, 2022, 7(3): 4176-4198. doi: 10.3934/math.2022232
[10]	Muhammad Imran Asjad, Waqas Ali Faridi, Mohammed M. Al-Shomrani, Abdullahi Yusuf . The generalization of Hermite-Hadamard type Inequality with exp-convexity involving non-singular fractional operator. AIMS Mathematics, 2022, 7(4): 7040-7055. doi: 10.3934/math.2022392

Abstract

1. Introduction

In 1882, Chebyshev proved on interesting and useful integral inequality as follows:

$\begin{equation} \frac{1}{b-a}\int_{a}^{b} f(x)g(x)dx\geq \bigg(\frac{1}{b-a}\int_{a}^{b} f(x)dx\bigg)\bigg(\frac{1}{b-a}\int_{a}^{b}g(x)dx\bigg) \end{equation}$

(1.1)

where $f$ and $g$ are two integrable and synchronous functions on $[a, b]$ . Here two functions $f$ and $g$ are called synchronous on $[a, b]$ , if

$\begin{equation*} \big(f(x)-f(y)\big)\big(g(x)-g(y)\big)\geq 0 \, \, \, \, \, \, \, \, (x, y \in [a, b]). \end{equation*}$

The inequality (1.1) that is well known as Chebyshev inequality has many applications in diverse research subjects such as numerical quadrature, transform theory, probability, existence of solutions of differential equations and statistical problems. Therefore, many researchers have given considerable attention to this inequality, (see ^{[3,10,11,12,18,19,20,21,22]}.

On the other hand, one of the methods used to generalize inequalities is fractional calculus. In this context, firstly, in 2009, Chebyshev inequality involving Riemann-Liouville fractional integrals is presented as the following:

Theorem 1.1. (^[2]) Let $f$ and $g$ be two synchronous function on $[0, \infty)$ . Then for all $t > 0, \, \, \, \alpha > 0$ , we have:

$\begin{equation} J^{\alpha}(f, g)(t) \geq\frac{\Gamma(\alpha+1)}{t^{\alpha}} J^{\alpha}f(t) J^{\alpha}g(t) \end{equation}$

(1.2)

where $J^{\alpha}f(t)$ denotes Riemann-Liouville fractional integral operator of a function $f(t)$ and $\Gamma$ is the Gamma function such that these are defined as follows (see e.g. ^[17]).

Let $f \in L[a, b]$ . The Riemann-Liouville fractional integrals $J_{a^{+}}^{\alpha} f$ and $J_{b^{-}}^{\alpha} f$ of order $\alpha > 0$ are defined by

$\begin{eqnarray*} J_{a^{+}}^{\alpha} f(x) & = &\frac{1}{\Gamma(\alpha)}\int_{a}^{x}(x-t)^{\alpha-1}f(t)dt, \, \, \, \, \, x \gt 0\\ J_{b^{-}}^{\alpha} f(x) & = &\frac{1}{\Gamma(\alpha)}\int_{x}^{b}(t-x)^{\alpha-1}f(t)dt , \, \, \, \, x \lt b \end{eqnarray*}$

respectively. Also the gamma function is defined by

$\begin{equation*} \Gamma(x) = \int_{0}^{\infty}e^{-t}t^{\alpha-1} dt. \end{equation*}$

Here is $J_{a^{+}}^{0}f(x) = J_{L}^{0}-f(x) = f(x)$ . In the case of $\alpha = 1$ , the this fractional integral reduces to the classical integral.

After this study of Belarbi and Dahmani, generalizations of Chebyshev inequality were obtained for the different types of fractional integral operator with similar technique. (see, e.g. ^{[2,5,6,7,14,23,24,25,28]}.

Recently, the new generalizations of the Riemann-Liouville fractional integral operator, have been described with the help of various extensions of the Mittag-Leffler function. Now lets give some of these operators which we will need in the second section.

Definition 1.1. (^[13]) Let $\alpha, \beta, \rho, \lambda \in \mathbb{C}$ , $Re(\alpha) > 0$ and $Re(\beta) > 0$ . Let $f \in L[a, b]$ and $x \in [a, b]$ . Then the fractional integral operator $\epsilon (\alpha, \beta, \rho, \lambda)$ defined by Prabhakar is as the following:

$\begin{equation*} \epsilon (\alpha, \beta, \rho, \lambda) f(x) = \int _a ^x (x-t)^{\beta-1} E^{\rho} _{\alpha, \beta} \lambda (x-t)^{\alpha} f(t) dt \end{equation*}$

where

$\begin{equation*} E^{\rho} _{\alpha, \beta} = \sum\limits ^{\infty} _{n = 0} \frac{(\rho)_n z^n}{\Gamma (\alpha n + \beta) n!} \end{equation*}$

and $\Gamma$ is the Gamma function.

Definition 1.2. (^[26]) Let $z, \beta, \gamma, \omega \in \mathbb{C}$ , $Re(\alpha) > max\{0, Re(\kappa)-1\}$ , $min\{Re(\beta), Re(\kappa)\} > 0$ . Let $f \in L[a, b]$ and $x \in [a, b]$ . Then the fractional integral operator $\epsilon_{a^{+}; \alpha, \beta}^{\omega; \gamma, \kappa}\varphi$ defined by Srivastava and Saxena is as the following:

$\begin{equation*} ( \epsilon_{a^{+};\alpha, \beta}^{\omega;\gamma, \kappa}\varphi)(x) = \int_{a}^{x}(x-t)^{\beta-1} E_{\alpha, \beta}^{\gamma, \kappa}[\omega(x-t)^{\alpha}] \varphi(t) dt \, \, \, \, \ (x \gt a) \end{equation*}$

where

$\begin{equation*} E_{\alpha, \beta}^{\gamma, \kappa} (z) = \sum \limits_{n = 0}^{\infty} \frac{(\gamma)_{\kappa n}}{\Gamma(\alpha n+\beta)} \frac{z^{n}}{n!} \end{equation*}$

and $\Gamma$ is the Gamma function.

Definition 1.3. (^[16]) Let $\alpha, \beta, \gamma, \delta \in \mathbb{C}$ , $min\{Re(\alpha), Re(\beta), Re(\gamma), Re(\delta)\} > 0$ , $p, q > 0$ and $q\leq Re(\alpha)+p$ . Let $f \in L[a, b]$ and $x \in [a, b]$ . Then the fractional integral operator $\epsilon_{\alpha, \beta, p, \omega, a^{+}}^{\gamma, \delta, q}$ defined by Salim and Faraj is as the following:

$\begin{equation*} \epsilon_{\alpha, \beta, p, \omega, a^{+}}^{\gamma, \delta, q}(x) = \int_{a}^{x}(x-t)^{\beta-1} E_{\alpha, \beta, p}^{\gamma, \delta, q}(\omega(x-t)^{\alpha})\varphi(t)dt \end{equation*}$

where

$\begin{equation*} E_{\alpha, \beta, p}^{\gamma, \delta, q}(z) = \sum\limits_{n = 0}^{\infty}\frac{\gamma_{qn}}{\Gamma(\alpha n+\beta)}\frac{z^{n}}{(\delta)_{pn}} \end{equation*}$

and $\Gamma$ is the Gamma function.

Definition 1.4 (^[15]) Let $p \geq 0$ , $q > 0$ , $\omega, \delta, \lambda, \sigma, c, \rho \in \mathbb{C}$ , $Re(c) > 0$ , $Re(\rho) > 0$ and $Re(\sigma) > 0$ . Let $f \in L[a, b]$ and $x \in [a, b]$ . Then the fractional integral operator $(\epsilon ^{\omega, \delta, q, c} _{a^{+}, \rho, \sigma} f)$ defined by Rahman et al. is as the following:

$\begin{equation*} (\epsilon ^{\omega, \delta, q, c} _{a^{+}, \rho, \sigma} f) (x) = \int ^x _a (x-\tau)^{\sigma-1} E^{\delta, q, c} _{p, \sigma} (\omega(x-\tau)^{\rho} ;p) f(\tau) d\tau \end{equation*}$

where

$\begin{equation*} E^{\delta, q, c} _{\rho, \sigma} (z;p) = \sum\limits ^{\infty} _{n = 0} \frac{B_p (\delta+nq, c-\delta)}{B(\delta, c-\delta)} \frac{(c)_{nq}}{\Gamma(\rho n+\sigma)}\frac{z^n}{n!} \end{equation*}$

and $B_p (x, y)$ is an extension of Beta function defined in ^[15]

$\begin{equation} B_{p}(x, y) = \int _0 ^1 t^{x-1} (1-t)^{y-1} e ^{- \frac{p}{t(1-t)}} dt \ \ \ \ x, y, p \gt 0 \end{equation}$

(1.3)

where $Re(p) > 0$ , $Re(x) > 0$ and $Re(y) > 0$ . Also, here $B$ is familiar Beta function as follows:

$\begin{equation} B \left( a, b\right) = \frac{\Gamma (a)\Gamma (b)}{\Gamma (a+b)} = \int_{0}^{1}t^{a-1}\left( 1-t\right) ^{b-1}dt, \ \ \ a, b \gt 0. \end{equation}$

(1.4)

Definition 1.5. (^[1]) Let $\omega, \alpha, \beta, \sigma, \delta, c \in \mathbb{C}$ , $Re(\alpha), Re(\beta), Re(\sigma), Re(\delta), Re(c) > 0$ with $p \geq 0$ , $q > 0$ and $0 < r\leq q+Re(\alpha)$ . Let $f \in L_1 [a, b]$ and $x \in [a, b]$ . Then the generalized fractional integral operator $\epsilon ^{\omega, \delta, q, r, c} _{a^+, \alpha, \beta, \sigma} f$ is defined by

$\begin{equation} \left(\epsilon ^{\omega, \delta, q, r, c} _{a^+, \alpha, \beta, \sigma} f\right) (x;p) = \int ^x _a (x-t)^{\beta-1} E^{\delta, q, r, c} _{\alpha, \beta, \sigma} (\omega (x-t)^\alpha; p) f(t) dt \end{equation}$

(1.5)

where

$\begin{equation*} E^{\delta, q, r, c} _{\alpha, \beta, \sigma}(z;p) = \sum\limits_{n = 0}^{\infty}\frac{B_{p}(\delta+nq, c-\delta)}{B(\delta, c-\delta)}\frac{(c)_{nq}}{\Gamma (\alpha n +\beta)}\frac{z^n}{(\sigma)_{nr}} \end{equation*}$

and $B_{p}$ and $B$ is as (1.3) and (1.4) respectively. For further information about this operator, (see ^[1,8,9,27]).

2. Main results

Theorem 2.1. Let $t$ be a positive function on $[0, \infty]$ and let $f$ and $g$ be two differentiable functions on $[0, \infty]$ . If $f'\in L_{r}$ $([0, \infty])$ , $g'\in L_{s}$ $([0, \infty])$ , $r > 1$ , $r^{-1}+s^{-1} = 1$ , then for all $x > 0$ , $\alpha > 0$ , $\beta > 0$ , we have

$\begin{eqnarray} &&2\bigg|(\epsilon_{0^{+}, \alpha, \beta, \sigma}^{\omega, \delta, q, r, c}t f g)(x;p)(\epsilon_{0^{+}, \alpha, \beta, \sigma}^{\omega, \delta, q, r, c}t)(x;p) -(\epsilon_{0^{+}, \alpha, \beta, \sigma}^{\omega, \delta, q, r, c}t f)(x;p)(\epsilon_{0^{+}, \alpha, \beta, \sigma}^{\omega, \delta, q, r, c}t g)(x;p)\bigg| \\ & \leq&||f'||_{r}||g'||_{s}\int_{0}^{x}\int_{0}^{x}(x-\tau)^{(\beta-1)} (x-\rho)^{(\beta-1)}|\tau-\rho| t(\tau)t(\rho) \\ &\times&E_{\alpha, \beta, \sigma}^{\delta, q, r, c}\Big(\omega(x-\tau)^{\alpha};p\Big)E_{\alpha, \beta, \sigma}^{\delta, q, r, c}\Big(\omega(x-\rho)^{\alpha};p\Big) d\tau d\rho\ \\ & \leq&||f'||_{r}||g'||_{s} x \big(\epsilon_{0^{+}, \alpha, \beta, \sigma}^{\omega, \delta, q, r, c}t(x;p)\big)^{2}. \end{eqnarray}$

(2.1)

Proof. Let $f$ and $g$ be two functions satisfying the conditions of Theorem 2.1 and let $t$ be a positive function on $[0, \infty]$ , Define

$\begin{equation} H(\tau, \rho): = \big(f(\tau)-f(\rho)\big)\big(g(\tau)-g(\rho)\big);\tau, \rho\in(0, x), x \gt 0. \end{equation}$

(2.2)

Multiplying (2.2) by

$\begin{equation*} (x-\tau)^{(\beta-1)} E_{\alpha, \beta, \sigma}^{\delta, q, r, c}\Big(\omega(x-\tau)^{\alpha};p\Big)\, \, t(\tau) ; \tau\in(0, x) \end{equation*}$

and integrating the resulting identity with respect to $\tau$ from $0$ $x$ , we can state that

$\begin{eqnarray} && \int_{0}^{x}(x-\tau)^{(\beta-1)} E_{\alpha, \beta, \sigma}^{\delta, q, r, c}\Big(\omega(x-\tau)^{\alpha};p\Big)t(\tau)H(\tau, \rho)d\tau \\ & = &(\epsilon_{0^{+}, \alpha, \beta, \sigma}^{\omega, \delta, q, r, c}t f g)(x;p)-f(\rho)(\epsilon_{0^{+}, \alpha, \beta, \sigma}^{\omega, \delta, q, r, c}t g)(x;p) \\ &-& g(\rho)(\epsilon_{0^{+}, \alpha, \beta, \sigma}^{\omega, \delta, q, r, c}t f)(x;p)+f(\rho)g(\rho) (\epsilon_{0^{+}, \alpha, \beta, \sigma}^{\omega, \delta, q, r, c}t)(x;p) \end{eqnarray}$

(2.3)

Now, multiplying (2.3) by

$\begin{equation*} (x-\rho)^{(\beta-1)} E_{\alpha, \beta, \sigma}^{\delta, q, r, c}\Big(\omega(x-\rho)^{\alpha};p\Big)t(\rho) ; \rho\in(0, x) \end{equation*}$

and integrating the resulting identity with respect to $\rho$ over $(0, x)$ , we can write

$\begin{eqnarray*} &&\int_{0}^{x}\int_{0}^{x}(x-\tau)^{(\beta-1)} (x-\rho)^{(\beta-1)}E_{\alpha, \beta, \sigma}^{\delta, q, r, c}\Big(\omega(x-\tau)^{\alpha};p\Big)\nonumber \\ &\times&E_{\alpha, \beta, \sigma}^{\delta, q, r, c}\Big(\omega(x-\rho)^{\alpha};p\Big)t(\tau)t(\rho)H(\tau, \rho)d\tau d\rho\nonumber \\ & = & (\epsilon_{0^{+}, \alpha, \beta, \sigma}^{\omega, \delta, q, r, c}t f g)(x;p)(\epsilon_{0^{+}, \alpha, \beta, \sigma}^{\omega, \delta, q, r, c}t)(x;p)-(\epsilon_{0^{+}, \alpha, \beta, \sigma}^{\omega, \delta, q, r, c}t f)(x;p)(\epsilon_{0^{+}, \alpha, \beta, \sigma}^{\omega, \delta, q, r, c}t g)(x;p) \nonumber \\ & - & (\epsilon_{0^{+}, \alpha, \beta, \sigma}^{\omega, \delta, q, r, c}t g)(x;p)(\epsilon_{0^{+}, \alpha, \beta, \sigma}^{\omega, \delta, q, r, c}t f)(x;p)+ (\epsilon_{0^{+}, \alpha, \beta, \sigma}^{\omega, \delta, q, r, c}t f g)(x;p)(\epsilon_{0^{+}, \alpha, \beta, \sigma}^{\omega, \delta, q, r, c}t)(x;p) \end{eqnarray*}$

Consequently,

(2.4)

On the other and, we have

$\begin{equation} H(\tau, \rho): = \int_{\tau}^{\rho}\int_{\tau}^{\rho}f'(y)g'(z)dydz. \end{equation}$

(2.5)

Using Hölder inequality for double integral, we can write

$\begin{equation} |H(\tau, \rho)|\leq \bigg|\int_{\tau}^{\rho}\int_{\tau}^{\rho}|f'(y)|^{r}dydz\bigg|^{r^{-1}}\bigg|\int_{\tau}^{\rho}\int_{\tau}^{\rho}|g'(z)|^{s}dydz\bigg|^{s^{-1}} \end{equation}$

(2.6)

Since,

$\begin{equation} \bigg|\int_{\tau}^{\rho}\int_{\tau}^{\rho}|f'(y)|^{r}dydz\bigg|^{r^{-1}} = |\tau-\rho|^{r^{-1}}\bigg|\int_{\tau}^{\rho}|f'(y)|^{r}dy\bigg|^{r^{-1}} \end{equation}$

(2.7)

and

$\begin{equation} \bigg|\int_{\tau}^{\rho}\int_{\tau}^{\rho}|g'(z)|^{s}dydz\bigg|^{s^{-1}} = |\tau-\rho|^{s^{-1}}\bigg|\int_{\tau}^{\rho}|g'(z)|^{s}dz\bigg|^{s^{-1}} \end{equation}$

(2.8)

then, we can estimate $H$ as follows:

$\begin{equation} |H(\tau, \rho)|\leq |\tau-\rho| \bigg|\int_{\tau}^{\rho}|f'(y)|^{r}dy\bigg|^{r^{-1}}\bigg|\int_{\tau}^{\rho}|g'(z)|^{s}dz\bigg|^{s^{-1}} \end{equation}$

(2.9)

On the other hand, we have

$\begin{eqnarray} && \int_{0}^{x}\int_{0}^{x}(x-\tau)^{(\beta-1)} (x-\rho)^{(\beta-1)}E_{\alpha, \beta, \sigma}^{\delta, q, r, c}\Big(\omega(x-\tau)^{\alpha};p\Big) \\ &\times &E_{\alpha, \beta, \sigma}^{\delta, q, r, c}\Big(\omega(x-\rho)^{\alpha};p\Big)t(\tau)t(\rho)\big|H(\tau, \rho)\big|d\tau d\rho \\ &\leq&\int_{0}^{x}\int_{0}^{x}(x-\tau)^{(\beta-1)} (x-\rho)^{(\beta-1)}|\tau-\rho|t(\tau)t(\rho)E_{\alpha, \beta, \sigma}^{\delta, q, r, c}\Big(\omega(x-\tau)^{\alpha};p\Big) \\ &\times& E_{\alpha, \beta, \sigma}^{\delta, q, r, c}\Big(\omega(x-\rho)^{\alpha};p\Big) \\ &\times& \bigg|\int_{\tau}^{\rho}|f'(y)|^{r}dy\bigg|^{r^{-1}}\bigg|\int_{\tau}^{\rho}|g'(z)|^{s}dz\bigg|^{s^{-1}}d\tau d\rho \end{eqnarray}$

(2.10)

Applying again Hölder inequality to the right-hand side of (2.10), we can state that

$\begin{eqnarray*} &&\int_{0}^{x}\int_{0}^{x}(x-\tau)^{(\beta-1)} (x-\rho)^{(\beta-1)}E_{\alpha, \beta, \sigma}^{\delta, q, r, c}\Big(\omega(x-\tau)^{\alpha};p\Big)\\ \times&& E_{\alpha, \beta, \sigma}^{\delta, q, r, c}\Big(\omega(x-\rho)^{\alpha};p\Big) t(\tau)t(\rho)\big|H(\tau, \rho)\big|d\tau d\rho \\ \leq&& \bigg[\int_{0}^{x}\int_{0}^{x}(x-\tau)^{(\beta-1)} (x-\rho)^{(\beta-1)}|\tau-\rho|t(\tau)t(\rho) E_{\alpha, \beta, \sigma}^{\delta, q, r, c}\Big(\omega(x-\tau)^{\alpha};p\Big) \\ \times && E_{\alpha, \beta, \sigma}^{\delta, q, r, c}\Big(\omega(x-\rho)^{\alpha};p\Big)\Big|\int_{\tau}^{\rho}|f'(y)|^{r}dy\Big|d\tau d\rho\bigg]^{r^{-1}} \\ \times && \bigg[\int_{0}^{x}\int_{0}^{x}(x-\tau)^{(\beta-1)} (x-\rho)^{(\beta-1)}|\tau-\rho|t(\tau)t(\rho) E_{\alpha, \beta, \sigma}^{\delta, q, r, c}\Big(\omega(x-\tau)^{\alpha};p\Big) \\ \times&& E_{\alpha, \beta, \sigma}^{\delta, q, r, c}\Big(\omega(x-\rho)^{\alpha};p\Big)\Big|\int_{\tau}^{\rho}|g'(z)|^{s}dz\Big|d\tau d\rho\bigg]^{s^{-1}}. \end{eqnarray*}$

Now, using the fact the

$\begin{equation} \Big|\int_{\tau}^{\rho}|f'(y)|^{r}dy\Big|\leq||f'||_{r^{, }}^{r}, \, \, \, \, \Big|\int_{\tau}^{\rho}|g'(z)|^{s}dz\Big|\leq||g'||_{s^{, }}^{s} \end{equation}$

(2.11)

we obtain

$\begin{eqnarray} && \int_{0}^{x}\int_{0}^{x}(x-\tau)^{(\beta-1)} (x-\rho)^{(\beta-1)}E_{\alpha, \beta, \sigma}^{\delta, q, r, c}\Big(\omega(x-\tau)^{\alpha};p\Big) \\ &\times &E_{\alpha, \beta, \sigma}^{\delta, q, r, c}\Big(\omega(x-\rho)^{\alpha};p\Big)t(\tau)t(\rho)\big|H(\tau, \rho)\big|d\tau d\rho \\ &\leq&\bigg[||f'||_{r^{, }}^{r}\int_{0}^{x}\int_{0}^{x}(x-\tau)^{(\beta-1)} (x-\rho)^{(\beta-1)}|\tau-\rho| t(\tau)t(\rho) \\ &\times&E_{\alpha, \beta, \sigma}^{\delta, q, r, c}\Big(\omega(x-\tau)^{\alpha};p\Big) E_{\alpha, \beta, \sigma}^{\delta, q, r, c}\Big(\omega(x-\rho)^{\alpha};p\Big) d\tau d\rho\bigg]^{r^{-1}} \\ &\times& \bigg[||g'||_{s^{, }}^{s} \int_{0}^{x}\int_{0}^{x}(x-\tau)^{(\beta-1)} (x-\rho)^{(\beta-1)}|\tau-\rho| t(\tau)t(\rho) \\ &\times&E_{\alpha, \beta, \sigma}^{\delta, q, r, c}\Big(\omega(x-\tau)^{\alpha};p\Big) E_{\alpha, \beta, \sigma}^{\delta, q, r, c}\Big(\omega(x-\rho)^{\alpha};p\Big)d\tau d\rho\bigg]^{s^{-1}}. \end{eqnarray}$

(2.12)

From (2.12), we get

$\begin{eqnarray} && \int_{0}^{x}\int_{0}^{x}(x-\tau)^{(\beta-1)} (x-\rho)^{(\beta-1)}E_{\alpha, \beta, \sigma}^{\delta, q, r, c}\Big(\omega(x-\tau)^{\alpha};p\Big) \\ &\times& E_{\alpha, \beta, \sigma}^{\delta, q, r, c}\Big(\omega(x-\rho)^{\alpha};p\Big)t(\tau)t(\rho)\big|H(\tau, \rho)\big|d\tau d\rho \\ &\leq&||f'||_{r}||g'||_{s}\bigg[\int_{0}^{x}\int_{0}^{x}(x-\tau)^{(\beta-1)} (x-\rho)^{(\beta-1)}|\tau-\rho| t(\tau)t(\rho) \\ &\times&E_{\alpha, \beta, \sigma}^{\delta, q, r, c}\Big(\omega(x-\tau)^{\alpha};p\Big) E_{\alpha, \beta, \sigma}^{\delta, q, r, c}\Big(\omega(x-\rho)^{\alpha};p\Big) d\tau d\rho\bigg]^{r^{-1}} \\ &\times& \bigg[\int_{0}^{x}\int_{0}^{x}(x-\tau)^{(\beta-1)} (x-\rho)^{(\beta-1)}|\tau-\rho|t(\tau)t(\rho) \\ &\times&E_{\alpha, \beta, \sigma}^{\delta, q, r, c}\Big(\omega(x-\tau)^{\alpha};p\Big) E_{\alpha, \beta, \sigma}^{\delta, q, r, c}\Big(\omega(x-\rho)^{\alpha};p\Big)d\tau d\rho\bigg]^{s^{-1}}. \end{eqnarray}$

(2.13)

Since ${r^{-1}}+{s^{-1}} = 1$ , then we have

(2.14)

By the relations (2.4) and (2.14) and using the properties of the modulus, we get the first inequality in (2.1). We have

$0\leq\tau\leq x, \, \, \, \, \, 0\leq\rho\leq x.$

Hence,

$0\leq|\tau-\rho|\leq x.$

Therefore, we have

$\begin{eqnarray} && \int_{0}^{x}\int_{0}^{x}(x-\tau)^{(\beta-1)} (x-\rho)^{(\beta-1)}E_{\alpha, \beta, \sigma}^{\delta, q, r, c}\Big(\omega(x-\tau)^{\alpha};p\Big) \\ &\times&E_{\alpha, \beta, \sigma}^{\delta, q, r, c}\Big(\omega(x-\rho)^{\alpha};p\Big)t(\tau)t(\rho)\big |H(\tau, \rho)\big|d\tau d\rho \\ &\leq&||f'||_{r}||g'||_{s}x\bigg[\int_{0}^{x}\int_{0}^{x}(x-\tau)^{(\beta-1)} (x-\rho)^{(\beta-1)} \\ &\times & E_{\alpha, \beta, \sigma}^{\delta, q, r, c}\Big(\omega(x-\tau)^{\alpha};p\Big) E_{\alpha, \beta, \sigma}^{\delta, q, r, c}\Big(\omega(x-\rho)^{\alpha};p\Big)t(\tau)t(\rho) d\tau d\rho\bigg] \\ & = & ||f'||_{r}||g'||_{s}\ x \big(\epsilon_{0^{+}, \alpha, \beta, \sigma}^{\omega, \delta, q, r, c}t(x)\big)^{2}. \end{eqnarray}$

(2.15)

Theorem (2.1) is thus proved.

In Theorem 2.1, if we set $t(x) = 1$ , we arrive at the following corollary :

Corollary 2.1. Let $f$ and g be two differentiable functions on $[0, \infty]$ . If $f'\in L_{r}$ $([0, \infty])$ , $g'\in L_{s}$ $([0, \infty])$ , $r > 1$ , $r^{-1}+s^{-1} = 1$ , then for all $x > 0$ , $\alpha > 0$ , $\beta > 0$ , we have:

$\begin{eqnarray} & &\bigg|(\epsilon_{0^{+}, \alpha, \beta, \sigma}^{\omega, \delta, q, r, c}f g)(x;p)-\frac{1}{(\epsilon_{0^{+}, \alpha, \beta, \sigma}^{\omega, \delta, q, r, c})(1)} (\epsilon_{0^{+}, \alpha, \beta, \sigma}^{\omega, \delta, q, r, c}f)(x;p)(\epsilon_{0^{+}, \alpha, \beta, \sigma}^{\omega, \delta, q, r, c} g)(x;p) \bigg| \\ & \leq& \frac{1}{2}\big(||f'||_{r}||g'||_{s} x (\epsilon_{0^{+}, \alpha, \beta, \sigma}^{\omega, \delta, q, r, c})(1)\big). \end{eqnarray}$

(2.16)

Corollary 2.2. For different choices of parameters in (2.1) we can establish the corresponding fractional integral inequalities such as

(i) setting $p = 0$ , we get Chebyshev inequality for the Salim-Faraj fractional integral operator, defined in ^[16],

(ii) setting $\sigma = r = 1$ , we get Chebyshev inequality for the fractional integral operator defined by Rahman et al. in ^[15],

(iii) setting $p = 0$ and $\sigma = r = 1$ , we get Chebyshev inequality for the Srivastava-Tomovski fractional integral operator defined in ^[26],

(iv) setting $p = 0$ and $\sigma = r = q = 1$ , we get Chebyshev inequality for the Prabhakar fractional integral operator defined in ^[13].

Remark 2.1. In (2.1) setting $p = \omega = 0$ , we get the inequality (3.1) in ^[4].

Theorem 2.2. Let $t$ be a positive function on $[0, \infty]$ and let $f$ and g be two differentiable functions on $[0, \infty]$ . If $f'\in L_{r}$ $([0, \infty])$ , $g'\in L_{s}$ $([0, \infty])$ , $r > 1$ , $r^{-1}+s^{-1} = 1$ , then for all $x > 0$ , $\alpha > 0$ , $\beta > 0$ , $\lambda > 0$ , $\theta > 0$ , we have

$\begin{eqnarray} &&\Big|(\epsilon_{0^{+}, \alpha, \beta, \sigma}^{\omega, \delta, q, r, c}t)(x;p)(\epsilon_{0^{+}, \lambda, \theta, p}^{\omega, \delta, q, r, c}t f g)(x;p)+(\epsilon_{0^{+}, \lambda, \theta, p}^{\omega, \delta, q, r, c}t)(x;p)(\epsilon_{0^{+}, \alpha, \beta, \sigma}^{\omega, \delta, q, r, c}t f g)(x;p) \\ -&&(\epsilon_{0^{+}, \alpha, \beta, \sigma}^{\omega, \delta, q, r, c}t f)(x;p)(\epsilon_{0^{+}, \lambda, \theta, p}^{\omega, \delta, q, r, c}t g)(x;p)- (\epsilon_{0^{+}, \lambda, \theta, p}^{\omega, \delta, q, r, c}t f)(x;p)(\epsilon_{0^{+}, \alpha, \beta, \sigma}^{\omega, \delta, q, r, c}t g)(x;p)\Big| \\ \leq&&||f'||_{r}||g'||_{s}\int_{0}^{x}\int_{0}^{x}(x-\tau)^{(\beta-1)} (x-\rho)^{(\theta-1)}|\tau-\rho| t(\tau)t(\rho) \\ \times&& E_{\alpha, \beta, \sigma}^{\delta, q, r, c}\Big(\omega(x-\tau)^{\alpha};p\Big) E_{\lambda, \theta, p}^{\delta, q, r, c}\Big(\omega(x-\rho)^{\lambda};p\Big) d\tau d\rho \\ \leq& & ||f'||_{r}||g'||_{s} x (\epsilon_{0^{+}, \alpha, \beta, \sigma}^{\omega, \delta, q, r, c}t)(x;p)(\epsilon_{0^{+}, \lambda, \theta, p}^{\omega, \delta, q, r, c}t)(x;p). \end{eqnarray}$

(2.17)

Proof. Using the identity (2.3), we can state that

$\begin{eqnarray} &&\int_{0}^{x}\int_{0}^{x}(x-\tau)^{(\beta-1)} (x-\rho)^{(\theta-1)}E_{\alpha, \beta, \sigma}^{\delta, q, r, c}\Big(\omega(x-\tau)^{\alpha};p\Big) \\ \times& & E_{\lambda, \theta, p}^{\delta, q, r, c}\Big(\omega(x-\rho)^{\lambda};p\Big)t(\tau)t(\rho)H(\tau, \rho)d\tau d\rho \\ = &&(\epsilon_{0^{+}, \alpha, \beta, \sigma}^{\omega, \delta, q, r, c}t)(x;p)(\epsilon_{0^{+}, \lambda, \theta, p}^{\omega, \delta, q, r, c}t f g)(x;p)+(\epsilon_{0^{+}, \lambda, \theta, p}^{\omega, \delta, q, r, c}t)(x;p)(\epsilon_{0^{+}, \alpha, \beta, \sigma}^{\omega, \delta, q, r, c}t f g)(x;p) \\ -&&(\epsilon_{0^{+}, \alpha, \beta, \sigma}^{\omega, \delta, q, r, c}t f)(x;p)(\epsilon_{0^{+}, \lambda, \theta, p}^{\omega, \delta, q, r, c}t g)(x;p)- (\epsilon_{0^{+}, \lambda, \theta, p}^{\omega, \delta, q, r, c}t f)(x;p)(\epsilon_{0^{+}, \alpha, \beta, \sigma}^{\omega, \delta, q, r, c}t g)(x;p). \end{eqnarray}$

(2.18)

From the relation (2.9), we can obtain the following estimation

$\begin{eqnarray} &&\int_{0}^{x}(x-\tau)^{(\beta-1)} E_{\alpha, \beta, \sigma}^{\delta, q, r, c}\Big(\omega(x-\tau)^{\alpha};p\Big)t(\tau)\big|H(\tau, \rho)\big|d\tau \\ \leq &&\int_{0}^{x}(x-\tau)^{(\beta-1)}|\tau-\rho| E_{\alpha, \beta, \sigma}^{\delta, q, r, c}\Big(\omega(x-\tau)^{\alpha};p\Big)t(\tau) \\ \times&&\bigg|\int_{\tau}^{\rho}|f'(y)|^{r}dy\bigg|^{r^{-1}}\bigg|\int_{\tau}^{\rho}|g'(z)|^{s}dz\bigg|^{s^{-1}} d\tau \end{eqnarray}$

(2.19)

Therefore, we have

(2.20)

Applying Hölder inequality for double integral to the right-hand side of (2.20), yields

$\begin{eqnarray} && \int_{0}^{x}\int_{0}^{x}(x-\tau)^{(\beta-1)} (x-\rho)^{(\theta-1)}E_{\alpha, \beta, \sigma}^{\delta, q, r, c}\Big(\omega(x-\tau)^{\alpha};p\Big) \\ \times&& E_{\lambda, \theta, p}^{\delta, q, r, c}\Big(\omega(x-\rho)^{\lambda};p\Big)t(\tau)t(\rho)\big|H(\tau, \rho)\big|d\tau d\rho \\ \leq &&\bigg[\int_{0}^{x}\int_{0}^{x}(x-\tau)^{(\beta-1)} (x-\rho)^{(\theta-1)}|\tau-\rho|E_{\alpha, \beta, \sigma}^{\delta, q, r, c}\Big(\omega(x-\tau)^{\alpha};p\Big)\\ \times&& E_{\lambda, \theta, p}^{\delta, q, r, c}\Big(\omega(x-\rho)^{\lambda};p\Big)t(\tau)t(\rho) \bigg|\int_{\tau}^{\rho}|f'(y)|^{r}dy\bigg|d\tau d\rho\bigg]^{r^{-1}} \\ \times&& \bigg[\int_{0}^{x}\int_{0}^{x}(x-\tau)^{(\beta-1)} (x-\rho)^{(\theta-1)}|\tau-\rho|E_{\alpha, \beta, \sigma}^{\delta, q, r, c}\Big(\omega(x-\tau)^{\alpha};p\Big)\\ \times&& E_{\lambda, \theta, p}^{\delta, q, r, c}\Big(\omega(x-\rho)^{\lambda};p\Big)t(\tau)t(\rho)\bigg|\int_{\tau}^{\rho}|g'(z)|^{s}dz\bigg|d\tau d\rho\bigg]^{s^{-1}} . \end{eqnarray}$

(2.21)

By (2.11) and (2.21), we get

(2.22)

Using (2.18) and (2.22) and the properties of modulus, we get the first inequality in (2.17).

Corollary 2.3. For different choices of parameters in (2.17) we can establish the corresponding fractional integral inequalities such as

(i) setting $p = 0$ , we get Chebyshev inequality for the Salim-Faraj fractional integral operator, defined in ^[16],

(ii) setting $\sigma = r = 1$ , we get Chebyshev inequality for the fractional integral operator defined by Rahman et al. in ^[15],

(iii) setting $p = 0$ and $\sigma = r = 1$ , we get Chebyshev inequality for the Srivastava-Tomovski fractional integral operator defined in ^[26],

(iv) setting $p = 0$ and $\sigma = r = q = 1$ , we get Chebyshev inequality for the Prabhakar fractional integral operator defined in ^[13].

Remark 2.2. Applying Theorem 2.2 for $\beta = \theta$ , $\alpha = \lambda$ , we obtain theorem 2.1.

Remark 2.3. In (2.17) setting $p = \omega = 0$ , we get the inequality (3.17) in ^[4].

Conflict of interest

The authors declare there is no conflicts of interest in this paper.

References

[1]	M. Andric, G. Farid, J. Pečarić, A further extension of Mittag-Leffler function, Fract. Calc. Appl. Anal., 21 (2018), 1377-1395. doi: 10.1515/fca-2018-0072
[2]	S. Belarbi, Z. Dahmani, On some new fractional integral inequalities, JIPAM, 10 (2009), 1-12.
[3]	Z. Dahmani, About some integral inequalities using Riemann-Liouville integrals, General Mathematics, 20 (2012), 63-69.
[4]	Z. Dahmani, O. Mechouar, S. Brahami, Certain inequalities related to the Chebyshev's functional involving a Riemann-Liouville operator, Bull. Math. Anal. Appl., 3 (2011), 38-44.
[5]	Z. Dahmani, New inequalities in fractional integrals, Int. J. Nonlinear Sci., 9 (2010), 493-497.
[6]	Z. Dahmani, Some results associated with fractional integrals involving the extended Chebyshev functional, Acta Universitatis Apulansis, 27 (2011), 217-224.
[7]	J. Daiya, J. Ram, R. K. Saxena, New fractional integral inequalities associated with Pathway operator, Acta Comment. Univ. Tartu. Math., 19 (2015), 121-126.
[8]	S. M. Kang, G. Farid, W. Nazeer, et al. Hadamard and Fejér-Hadamard inequalities for extended generalized fractional integrals involving special functions, J. Ineq. Appl., 2018 (2018), 119.
[9]	S. M. Kang, G. Farid, W. Nazeer, et al. (h-m)-convex functions and associated fractional Hadamard and Fejér-Hadamard inequalities via an extended generalized Mittag-Leffler function, J. Ineq. Appl., 2019 (2019), 78.
[10]	C. P. Niculescu, I. Roventa, An extention of Chebyshev's algebric inequality, Math. Reports, 15 (2013), 91-95.
[11]	M. E. Özdemir, E. Set, A. O. Akdemir, et al. Some new Chebyshev type inequalities for functions whose derivatives belongs to spaces, Afrika Matematika, 26 (2015), 1609-1619. doi: 10.1007/s13370-014-0312-5
[12]	B. G. Pachpatte, A note on Chebyshev-Grüss type inequalities for diferential functions, Tamsui Oxford Journal of Mathematical Sciences, 22 (2006), 29-36.
[13]	T. R. Prabhakar, A singular integral equation with generalized Mittag-Leffler function in the kernel, Yokohama Math. J., 19 (1971), 7-15.
[14]	S. D. Purohit, S. L. Kalla, Certain inequalities related to the Chebyshev's functional involving Erdelyi-Kober operators, Scientia Mathematical Sciences, 25 (2014), 55-63
[15]	G. Rahman, D. Baleanu, M. A. Qurashi, et al. The extended Mittag-Leffler function via fractional calculus, J. Nonlinear Sci. Appl., 10 (2017), 4244-4253. doi: 10.22436/jnsa.010.08.19
[16]	T. O. Salim, A. W. Faraj, A generalization of Mittag-Leffler function and integral operator associated with fractional calculus, J. Fract. Calc. Appl., 3 (2012), 1-13. doi: 10.1142/9789814355216_0001
[17]	S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional Integral and Derivatives: Theory and Applications, Gordon and Breach, 1993.
[18]	M. Z. Sarıkaya, N. Aktan, H. Yıldırım, On weighted Chebyshev-Grüss like inequalities on time scales, J. Math. Ineq., 2 (2008), 185-195. doi: 10.7153/jmi-02-17
[19]	M. Z. Sarıkaya, A. Saglam, H. Yıldırım, On generalization of Chebyshev type inequalities, Iranian J. Math. Sci. Inform., 5 (2010), 41-48.
[20]	M. Z. Sarıkaya, M. E. Kiriş, On Ostrowski type inequalities and Chebyshev type inequalities with applications, Filomat, 29 (2015), 123-130. doi: 10.2298/FIL1506307S
[21]	E. Set, M. Z. Sarıkaya, F. Ahmad, A generalization of Chebyshev type inequalities for first differentiable mappings, Miskolc Mathematical Notes, 12 (2011), 245-253. doi: 10.18514/MMN.2011.338
[22]	E. Set, Z. Dahmani and İ. Mumcu, New extensions of Chebyshev type inequalities using generalized Katugampola integrals via Polya-Szegö inequality, An International Journal of Optimization and Control: Theories Applications, 8 (2018), 137-144.
[23]	E. Set, J. Choi, İ. Mumcu, Chebyshev type inequalities involving generalized Katugampola fractional integral operators, Tamkang J. Math., 50 (2019), 381-390. doi: 10.5556/j.tkjm.50.2019.2791
[24]	E. Set, A. O. Akdemir, İ. Mumcu, Chebyshev type inequalities for conformable fractional integrals, Miskolc Mathematical Notes, 20 (2019).
[25]	E. Set, İ. Mumcu, S. Demirbaş, Chebyshev type inequalities involving new conformable fractional integral operators, RACSAM, 113 (2018), 2253-2259. doi: 10.1007/s13398-018-0614-9
[26]	H. M. Srivastava, Z. Tomovski, Fractional calculus with an integral operator containing a generalized Mittag-Leffler function in the kernel, Appl. Math. Comput., 211 (2009), 198-210.
[27]	S. Ullah, G. Farid, K. A. Khan, et al. Generalized fractional inequalities for quasi-convex functions, Adv. Difference Equ., 2019 (2019), 1-16. doi: 10.1186/s13662-018-1939-6
[28]	F. Usta, H. Budak, M. Z. Sarıkaya, On Chebyshev Type Inequalities for Fractional Integral Operators, AIP Conference Proceedings, 1833 (2017), 020045.

This article has been cited by:

1.	Barış ÇELİK, Erhan SET, On new integral inequalities using mixed conformable fractional integrals, 2020, 1303-5991, 63, 10.31801/cfsuasmas.698841
2.	Ahmet Ocak Akdemir, Saad Ihsan Butt, Muhammad Nadeem, Maria Alessandra Ragusa, New General Variants of Chebyshev Type Inequalities via Generalized Fractional Integral Operators, 2021, 9, 2227-7390, 122, 10.3390/math9020122
3.	Wengui Yang, Certain New Chebyshev and Grüss-Type Inequalities for Unified Fractional Integral Operators via an Extended Generalized Mittag-Leffler Function, 2022, 6, 2504-3110, 182, 10.3390/fractalfract6040182
4.	Rozana Liko, Pshtiwan Othman Mohammed, Artion Kashuri, Y. S. Hamed, Reverse Minkowski Inequalities Pertaining to New Weighted Generalized Fractional Integral Operators, 2022, 6, 2504-3110, 131, 10.3390/fractalfract6030131
5.	Hari Mohan Srivastava, Artion Kashuri, Pshtiwan Othman Mohammed, Kamsing Nonlaopon, Certain Inequalities Pertaining to Some New Generalized Fractional Integral Operators, 2021, 5, 2504-3110, 160, 10.3390/fractalfract5040160
6.	MAYSAA AL-QURASHI, SAIMA RASHID, YELIZ KARACA, ZAKIA HAMMOUCH, DUMITRU BALEANU, YU-MING CHU, ACHIEVING MORE PRECISE BOUNDS BASED ON DOUBLE AND TRIPLE INTEGRAL AS PROPOSED BY GENERALIZED PROPORTIONAL FRACTIONAL OPERATORS IN THE HILFER SENSE, 2021, 29, 0218-348X, 2140027, 10.1142/S0218348X21400272
7.	Saad Ihsan Butt, Ahmet Ocak Akdemir, Muhammad Nadeem, Malik Ali Raza, Grüss type inequalities via generalized fractional operators, 2021, 44, 0170-4214, 12559, 10.1002/mma.7563
8.	Bhagwat R. Yewale, Deepak B. Pachpatte, Tariq A. Aljaaidi, Wilfredo Urbina, Chebyshev-Type Inequalities Involving ( k , ψ )-Proportional Fractional Integral Operators, 2022, 2022, 2314-8888, 1, 10.1155/2022/3966177
9.	Saima Rashid, Zakia Hammouch, Rehana Ashraf, Yu-Ming Chu, New Computation of Unified Bounds via a More General Fractional Operator Using Generalized Mittag–Leffler Function in the Kernel, 2021, 126, 1526-1506, 359, 10.32604/cmes.2021.011782
10.	Erhan Set, Ahmet Ocak Akdemi̇r, Ali̇ Karaoğlan, New integral inequalities for synchronous functions via Atangana–Baleanu fractional integral operators, 2024, 186, 09600779, 115193, 10.1016/j.chaos.2024.115193
11.	Bhagwat R. Yewale, Deepak B. Pachpatte, 2023, 9781119879671, 45, 10.1002/9781119879831.ch3
12.	Maryam Saddiqa, Saleem Ullah, Ferdous M. O. Tawfiq, Jong-Suk Ro, Ghulam Farid, Saira Zainab, $k$ -Fractional inequalities associated with a generalized convexity, 2023, 8, 2473-6988, 28540, 10.3934/math.20231460

Reader Comments

Your name:*

Email:*
© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)

通讯作者: 陈斌, bchen63@163.com

1.
沈阳化工大学材料科学与工程学院沈阳 110142

AIMS Mathematics

1.8 3.4

Metrics

Article views(3922) PDF downloads(339) Cited by(12)

Preview PDF

Download XML

Export Citation

Article outline

Show full outline

AIMS Mathematics

Chebyshev type inequalities involving extended generalized fractional integral operators

Related Papers:

Abstract

1. Introduction

2. Main results

Conflict of interest

References

This article has been cited by:

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Other Articles By Authors

Catalog

AIMS Mathematics

Chebyshev type inequalities involving extended generalized fractional integral operators

Related Papers:

Abstract

1. Introduction

2. Main results

Conflict of interest

References

This article has been cited by:

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Other Articles By Authors

Related pages

Tools

Export File

Citation

Format

Content

Catalog