Instabilities in internal gravity waves

1.   Introduction

Fractional calculus signifies the identity of the distinguished materials in the modern research field due to its integrated applications in diverse regions such as mathematical physics, fluid dynamics, mathematical biology, etc. Convex function, exponentially convex function ^[1,2,3,4,5], related inequalities like as trapezium inequality, Ostrowski's inequality and Hermite Hadamard inequality, integrals ^[6,7,8,9,10] having succeed in mathematical analysis, approximation theory due to immense applications ^[11,12] have great importance in mathematics theory. Many authors established quadrature rules in numerical analysis for approximate definite integrals. Recently, Pólya-Szegö and Chebyshev inequalities occupied immense space in the field analysis. Chebyshev ^[13] was introduced the well-known inequality called Chebyshev inequality.

In the literature of convex function, the Jensen inequality has gained much importance which describes a connection between an integral of the convex function and the value of the convex function of an interval ^[14,15,16]. Pshtiwan and Thabet ^[17] considered the modified Hermite Hadamard inequality in the context of fractional calculus using the Riemann-Liouville fractional integrals. Arran and Pshtiwan ^[18] discussed the Hermite Hadamard inequality results with fractional integrals and derivatives using Mittag-Leffler kernel. Pshtiwan and Thabet ^[19] constructed a connection between the Riemann-Liouville fractional integrals of a function concerning a monotone function with nonsingular kernel and Atangana-Baleanu. Pshtiwan and Brevik ^[20] obtained an inequality of Hermite Hadamard type for Riemann-Liouville fractional integrals, and proved the application of obtained inequalities on modified Bessel functions and $q$-digamma function. In ^[21], Set et al. introduced Grüss type inequalities by employing generalized $k$-fractional integrals. Recently, Nisar et al. ^[22] gave some new generalized fractional integral inequalities.

Very recently, the fractional conformable and proportional fractional integral operators were given in ^[23,24]. Later on, Huang et al. ^[25] gave Hermite–Hadamard type inequalities by using fractional conformable integrals (FCI). Qi et al. ^[26] investigated Čebyšev type inequalities involving FCI. The Chebyshev type inequalities and certain Minkowski's type inequalities are found in ^[27,28,29]. Nisar et al. ^[30] have investigated some new inequalities for a class of $n\ \ (n\in\mathbb{N})$ positive, continuous, and decreasing functions by employing FCI. Rahman et al. ^[31] introduced Grüss type inequalities for $k$-fractional conformable integrals.

Some significant inequalities are given as applications of fractional integrals ^{[32,33,34,35,36,37,38]}. Recently, Rahman et al. ^[39,40] presented fractional integral inequalities involving tempered fractional integrals. Qiang et al. ^[41] discussed a fractional integral containing the Mittag-Leffler function in inequality theory and contributed Hadamard type inequality, continuity, and boundedness, upper bounds of that integral. Nisar et al. ^[42] established weighted fractional Pólya-Szegö and Chebyshev type integral inequalities by operating the generalized weighted fractional integral involving kernel function. The dynamical approach of fractional calculus ^{[43,44,45,46,47,48,49]} in the field of inequalities.

Grüss inequality ^[50] established for two integrable function as follows

where the $h$ and $l$ are two integrable functions which are synchronous on $[a, b]$ and satisfy:

for some $s, k, S, K \in \mathbb{R}$.

Pólya and Szegö ^[51] proved the inequalities

Dragomir and Diamond ^[52], proves the inequality by using the Pólya-szegö inequality

where $h$ and $l$ are two integrable functions which are synchronous on $[a, b]$, and

for some $s, k, S, K \in \mathbb{R}$.

The aim of this paper is to estimate a new version of Pólya-Szegö inequality, Chebyshev integral inequality, and Hermite Hadamard type integral inequality by a fractional integral operator having a nonsingular function (generalized multi-index Bessel function) as a kernel, and these established results have great contribution in the field of inequalities. The Hermite Hadamard type integral inequality provides the upper and lower estimate to find the average integral for the convex function of any defined interval.

The structure of the paper follows:

In section 2, we present some well-known definitions and mathematical preliminaries. The new generalized fractional integral with nonsingular function as a kernel is defined in section 3. In section 4, we present Hermite Hadamard type Mercer inequality of new designed fractional integral operator with nonsingular function (generalized multi-index Bessel function) as a kernel. some inequalities of $(s-m)$-preinvex function involving new designed fractional integral operator with nonsingular function (generalized multi-index Bessel function) as a kernel are presented in section 5. Here section 6 and 7, we present Pólya-Szegö and Chebyshev integral inequalities involving generalized fractional integral operator with nonsingular function as a kernel, respectively.

2.   Preliminaries

Definition 2.1. The inequality holds for the convex function if a mapping $g: K \rightarrow \mathbb{R}$ exist as

where $\forall y_{1}, y_{2}\in K$ and $\delta\in[0, 1]$.

Definition 2.2. The inequality derived by Hermite ^[53] call as Hermite Hadamard inequality

where $y_{1}, y_{2} \in I$, with $y_{2}\neq y_{1}$, if $g:I \subseteq \mathbb{R} \rightarrow \mathbb{R}$ is a convex function.

Definition 2.3. Let $y_{j}\in K$ for all $j \in I_{n}$, $\omega_{j} > 0$ such that $\sum_{j \in I_{n}} \omega_{j} = 1$. Then the Jensen inequality holds

exist if $g:k \rightarrow \mathbb{R}$ is convex function.

Mercer ^[54] derived the Mercer inequality by applying the Jensen inequality and properties of convex function.

Definition 2.4. Let $y_{j}\in K$ for all $j \in I_{n}$, $\omega_{j} > 0$ such that $\sum_{j \in I_{n}} \omega_{j} = 1$, $m = \min_{j \in I_{n}}\{y_{j}\}$ and $n = \max_{j \in I_{n}}\{y_{j}\}$. Then the inequality holds for convex function as

if $g:k \rightarrow \mathbb{R}$ is convex function.

Definition 2.5. ^[55] The inequality holds for exponentially convex function, if a real valued mapping $g: K \rightarrow \mathbb{R}$ exist as

where $\forall y_{1}, y_{2}\in K$ and $\delta\in[0, 1]$ and $\theta\in \mathbb{R}$.

Suppose that $\Omega\subseteq \mathbb{R}^{n}$ is a set. Let $g:\Omega\rightarrow \mathbb{R}$ continuous function and let $\xi:\Omega\times\Omega\rightarrow \mathbb{R}^{n}$ be continuous function:

Definition 2.6. ^[56] With respect to bifunction $\xi(., .)$ a set $\Omega$ is called a invex set, if

where $\forall y_{1}, y_{2} \in \Omega, \delta\in[0, 1]$.

Definition 2.7. ^[57] A invex set $\Omega$ and a mapping $g$ with respect to $\xi(., .)$ is called a preinvex function, as

where $\forall$ $y_{1}, y_{2}+\xi(y_{2}, y_{1})\in \Omega, \delta \in [0, 1]$.

Definition 2.8. A invex set $\Omega$ with real valued mapping $g$ and respect to $\xi(., .)$ is called a exponentially preinvex, if the inequality

where for all $y_{1}, y_{2}+\xi(y_{2}, y_{1})\in \Omega, \delta \in [0, 1]$ and $\theta \in \mathbb{R}$.

Definition 2.9. A invex set $\Omega$ with real valued mapping $g$ and respect to $\xi(., .)$ is called a exponentially s-preinvex, if

where for all $y_{1}, y_{2}+\xi(y_{2}, y_{1})\in \Omega, \delta \in [0, 1]$, $s\in(0, 1]$ and $\theta \in \mathbb{R}$.

Definition 2.10. A invex set $\Omega$ with real valued mapping $g$ and respect to $\xi(., .)$ is called exponentially (s-m)-preinvex, if

where for all $y_{1}, y_{2}+\xi(y_{2}, y_{1})\in \Omega$, $\delta, m \in [0, 1]$ and $\theta \in \mathbb{R}$.

Definition 2.11. ^[58] Generalized multi-index Bessel function is defined by Choi et al as follows

where $\xi_j, \delta_j, \lambda \in\mathbb{C}$, $(j = 1, \cdots, m)$, $\Re(\lambda) > 0, \Re(\delta_{j}) > -1, \sum^{m}_{j = 1} \Re(\xi)_j > max\{0: \Re(\sigma)-1\}, \sigma > 0$.

Definition 2.12. ^[58] Pohhammer symbol is defined for $\lambda\in \mathbb{C}$ as follows

where $\Gamma$ being the Gamma function.

3.   Fractional integral operator with nonsingular function

This section presents a generalized fractional integral operator with a nonsingular function (multi-index Bessel function) as a kernel.

Definition 3.1. Let $\xi_j, \delta_j, \lambda, \zeta \in \mathbb{C}, (j = 1, \cdots, m), \Re(\lambda) > 0, \Re(\delta_{j}) > -1, \sum^{m}_{j = 1} \Re(\xi_{j}) > max\{0: \Re(\sigma)-1\}, \sigma > 0$. Let $g \in L \ \ [y_{1}, y_{2}]$ and $t\in[y_{1}, y_{2}]$. Then the corresponding left sided and right sided generalized integral operators having generalized multi-index Bessel function defined as:

and

Remark 3.1. The special cases of generalized fractional integrals with nonsingular kernel are given below:

1. If set $j = m = 1$, $\sigma = 0$ and limits from $[0, z]$ in Eq (3.1), we get a fractional integral defined by Srivastava and Singh in ^[59] as

2. If set $j = m = 1$, $\delta_{1} = \delta_{1}-1$ in Eq (3.1), we have a fractional integral defined by Srivastava and Tomovski in ^[60] as

3. If set $j = m = 1$, $\delta_{1} = \delta_{1}-1$, $\zeta = 0$ in Eq (3.1), we get a Riemann-Liouville fractional integral operator defined in ^[61] as

4. If set $j = m = 1$, $\sigma = 1$, $\delta_{1} = \delta_{1}-1$, in Eq (3.1) and Eq (3.2), we get the fractional integral operator defined by Prabhakar in ^[62] as follows

Lemma 3.1. From generalized fractional integral operator, we have

Hence, the Eq (3.8) becomes

and similarly we have

4.   Hadamard type Mercer inequality with fractional operators

In this section, we derive Hermite Hadamard type Mercer inequality of new designed fractional integral operator in a generalized multi-index Bessel function using a kernel.

Theorem 4.1. Let $g:[m, n] \rightarrow (0, \infty)$ is convex function such that $g\in\chi_{c}(m, n)$, $\forall x, y \in [m, n]$ and the operator defined in Eq (5.2) in the form of left sense operator and Eq (3.2) in the form of right sense operator then we have

Proof. Consider the mercer inequality

Let $x, y \in [m, n]$, $t \in [z-1, z]$, $y_{1} = (z-t)x+(1-z+t)y$ and $y_{2} = (1-z+t)x+(z-t)y$ then inequality (4.3) becomes

Multiply both sides of Eq (4.4) by $(z-t)^{\delta_j} {\mathrm{J}^{(\xi_j)_m, \lambda}_{(\delta_j)_m, \sigma}(\zeta(z-t)^{\xi_{j}})}$ and integrating with respect to $t$ from $[z-1, z]$, we get

we get the desired inequality, as

Thus, we get the inequality (4.1). Let $t \in[z-1, z]$. From the convexity of function $g$ we have

Both sides multiply of Eq (4.6) by $(z-t)^{\delta_j} {\mathrm{J}^{(\xi_j)_m, \lambda}_{(\delta_j)_m, \sigma}(\zeta(z-t)^{\xi_{j}})}$ and integrating with respect to $t$ from $[z-1, z]$, we obtain

We get the inequality of negative sign

By adding $g(m)+g(n)$ of both sides of inequality (4.7), we have

Hence, we get the inequality (4.2).

Theorem 4.2. Let $g:[m, n] \rightarrow (0, \infty)$ is convex function such that $g\in\chi_{c}(m, n)$ then we have the following inequalities:

Where $\forall x, y \in [m, n]$.

Proof. We see that from the convexity of $g$ as

Let $x, y \in[m, n]$, $t \in [z-1, z]$, $m+n-y_{1} = (z-t)(m+n-x)+(1-z+t)(m+n-y)$, $m+n-y_{2} = (1-z+t)(m+n-x)+(z-t)(m+n-y)$, then inequality (4.10) gives

multiply of both sides of inequality (4.11) by $(z-t)^{\delta_j} {\mathrm{J}^{(\xi_j)_m, \lambda}_{(\delta_j)_m, \sigma}(\zeta(z-t)^{\xi_{j}})}$ then integrate with respect to $t$ from $[z-1, z]$, we get

Thus, we get the inequality (4.8)

From the convexity of $g$, we obtain

and

Adding up the above inequalities and applying Jensen-Mercer inequality, we get

Multiply both sides of inequality (4.14) by $(z-t)^{\delta_j} {\mathrm{J}^{(\xi_j)_m, \lambda}_{(\delta_j)_m, \sigma}(\zeta(z-t)^{\xi_{j}})}$ and then integrating with respect to $t$ from $[z-1, z]$ we obtain the two inequalities (4.9).

5.   $(s-m)$ preinvex inequalities involving fractional operators

In this section, we derive some inequalities of $(s-m)$ preinvex function involving new designed fractional integral operator $Œ^{(\xi_j, \delta_j)_m}_{\lambda, \sigma, \zeta} g)(z)$ having generalized multi-index Bessel function as its kernel in the form of theorems.

Theorem 5.1. Suppose a real valued function $g:[y_{1}, y_{1}+\xi(y_{2}, y_{1})]\rightarrow R$ be exponentially (s-m) preinvex function, then the following fractional inequality holds:

$\forall$ $z\in[y_{1}, y_{1}+\xi(y_{2}, y_{1})]$, $\theta_{1}, \theta_{2} \in R$.

Proof. Let $z\in[y_{1}, y_{1}+\xi(y_{2}, y_{1})]$, and then for $t\in [y_{1}, z)$ and $\delta_{j} > -1$, we have the subsequent inequality

For $g$ is exponentially (s-m)-preinvex function, we obtain

Taking product (5.1) and (5.2), and integrating with respect to $t$ from $y_{1}$ to $z$, we get

apply definition (13) in Eq (5.3), we have

Analogously for $t\in(z, y_{1}+\xi(y_{2}, y_{1})]$ and $\mu_{j} > -1$, we have

Further, the exponentially (s-m) convexity of $g$, we get

Taking product of (5.5) and (5.6) and integrating with respect to $t$ from $z$ to $y_{1}+\xi(y_{2}, y_{1})$, we have

apply the definition (13) in inequality (5.7), we have

Now, add the inequalities (5.4) and (5.8), we get the result

Corollary 5.1. If $g\in L_{\infty}[y_{1}, y_{1}+\xi(y_{2}, y_{1})]$, then under the assumption of theorem (5.1), we have

Corollary 5.2. Setting $m = 1$ and $g\in L_{\infty}[y_{1}, y_{1}+\xi(y_{2}, y_{1})]$, then under the assumption of theorem (5.1), we have

Corollary 5.3. Setting $m = s = 1$ and $g\in L_{\infty}[y_{1}, y_{1}+\xi(y_{2}, y_{1})]$, then under the assumption of theorem (5.1), we have

Corollary 5.4. Setting $\xi(y_{2}, y_{1}) = y_{2}-y_{1}$ and $g\in L_{\infty}[y_{1}, y_{2}]$, then under the assumption of theorem (5.1), we have

Theorem 5.2. Suppose a real value function $g:[y_{1}, y_{1}+\xi(y_{2}, y_{1})]\rightarrow R$ is differentiable and $|g|^{\prime}$ is exponentially (s-m) preinvex, then the following fractional inequality for (3.1) and (3.2) holds:

$\forall$ $z\in[y_{1}, y_{1}+\xi(y_{2}, y_{1})]$, $\theta_{1}, \theta_{2} \in \mathbb{R}$.

Proof. Let $z\in[y_{1}, y_{1}+\xi(y_{2}, y_{1})]$, $t\in[y_{1}, z)$, and applying exponentially (s-m) preinvex of $|g|^{\prime}$, we get

Get the inequality (5.9), we have

Subsequently inequality as:

Conducting product of inequality (5.10) and (5.11), we have

integrating before mention inequality with respect to $t$ from $y_{1}$ to $z$, we have

Now, solving left side of (5.13) by putting $z-t = \alpha$, then we have

Now, again subsisting $z-\alpha = t$, we get

Therefore, the inequality (5.13) have the following form

Also from (5.9), we get

Adopting the same procedure as we have done for (5.10), we obtain

From (5.14) and (5.16), we get

Now, we let $z\in[y_{1}, y_{1}+\eta(y_{2}, y_{1})]$ and $t\in(z, y_{1}+\xi(y_{2}, y_{1})]$, and by exponentially (s-m) preinvex of $|g^{\prime}|$, we get

repeat the same procedure from Eq (5.9) to Eq (5.17), we get

From inequalities (5.17) and (5.19), we have

Corollary 5.5. Setting $\xi(y_{2}, y_{1}) = y_{2}-y_{1}$, then under the assumption of theorem (5.2), we have

$\forall$ $t\in[y_{1}, y_{2}]$, $\theta_{1}, \theta_{2} \in \mathbb{R}$.

Corollary 5.6. Setting $\xi(y_{2}, y_{1}) = y_{2}-y_{1}$, along with $m = s = 1$ then under the assumption of theorem (5.2), we have

$\forall$ $t\in[y_{1}, y_{2}]$, $\theta_{1}, \theta_{2} \in \mathbb{R}$.

Definition 5.1. Let $g: [y_{1}, y_{1}+\xi(y_{2}, y_{1})]\rightarrow R$ is a function, and $g$ is exponentially symmetric about $\frac{2y_{1}+\xi(y_{2}, y_{1})}{2}$ if

Lemma 5.1. Let $g:[y_{1}, y_{1}+\xi(y_{2}, y_{1})]\rightarrow R$ be exponentially symmetric, then

Proof. For $g$ is exponentially (s-m) preinvex, therefore

Let $t = y_{1}+\delta\xi(y_{2}, y_{1})$, where $t\in[y_{1}, y_{1}+\xi(y_{2}, y_{1})]$, and then $2y_{1}+\xi(y_{2}, y_{1}) = y_{1}+(1-\delta)\xi(y_{2}, y_{1})$, we have

applying that $g$ is exponentially symmetric, we obtain

Theorem 5.3. Suppose a real valued function $g:[y_{1}, y_{1}+\xi(y_{2}, y_{1})]\rightarrow R$ is exponentially (s-m) preinvex and symmetric about exponentially $\frac{2y_{1}+\xi(y_{2}, y_{1})}{2}$, then the following integral inequality for (3.1) and (3.2) holds:

Proof. For $z\in[y_{1}, y_{1}+\xi(y_{2}, y_{1})]$, we have

the real value function $g$ is exponentially (s-m) preinvex, then for $z\in[y_{1}, y_{1}+\xi(y_{2}, y_{1})]$, we get

Conducting product of (5.26) and (5.27), and integrating with respect to $z$ from $y_{1}$ to $y_{2}$, we get

then we have

Analogously for $z\in[y_{1}, y_{1}+\xi(y_{2}, y_{1})]$, we have

Conducting product of (5.27) and (5.30), and integrating with respect to $z$ from $y_{1}$ to $y_{2}$, we have

then

Summing (5.29) and (5.31), we obtain

Take the product of Eq (5.21) with $(z-y_{1})^{\tau_{j}}{\mathrm{J}^{(\mu_j)_m, \lambda}_{(\tau_j)_m, \sigma}(\zeta(z-y_{1})^{\mu_{j}})}$ and integrating with respect to $t$ from $y_{1}$ to $y_{2}$, we have

using definition (13), we have

Taking product (5.21) with $(y_{1}+\xi(y_{2}, y_{1})-z)^{\delta_{j}}{\mathrm{J}^{(\mu_j)_m, \lambda}_{(\delta_j)_m, \sigma}(\zeta(y_{1}+\xi(y_{2}, y_{1})-z)^{\mu_{j}})}$ and integrating with respect to variable $z$ from $y_{1}$ to $y_{2}$, we have

Summing up (5.34) and (5.35), we get

Now, combining (5.32) and (5.36), we get inequality

Corollary 5.7. Setting $\xi(y_{2}, y_{1}) = y_{2}-y_{1}$, then under the assumption of theorem (5.3), we have

6.   Some Pólya-Szegö type integral inequalities of fractional operators

In this section, we derive some Pólya-Szegö inequalities for four positive integrable functions having fractional operator $Œ^{(\xi_j, \delta_j)_m}_{\lambda, \sigma}(z)$ in the form of theorems.

Theorem 6.1. Let $h$ and $l$ are integrable functions on $[y_{1}, \infty)$. Suppose that there exist integrable functions $\theta_{1}, \theta_{2}, \psi_{1}$ and $\psi_{2}$ on $[y_{1}, \infty)$ such that:

Then, for $z > y_{1}, y_{1}\geq0$, $\xi_j, \delta_j, \lambda \in\mathbb{C}, (j = 1, \cdots, m), \Re(\lambda) > 0, \Re(\delta_{j}) > -1, \sum^{m}_{j = 1} \Re(\xi)_j > max\{0: \Re(\sigma)-1\}, \sigma > 0$ and $(z-b)\in \Omega$, then the following inequalities hold:

Proof. From $(R1)$, for $b\in[y_{1}, z]$, $z > y_{1}$, we have

the inequality write as

Similarly, we get

thus

Multiplying Eq (6.3) and Eq (6.5), it follows

i.e.

The last inequality can be written as

Consequently, multiply both sides of (6.8) by $(y_{1}-b)^{\delta_{j}}\mathrm{J}^{(\xi_j)_m, \lambda}_{(\delta_j)_m, \sigma}(\zeta(y_{1}-b)^{\xi_{j}})$, $(z-b) \in \Omega$ and integrating with respect to $b$ from $y_{1}$ to $z$, we get

Besides, by AM-GM (arithmetic mean- geometric mean) inequality, i.e., $a_{1}+b_{1}\geq2\sqrt{a_{1}b_{1}}$ $a_{1}, b_{1} \in \Re^{+}$, we get

and it follows straightforward the statement of Eq (6.1).

Corollary 6.1.. Let $h$ and $l$ be two integrable functions on $[0, \infty)$ and satisfying the inequality

For $z > y_{1}, y_{1}\geq0$, $\xi_j, \delta_j, \lambda \in\mathbb{C}, (j = 1, \cdots, m), \Re(\lambda) > 0, \Re(\delta_{j}) > -1, \sum^{m}_{j = 1} \Re(\xi)_j > max\{0: \Re(\sigma)-1\}, \sigma > 0$ and $(z-b)\in \Omega$, then the following inequalities hold:

Theorem 6.2. Let $h$ and $l$ are positive integrable functions on $[y_{1}, \infty)$. Suppose that there exist integrable functions $\theta_{1}, \theta_{2}, \psi_{1}$ and $\psi_{2}$ on $[y_{1}, \infty)$ satisfying $(R1)$ on $[y_{1}, \infty)$. Then, for $z > y_{1}, y_{1}\geq0$, $\xi_j, \delta_j, \lambda \in\mathbb{C}, (j = 1, \cdots, m), \Re(\lambda) > 0, \Re(\delta_{j}) > -1, \sum^{m}_{j = 1} \Re(\xi)_j > max\{0: \Re(\sigma)-1\}, \sigma > 0$ and $(z-b), (\tau-z)\in \Omega$, then the following inequalities hold:

Proof. By condition $(R1)$, it is clear that

and

these inequalities implies that

The Eq (6.16), multiply by $\psi_{1}(\alpha)\psi_{2}(\alpha)l^{2}(\alpha)$ of both sides, we have

Hence, the Eq (6.17) multiply both sides by

And integrating double with respect to $b$ and $\alpha$ from $y_{1}$ to $z$ and $z$ to $y_{2}$ respectively, we have

At last, we come to Eq (6.13) by using the arithmetic and geometric mean inequality to the upper inequality.

Theorem 6.3. Let $h$ and $l$ are integrable functions on $[y_{1}, \infty)$. Suppose that there exist integrable functions $\theta_{1}, \theta_{2}, \psi_{1}$ and $\psi_{2}$ on $[y_{1}, \infty)$ satisfying $(R1)$ on $[y_{1}, \infty)$. Then, for $z > y_{1}, y_{1}\geq0$, $\xi_j, \delta_j, \lambda \in\mathbb{C}, (j = 1, \cdots, m), \Re(\lambda) > 0, \Re(\delta_{j}) > -1, \sum^{m}_{j = 1} \Re(\xi)_j > max\{0: \Re(\sigma)-1\}, \sigma > 0$ and $(z-b), (\alpha-z)\in \Omega$, then the following inequalities hold:

Proof. We have for any $(z-b), (\alpha-z)\in \Omega$, from Eq (6.2), thus

which implies

and analogously, by Eq (6.4), we get

hence, by multiplying Eq (6.21) and Eq (6.22), follow Eq (6.20).

Corollary 6.2. Let $h$ and $l$ be integrable functions on $[y_{1}, \infty)$ satisfying $(R2)$. Then, for $z > y_{1}, y_{1}\geq0$, $\xi_j, \delta_j, \lambda \in\mathbb{C}, (j = 1, \cdots, m), \Re(\lambda) > 0, \Re(\delta_{j}) > -1, \sum^{m}_{j = 1} \Re(\xi)_j > max\{0: \Re(\sigma)-1\}, \sigma > 0$ and $(z-b), (\alpha-z)\in \Omega$, we obtain

7.   Chebyshev type integral inequalities of fractional operator

In this section, Chebyshev type integral inequalities established involving the fractional operator $Œ^{(\xi_j, \delta_j)_m}_{\lambda, \sigma}(z)$ and using the Pólya-Szegö fractional integral inequalities of theorem (6.1) in the form of theorem, and then discuss its corollary.

Theorem 7.1. Let $h$ and $l$ be integrable functions on $[y_{1}, \infty)$, and suppose that there exist integrable functions $\theta_{1}, \theta_{2}, \psi_{1}$ and $\psi_{2}$ on $[y_{1}, \infty)$ satisfying $(R1)$. Then, for $z > y_{1}, y_{1}\geq0$, $\xi_j, \delta_j, \lambda \in\mathbb{C}, (j = 1, \cdots, m), \Re(\lambda) > 0, \Re(\delta_{j}) > -1, \sum^{m}_{j = 1} \Re(\xi)_j > max\{0: \Re(\sigma)-1\}, \sigma > 0$ and $(z-b)(\alpha-z)\in \Omega$ the following inequality hold:

where

Proof. For $(b, \alpha)\in(y_{1}, z)$ $(z > y_{1})$, we defined $A(b, \alpha) = (h(b)-h(\alpha))(l(b)-l(\alpha))$ which is the same

Further, the Eq (7.2), multiply both sides by

and integrating double with respect to $b$ and $\alpha$ from $y_{1}$ to $z$ and $z$ to $y_{2}$ respectively, we get

Now, applying Cauchy-Schwartz inequality for integrals, we get

it follow as

By applying lemma (6.1) for $\psi_{1}(z) = \psi_{2}(z) = l(z) = 1$, we get for any $\mathrm{J}^{(\xi_j, \delta_j)_m}_{\lambda, \sigma}(z)^{\delta_{j}}\in \Omega$

this implies

Analogously, it is clear when $\theta_{1}(z) = \theta_{2}(z) = h(z) = 1$, according to Lemma (6.1), we get

Thus, by resulting Eqs (7.4), (7.6), (7.8) and (7.9), we get the desired inequality (7.1).

Corollary 7.1. Let $h$ and $l$ be integrable functions on $[y_{1}, \infty)$, suppose that there exist integrable functions $\theta_{1}, \theta_{2}, \psi_{1}$ and $\psi_{2}$ on $[y_{1}, \infty)$ satisfying $(R1)$. Then, for $z > y_{1}, y_{1}\geq0$, $\xi_j, \delta_j, \lambda \in\mathbb{C}, (j = 1, \cdots, m), \Re(\lambda) > 0, \Re(\delta_{j}) > -1, \sum^{m}_{j = 1} \Re(\xi)_j > max\{0: \Re(\sigma)-1\}, \sigma > 0$ and $(z-b), (\alpha-z)\in \Omega$ the following inequalities hold:

where

8.   Conclusions

This article analyzed the generalized fractional integral operator having nonsingular function (generalized multi-index Bessel function) as kernel and developed a new version of inequalities. We estimate some inequalities (Hermite Hadamard type Mercer inequality, exponentially $(s-m)$ preinvex inequality, Pólya-Szegö type integral inequality and the Chebyshev type inequality) with the generalized fractional integral operator in which nonsingular function as the kernel. Introducing the new version of inequalities of newly constricted operators have strengthened the idea and results.

Conflict of interest

The authors declare that they have no competing interest.