Efficiency of honey–grape blend in reducing radiation-induced mucositis in locally advanced head and neck squamous cell carcinoma

1.   Introduction

Fractional calculus is one of the renowned fields in recent research due to its inherent applications in various areas such as mathematical physics, fluid dynamics, mathematical biology etc. ^{[1,2,3,4,5,6]}. On the other hand, the fractional integral inequalities with the fractional operators are developed by many researchers because these inequalities are used to verify various results of applied problems ^[7,8]. In particular, the researchers ^{[30,31,32,33]} have recently studied many remarkable fractional integral inequalities and their applications. In ^[39], Mehmood et al. discussed the Hermite-Hadamard-Fejér inequality for fractional integrals involving preinvex functions. Mehreen and Anwar ^[40] estimated he Hermite-Hadamard and Hermite-Hadamard-Fejér type inequalities for $p$-convex functions by utilizing conformable fractional integrals. In ^[41], Almutairi and Adem Kılıçman discussed new integral inequalities of Hermite-Hadamard type involving $s$-convexity and studied their properties. Budak ^[42] establish Hermite-Hadamard-Fejér type inequalities for convex function involving fractional integrals with respect to another function. The Hermite-Hadamard inequality is defined can be found in ^[9] for convex function by

$\Psi:I \rightarrow \mathbb{R}, \, m, n \in I, m < n, m, n \in\mathbb{R}, I\subseteq \mathbb{R}$ and is playing a significant role in the field of inequalities and are widely used by the researchers ^[10].

Fejér type integral inequalities can be found in ^[27,28,29] by

for convex function $\Psi:[m, n]\rightarrow \mathbb{R}$ and $\Phi:[m, n]\rightarrow \mathbb{R}^{+}, m, n \in \mathbb{R}$ where the function $\Phi$ is integrable and is symmetric about $x = \frac{m+n}{2}$. Note that the Hermite- Hadamard inequality is obtained if $\Phi = 1$ in Fejér inequality (1.1).

The $(\eta_{1}, \eta_{2})$-convex function has been presented ^[11,12,13] by obtaining the generalization of $\eta$-convex function ^{[14,15,16,17]} and preinvex function ^[18,19,20].

Sarikaya ^[25] discussed the Hermite and trapezoid inequalities related to the Hermite-Hadamard inequality, and Rostamian Delavar ^[13] discussed Fejér, midpoint and trapezoid type inequalities related to the Hermite-Hadamard inequalities.

Definition 1.1. ^[26] The convex function $\Psi:I\rightarrow \mathbb{R}$ is defined for $t\in[0, 1], \forall u, v \in I$ as follows:

Definition 1.2. ^[13] An invex set $I \subseteq \mathbb{R}$ with respect to a real bifunction $\theta:I \times I \rightarrow \mathbb{R}$, is defined for $m, n\in I, \lambda\in [0, 1]$ as follows

Definition 1.3. ^[13] The preinvex function $\Psi: I \rightarrow \mathbb{R}$ is defined for $m, n \in I$ and $\lambda \in [0, 1]$ as follows

where $I$ is an invex set with respect to $\theta$.

Definition 1.4. ^[13] A function $\Psi:I\rightarrow \mathbb{R}$ is said to be convex with respect to $\eta$ i.e $(\eta-convex)$ if it satisfies

for all $m, n \in I$ and $\lambda \in [0, 1]$ and $I\subseteq \mathbb{R}$ is a convex function and $\eta:\Psi(I)\times\Psi(I)\rightarrow \mathbb{R}$ is a bifunction.

Definition 1.5. ^[13] Let $\Psi:I \rightarrow \mathbb{R}$, $\eta_{1}:I \times I\longrightarrow \mathbb{R}$, and $\eta_{2}:\Psi(I)\times\Psi(I)\rightarrow \mathbb{R}$, then $\Psi$ is called $(\eta_{1}, \eta_{2})$-convex function if

holds for all $x, y \in I$ and $\lambda \in[0, 1]$.

Example 1.1. ^[12] Let $\Psi$ be the function such that $\Psi:\mathbb{R}^{+}\rightarrow \mathbb{R}^{+}$ defined by

Let the two bifunctions $\eta_{1}:\mathbb{R}^{+}\times\mathbb{R}^{+}\rightarrow \mathbb{R}$ and $\eta_{2}:\mathbb{R}^{+}\times \mathbb{R}^{+}\rightarrow \mathbb{R}$ defined by

Then $\Psi$ is $(\eta_{1}, \eta_{2})$-convex function.

Definition 1.6. ^[43] The Pochhammer's symbol is defined for $t\in \mathbb{N}$ as

for $\Im\in \mathbb{C}$, where $\Gamma$ being the gamma function.

Definition 1.7. ^[43] The integral representation of gamma function is defined as

for $, \Re(t) > 0$.

Definition 1.8. ^[35,36,37] The Classical beta function is defined for $\Re(m) > 0$ and $\Re(n) > 0$,

Definition 1.9. ^[38] Extended beta functions is defined for $\Re(m) > 0, \Re(n) > 0, \Re(p) > 0$ is

Definition 1.10. ^[34] Ali et al. defined and investigated the generalized Bessel-Maitland function (eight-parameters) with a new fractional integral operator and discussed its properties and relations with Mittag-Leffler functions. The function of generalized Bessel-Maitland as follows:

where $\phi, \psi, \theta, \delta, \vartheta \in \mathbb{C}$, $\Re(\phi) > 0$, $\Re(\nu)\geq-1$, $\Re(\theta) > 0$, $\Re(\delta) > 0$, $\Re(\vartheta) > 0$; $\xi, m, \sigma \geq0$ and $m, \xi > \Re(\phi)+\sigma$.

Definition 1.11. The extended generalized Bessel-Maitland function is defined for $\mu, \nu, \eta, \rho, \gamma, c \in \mathbb{C}$, $\Re(\mu) > 0$, $\Re(\nu)\geq-1$, $\Re(\eta) > 0$, $\Re(\rho) > 0$, $\Re(\gamma) > 0$; $\xi, m, \sigma \geq0$ and $m, \xi > \Re(\mu)+\sigma$ by

In the recent era of research, the field of fractional calculus has gained more recognition due to its wide range of applications in different sciences ^[44,45]. Such new developments in fractional calculus motivate the researchers to establish some new innovative ideas to unify the fractional operators and propose new inequalities involving new fractional operators.

The Hermite-Hadamard integral inequalities and their extensions have been widely studied for a different type of convexities ^{[46,47,48,49,50]}. Here, we defined the following generalized fractional integral operators containing generalized Bessel-Maitland function as its kernel defined, which are the generalization of many well-known fractional integrals:

Definition 1.12. The generalized fractional integral operators with extended generalized Bessel-Maitland function acting as kernel, is defined for $\mu, \nu, \eta, \rho, \gamma, c \in \mathbb{C}$, $\Re(\mu) > 0$, $\Re(\nu)\geq-1$, $\Re(\eta) > 0$, $\Re(\rho) > 0$, $\Re(\gamma) > 0$; $\xi, m, \sigma \geq0$ and $m, \xi > \Re(\mu)+\sigma$ as follows

and

Remark 1.1. 1. If we put $r = 0$, $w = 0$ and replacing $v$ by $v-1$ in definition 1.12, we get the Riemann-Liouville fractional integral operators ^[21].

2. If we put $\sigma = 0$ and replace $v$ by $v-1$ in definition 1.12, $\omega$ by $-\omega$, we get generalized fractional integral operator containing extended generalized Mittag-Leffler function as their kernels defined by Andric et al.^[22].

3. If we put $r = 0$, $\xi = 0$ and replacing $v$ by $v-1$, $\omega$ by $-\omega$ in definition 1.12, we get generalized fractional integral operator containing generalized Mittag-Leffler function as their kernels defined in ^[23].

4. If we put $r = 0$, $\xi = 0$, $\sigma = 0$, $\rho = m = 1$ in definition 1.12, we get the Srivastava fractional integral operator ^[24].

This paper aims to obtain Hermite Hadamard and Fejér inequalities using generalized fractional integral having extended generalized Bessel-Maitland function as its kernel.

The structure of the paper follows: In section 2, we present Hermite-Hadamard inequalities for convex function using generalized fractional operator. Section 3 is devoted to Trapezoid type inequalities related to Hermite-Hadamard inequalities. Fejér type inequalities for $(\eta_{1}, \eta_{2})$- convex function using the generalized fractional operator are presented in section 4.

2.   Hermite-Hadamard inequalities

In this section, we obtain the Hermite-Hadamard inequalities for convex function using generalized fractional operator as follows:

Theorem 2.1. Let $\Psi:[u, v] \rightarrow \mathbb{R}$ be a convex function where $0 < u < v$ and $\Psi \in L_{1}[u, v]$. If $\Psi$ is an increasing function on $[u, v]$, then for the generalized fractional integrals defined in definition 1.12, we have

Proof. By the convexity of $\Psi$ on the interval $[u, v]$, let $x, y \in [u, v]$ with $t = \frac{1}{2}$, we have

where if we takes

leads to

Multiplying both sides by $(1-t)^{v^\prime}\mathfrak{J}_{v^\prime, \eta, \rho, \gamma}^{\mu, \xi, m, \sigma, c}(\omega(1-t)^{\mu}; p)$ and integrating the resulting inequality on $[0, 1]$ with respect to t, we have

By making suitable substitutions in inequality (2.1), we obtain

For second part of inequality, again using the convexity of $\Psi$,

and

Addition of these inequalities, gives

Multiplying both sides by $(1-t)^{v^\prime}\mathfrak{J}_{v^\prime, \eta, \rho, \gamma}^{\mu, \xi, m, \sigma, c}(\omega(1-t)^{\mu}; p)$ and integrating the resulting inequality on $[0, 1]$ with respect to $t$, we get

Making substitution in the integrals involved leads to

combining (2.2) and (2.3), we get the desired result.

3.   Trapezoid inequalities related to the Hermite-Hadamard inequalities

The Trapezoid type inequalities related to the Hermite-Hadamard inequalities are presented in this section.

Lemma 3.1. Let a function $\Psi:I \rightarrow \mathbb{R}$ with $I = [u, v] \subseteq \mathbb{R}$, $\Psi' \in L_{1}[u, v]$ be a differentiable function on $(u, v)$. Then for the generalized fractional integrals defined in definition (1.12), we have

Proof. If we consider the integral

Let

Firstly, we consider $I_{1}$

Integrating by parts, we have

On the same lines, we get

Multiplying by $\frac{v-u}{2}$, we have the required result.

By using Lemma 3.1, we present the following theorem.

Theorem 3.1. Let a function $\Psi:I = [u, v]\rightarrow \mathbb{R}$ with $I \in \mathbb{R}$ be a differentiable function on $(u, v)$. Also, suppose that $|\Psi'|$ is a convex function on $I$, then for the generalized fractional integrals in definition 1.12, we have

where $v^\prime \geq 0$.

Proof. If we consider the following integral expression

Solving the integrals involved by using integrating by parts method, we obtain

4.   Fejér type inequalities for $(\eta_{1}, \eta_{2})$-convex function

Here, we present Fejér type inequalities for $(\eta_{1}, \eta_{2})$-convex function by using the generalized fractional operator in definition 1.12.

Theorem 4.1. Let $\Psi:I \rightarrow \mathbb{R}$, be an $(\eta_{1}, \eta_{2})$-convex functions such that $\eta_{2}$ is an integrable bi-function on $\Psi(I)\times\Psi(I)$ and for any $u, v \in I, \eta_{1}(v, u) > 0$ with $\Psi \in L_{1}[u, u+\eta_{1}(v, u)]$ and the function $\Phi:[u, u+\eta_{1}(v, u)]\rightarrow \mathbb{ R}^{+}$ is integrable and symmetric to $u+\frac{1}{2}\eta_{1}(v, u)$ i.e., $\Phi(2u+\eta_{1}(v, u)-x) = \Phi(x)$, where $I \subseteq\mathbb{R}$ be an invex set with respect to $\eta_{1}$ such that

for all $t_{1}, t_{2} \in [0, 1]$. Then for the generalized fractional integrals defined in definition (1.12), the following Fejér type inequality holds:

Proof. By the $(\eta_{1}, \eta_{2})$-convexity of the function $\Psi$ and using (4.1), we get

or

Now by adapting the same procedure as above, we obtain

Using the generalized fractional integral operators defined in 12, we have

By making the substitution $x = \frac{2u+(1-t) \eta_{1}(v, u)}{2}$ and $x = \frac{2u+(1+t)\eta_{1}(v, u)}{2}$ respectively in the integrals appearing in (4.4), we have

By using the inequalities (4.2) and (4.3), we proceed

or

Again by simplification and using the above mentioned substitution as well as the symmetry of $\Phi$ to $u+\frac{1}{2}\eta_{1}(v, u)$ leads to the following

It follows that

Now, consider

Solving on the same pattern as used above, we get

By adding $I_{1}$ and $I_{2}$, we get the required inequality.

Lemma 4.1. Let I be an invex subset of $\mathbb{R}$ with respect to $\eta_{1}:I\times I\longrightarrow \mathbb{R}$. Let $u, v \in I$ satisfying $\eta_{1}(v, u) > 0$ and $\Phi:[u, u+\eta_{1}(v, u)]\longrightarrow\mathbb{R}$ be integrable and symmetric about $u+\frac{1}{2}\eta_{1}(v, u)$. Then for the integrals defined in definition (1.12), the following holds;

Proof. If we consider

By substituting ${x = 2u+\eta_{1}(v, u)-t}$ in (4.5), we get

Addition of $\Big(\mathfrak{T}_{v^\prime, \eta, \rho, \gamma; (u+\eta_{1}(v, u))^{-}}^{\mu, \xi, m, \sigma, c} {\Psi}\Big)(u; p)$ in Eq (4.6) on both sides, leads to the required result.

Theorem 4.2. Let $\Psi:I \rightarrow \mathbb{R}$, be an $(\eta_{1}, \eta_{2})$-convex functions such that $\eta_{2}$ is an integrable bifunction on $\Psi(I)\times\Psi(I)$ and for any $u, v \in I$, $\eta_{1}(v, u) > 0$ with $\Psi \in L_{1}[u, u+\eta_{1}(v, u)]$ and the function $\Phi:[u, u+\eta_{1}(v, u)]\rightarrow \mathbb{R}^{+}$ is integrable and symmetric to $u+\frac{1}{2}\eta_{1}(v, u)$ i.e $\Phi(2u+\eta_{1}(v, u)-x) = \Phi(x)$, where $I \subseteq\mathbb{R}$ be an invex set with respect to $\eta_{1}$. Then for the generalized fractional integrals defined in definition (1.12), the following Fejér type inequality holds:

Proof. If we consider the integral

Making substitution $x = u+t\eta_{1}(v, u)$ leads to following integral,

Using the $(\eta_{1}, \eta_{2})$-convexity of $\Psi$, we get

Now, consider

Adding Eqs (4.7) and (4.8) and using the symmetry of $\Phi$ about $u+\frac{1}{2}\eta_{1}(v, u)$, we have

By using lemma 4.1, we have

Corollary 4.1. In Fejér type inequalities defined in Theorems 4.1 and 4.2, if we take ${ \eta_{1}(u, v) = u-v}$, $\forall u, v\in I$,

which is Fejér inequality for generalized fractional integral can be obtained by considering the function $\Psi$ to be $\eta$-convex.

Corollary 4.2. In Theorems 4.1 and 4.2, if we put $\eta_{2}(u, v) = u-v$, $for\; all\; u, v \in \Psi(I)$, then

which is Fejér inequality for generalized fractional integral can be obtained by considering the function $\Psi$ to be preinvex convex.

Corollary 4.3. In Fejér type inequalities 4.1 and 4.2, if we put $\eta_{1}(u, v) = u-v, \forall u, v \in I$ and $\eta_{2}(x, y) = x-y, \forall x, y \in \Psi(I)$,

which is Fejér type inequality for generalized fractional integral can be obtained by considering the function $\Psi$ to be convex.

Corollary 4.4. In corollary 4.3, if we take $\Phi = 1$, we get Hermite-Hadamard type inequality for convex function discussed in 2.1.

Corollary 4.5. In Fejér type inequalities defined in 4.1 and 4.2, if we take $\Phi = 1$ then can obtain the Hermite-Hadamard type inequality for $(\eta_{1}, \eta_{2})$-convex function as

5.   Mid point and trapezoid type inequalities related to Hermite-Hadamard type inequalities

In the section, we discuss the midpoint and trapezoid type inequalities connected to Hermite-Hadamard inequalities for the function whose absolute value of the derivative is $(\eta_{1}, \eta_{2})$-convex function. The following lemma will help us in the next result.

Lemma 5.1. Let a function $\Psi:I\rightarrow \mathbb{R}$ with $I \in \mathbb{R}$, $\Psi \in L_{1}[u, u+\eta_{1}(v, u)]$ be a differentiable function where $I$ is taken to be an open invex set with respect to $\eta_{1}:I\times I\rightarrow \mathbb{R}$ with $\eta_{1}(v, u) > 0$, for $u, v\in I$. Then for the generalized fractional integrals defined in definition 1.12, we have

Proof. If we consider

Solving the integrals by using integrating by parts method leads to

Similarly,

Now

Similarly,

Adding $I_{1}, I_{2}, I_{3}$ and $I_{4}$, we proceed to the desired result as,

Next, we present mid-point type inequalities related to Hermite-Hadamard inequalities:

Theorem 5.1. Consider a function $\Psi:I\rightarrow \mathbb{R}$ with $I \in \mathbb{R}$, $\Psi' \in L_{1}[u, u+\eta_{1}(v, u)]$ be a differentiable function where $I$ is taken to be an open invex set with respect to $\eta_{1}:I\times I\rightarrow \mathbb{R}$ with $\eta_{1}(v, u) > 0$ for $u, v\in I$. Then for the generalized fractional integrals defined in definition 1.12, we have

for $0 < v^\prime+\mu n\leq 1$.

Proof. By using lemma 5.1 and using the property of absolute function for addition, leads to

To solve $|I_{k}|, k = 1, 2, 3, 4$, we further move by using $(\eta_{1}, \eta_{2})$-convexity of $|\Psi'|$

For $|I_{k}|, k = 3, 4$. We will use the fact that for all $j \in(0, 1]$ and $u_{1}, u_{2} \in[0, 1]$. Therefore, we have

and

Using the above evaluated absolute values in (5.1), we have

Corollary 5.1. In Theorem 5.1, if we take $\eta_{1}(u, v) = u-v$, $\forall u, v\in I$, then

Corollary 5.2. In Theorem 5.1, if we take $\eta_{1}(u, v) = u-v$, $\forall u, v \in I$ and $\eta_{2}(x, y) = x-y$, $\forall x, y\in \Psi(I)$, then

Lemma 5.2. If we consider a function $\Psi:I\rightarrow \mathbb{R}$ with $I \in \mathbb{R}$, $\Psi\in L_{1}[u, u+\eta_{1}(v, u)]$ be a differentiable function where $I$ is taken to be an open invex set with respect to $\eta_{1}:I\times I\rightarrow \mathbb{R}$ with $\eta_{1}(v, u) > 0$ for $u, v\in I$. Then for the generalized fractional integrals defined in definition (1.12), we have

Proof. We consider the fractional integral

Let

First, we consider $I_{1}$

Integrating by parts, we have

On the same lines, we get

Multiplying by $\frac{\eta_{1}(v, u)}{2}$, we get the required result.

Here, we are able to give trapezoid-type inequalities related to Hermite-Hadamard inequalities:

Theorem 5.2. If we consider a function $\Psi:I\rightarrow \mathbb{R}$ with $I \in \mathbb{R}$, $\Psi'\in L_{1}[u, u+\eta_{1}(v, u)]$ be a differentiable function where $I$ is taken to be an open invex set with respect to $\eta_{1}:I\times I\rightarrow \mathbb{R}$ with $\eta_{1}(v, u) > 0$ for $u, v\in I$. Suppose also that $|\Psi'|$ is an $(\eta_{1}, \eta_{2})$-convex function on $I$. Then for the generalized fractional integrals defined in definition 1.12, we have

where $v^\prime\geq0$.

Proof.

Solving the integrals involved by using integrating by parts method, we obtain the desired result

Corollary 5.3. In theorem 5.2, $\eta_{1}(u, v) = u-v$, $\forall u, v \in I$ gives

Corollary 5.4. In theorem (5.2), $\eta_{1}(u, v) = u-v$, $\forall u, v \in I$, and $\eta_{2}(u, v) = u-v$, $\forall u, v \in \Psi(I)$ gives

6.   Concluding remarks

Various researchers have studied integral inequalities due to their wide applications in both pure and applied mathematics. This paper discussed the new version of integral inequalities such as Hermite-Hadamard type and trapezoid type inequalities for the convex function by utilizing generalized fractional integrals concerning the extended Wright generalized Bessel function as a kernel. Also, we established new mid-point type and trapezoidal type integral inequalities for $(\eta_{1}, \eta_{2})$-convex function related to Hermite-Hadamard and Fejér type inequalities. All the inequalities presented in this paper are more general than the inequalities available in the literature, which can easily observe from the corollaries.

Acknowledgments

The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through Research Group Program under Grant No. RGP. 2/51/42.

Conflict of interest

The authors declare that they have no competing interest.