New Fractional Integral Inequalities Involving the Fox-H and Meijer-G Functions for Convex and Synchronous Functions

Abstract

On the one hand, convex functions are important to derive rigorous convergence rates, and on the other, synchronous functions are significant to solve statistical problems using Chebyshev inequalities. Therefore, fractional integral inequalities involving such functions play a crucial role in creating new models and methods. Although a large class of fractional operators have been used to establish inequalities, nevertheless, these operators having the Fox-H and the Meijer-G functions in their kernel have been applied to establish fractional integral inequalities for such important classes of functions. Taking motivation from these facts, the primary objective of this work is to develop fractional inequalities involving the Fox-H function for convex and synchronous functions. Since the Fox-H function generalizes several important special functions of fractional calculus, our results are significant to innovate the existing literature. The inventive features of these functions compel researchers to formulate deeper results involving them. Therefore, compared with the ongoing research in this field, our results are general enough to yield novel and inventive fractional inequalities. For instance, new inequalities involving the Meijer-G function are obtained as the special cases of these outcomes, and certain generalizations of Chebyshev inequality are also included in this article.

1. Introduction and Motivation

Based on the mathematical theory of convex analysis [1], it is possible to derive rigorous convergence rates for optimization methods [2,3] used in emerging areas such as machine learning [4], artificial intelligence [5], and data science. Convex functions have a distinct global minimum, which facilitates and increases the reliability of optimization. Moreover, the developed theory can often be transferred to general non-convex problems. An intriguing field of study that links the ideas of classical calculus and mathematical analysis with more recent developments in fractional calculus is the relationship between convexity and fractional calculus [6]. Fractional calculus produces new directions in mathematical modeling and optimization using a framework for capturing non-local effects and applying them to convex functions. Therefore, the literature for the study of fractional inequalities is very rich, and numerous authors have examined various forms of inequalities in recent years (see [7] and references therein). For instance, subsequent classical inequalities [8]

$${\displaystyle \frac{{\int }_a^{b}f\left(x\right)dx}{{\int }_a^{b}h\left(x\right)dx}}\ge {\displaystyle \frac{{\int }_a^b\eta \left(f\left(x\right)\right)dx}{{\int }_a^b\eta \left(h\left(x\right)\right)dx}}$$

(1)

and

$${\displaystyle \frac{{\int }_a^{b}f\left(x\right)dx}{{\int }_a^{b}h\left(x\right)dx}}\ge {\displaystyle \frac{{\int }_a^{b}g\left(x\right)\eta \left(f\left(x\right)\right)dx}{{\int }_a^{b}g\left(x\right)\eta \left(h\left(x\right)\right)dx}},$$

(2)

for convex functions gained considerable interest from researchers for further study. The Riemann–Liouville (R-L) operator is used to generalize the results of [8] in [9]. Those results were generalized in [10,11,12] using Saigo and the Marichev–Saigo–Maeda (M-S-M) operators. Similarly, numerous academics have been interested in the Chebyshev inequality defined for integrable functions as [13,14]

$${\displaystyle \frac{1}{b-a}}{\int }_a^{b}f\left(x\right)g\left(x\right)dx\ge \left({\displaystyle \frac{1}{b-a}}{\int }_a^{b}f\left(x\right)dx\right)\left({\displaystyle \frac{1}{b-a}}{\int }_a^{b}g\left(x\right)dx\right).$$

(3)

This is because of its uses in statistics, probability, numerical quadrature, and transform theory. For example, let f and g be increasing functions and x be a probability measure on the real line, then according to (3), there is a positive correlation between the random variables f and g, i.e.,

$$E\left[f\left(x\right)g\left(x\right)\right]\ge E\left[f\left(x\right)\right]E\left[g\left(x\right)\right].$$

(4)

Hence, due to their tremendous applications, various well-known fractional integral operators have been applied to generalize the inequality (3); see [15,16,17,18,19] and the citations therein. Interestingly, we use multiple Erdélyi–Kober (m-E-K) fractional operators [20,21,22] to establish the generalized forms of (1)–(3), and several others from which the various special cases can be generated. Most of them are new and not found in the existing literature, but the known ones are also validated with them. We also establish the weighted and extended Chebyshev inequalities. This is because these multiple operators have been used only to establish inequalities in [23,24]. Since these are in the most general form of many applicable operators, the results explored in this research produce a large class of fractional inequalities. Among them, we investigate the novel inequalities generalizing the previous works investigated and cited in [11,18,19].

Furthermore, while generalized Ostrowski-type inequalities using local fractional transforms are developed in [25], generalized Hermite–Hadamard-type inequalities containing fractional transforms are identified in [26]. The authors of [10] demonstrated a general class of inequalities over $j\in N$ decreasing as well as positive functions, applying the Saigo fractional order operator. For inequalities containing fractional integrals of M-S-M, some articles are available [11,12,19,27,28]. This review of the literature shows that many expansions of the R-L along with E-K operators have been examined so far. It can be noted that various contemporary forms of fractional integrals (proportional and conformable) are employed in the existing literature. The Bessel functions $J_{\mu }$ and the $G,H$-functions in the integrand [20,21] are examples of these extensions. Such generalized fractional operators can frequently be written in the manner described by Kalla ([22], Equation 33):

$$I\left(f\left(z\right)\right)={\int }_0^1\Psi \left(\sigma \right){\sigma }^{\gamma }f\left(z\sigma \right)d\sigma =z^{-\gamma -1}{\int }_0^z\Psi \left({\displaystyle \frac{\xi }{z}}\right){\xi }^{\gamma }f\left(\xi \right)d\xi .$$

(5)

The integral in (5) becomes meaningful for large functional spaces due to a good choice of function $\Psi$. We must note that only a few applicable laws for (5) may be given for a relatively broad or limited choice of the special function $\Psi$. Consequently, it is essential to select the kernel functions $\Psi$ as $G_{m,m}^{m,0}$ - and $H_{m,m}^{m,0}$ -functions [22], namely, the m-E-K operators [20,21], which can be decomposed into Erdélyi–Kober (E-K) operators as a commutable product of ordinary operators. Likewise, several compositions of R-L and E-K operators are also taken into consideration [29,30]. Such compositions have kernels $H_{m,m}^{m,0}$ (or $H_{n+n,l+n}^{m,m}$ if compositions of left and right operators are chosen). These operators are used to handle several topics containing the theory of integral and differential transforms, operational calculus, and geometric function theory [20], and possess several proven applications and a comprehensive, well-developed theory [20,21,22]. The authors of [22] also examine the Riemann–Liouville and Caputo versions of these operators in connection with the spaces defined in [31]. Consequently, by taking into account that for some special values of the involved parameters, the H-function simplifies to the Meijer G-function [20], which is related to the Mittag–Leffler of different parameters, generalized hypergeometric, Fox–Wright, and numerous other special functions, a class of popular fractional transforms, including M-S-M [32,33], Saigo [34,35], R-L [20,21,22], Katugampola, Weyl, and Hadamard, is obtainable from these operators (see [20,21,22] and cited bibliography therein). In numerous recent works [36,37,38], this operator has been used in connection with the special functions. Such multiple operators also hold the class of weighted analytic functions [31], meet the semigroup property, and are commutable, bilinear, as well as invertible [22]. It is also demonstrated that they behave as bounded linear operators over $L_{\vartheta }^p$ under certain conditions [20,21,22] and have a positive kernel [39,40,41]. Further extensions of these m-E-K operators are also discussed in [42].

The primary purpose of this work is to develop certain generalized inequalities for convex and synchronous functions using m-E-K transforms. Therefore, before going into greater detail about our discoveries, we first give the relevant background information and fundamental preliminary information in Section 2 that follows. The outline of this paper is stated as follows: after presenting the necessary preliminaries and basic concepts in Section 2, we apply m-E-K transforms to analyze the new inequalities in Section 3. Here, we provide a new class of Fox-H function inequalities for a collection of convex functions. Likewise, in Section 4, we use synchronous functions to extend the results for the generalized Chebyshev inequality. The results are summarized in the final Section 5.

2. Preliminaries

This section contains and comprehends the basic required information about the classes of functions and the fractional integral transforms. The sets of natural, real, and complex numbers are denoted by $\mathbb{N},\mathbb{R}$ and $\mathbb{C}$, respectively.

Definition 1.

Any function η is convex [1], if for any two points $t_1$ and $t_2$ in its domain and any α in the interval $[0,1]$, we have

$$\eta (\alpha t_1+(1-\alpha )t_2)\le \alpha \eta \left(t_1\right)+(1-\alpha )\eta \left(t_2\right).$$

(6)

Secondly, the function η has no finite values of negative infinity, which means that there does not exist any t in its domain such that $\eta \left(t\right)=-\infty$. The first condition of convexity ensures that the function’s graph lies above any line segment connecting two points in its domain. In contrast, the second condition ensures that the function does not have an arbitrarily large negative value. A function that is not convex is called non-convex. In other words, the function “curves upwards” and does not have any “dips” or “valleys” in its graph.

Definition 2.

Two functions ${\eta }_1\left(t\right)$ and ${\eta }_2\left(t\right)$ are synchronous functions [15,16,17,18,19] over the interval $(0\le x,y<\infty )$, if

$$({\eta }_1\left(x\right)-{\eta }_1\left(y\right))({\eta }_2\left(x\right)-{\eta }_2\left(y\right))\ge 0.$$

(7)

Definition 3.

For a pair of integrable functions ${\eta }_1\left(t\right)$ and ${\eta }_2\left(t\right)$ defined over the interval $[a,b]$ and a pair of positive functions $p\left(t\right),q\left(t\right)$, the extended Chebyshev inequality is stated as [15,16,17,19]

$$\begin{array}{c}\hfill {\int }_a^{b}q\left(x\right)dx{\int }_a^{b}p\left(x\right){\eta }_1\left(x\right){\eta }_2\left(x\right)dx+{\int }_a^{b}p\left(x\right)dx{\int }_a^{b}q\left(x\right){\eta }_1\left(x\right){\eta }_2\left(x\right)dx\\ \hfill \ge {\int }_a^{b}p\left(x\right){\eta }_1\left(x\right)dx{\int }_a^{b}q\left(x\right){\eta }_2\left(x\right)dx+{\int }_a^{b}q\left(x\right){\eta }_1\left(x\right)dx{\int }_a^{b}p\left(x\right){\eta }_2\left(x\right)dx.\end{array}$$

(8)

Likewise, the weighted Chebyshev inequality is stated as [15,16,17,19]

$${\int }_a^{b}p\left(x\right)dx{\int }_a^b{\eta }_1\left(x\right){\eta }_2\left(x\right)dx\ge \left({\int }_a^{b}p\left(x\right)dx{\int }_a^b{\eta }_1\left(x\right)dx\right)\left({\int }_a^{b}p\left(x\right)dx{\int }_a^b{\eta }_2\left(x\right)dx\right).$$

(9)

Remark 1.

For $p\left(x\right)=q\left(x\right)$ in (8), one obtains (9), and for $p\left(x\right)=1;{\eta }_1=f;{\eta }_2=g$ in (9), we obtain (3).

Definition 4.

Here, we start defining the Fox-H function as follows (see [20,21,22]):

$$\begin{gathered} \begin{array}{c}\hfill H_{p,q}^{l,m}\left(\theta \right)=H_{p,q}^{l,m}\left[\theta |\begin{array}{c}\hfill (a_i,A_i)\\ \hfill (b_j,B_j)\end{array}\right]={\displaystyle \frac{1}{2\pi \iota }}{\int }_L{\displaystyle \frac{{\prod }_{j=1}^l\Gamma (b_j,+B_{j}s){\prod }_{i=1}^m\Gamma (1-a_i-A_{i}s)}{{\prod }_{j=l+1}^q\Gamma (1-b_j-B_{j}s){\prod }_{i=m+1}^p\Gamma (a_i+A_{i}s)}}{\theta }^{-s}ds\end{array} \\ \begin{array}{c}\hfill (i=1,\cdots ,p;j=1,\cdots ,q;A_i,B_j>0;a_i,b_j\in \mathbb{C};1\le m\le p;1\le l\le q).\end{array} \end{gathered}$$

(10)

An appropriate contour L is used to split the singularities from the involved ratios of the gamma function in Equation (10).

Remark 2.

Additionally, we take into account that if the values of $A_i,B_j$ coincide with unity, then the H-function simplifies to the Meijer G-function [20], which is related to the Mittag–Leffler of different parameters, generalized hypergeometric, Fox–Wright, and numerous other special functions.

Definition 5.

For $t>0$, m-E-K operators of non-integer order ${\mu }_m$ denoted by ${\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}$ of multiplicity m are defined as [21]

$${\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[f\left(t\right)\right]={\displaystyle \frac{1}{t}}{\int }_0^t{H}_{m,m}^{m,0}\left[{\displaystyle \frac{x}{t}}|\begin{array}{c}\hfill {({\mu }_m+{\xi }_m+1-{\displaystyle \frac{1}{{\delta }_m}},{\displaystyle \frac{1}{{\delta }_m}})}_1^k\\ \hfill {({\mu }_m+1-{\displaystyle \frac{1}{{\delta }_m}},{\displaystyle \frac{1}{{\delta }_m}})}_1^k\end{array}\right]f\left(x\right)dx.$$

(11)

The order of the fractional integration is provided by the different parameters employed in Equation (11) above, i.e., $({\mu }_m\ge 0)$, $({\xi }_m\ge 0)$ are multi-weights and other multi-parameters $({\delta }_m>0)$. Moreover, we can observe that ${\sum }_{m=1}^k{\xi }_m>0$ and for ${\xi }_m=0$, $I_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),(0,\cdots ,0)}f\left(t\right):=f\left(t\right)$ can be obtained from Equation (11).

Definition 6.

The spaces $C_{\vartheta }\left([0,\infty ]\right)$ over real variable $x>0$ of good functions were introduced by Dimovski in [31] taking $\tilde{f}\left(x\right)=x^{p}f\left(x\right),p>\vartheta$ for $x>0,\vartheta \in \mathbb{R}$ and $f\in C_{\vartheta }$, where $\tilde{f}\in C[0,\infty ]$ is continuous, and $f\in C_{\vartheta }^{\left(n\right)},n\in \mathbb{N}$ with $\tilde{f}\in C^n\left([0,\infty )\right)$ exhibiting a similar mapping.

Remark 3.

Other features of these areas are highlighted in [20]. Lebesgue integrable spaces ($L_{\vartheta }^p(0,\infty ),1\le p<\infty$) that fulfill $f_{\vartheta ,p}={\int }_0^{\infty }{x}^{\vartheta -1}{\left|f\left(x\right)\right|}^p{dx]}^{1/p}<\infty$ also maintain the power function. Additionally, our findings are predicated on the following hypotheses:

$$\begin{array}{c}\hfill {\xi }_m\ge 0;m=1,\cdots ,k;\\ \hfill {\delta }_m({\mu }_m+1)>-\vartheta ,(f\in C_{\vartheta }\left([0,\infty )\right);\\ \hfill {\delta }_m({\mu }_m+1)>\vartheta /\rho ;(f\in L_{\vartheta ,\rho }\left([0,\infty )\right).\end{array}$$

(12)

Lemma 1.

Since $x^{\vartheta }=f\in C_{\vartheta }$, therefore we have the subsequent formula

$${\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[x^{\vartheta -1}\right]=\prod _{m=1}^k{\displaystyle \frac{\Gamma ({\mu }_m+1+\frac{\vartheta -1}{{\delta }_m})}{\Gamma ({\mu }_m+{\xi }_m+\frac{\vartheta -1}{{\delta }_m})}}{x}^{\vartheta -1},({\mu }_m\ge 0;\vartheta -1>-{\delta }_m({\mu }_m+1);m=1,\cdots ,k).$$

(13)

Taking $\vartheta =1$ in the above Equation (13) leads to the following:

$${\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[1\right]=\prod _{m=1}^k{\displaystyle \frac{\Gamma ({\mu }_m+1)}{\Gamma ({\mu }_m+{\xi }_m+1)}},({\mu }_m\ge 0,m=1,\cdots ,k).$$

(14)

Remark 4.

In the Formulation (11) of the m-E-K operators of non-integer order, it must be noted that the kernel (special Fox-H function) stays positive [39,40,41].

Remark 5.

When all ${\beta }_k^{\prime }s$ are equal, that is, when ${\delta }_m=\delta$, these operators are also connected with a variety of commonly used integral operators of fractional order as follows [20,21]:

• For ${\delta }_m=\delta >0$, m-E-K integrals of a non-integer order can be more easily described by the Meijer G-function [21]; see Equation (19) (see also [20], Ch.1]). Using this point, we have from (11)

  $${\mathcal{G}}_{\left(\delta \right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[f\left(t\right)\right]={\displaystyle \frac{\delta }{x^{\delta }}}{\int }_0^t{t}^{\delta -1}{G}_{m,m}^{m,0}\left[{\left({\displaystyle \frac{x}{t}}\right)}^{\delta }|\begin{array}{c}\hfill {({\mu }_m+{\xi }_m+1-{\displaystyle \frac{1}{\delta }},{\displaystyle \frac{1}{\delta }})}_1^k\\ \hfill {({\mu }_m+1-{\displaystyle \frac{1}{\delta }},{\displaystyle \frac{1}{\delta }})}_1^k\end{array}\right]f\left(x\right)dx.$$

  (15)
• The M-S-M operator of fractional order is obtained by reducing m-E-K for $m=3$ and ${\delta }_m=\delta =1$, [32,33].
• The Saigo fractional operator can be obtained by reducing m-E-K for $m=2$ and ${\delta }_m=\delta >0$ [34,35].
• E-K fractional operator can be obtained by reducing m-E-K for $m=1;{\delta }_m=\delta >0$.
• R-L fractional operator can be obtained by reducing m-E-K for $(m=1=\delta )$; it reduces to [20,21].
• Hyper Bessel as well as Bessel operators [20,21] can be obtained for $m\ge 0;({\xi }_1={\xi }_2={\xi }_3=1)$.

3. New Fractional Integral Inequalities Involving Fox-H Functions for Convex Functions

In this part, we use the m-E-K transform to build new and novel inequalities for a class of convex functions. These inequalities generalize the inequality (1) and build such inequalities for weighted and extended functionals from (2).

Theorem 1.

Consider two positive continuous functions $g,h$ such that $g\le h$ on $[0,\infty )$. Furthermore, for an arbitrary convex function $\eta ,\eta \left(0\right)=0$, if g is increasing and ${\displaystyle \frac{g}{h}}$ is decreasing on $[0,\infty )$, then prove that

$${\displaystyle \frac{{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[g\left(t\right)\right]}{{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[h\left(t\right)\right]}}\ge {\displaystyle \frac{{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[\eta \left(g\left(t\right)\right)\right]}{{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[\eta \left(h\left(t\right)\right)\right]}},$$

(16)

provided $t>0,k>1,{\xi }_m\ge 0,{\delta }_m>0$, and ${\mu }_k>-1-{\displaystyle \frac{\vartheta }{{\delta }_m}}$.

Proof. 

Due to the convexity of $\eta$ along with $\eta \left(0\right)=0$, the ratio ${\displaystyle \frac{\eta \left(t\right)}{t}}$ has increasing behavior. Such behaviour of g indicates that ${\displaystyle \frac{\eta \left(g\right(t\left)\right)}{g\left(t\right)}}$ is also increasing. Clearly, ${\displaystyle \frac{g\left(t\right)}{h\left(t\right)}}$ is decreasing, $\forall x,y\in [0,\infty )$, and

$$\left({\displaystyle \frac{\eta \left(g\right(x\left)\right)}{g\left(x\right)}}-{\displaystyle \frac{\eta \left(g\right(y\left)\right)}{g\left(y\right)}}\right)\left({\displaystyle \frac{g\left(y\right)}{h\left(y\right)}}-{\displaystyle \frac{g\left(x\right)}{h\left(x\right)}}\right)\ge 0.$$

(17)

This implies that

$${\displaystyle \frac{\eta \left(g\right(x\left)\right)}{g\left(x\right)}}{\displaystyle \frac{g\left(y\right)}{h\left(y\right)}}+{\displaystyle \frac{\eta \left(g\right(y\left)\right)}{g\left(y\right)}}{\displaystyle \frac{g\left(x\right)}{h\left(x\right)}}-{\displaystyle \frac{\eta \left(g\right(x\left)\right)}{g\left(x\right)}}{\displaystyle \frac{g\left(x\right)}{h\left(x\right)}}-{\displaystyle \frac{\eta \left(g\right(y\left)\right)}{g\left(y\right)}}{\displaystyle \frac{g\left(y\right)}{h\left(y\right)}}\ge 0.$$

(18)

Multiplying the inequality (18) by $h\left(x\right)h\left(y\right)$, we have

$${\displaystyle \frac{\eta \left(g\right(x\left)\right)}{g\left(x\right)}}g\left(y\right)h\left(x\right)+{\displaystyle \frac{\eta \left(g\right(y\left)\right)}{g\left(y\right)}}g\left(x\right)h\left(y\right)-{\displaystyle \frac{\eta \left(g\right(x\left)\right)}{g\left(x\right)}}g\left(x\right)h\left(y\right)-{\displaystyle \frac{\eta \left(g\right(y\left)\right)}{g\left(y\right)}}g\left(y\right)h\left(x\right)\ge 0.$$

(19)

Multiplying both sides of (19) by ${\displaystyle \frac{1}{t}}{H}_{m,m}^{m,0}\left[{\displaystyle \frac{x}{t}}|\begin{array}{c}\hfill {({\mu }_m+{\xi }_m+1-{\displaystyle \frac{1}{{\delta }_m}},{\displaystyle \frac{1}{{\delta }_m}})}_1^k\\ \hfill {({\mu }_m+1-{\displaystyle \frac{1}{{\delta }_m}},{\displaystyle \frac{1}{{\delta }_m}})}_1^k\end{array}\right]$ and integrating for $x\in (0,t)$,

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{t}}{\int }_0^t{H}_{m,m}^{m,0}\left[{\displaystyle \frac{x}{t}}|\begin{array}{c}\hfill {({\mu }_m+{\xi }_m+1-{\displaystyle \frac{1}{{\delta }_m}},{\displaystyle \frac{1}{{\delta }_m}})}_1^k\\ \hfill {({\mu }_m+1-{\displaystyle \frac{1}{{\delta }_m}},{\displaystyle \frac{1}{{\delta }_m}})}_1^k\end{array}\right]{\displaystyle \frac{\eta \left(g\right(x\left)\right)}{g\left(x\right)}}g\left(y\right)h\left(x\right)dx\\ \hfill +{\displaystyle \frac{1}{t}}{\int }_0^t{H}_{m,m}^{m,0}\left[{\displaystyle \frac{x}{t}}|\begin{array}{c}\hfill {({\mu }_m+{\xi }_m+1-{\displaystyle \frac{1}{{\delta }_m}},{\displaystyle \frac{1}{{\delta }_m}})}_1^k\\ \hfill {({\mu }_m+1-{\displaystyle \frac{1}{{\delta }_m}},{\displaystyle \frac{1}{{\delta }_m}})}_1^k\end{array}\right]{\displaystyle \frac{\eta \left(g\right(y\left)\right)}{g\left(y\right)}}g\left(x\right)h\left(y\right)dx\\ \hfill -{\displaystyle \frac{1}{t}}{\int }_0^t{H}_{m,m}^{m,0}\left[{\displaystyle \frac{x}{t}}|\begin{array}{c}\hfill {({\mu }_m+{\xi }_m+1-{\displaystyle \frac{1}{{\delta }_m}},{\displaystyle \frac{1}{{\delta }_m}})}_1^k\\ \hfill {({\mu }_m+1-{\displaystyle \frac{1}{{\delta }_m}},{\displaystyle \frac{1}{{\delta }_m}})}_1^k\end{array}\right]{\displaystyle \frac{\eta \left(g\right(x\left)\right)}{g\left(x\right)}}g\left(x\right)h\left(y\right)dx\\ \hfill -{\displaystyle \frac{1}{t}}{\int }_0^t{H}_{m,m}^{m,0}\left[{\displaystyle \frac{x}{t}}|\begin{array}{c}\hfill {({\mu }_m+{\xi }_m+1-{\displaystyle \frac{1}{{\delta }_m}},{\displaystyle \frac{1}{{\delta }_m}})}_1^k\\ \hfill {({\mu }_m+1-{\displaystyle \frac{1}{{\delta }_m}},{\displaystyle \frac{1}{{\delta }_m}})}_1^k\end{array}\right]{\displaystyle \frac{\eta \left(g\right(y\left)\right)}{g\left(y\right)}}g\left(y\right)h\left(x\right)dx\ge 0.\end{array}$$

(20)

Then, we obtain the following

$$\begin{array}{cc}\hfill & g\left(y\right){\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[{\displaystyle \frac{\eta \left(g\right(t\left)\right)}{g\left(t\right)}}h\left(t\right)\right]+{\displaystyle \frac{\eta \left(g\right(y\left)\right)}{g\left(y\right)}}h\left(y\right){\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[g\left(t\right)\right]\hfill \\ \hfill & -h\left(y\right){\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[{\displaystyle \frac{\eta \left(g\right(t\left)\right)}{g\left(t\right)}}g\left(t\right)\right]-{\displaystyle \frac{\eta \left(g\right(y\left)\right)}{g\left(y\right)}}g\left(y\right){\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[h\left(t\right)\right]\ge 0.\hfill \end{array}$$

(21)

Next, multiply both sides of (21) by ${\displaystyle \frac{1}{t}}{H}_{m,m}^{m,0}\left[{\displaystyle \frac{y}{t}}|\begin{array}{c}\hfill {({\mu }_m+{\xi }_m+1-{\displaystyle \frac{1}{{\delta }_m}},{\displaystyle \frac{1}{{\delta }_m}})}_1^k\\ \hfill {({\mu }_m+1-{\displaystyle \frac{1}{{\delta }_m}},{\displaystyle \frac{1}{{\delta }_m}})}_1^k\end{array}\right]$ as well as integrating the resultant form over $y\in (0,t)$ as follows:

$$\begin{array}{c}\hfill {\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[{\displaystyle \frac{\eta \left(g\right(t\left)\right)}{g\left(t\right)}}h\left(t\right)\right]{\displaystyle \frac{1}{t}}{\int }_0^t{H}_{m,m}^{m,0}\left[{\displaystyle \frac{y}{t}}|\begin{array}{c}\hfill {({\mu }_m+{\xi }_m+1-{\displaystyle \frac{1}{{\delta }_m}},{\displaystyle \frac{1}{{\delta }_m}})}_1^k\\ \hfill {({\mu }_m+1-{\displaystyle \frac{1}{{\delta }_m}},{\displaystyle \frac{1}{{\delta }_m}})}_1^k\end{array}\right]g\left(y\right)dy\\ \hfill +{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[g\left(t\right)\right]{\displaystyle \frac{1}{t}}{\int }_0^t{H}_{m,m}^{m,0}\left[{\displaystyle \frac{y}{t}}|\begin{array}{c}\hfill {({\mu }_m+{\xi }_m+1-{\displaystyle \frac{1}{{\delta }_m}},{\displaystyle \frac{1}{{\delta }_m}})}_1^k\\ \hfill {({\mu }_m+1-{\displaystyle \frac{1}{{\delta }_m}},{\displaystyle \frac{1}{{\delta }_m}})}_1^k\end{array}\right]{\displaystyle \frac{\eta \left(g\right(y\left)\right)}{g\left(y\right)}}h\left(y\right)dy\\ \hfill -{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[{\displaystyle \frac{\eta \left(g\right(t\left)\right)}{g\left(t\right)}}g\left(t\right)\right]{\displaystyle \frac{1}{t}}{\int }_0^t{H}_{m,m}^{m,0}\left[{\displaystyle \frac{y}{t}}|\begin{array}{c}\hfill {({\mu }_m+{\xi }_m+1-{\displaystyle \frac{1}{{\delta }_m}},{\displaystyle \frac{1}{{\delta }_m}})}_1^k\\ \hfill {({\mu }_m+1-{\displaystyle \frac{1}{{\delta }_m}},{\displaystyle \frac{1}{{\delta }_m}})}_1^k\end{array}\right]h\left(y\right)dy\\ \hfill -{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[h\left(t\right)\right]{\displaystyle \frac{1}{t}}{\int }_0^t{H}_{m,m}^{m,0}\left[{\displaystyle \frac{y}{t}}|\begin{array}{c}\hfill {({\mu }_m+{\xi }_m+1-{\displaystyle \frac{1}{{\delta }_m}},{\displaystyle \frac{1}{{\delta }_m}})}_1^k\\ \hfill {({\mu }_m+1-{\displaystyle \frac{1}{{\delta }_m}},{\displaystyle \frac{1}{{\delta }_m}})}_1^k\end{array}\right]{\displaystyle \frac{\eta \left(g\right(y\left)\right)}{g\left(y\right)}}g\left(y\right)dy\ge 0,\end{array}$$

(22)

which implies that

$$\begin{array}{cc}\hfill & {\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[g\left(t\right)\right]{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[{\displaystyle \frac{\eta \left(g\right(t\left)\right)}{g\left(t\right)}}h\left(t\right)\right]+{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[{\displaystyle \frac{\eta \left(g\right(t\left)\right)}{g\left(t\right)}}h\left(t\right)\right]{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[g\left(t\right)\right]\hfill \\ \hfill & -{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[h\left(t\right)\right]{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[{\displaystyle \frac{\eta \left(g\right(t\left)\right)}{g\left(t\right)}}g\left(t\right)\right]-{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[{\displaystyle \frac{\eta \left(g\right(t\left)\right)}{g\left(t\right)}}g\left(t\right)\right]{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[h\left(t\right)\right]\ge 0,\hfill \end{array}$$

(23)

and

$${\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[g\left(t\right)\right]{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[{\displaystyle \frac{\eta \left(g\right(t\left)\right)}{g\left(t\right)}}h\left(t\right)\right]\ge {\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[h\left(t\right)\right]{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[{\displaystyle \frac{\eta \left(g\right(t\left)\right)}{g\left(t\right)}}g\left(t\right)\right],$$

(24)

$${\displaystyle \frac{{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[g\left(t\right)\right]}{{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[h\left(t\right)\right]}}\ge {\displaystyle \frac{{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[\frac{\eta \left(g\right(t\left)\right)}{g\left(t\right)}g\left(t\right)\right]}{{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[\frac{\eta \left(g\right(t\left)\right)}{g\left(t\right)}h\left(t\right)\right]}}.$$

(25)

Given that the ratio ${\displaystyle \frac{\eta \left(t\right)}{t}}$ is increasing, and $g\ge h$ on $[0,\infty )$, therefore we state the following:

$${\displaystyle \frac{\eta \left(g\right(x\left)\right)}{g\left(x\right)}}\le {\displaystyle \frac{\eta \left(h\right(x\left)\right)}{h\left(x\right)}}.$$

(26)

Hence, integrate over $x\in (0,t)$ to obtain the subsequent inequality after multiplying (26) with the Fox-H function ${\displaystyle \frac{h\left(x\right)}{t}}{H}_{m,m}^{m,0}\left[{\displaystyle \frac{x}{t}}|\begin{array}{c}\hfill {({\mu }_m+{\xi }_m+1-{\displaystyle \frac{1}{{\delta }_m}},{\displaystyle \frac{1}{{\delta }_m}})}_1^k\\ \hfill {({\mu }_m+1-{\displaystyle \frac{1}{{\delta }_m}},{\displaystyle \frac{1}{{\delta }_m}})}_1^k\end{array}\right]$ to obtain

$${\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[{\displaystyle \frac{\eta \left(g\right(t\left)\right)}{g\left(t\right)}}h\left(t\right)\right]\le {\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[{\displaystyle \frac{\eta \left(h\right(t\left)\right)}{h\left(t\right)}}h\left(t\right)\right].$$

(27)

Combining the inequalities (25) and (27), the required form stated in (16) can be obtained. □

Corollary 1.

Consider two positive continuous functions $g,h$ such that $g\le h$ on $[0,\infty )$. Furthermore, for an arbitrary convex function $\eta ,\eta \left(0\right)=0$, if g is increasing and ${\displaystyle \frac{g}{h}}$ decreasing, then prove that

$${\displaystyle \frac{{\mathcal{G}}_{\left(\delta \right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[g\left(t\right)\right]}{{\mathcal{G}}_{\left(\delta \right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[h\left(t\right)\right]}}\ge {\displaystyle \frac{{\mathcal{G}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[\eta \left(g\left(t\right)\right)\right]}{{\mathcal{G}}_{\left(\delta \right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[\eta \left(h\left(t\right)\right)\right]}},$$

(28)

provided $t>0,k>1,{\xi }_m\ge 0,\delta >0$, and ${\mu }_k>-1-{\displaystyle \frac{\vartheta }{\delta }}$.

Proof. 

This new inclusion involving the Meijer-G function can be proved by taking ${\delta }_m=\delta$ in (16) and using the relation (15). □

Theorem 2.

Consider two positive continuous functions g and h defined on $[0,\infty )$ such that $g\le h$. Furthermore, for any convex function $\eta ,\eta \left(0\right)=0$, if g is increasing along with ${\displaystyle \frac{g}{h}}$ decreasing, then prove that

$${\displaystyle \frac{{\mathcal{I}}_{\left({\tau }_m\right),k}^{\left({\lambda }_m\right),\left({\beta }_m\right)}\left[g\left(t\right)\right]{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[\eta \left(h\left(t\right)\right)\right]+{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[g\left(t\right)\right]{\mathcal{I}}_{\left({\tau }_m\right),k}^{\left({\lambda }_m\right),\left({\beta }_m\right)}\left[\eta \left(h\left(t\right)\right)\right]}{{\mathcal{I}}_{\left({\tau }_m\right),k}^{\left({\lambda }_m\right),\left({\beta }_m\right)}\left[h\left(t\right)\right]{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[\eta \left(g\left(t\right)\right)\right]+{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[h\left(t\right)\right]{\mathcal{I}}_{\left({\tau }_m\right),k}^{\left({\lambda }_m\right),\left({\beta }_m\right)}\left[\eta \left(g\left(t\right)\right)\right]}}\ge 1$$

(29)

provided $t>0,k>1,{\beta }_m;{\xi }_m\ge 0,{\tau }_m;{\delta }_m>0$, and ${\mu }_m>-1-{\displaystyle \frac{\vartheta }{{\delta }_m}};{\lambda }_m>-1-{\displaystyle \frac{\vartheta }{{\tau }_m}}$.

Proof. 

Since the ratio ${\displaystyle \frac{\eta \left(t\right)}{t}}$ is increasing, once the function $\eta$ is convex with $\eta \left(0\right)=0$. Furthermore, ${\displaystyle \frac{\eta \left(g\right(t\left)\right)}{g\left(t\right)}}$ is increasing as well because g is increasing. Since $t>0$, it is evident that ${\displaystyle \frac{g\left(t\right)}{h\left(t\right)}}$ is decreasing for any $x,y\in (0,t)$.

The multiplication of inequality (21), with ${\displaystyle \frac{1}{t}}{H}_{m,m}^{m,0}\left[{\displaystyle \frac{y}{t}}|\begin{array}{c}\hfill {({\lambda }_m+{\beta }_m+1-{\displaystyle \frac{1}{{\tau }_m}},{\displaystyle \frac{1}{{\tau }_m}})}_1^k\\ \hfill {({\lambda }_m+1-{\displaystyle \frac{1}{{\tau }_m}},{\displaystyle \frac{1}{{\tau }_m}})}_1^k\end{array}\right]$ along with integration over $y\in (0,t)$, leads to the following:

$$\begin{array}{cc}\hfill & {\mathcal{I}}_{\left({\tau }_m\right),k}^{\left({\lambda }_m\right),\left({\beta }_m\right)}\left[g\left(t\right)\right]{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[{\displaystyle \frac{\eta \left(g\right(t\left)\right)}{g\left(t\right)}}h\left(t\right)\right]+{\mathcal{I}}_{\left({\tau }_m\right),k}^{\left({\lambda }_m\right),\left({\beta }_m\right)}\left[{\displaystyle \frac{\eta \left(g\right(t\left)\right)}{g\left(t\right)}}h\left(t\right)\right]{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[g\left(t\right)\right]\hfill \\ \hfill & \ge {\mathcal{I}}_{\left({\tau }_m\right),k}^{\left({\lambda }_m\right),\left({\beta }_m\right)}\left[h\left(t\right)\right]{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[\eta \left(g\left(t\right)\right)\right]+{\mathcal{I}}_{\left({\tau }_m\right),k}^{\left({\lambda }_m\right),\left({\beta }_m\right)}\left[\eta \left(g\left(t\right)\right)\right]{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[h\left(t\right)\right].\hfill \end{array}$$

(30)

Given that for $x,y\in [0,\infty )$, the function ${\displaystyle \frac{\eta \left(t\right)}{t}}$ is increasing as well as $g\le h$. This implies that

$${\displaystyle \frac{\eta \left(g\right(x\left)\right)}{g\left(x\right)}}\le {\displaystyle \frac{\eta \left(h\right(x\left)\right)}{h\left(x\right)}}.$$

(31)

Multiply the inequality (31) by ${\displaystyle \frac{h\left(x\right)}{t}}{H}_{m,m}^{m,0}\left[{\displaystyle \frac{x}{t}}|\begin{array}{c}\hfill {({\lambda }_m+{\beta }_m+1-{\displaystyle \frac{1}{{\tau }_m}},{\displaystyle \frac{1}{{\tau }_m}})}_1^k\\ \hfill {({\lambda }_m+1-{\displaystyle \frac{1}{{\tau }_m}},{\displaystyle \frac{1}{{\tau }_m}})}_1^k\end{array}\right]$, and integrate over $x\in (0,t),t>0$, we obtain the following inequality:

$${\mathcal{I}}_{\left({\tau }_m\right),k}^{\left({\lambda }_m\right),\left({\beta }_m\right)}\left[{\displaystyle \frac{\eta \left(g\right(t\left)\right)}{g\left(t\right)}}h\left(t\right)\right]\le {\mathcal{I}}_{\left({\tau }_m\right),k}^{\left({\lambda }_m\right),\left({\beta }_m\right)}\left[\eta \left(h\left(t\right)\right)\right].$$

(32)

A proper merge of the inequalities (30) and (32) leads to the required form, i.e., (29). □

Remark 6.

If we put ${\mu }_m={\lambda }_m,{\tau }_m={\delta }_m$ in (29), we obtain (16).

Corollary 2.

Consider g and h, a pair of positive continuous functions such that $g\le h$ on $[0,\infty )$. Furthermore, for an arbitrary convex function $\eta ,\eta \left(0\right)=0$, if g is increasing and ${\displaystyle \frac{g}{h}}$ is decreasing, on $[0,\infty )$ for $t>0$, then prove that

$${\displaystyle \frac{{\mathcal{G}}_{\left(\tau \right),k}^{\left({\lambda }_m\right),\left({\beta }_m\right)}\left[g\left(t\right)\right]{\mathcal{G}}_{\left(\delta \right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[\eta \left(h\left(t\right)\right)\right]+{\mathcal{G}}_{\left(\delta \right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[g\left(t\right)\right]{\mathcal{G}}_{\left(\tau \right),k}^{\left({\lambda }_m\right),\left({\beta }_m\right)}\left[\eta \left(h\left(t\right)\right)\right]}{{\mathcal{G}}_{\left(\tau \right),k}^{\left({\lambda }_m\right),\left({\beta }_m\right)}\left[h\left(t\right)\right]{\mathcal{G}}_{\left(\delta \right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[\eta \left(g\left(t\right)\right)\right]+{\mathcal{G}}_{\left(\delta \right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[h\left(t\right)\right]{\mathcal{G}}_{\left(\tau \right),k}^{\left({\lambda }_m\right),\left({\beta }_m\right)}\left[\eta \left(g\left(t\right)\right)\right]}}\ge 1$$

(33)

provided $t>0,k>1,{\beta }_m;{\xi }_m\ge 0,\tau ;\delta >0$, and ${\mu }_m>-1-{\displaystyle \frac{\vartheta }{\delta }};{\lambda }_m>-1-{\displaystyle \frac{\vartheta }{\tau }}$.

Proof. 

This new inclusion involving the Meijer-G function can be proved by taking ${\delta }_m=\delta ,{\tau }_m=\tau$ in (29) and using the relation (15). □

Theorem 3.

Consider three positive continuous functions $f,g,h$ with $g\le h$ on $[0,\infty )$. Suppose ${\displaystyle \frac{g}{h}}$ is decreasing, but g and f are increasing functions on $[0,\infty )$ along with $\eta ,\eta \left(0\right)=0$, then prove that

$${\displaystyle \frac{{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[f\left(t\right)\eta \left(g\left(t\right)\right)\right]}{{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[f\left(t\right)\eta \left(h\left(t\right)\right)\right]}}\le {\displaystyle \frac{{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[g\left(t\right)\right]}{{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[h\left(t\right)\right]}}$$

(34)

provided $t>0;k>1,{\xi }_m\ge 0,{\delta }_m>0$, and ${\mu }_m>-1-{\displaystyle \frac{\vartheta }{{\delta }_m}}$.

Proof. 

Since $g\le h$ along with the increasing behavior of ${\displaystyle \frac{\eta \left(t\right)}{t}}$, which implies that

$${\displaystyle \frac{\eta \left(g\right(x\left)\right)}{g\left(x\right)}}\le {\displaystyle \frac{\eta \left(h\right(x\left)\right)}{h\left(x\right)}};(x,y\in [0,\infty )).$$

(35)

Multiplying the inequality (35) by ${\displaystyle \frac{h\left(x\right)f\left(x\right)}{t}}{H}_{m,m}^{m,0}\left[{\displaystyle \frac{x}{t}}|\begin{array}{c}\hfill {({\mu }_m+{\xi }_m+1-{\displaystyle \frac{1}{{\delta }_m}},{\displaystyle \frac{1}{{\delta }_m}})}_1^k\\ \hfill {({\mu }_m+1-{\displaystyle \frac{1}{{\delta }_m}},{\displaystyle \frac{1}{{\delta }_m}})}_1^k\end{array}\right]$, and integrating over $x\in (0,t),t>0$, we obtain the following inequality using (11):

$${\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[{\displaystyle \frac{\eta \left(g\right(t\left)\right)}{g\left(t\right)}}h\left(t\right)f\left(t\right)\right]\le {\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[f\left(t\right)\eta \left(h\left(t\right)\right)\right].$$

(36)

Also the function ${\displaystyle \frac{\eta \left(t\right)}{t}}$ is increasing. Since g is increasing, ${\displaystyle \frac{\eta \left(g\right(t\left)\right)}{g\left(t\right)}}$ is also increasing. Clearly, we can say that ${\displaystyle \frac{\eta \left(t\right)f\left(t\right)}{t}}$ is increasing, for all $x,y\in [0,t),t>0$:

$$\left({\displaystyle \frac{\eta \left(g\right(x\left)\right)}{g\left(x\right)}}f\left(x\right)-{\displaystyle \frac{\eta \left(g\right(y\left)\right)}{g\left(y\right)}}f\left(y\right)\right)(g\left(y\right)h\left(x\right)-g\left(x\right)h\left(y\right))\ge 0,$$

(37)

which implies that

$${\displaystyle \frac{f\left(x\right)\eta \left(g\right(x\left)\right)}{g\left(x\right)}}g\left(y\right)h\left(x\right)+{\displaystyle \frac{\eta \left(g\right(y\left)\right)f\left(y\right)}{g\left(y\right)}}g\left(x\right)h\left(y\right)-{\displaystyle \frac{f\left(x\right)\eta \left(g\right(x\left)\right)}{g\left(x\right)}}g\left(x\right)h\left(y\right)-{\displaystyle \frac{\eta \left(g\right(y\left)\right)f\left(y\right)}{g\left(y\right)}}g\left(y\right)h\left(x\right)\ge 0.$$

(38)

Multiplying both sides of (38) by ${\displaystyle \frac{1}{t}}{H}_{m,m}^{m,0}\left[{\displaystyle \frac{x}{t}}|\begin{array}{c}\hfill {({\mu }_m+{\xi }_m+1-{\displaystyle \frac{1}{{\delta }_m}},{\displaystyle \frac{1}{{\delta }_m}})}_1^k\\ \hfill {({\mu }_m+1-{\displaystyle \frac{1}{{\delta }_m}},{\displaystyle \frac{1}{{\delta }_m}})}_1^k\end{array}\right]$, and integrating over $x\in (0,t),t>0$, we obtain

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{t}}{\int }_0^t{H}_{m,m}^{m,0}\left[{\displaystyle \frac{x}{t}}|\begin{array}{c}\hfill {({\mu }_m+{\xi }_m+1-{\displaystyle \frac{1}{{\delta }_m}},{\displaystyle \frac{1}{{\delta }_m}})}_1^k\\ \hfill {({\mu }_m+1-{\displaystyle \frac{1}{{\delta }_m}},{\displaystyle \frac{1}{{\delta }_m}})}_1^k\end{array}\right]{\displaystyle \frac{f\left(x\right)\eta \left(g\right(x\left)\right)}{g\left(x\right)}}g\left(y\right)h\left(x\right)dx\\ \hfill +{\displaystyle \frac{1}{t}}{\int }_0^t{H}_{m,m}^{m,0}\left[{\displaystyle \frac{x}{t}}|\begin{array}{c}\hfill {({\mu }_m+{\xi }_m+1-{\displaystyle \frac{1}{{\delta }_m}},{\displaystyle \frac{1}{{\delta }_m}})}_1^k\\ \hfill {({\mu }_m+1-{\displaystyle \frac{1}{{\delta }_m}},{\displaystyle \frac{1}{{\delta }_m}})}_1^k\end{array}\right]{\displaystyle \frac{\eta \left(g\right(y\left)\right)f\left(y\right)}{g\left(y\right)}}g\left(x\right)h\left(y\right)dx\\ \hfill -{\displaystyle \frac{1}{t}}{\int }_0^t{H}_{m,m}^{m,0}\left[{\displaystyle \frac{x}{t}}|\begin{array}{c}\hfill {({\mu }_m+{\xi }_m+1-{\displaystyle \frac{1}{{\delta }_m}},{\displaystyle \frac{1}{{\delta }_m}})}_1^k\\ \hfill {({\mu }_m+1-{\displaystyle \frac{1}{{\delta }_m}},{\displaystyle \frac{1}{{\delta }_m}})}_1^k\end{array}\right]{\displaystyle \frac{f\left(x\right)\eta \left(g\right(x\left)\right)}{g\left(x\right)}}g\left(x\right)h\left(y\right)dx\\ \hfill -{\displaystyle \frac{1}{t}}{\int }_0^t{H}_{m,m}^{m,0}\left[{\displaystyle \frac{x}{t}}|\begin{array}{c}\hfill {({\mu }_m+{\xi }_m+1-{\displaystyle \frac{1}{{\delta }_m}},{\displaystyle \frac{1}{{\delta }_m}})}_1^k\\ \hfill {({\mu }_m+1-{\displaystyle \frac{1}{{\delta }_m}},{\displaystyle \frac{1}{{\delta }_m}})}_1^k\end{array}\right]{\displaystyle \frac{\eta \left(g\right(y\left)\right)f\left(y\right)}{g\left(y\right)}}g\left(y\right)h\left(x\right)dx\ge 0.\end{array}$$

(39)

Using (11), this implies that

$$\begin{array}{cc}\hfill & g\left(y\right){\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[{\displaystyle \frac{f\left(t\right)\eta \left(g\right(t\left)\right)}{g\left(t\right)}}h\left(t\right)\right]+{\displaystyle \frac{\eta \left(g\right(y\left)\right)f\left(y\right)}{g\left(y\right)}}h\left(y\right){\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[g\left(t\right)\right]\hfill \\ \hfill & -h\left(y\right){\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[f\left(t\right)\eta \left(g\left(t\right)\right)\right]-{\displaystyle \frac{\eta \left(g\right(y\left)\right)f\left(y\right)}{g\left(y\right)}}g\left(y\right){\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[h\left(t\right)\right]\ge 0.\hfill \end{array}$$

(40)

Again multiplying both sides of (40) by ${\displaystyle \frac{1}{t}}{H}_{m,m}^{m,0}\left[{\displaystyle \frac{y}{t}}|\begin{array}{c}\hfill {({\mu }_m+{\xi }_m+1-{\displaystyle \frac{1}{{\delta }_m}},{\displaystyle \frac{1}{{\delta }_m}})}_1^k\\ \hfill {({\mu }_m+1-{\displaystyle \frac{1}{{\delta }_m}},{\displaystyle \frac{1}{{\delta }_m}})}_1^k\end{array}\right]$, and integrating the product over $y\in (0,t),t>0$, we obtain

$$\begin{array}{c}\hfill {\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[{\displaystyle \frac{f\left(t\right)\eta \left(g\right(t\left)\right)}{g\left(t\right)}}h\left(t\right)\right]{\displaystyle \frac{1}{t}}{\int }_0^t{H}_{m,m}^{m,0}\left[{\displaystyle \frac{y}{t}}|\begin{array}{c}\hfill {({\mu }_m+{\xi }_m+1-{\displaystyle \frac{1}{{\delta }_m}},{\displaystyle \frac{1}{{\delta }_m}})}_1^k\\ \hfill {({\mu }_m+1-{\displaystyle \frac{1}{{\delta }_m}},{\displaystyle \frac{1}{{\delta }_m}})}_1^k\end{array}\right]g\left(y\right)dy\\ \hfill +{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[g\left(t\right)\right]{\displaystyle \frac{1}{t}}{\int }_0^t{H}_{m,m}^{m,0}\left[{\displaystyle \frac{y}{t}}|\begin{array}{c}\hfill {({\mu }_m+{\xi }_m+1-{\displaystyle \frac{1}{{\delta }_m}},{\displaystyle \frac{1}{{\delta }_m}})}_1^k\\ \hfill {({\mu }_m+1-{\displaystyle \frac{1}{{\delta }_m}},{\displaystyle \frac{1}{{\delta }_m}})}_1^k\end{array}\right]{\displaystyle \frac{\eta \left(g\right(y\left)\right)f\left(y\right)}{g\left(y\right)}}h\left(y\right)dy\\ \hfill -{\displaystyle \frac{1}{t}}{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[f\left(t\right)\eta \left(g\left(t\right)\right)\right]{\int }_0^t{H}_{m,m}^{m,0}\left[{\displaystyle \frac{y}{t}}|\begin{array}{c}\hfill {({\mu }_m+{\xi }_m+1-{\displaystyle \frac{1}{{\delta }_m}},{\displaystyle \frac{1}{{\delta }_m}})}_1^k\\ \hfill {({\mu }_m+1-{\displaystyle \frac{1}{{\delta }_m}},{\displaystyle \frac{1}{{\delta }_m}})}_1^k\end{array}\right]h\left(y\right)dy\\ \hfill -{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[h\left(t\right)\right]{\displaystyle \frac{1}{t}}{\int }_0^t{H}_{m,m}^{m,0}\left[{\displaystyle \frac{y}{t}}|\begin{array}{c}\hfill {({\mu }_m+{\xi }_m+1-{\displaystyle \frac{1}{{\delta }_m}},{\displaystyle \frac{1}{{\delta }_m}})}_1^k\\ \hfill {({\mu }_m+1-{\displaystyle \frac{1}{{\delta }_m}},{\displaystyle \frac{1}{{\delta }_m}})}_1^k\end{array}\right]{\displaystyle \frac{\eta \left(g\right(y\left)\right)f\left(y\right)}{g\left(y\right)}}g\left(y\right)dy\ge 0.\end{array}$$

(41)

This leads to the following using (11)

$$\begin{array}{cc}\hfill & {\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[g\left(t\right)\right]{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[{\displaystyle \frac{f\left(t\right)\eta \left(g\right(t\left)\right)}{g\left(t\right)}}h\left(t\right)\right]+{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[{\displaystyle \frac{f\left(t\right)\eta \left(g\right(t\left)\right)}{g\left(t\right)}}h\left(t\right)\right]{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[g\left(t\right)\right]\hfill \\ \hfill & -{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[h\left(t\right)\right]{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[f\left(t\right)\eta \left(g\left(t\right)\right)\right]-{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[{\displaystyle \frac{f\left(t\right)\eta \left(g\right(t\left)\right)}{g\left(t\right)}}g\left(t\right)\right]{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[h\left(t\right)\right]\ge 0,\hfill \end{array}$$

(42)

which implies that

$${\displaystyle \frac{{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[f\left(t\right)\eta \left(g\left(t\right)\right)\right]}{{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[\frac{f\left(t\right)\eta \left(g\right(t\left)\right)}{g\left(t\right)}h\left(t\right)\right]}}\le {\displaystyle \frac{{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[g\left(t\right)\right]}{{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[h\left(t\right)\right]}}.$$

(43)

The result (34) is proved using (36) and (43). □

Corollary 3.

Let g, h, and f be three positive continuous functions with $g\le h$ on $[0,\infty )$. If ${\displaystyle \frac{g}{h}}$ is decreasing and $f,g$ are increasing functions on $[0,\infty )$, then for any convex function $\eta ,\eta \left(0\right)=0$, prove that

$${\displaystyle \frac{{\mathcal{G}}_{\left(\delta \right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[f\left(t\right)\eta \left(g\left(t\right)\right)\right]}{{\mathcal{G}}_{\left(\delta \right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[f\left(t\right)\eta \left(h\left(t\right)\right)\right]}}\le {\displaystyle \frac{{\mathcal{G}}_{\left(\delta \right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[g\left(t\right)\right]}{{\mathcal{G}}_{\left(\delta \right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[h\left(t\right)\right]}}$$

(44)

provided $t>0,k>1,{\xi }_m\ge 0,\delta >0$, and ${\mu }_m>-1-{\displaystyle \frac{\vartheta }{\delta }}$.

Proof. 

This new inclusion involving the Meijer-G function can be proved by taking ${\delta }_m=\delta$ in (34) and using the relation (15). □

Theorem 4.

Consider $f,g,h$ as three positive continuous functions with $g\le h$ on $[0,\infty )$. If ${\displaystyle \frac{g}{h}}$ is decreasing and $f,g$ are increasing functions on $[0,\infty )$, then for any convex function $\eta ,\eta \left(0\right)=0$, prove that

$${\displaystyle \frac{{\mathcal{I}}_{\left({\tau }_m\right),k}^{\left({\lambda }_m\right),\left({\beta }_m\right)}\left[g\left(t\right)\right]{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[f\left(t\right)\eta \left(h\left(t\right)\right)\right]+{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[g\left(t\right)\right]{\mathcal{I}}_{\left({\tau }_m\right),k}^{\left({\lambda }_m\right),\left({\beta }_m\right)}\left[f\left(t\right)\eta \left(h\left(t\right)\right)\right]}{{\mathcal{I}}_{\left({\tau }_m\right),k}^{\left({\lambda }_m\right),\left({\beta }_m\right)}\left[h\left(t\right)\right]{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[f\left(t\right)\eta \left(g\left(t\right)\right)\right]+{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[h\left(t\right)\right]{\mathcal{I}}_{\left({\tau }_m\right),k}^{\left({\lambda }_m\right),\left({\beta }_m\right)}\left[f\left(t\right)\eta \left(g\left(t\right)\right)\right]}}\ge 1$$

(45)

provided $t>0,k>1,{\beta }_m;{\xi }_m\ge 0,{\tau }_m;{\delta }_m>0$, and ${\mu }_m>-1-{\displaystyle \frac{\vartheta }{{\delta }_m}};{\lambda }_m>-1-{\displaystyle \frac{\vartheta }{{\tau }_m}}$.

Proof. 

Multiplying the inequality (40) by ${\displaystyle \frac{v\left(y\right)}{t}}{H}_{m,m}^{m,0}\left[{\displaystyle \frac{y}{t}}|\begin{array}{c}\hfill {({\lambda }_m+{\beta }_m+1-{\displaystyle \frac{1}{{\tau }_m}},{\displaystyle \frac{1}{{\tau }_m}})}_1^k\\ \hfill {({\lambda }_m+1-{\displaystyle \frac{1}{{\tau }_m}},{\displaystyle \frac{1}{{\tau }_m}})}_1^k\end{array}\right]$, and integrating over $y\in (0,t)$, we obtain

$$\begin{array}{cc}\hfill & {\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[g\left(t\right)\right]{\mathcal{I}}_{\left({\tau }_m\right),k}^{\left({\lambda }_m\right),\left({\beta }_m\right)}\left[{\displaystyle \frac{f\left(t\right)\eta \left(g\right(t\left)\right)}{g\left(t\right)}}h\left(t\right)\right]+{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[{\displaystyle \frac{f\left(t\right)\eta \left(g\right(t\left)\right)}{g\left(t\right)}}h\left(t\right)\right]{\mathcal{I}}_{\left({\tau }_m\right),k}^{\left({\lambda }_m\right),\left({\beta }_m\right)}\left[g\left(t\right)\right]\hfill \\ \hfill & \ge {\mathcal{I}}_{\left({\tau }_m\right),k}^{\left({\lambda }_m\right),\left({\beta }_m\right)}\left[h\left(t\right)\right]{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[f\left(t\right)\eta \left(g\left(t\right)\right)\right]+{\mathcal{I}}_{\left({\tau }_m\right),k}^{\left({\lambda }_m\right),\left({\beta }_m\right)}\left[f\left(t\right)\eta \left(g\left(t\right)\right)\right]{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[h\left(t\right)\right].\hfill \end{array}$$

(46)

According to the assumptions of the stated theorem, ${\displaystyle \frac{\eta \left(t\right)f\left(t\right)}{t}}$ is an increasing function along with $g\le h$ over the interval $[0,\infty )$. Therefore, we have

$${\mathcal{I}}_{\left({\tau }_m\right),k}^{\left({\lambda }_m\right),\left({\beta }_m\right)}\left[{\displaystyle \frac{f\left(t\right)\eta \left(g\right(t\left)\right)}{g\left(t\right)}}h\left(t\right)\right]\le {\mathcal{I}}_{\left({\tau }_m\right),k}^{\left({\lambda }_m\right),\left({\beta }_m\right)}\left[f\left(t\right)\eta \left(h\left(t\right)\right)\right]$$

(47)

and

$${\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[{\displaystyle \frac{f\left(t\right)\eta \left(g\right(t\left)\right)}{g\left(t\right)}}h\left(t\right)\right]\le {\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[f\left(t\right)\eta \left(h\left(t\right)\right)\right].$$

(48)

Finally, combining the inequalities (46)–(48), we obtain the required result (45). □

Remark 7.

By making use of the substitution ${\mu }_m={\lambda }_m,{\tau }_m={\delta }_m$ in (45), we obtain Inequality (34).

Corollary 4.

Consider $f,g,h$ to be three positive continuous functions with $g\le h$ on $[0,\infty )$. If ${\displaystyle \frac{g}{h}}$ is decreasing and $f,g$ are increasing functions on $[0,\infty )$, then for any convex function $\eta ,\eta \left(0\right)=0$, prove that

$${\displaystyle \frac{{\mathcal{G}}_{\left(\tau \right),k}^{\left({\lambda }_m\right),\left({\beta }_m\right)}\left[g\left(t\right)\right]{\mathcal{G}}_{\left(\delta \right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[f\left(t\right)\eta \left(h\left(t\right)\right)\right]+{\mathcal{G}}_{\left(\delta \right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[g\left(t\right)\right]{\mathcal{G}}_{\left(\tau \right),k}^{\left({\lambda }_m\right),\left({\beta }_m\right)}\left[f\left(t\right)\eta \left(h\left(t\right)\right)\right]}{{\mathcal{G}}_{\left(\tau \right),k}^{\left({\lambda }_m\right),\left({\beta }_m\right)}\left[h\left(t\right)\right]{\mathcal{G}}_{\left(\delta \right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[f\left(t\right)\eta \left(g\left(t\right)\right)\right]+{\mathcal{G}}_{\left(\delta \right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[h\left(t\right)\right]{\mathcal{G}}_{\left(\tau \right),k}^{\left({\lambda }_m\right),\left({\beta }_m\right)}\left[\eta \left(g\left(t\right)\right)f\left(t\right)\right]}}\ge 1$$

(49)

provided $t>0,k>1,{\beta }_m;{\xi }_m\ge 0,\tau ;\delta >0$, and ${\mu }_m>-1-{\displaystyle \frac{\vartheta }{\delta }};{\lambda }_m>-1-{\displaystyle \frac{\vartheta }{\tau }}$.

Proof. 

This new inclusion involving the Meijer-G function can be proved by taking ${\delta }_m=\delta ,{\tau }_m=\tau$ in (45) and using the relation (15). □

4. New Fractional Integral Inequalities Involving Fox-H Functions for Synchronous Functions

In this part, we use the m-E-K transform for synchronous functions to build novel inequalities. These inequalities generalize the inequality (3) and build such inequalities for weighted and extended functionals from (8) and (9).

Theorem 5.

If ϕ and ψ are two integrable and synchronous functions over the interval $[0,\infty )$, then we establish the following inequality:

$$\begin{array}{c}\hfill {\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[u\left(t\right)\right]{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[v\left(t\right)\varphi \left(t\right)\psi \left(t\right)\right]+{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[v\left(t\right)\right]{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[u\left(t\right)\varphi \left(t\right)\psi \left(t\right)\right]\\ \hfill \ge {\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[u\left(t\right)\varphi \left(t\right)\right]{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[v\left(t\right)\psi \left(t\right)\right]+{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[v\left(t\right)\varphi \left(t\right)\right]{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[u\left(t\right)\psi \left(t\right)\right]\end{array}$$

(50)

provided $t>0,k>1,{\xi }_m\ge 0,{\delta }_m>0$, and ${\mu }_k>-1-{\displaystyle \frac{\vartheta }{{\delta }_m}}$.

Proof. 

Because $\varphi \left(t\right)$ and $\psi \left(t\right)$ are synchronous functions over the interval $(0\le x,y<\infty )$, the following hold true:

$$\left(\varphi \right(x)-\varphi (y\left)\right)\left(\psi \right(x)-\psi (y\left)\right)\ge 0.$$

(51)

$$\varphi \left(x\right)\psi \left(x\right)-\varphi \left(x\right)\psi \left(y\right)-\varphi \left(y\right)\psi \left(x\right)+\varphi \left(y\right)\psi \left(y\right)\ge 0.$$

(52)

$$\varphi \left(x\right)\psi \left(x\right)+\varphi \left(y\right)\psi \left(y\right)\ge \varphi \left(x\right)\psi \left(y\right)+\varphi \left(y\right)\psi \left(x\right).$$

(53)

Multiplying both sides of the inequality (53) by ${\displaystyle \frac{u\left(x\right)}{t}}{H}_{m,m}^{m,0}\left[{\displaystyle \frac{x}{t}}|\begin{array}{c}\hfill {({\mu }_m+{\xi }_m+1-{\displaystyle \frac{1}{{\delta }_m}},{\displaystyle \frac{1}{{\delta }_m}})}_1^k\\ \hfill {({\mu }_m+1-{\displaystyle \frac{u\left(x\right)}{{\delta }_m}},{\displaystyle \frac{1}{{\delta }_m}})}_1^k\end{array}\right]$ and integrating the resulting inequality over $x\in (0,t)$,

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{t}}{\int }_0^t{H}_{m,m}^{m,0}\left[{\displaystyle \frac{x}{t}}|\begin{array}{c}\hfill {({\mu }_m+{\xi }_m+1-{\displaystyle \frac{1}{{\delta }_m}},{\displaystyle \frac{1}{{\delta }_m}})}_1^k\\ \hfill {({\mu }_m+1-{\displaystyle \frac{1}{{\delta }_m}},{\displaystyle \frac{1}{{\delta }_m}})}_1^k\end{array}\right]u\left(x\right)\varphi \left(x\right)\psi \left(x\right)dx\\ \hfill -{\displaystyle \frac{1}{t}}{\int }_0^t{H}_{m,m}^{m,0}\left[{\displaystyle \frac{x}{t}}|\begin{array}{c}\hfill {({\mu }_m+{\xi }_m+1-{\displaystyle \frac{1}{{\delta }_m}},{\displaystyle \frac{1}{{\delta }_m}})}_1^k\\ \hfill {({\mu }_m+1-{\displaystyle \frac{1}{{\delta }_m}},{\displaystyle \frac{1}{{\delta }_m}})}_1^k\end{array}\right]u\left(x\right)\varphi \left(y\right)\psi \left(y\right)dx\\ \hfill -{\displaystyle \frac{1}{t}}{\int }_0^t{H}_{m,m}^{m,0}\left[{\displaystyle \frac{x}{t}}|\begin{array}{c}\hfill {({\mu }_m+{\xi }_m+1-{\displaystyle \frac{1}{{\delta }_m}},{\displaystyle \frac{1}{{\delta }_m}})}_1^k\\ \hfill {({\mu }_m+1-{\displaystyle \frac{1}{{\delta }_m}},{\displaystyle \frac{1}{{\delta }_m}})}_1^k\end{array}\right]u\left(x\right)\varphi \left(x\right)\psi \left(y\right)dx\\ \hfill +{\displaystyle \frac{1}{t}}{\int }_0^t{H}_{m,m}^{m,0}\left[{\displaystyle \frac{x}{t}}|\begin{array}{c}\hfill {({\mu }_m+{\xi }_m+1-{\displaystyle \frac{1}{{\delta }_m}},{\displaystyle \frac{1}{{\delta }_m}})}_1^k\\ \hfill {({\mu }_m+1-{\displaystyle \frac{1}{{\delta }_m}},{\displaystyle \frac{1}{{\delta }_m}})}_1^k\end{array}\right]u\left(x\right)\varphi \left(y\right)\psi \left(x\right)dx\ge 0,\end{array}$$

(54)

and we have the subsequent inequality by using (11)

$$\begin{array}{c}\hfill {\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[u\left(t\right)\varphi \left(t\right)\psi \left(t\right)\right]+\varphi \left(y\right)\psi \left(y\right){\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[u\left(t\right)\right]\\ \hfill \ge \psi \left(y\right){\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[u\left(t\right)\varphi \left(t\right)\right]+\varphi \left(y\right){\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[u\left(t\right)\psi \left(t\right)\right].\end{array}$$

(55)

Multiplying both sides of inequality (55) by ${\displaystyle \frac{v\left(y\right)}{t}}{H}_{m,m}^{m,0}\left[{\displaystyle \frac{y}{t}}|\begin{array}{c}\hfill {({\mu }_m+{\xi }_m+1-{\displaystyle \frac{1}{{\delta }_m}},{\displaystyle \frac{1}{{\delta }_m}})}_1^k\\ \hfill {({\mu }_m+1-{\displaystyle \frac{1}{{\delta }_m}},{\displaystyle \frac{1}{{\delta }_m}})}_1^k\end{array}\right]$ and integrating the result over the interval $0<y<t$, we compute the following:

$$\begin{array}{c}\hfill {\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[u\left(t\right)\varphi \left(t\right)\psi \left(t\right)\right]{\displaystyle \frac{1}{t}}{\int }_0^t{H}_{m,m}^{m,0}\left[{\displaystyle \frac{y}{t}}|\begin{array}{c}\hfill {({\mu }_m+{\xi }_m+1-{\displaystyle \frac{1}{{\delta }_m}},{\displaystyle \frac{1}{{\delta }_m}})}_1^k\\ \hfill {({\mu }_m+1-{\displaystyle \frac{1}{{\delta }_m}},{\displaystyle \frac{1}{{\delta }_m}})}_1^k\end{array}\right]v\left(y\right)dy\\ \hfill +{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[u\left(t\right)\right]{\displaystyle \frac{1}{t}}{\int }_0^t{H}_{m,m}^{m,0}\left[{\displaystyle \frac{y}{t}}|\begin{array}{c}\hfill {({\mu }_m+{\xi }_m+1-{\displaystyle \frac{1}{{\delta }_m}},{\displaystyle \frac{1}{{\delta }_m}})}_1^k\\ \hfill {({\mu }_m+1-{\displaystyle \frac{1}{{\delta }_m}},{\displaystyle \frac{1}{{\delta }_m}})}_1^k\end{array}\right]v\left(y\right)\varphi \left(y\right)\psi \left(y\right)dy\\ \hfill \ge {\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[u\left(t\right)\varphi \left(t\right)\right]{\displaystyle \frac{1}{t}}{\int }_0^t{H}_{m,m}^{m,0}\left[{\displaystyle \frac{y}{t}}|\begin{array}{c}\hfill {({\mu }_m+{\xi }_m+1-{\displaystyle \frac{1}{{\delta }_m}},{\displaystyle \frac{1}{{\delta }_m}})}_1^k\\ \hfill {({\mu }_m+1-{\displaystyle \frac{1}{{\delta }_m}},{\displaystyle \frac{1}{{\delta }_m}})}_1^k\end{array}\right]\psi \left(y\right)dy\\ \hfill +{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[u\left(t\right)\psi \left(t\right)\right]{\displaystyle \frac{1}{t}}{\int }_0^t{H}_{m,m}^{m,0}\left[{\displaystyle \frac{y}{t}}|\begin{array}{c}\hfill {({\mu }_m+{\xi }_m+1-{\displaystyle \frac{1}{{\delta }_m}},{\displaystyle \frac{1}{{\delta }_m}})}_1^k\\ \hfill {({\mu }_m+1-{\displaystyle \frac{1}{{\delta }_m}},{\displaystyle \frac{1}{{\delta }_m}})}_1^k\end{array}\right]v\left(y\right)\varphi \left(y\right)dy,\end{array}$$

(56)

which leads to the following by making use of (11)

$$\begin{array}{c}\hfill {\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[u\left(t\right)\varphi \left(t\right)\psi \left(t\right)\right]{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[v\left(t\right)\right]+{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[u\left(t\right)\right]{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\varphi \left(t\right)\psi \left(t\right)v\left(t\right)]\\ \hfill \ge {\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[u\left(t\right)\varphi \left(t\right)\right]{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[v\left(t\right)\psi \left(t\right)\right]+{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[u\left(t\right)\psi \left(t\right)\right]{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[v\left(t\right)\varphi \left(t\right)\right],\end{array}$$

(57)

and we establish the required result. □

Corollary 5.

If $\varphi ,\psi$ are integrable as well as synchronous functions over the interval $[0,\infty )$, then we establish the following inequality:

$$\begin{array}{c}\hfill {\mathcal{G}}_{\left(\delta \right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[u\left(t\right)\right]{\mathcal{G}}_{\left(\delta \right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[v\left(t\right)\varphi \left(t\right)\psi \left(t\right)\right]+{\mathcal{G}}_{\left(\delta \right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[v\left(t\right)\right]{\mathcal{G}}_{\left(\delta \right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[u\left(t\right)\varphi \left(t\right)\psi \left(t\right)\right]\\ \hfill \ge {\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[u\left(t\right)\varphi \left(t\right)\right]{\mathcal{G}}_{\left(\delta \right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[v\left(t\right)\psi \left(t\right)\right]+{\mathcal{G}}_{\left(\delta \right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[v\left(t\right)\varphi \left(t\right)\right]{\mathcal{G}}_{\left(\delta \right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[u\left(t\right)\psi \left(t\right)\right]\end{array}$$

(58)

provided $t>0,k>1,{\xi }_m\ge 0,\delta >0$, and ${\mu }_k>-1-{\displaystyle \frac{\vartheta }{\delta }}$.

Proof. 

This new inclusion involving the Meijer-G function can be proved by taking ${\delta }_m=\delta$ in (50) and using the relation (15). □

Theorem 6.

If $\varphi ,\psi$ are two integrable as well as synchronous functions over the interval $t\in [0,\infty )$, then we establish the following inequality:

$$\begin{array}{c}\hfill 2{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[r\left(t\right)\right]\left[{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[q\left(t\right)\varphi \left(t\right)\psi \left(t\right)\right]{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[p\left(t\right)\right]+{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[\varphi \left(t\right)\psi \left(t\right)p\left(t\right)\right]{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[q\left(t\right)\right]\right]\\ \hfill +2{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[p\left(t\right)\right]\left[{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[\varphi \left(t\right)\psi \left(t\right)r\left(t\right)\right]{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[q\left(t\right)\right]\right]\\ \hfill \ge {\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[r\left(t\right)\right]\left[{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[p\left(t\right)\varphi \left(t\right)\right]{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[q\left(t\right)\psi \left(t\right)\right]+{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[\varphi \left(t\right)q\left(t\right)\right]{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[p\left(t\right)\psi \left(t\right)\right]\right]\\ \hfill +{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[p\left(t\right)\right]\left[{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[r\left(t\right)\varphi \left(t\right)\right]{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[q\left(t\right)\psi \left(t\right)\right]+{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[\varphi \left(t\right)q\left(t\right)\right]{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[r\left(t\right)\psi \left(t\right)\right]\right]\\ \hfill +{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[q\left(t\right)\right]\left[{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[r\left(t\right)\varphi \left(t\right)\right]{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[p\left(t\right)\psi \left(t\right)\right]+{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[\varphi \left(t\right)p\left(t\right)\right]{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[r\left(t\right)\psi \left(t\right)\right]\right]\end{array}$$

(59)

provided $t>0,k>1,{\xi }_m\ge 0,{\delta }_m>0$, and ${\mu }_k>-1-{\displaystyle \frac{\vartheta }{{\delta }_m}}$.

Proof. 

Using $u\left(t\right)=p\left(t\right),v\left(t\right)=q\left(t\right)$, along with Theorem 5, we obtain the following:

$$\begin{array}{c}\hfill {\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[q\left(t\right)\varphi \left(t\right)\psi \left(t\right)\right]{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[p\left(t\right)\right]+{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[\varphi \left(t\right)\psi \left(t\right)p\left(t\right)\right]{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[q\left(t\right)\right]\\ \hfill \ge {\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[p\left(t\right)\varphi \left(t\right)\right]{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[q\left(t\right)\psi \left(t\right)\right]+{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[p\left(t\right)\psi \left(t\right)\right]{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[q\left(t\right)\varphi \left(t\right)\right].\end{array}$$

(60)

Next, if we multiply the factor ${\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[r\left(t\right)\right]$, with the expression (60), we then achieve the following:

$$\begin{array}{c}\hfill {\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[r\left(t\right)\right]\left[{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[q\left(t\right)\varphi \left(t\right)\psi \left(t\right)\right]{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[p\left(t\right)\right]+{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\varphi \left(t\right)\psi \left(t\right)p\left(t\right)]{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[q\left(t\right)\right]\right]\\ \hfill \ge {\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[r\left(t\right)\right]\left[{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[p\left(t\right)\varphi \left(t\right)\right]{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[q\left(t\right)\psi \left(t\right)\right]+{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[p\left(t\right)\psi \left(t\right)\right]{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[q\left(t\right)\varphi \left(t\right)\right]\right].\end{array}$$

(61)

Next, using $u\left(t\right)=r\left(t\right),v\left(t\right)=q\left(t\right)$, in Theorem 5, we obtain

$$\begin{array}{c}\hfill {\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[q\left(t\right)\varphi \left(t\right)\psi \left(t\right)\right]{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[r\left(t\right)\right]+{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[\varphi \left(t\right)\psi \left(t\right)r\left(t\right)\right]{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[q\left(t\right)\right]\\ \hfill \ge {\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[r\left(t\right)\varphi \left(t\right)\right]{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[q\left(t\right)\psi \left(t\right)\right]+{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[r\left(t\right)\psi \left(t\right)\right]{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[q\left(t\right)\varphi \left(t\right)\right].\end{array}$$

(62)

Then, multiply the factor ${\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[p\left(x\right)\right]$ with the inequality (62) to achieve

$$\begin{array}{c}\hfill {\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[p\left(x\right)\right]\left[{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[q\left(t\right)\varphi \left(t\right)\psi \left(t\right)\right]{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[r\left(t\right)\right]+{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[\varphi \left(t\right)\psi \left(t\right)r\left(t\right)\right]{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[q\left(t\right)\right]\right]\\ \hfill \ge {\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[p\left(x\right)\right]\left[{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[r\left(t\right)\varphi \left(t\right)\right]{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[r\left(t\right)\psi \left(t\right)\right]+{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[p\left(t\right)\psi \left(t\right)\right]{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[q\left(t\right)\varphi \left(t\right)\right]\right.\end{array}$$

(63)

Following the same steps as for the inequality (61) and the inequality (63), we state that

$$\begin{array}{c}\hfill {\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[q\left(x\right)\right]\left[{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[p\left(t\right)\varphi \left(t\right)\psi \left(t\right)\right]{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[r\left(t\right)\right]+{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[\varphi \left(t\right)\psi \left(t\right)r\left(t\right)\right]{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[p\left(t\right)\right]\right]\\ \hfill \ge {\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[q\left(x\right)\right]\left[{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[r\left(t\right)\varphi \left(t\right)\right]{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[p\left(t\right)\psi \left(t\right)\right]+{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[r\left(t\right)\psi \left(t\right)\right]{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[p\left(t\right)\varphi \left(t\right)\right]\right.\end{array}$$

(64)

The addition of the results (61), (63), and (64) leads to the required stated form (59). □

Corollary 6.

If $\varphi ,\psi$ are integrable as well as synchronous functions over the interval $t\in [0,\infty )$, then establish the following inequality:

$$\begin{array}{c}\hfill 2{\mathcal{G}}_{\left(\delta \right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[r\left(t\right)\right]\left[{\mathcal{G}}_{\left(\delta \right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[q\left(t\right)\varphi \left(t\right)\psi \left(t\right)\right]{\mathcal{G}}_{\left(\delta \right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[p\left(t\right)\right]+{\mathcal{G}}_{\left(\delta \right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[\varphi \left(t\right)\psi \left(t\right)p\left(t\right)\right]{\mathcal{G}}_{\left(\delta \right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[q\left(t\right)\right]\right]\\ \hfill +2{\mathcal{G}}_{\left(\delta \right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[p\left(t\right)\right]\left[{\mathcal{G}}_{\left(\delta \right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[\varphi \left(t\right)\psi \left(t\right)r\left(t\right)\right]{\mathcal{G}}_{\left(\delta \right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[q\left(t\right)\right]\right]\\ \hfill \ge {\mathcal{G}}_{\left(\delta \right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[r\left(t\right)\right]\left[{\mathcal{G}}_{\left(\delta \right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[p\left(t\right)\varphi \left(t\right)\right]{\mathcal{G}}_{\left(\delta \right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[q\left(t\right)\psi \left(t\right)\right]+{\mathcal{G}}_{\left(\delta \right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[\varphi \left(t\right)q\left(t\right)\right]{\mathcal{G}}_{\left(\delta \right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[p\left(t\right)\psi \left(t\right)\right]\right]\\ \hfill +{\mathcal{G}}_{\left(\delta \right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[p\left(t\right)\right]\left[{\mathcal{G}}_{\left(\delta \right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[r\left(t\right)\varphi \left(t\right)\right]{\mathcal{G}}_{\left(\delta \right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[q\left(t\right)\psi \left(t\right)\right]+{\mathcal{G}}_{\left(\delta \right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[\varphi \left(t\right)q\left(t\right)\right]{\mathcal{G}}_{\left(\delta \right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[r\left(t\right)\psi \left(t\right)\right]\right]\\ \hfill +{\mathcal{G}}_{\left(\delta \right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[q\left(t\right)\right]\left[{\mathcal{G}}_{\left(\delta \right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[r\left(t\right)\varphi \left(t\right)\right]{\mathcal{G}}_{\left(\delta \right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[p\left(t\right)\psi \left(t\right)\right]+{\mathcal{G}}_{\left(\delta \right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[\varphi \left(t\right)p\left(t\right)\right]{\mathcal{G}}_{\left(\delta \right),k}^{\left({\mu }_m\right)\left({\xi }_m\right)}\left[r\left(t\right)\psi \left(t\right)\right]\right]\end{array}$$

(65)

provided $t>0,k>1,{\xi }_m\ge 0,\delta >0$, and ${\mu }_k>-1-{\displaystyle \frac{\vartheta }{\delta }}$.

Proof. 

This new inclusion involving the Meijer-G function can be proved by taking ${\delta }_m=\delta$ in (59) and using the relation (15). □

Remark 8.

If the following requirements are met by the five functions $\varphi \left(t\right),\psi \left(t\right),p\left(t\right),q\left(t\right),r\left(t\right)$:

• Over the interval $[0,\infty )$, the functions $\varphi \left(t\right)$ and $\psi \left(t\right)$ operate asynchronously.
• Over the interval $[0,\infty )$, the functions $r\left(t\right),p\left(t\right),q\left(t\right)$ are negative.
• Over the interval $[0,\infty )$, a pair of functions from $r\left(t\right),p\left(t\right),q\left(t\right)$ behave positively, while the third one is negative.

Formerly, the inequalities stated in (59) and (65) were inverted.

Theorem 7.

Consider two integrable as well as synchronous functions denoted as $\varphi \left(t\right)$ and $\psi \left(t\right)$ and defined on the interval $[0,\infty )$. Then, for $t>0$, prove that

$$\begin{array}{c}\hfill {\mathcal{I}}_{\left({\tau }_m\right),k}^{\left({\lambda }_m\right),\left({\beta }_m\right)}\left[u\left(t\right)\varphi \left(t\right)\psi \left(t\right)\right]{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[v\left(t\right)\right]+{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[\varphi \left(t\right)\psi \left(t\right)v\left(t\right)\right]{\mathcal{I}}_{\left({\tau }_m\right),k}^{\left({\lambda }_m\right),\left({\beta }_m\right)}\left[u\left(t\right)\right]\\ \hfill \ge {\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[\psi \left(t\right)v\left(t\right)\right]{\mathcal{I}}_{\left({\tau }_m\right),k}^{\left({\lambda }_m\right),\left({\beta }_m\right)}\left[u\left(t\right)\varphi \left(t\right)\right]+{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}[\varphi \left(t\right)v\left(t\right){\mathcal{I}}_{\left({\tau }_m\right),k}^{\left({\lambda }_m\right),\left({\beta }_m\right)}\left[u\left(t\right)\psi \left(t\right)\right]\end{array}$$

(66)

provided $k>1,{\beta }_m;{\xi }_m\ge 0,{\tau }_m;{\delta }_m>0$, and ${\mu }_m>-1-{\displaystyle \frac{\vartheta }{{\delta }_m}};{\lambda }_m>-1-{\displaystyle \frac{\vartheta }{{\tau }_m}}.$

Proof. 

This can be achieved by integrating over the interval $y\in (0,t),t>0$, after multiplying both sides of (55) by ${\displaystyle \frac{v\left(y\right)}{t}}{H}_{m,m}^{m,0}\left[{\displaystyle \frac{y}{t}}|\begin{array}{c}\hfill {({\lambda }_m+{\beta }_m+1-{\displaystyle \frac{1}{{\tau }_m}},{\displaystyle \frac{1}{{\tau }_m}})}_1^k\\ \hfill {({\lambda }_m+1-{\displaystyle \frac{1}{{\tau }_m}},{\displaystyle \frac{1}{{\tau }_m}})}_1^k\end{array}\right]$ as follows:

$$\begin{array}{c}\hfill {\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[u\left(t\right)\varphi \left(t\right)\psi \left(t\right)\right]{\displaystyle \frac{1}{t}}{\int }_0^t{H}_{m,m}^{m,0}\left[{\displaystyle \frac{y}{t}}|\begin{array}{c}\hfill {({\lambda }_m+{\beta }_m+1-{\displaystyle \frac{1}{{\tau }_m}},{\displaystyle \frac{1}{{\tau }_m}})}_1^k\\ \hfill {({\lambda }_m+1-{\displaystyle \frac{1}{{\tau }_m}},{\displaystyle \frac{1}{{\tau }_m}})}_1^k\end{array}\right]v\left(y\right)dy\\ \hfill +{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[u\left(t\right)\right]{\displaystyle \frac{1}{t}}{\int }_0^t{H}_{m,m}^{m,0}\left[{\displaystyle \frac{y}{t}}|\begin{array}{c}\hfill {({\lambda }_m+{\beta }_m+1-{\displaystyle \frac{1}{{\tau }_m}},{\displaystyle \frac{1}{{\tau }_m}})}_1^k\\ \hfill {({\lambda }_m+1-{\displaystyle \frac{1}{{\tau }_m}},{\displaystyle \frac{1}{{\tau }_m}})}_1^k\end{array}\right]\varphi \left(y\right)\psi \left(y\right)v\left(y\right)dy\\ \hfill \ge {\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[u\left(t\right)\varphi \left(t\right)\right]{\displaystyle \frac{1}{t}}{\int }_0^t{H}_{m,m}^{m,0}\left[{\displaystyle \frac{y}{t}}|\begin{array}{c}\hfill {({\lambda }_m+{\beta }_m+1-{\displaystyle \frac{1}{{\tau }_m}},{\displaystyle \frac{1}{{\tau }_m}})}_1^k\\ \hfill {({\lambda }_m+1-{\displaystyle \frac{1}{{\tau }_m}},{\displaystyle \frac{1}{{\tau }_m}})}_1^k\end{array}\right]v\left(y\right)\psi \left(y\right)dy\\ \hfill +{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[u\left(t\right)\psi \left(t\right)\right]{\displaystyle \frac{1}{t}}{\int }_0^t{H}_{m,m}^{m,0}\left[{\displaystyle \frac{y}{t}}|\begin{array}{c}\hfill {({\lambda }_m+{\beta }_m+1-{\displaystyle \frac{1}{{\tau }_m}},{\displaystyle \frac{1}{{\tau }_m}})}_1^k\\ \hfill {({\lambda }_m+1-{\displaystyle \frac{1}{{\tau }_m}},{\displaystyle \frac{1}{{\tau }_m}})}_1^k\end{array}\right]v\left(y\right)\varphi \left(y\right)dy.\end{array}$$

(67)

Hence, the required inequality is established using (11). □

Corollary 7.

Consider two integrable as well as synchronous functions denoted as $\varphi \left(t\right)$ and $\psi \left(t\right)$ and defined on the interval $[0,\infty )$. Then, for $t>0$, prove that

$$\begin{array}{c}\hfill {\mathcal{G}}_{\left({\tau }_m\right),k}^{\left({\lambda }_m\right),\left({\beta }_m\right)}\left[u\left(t\right)\varphi \left(t\right)\psi \left(t\right)\right]{\mathcal{G}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[v\left(t\right)\right]+{\mathcal{G}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[\varphi \left(t\right)\psi \left(t\right)v\left(t\right)\right]{\mathcal{G}}_{\left({\tau }_m\right),k}^{\left({\lambda }_m\right),\left({\beta }_m\right)}\left[u\left(t\right)\right]\\ \hfill \ge {\mathcal{G}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[\psi \left(t\right)v\left(t\right)\right]{\mathcal{I}}_{\left({\tau }_m\right),k}^{\left({\lambda }_m\right),\left({\beta }_m\right)}\left[u\left(t\right)\varphi \left(t\right)\right]+{\mathcal{G}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}[\varphi \left(t\right)v\left(t\right){\mathcal{G}}_{\left({\tau }_m\right),k}^{\left({\lambda }_m\right),\left({\beta }_m\right)}\left[u\left(t\right)\psi \left(t\right)\right].\end{array}$$

(68)

provided $k>1,{\beta }_m;{\xi }_m\ge 0,\tau ;\delta >0$, and ${\mu }_m>-1-{\displaystyle \frac{\vartheta }{\delta }};{\lambda }_m>-1-{\displaystyle \frac{\vartheta }{\tau }}$.

Proof. 

This new inclusion involving the Meijer-G function can be proved by taking ${\delta }_m=\delta ,{\tau }_m=\tau$ in (66) and using the relation (15). □

Theorem 8.

Consider two integrable as well as synchronous functions denoted as $\varphi \left(t\right)$ and $\psi \left(t\right)$ and defined on the interval $[0,\infty )$. Then, for $t>0$, we state and prove the following result:

$$\begin{array}{c}\hfill {\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[r\left(t\right)\right]\left[{\mathcal{I}}_{\left({\tau }_m\right),k}^{\left({\lambda }_m\right),\left({\beta }_m\right)}\left[p\left(t\right)\varphi \left(t\right)\psi \left(t\right)\right]{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[q\left(t\right)\right]+2{\mathcal{I}}_{\left({\tau }_m\right),k}^{\left({\lambda }_m\right),\left({\beta }_m\right)}\left[\varphi \left(t\right)\psi \left(t\right)q\left(t\right)\right]{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[p\left(t\right)\right]\right.\\ \hfill +\left.{\mathcal{I}}_{\left({\tau }_m\right),k}^{\left({\lambda }_m\right),\left({\xi }_m\right)}\left[\varphi \left(t\right)\psi \left(t\right)p\left(t\right)\right]{\mathcal{I}}_{\left({\tau }_m\right),k}^{\left({\lambda }_m\right),\left({\beta }_m\right)}\left[q\left(t\right)\right]\right]\\ \hfill +\left[{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[p\left(t\right)\right]{\mathcal{I}}_{\left({\tau }_m\right),k}^{\left({\lambda }_m\right),\left({\beta }_m\right)}\left[q\left(t\right)\right]+{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[q\left(t\right)\right]{\mathcal{I}}_{\left({\tau }_m\right),k}^{\left({\lambda }_m\right),\left({\beta }_m\right)}\left[p\left(t\right)\right]\right]{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[r\left(t\right)f\left(t\right)g\left(t\right)\right]\\ \hfill \ge {\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[r\left(t\right)\right]\left[{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[p\left(t\right)\varphi \left(t\right)\right]{\mathcal{I}}_{\left({\tau }_m\right),k}^{\left({\lambda }_m\right),\left({\beta }_m\right)}\left[q\left(t\right)\psi \left(t\right)\right]+{\mathcal{I}}_{\left({\tau }_m\right),k}^{\left({\lambda }_m\right),\left({\beta }_m\right)}\left[\varphi \left(t\right)q\left(t\right)\right]{\mathcal{I}}_{\left({\tau }_m\right),k}^{\left({\lambda }_m\right),\left({\beta }_m\right)}\left[p\left(t\right)\psi \left(t\right)\right]\right]\\ \hfill +{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[p\left(t\right)\right]\left[{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[r\left(t\right)\varphi \left(t\right)\right]{\mathcal{I}}_{\left({\tau }_m\right),k}^{\left({\lambda }_m\right),\left({\xi }_m\right)}\left[q\left(t\right)\psi \left(t\right)\right]+{\mathcal{I}}_{\left({\tau }_m\right),k}^{\left({\lambda }_m\right),\left({\beta }_m\right)}\left[\varphi \left(t\right)q\left(t\right)\right]{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[r\left(t\right)\psi \left(t\right)\right]\right]\\ \hfill +{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[q\left(t\right)\right]\left[{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[r\left(t\right)\varphi \left(t\right)\right]{\mathcal{I}}_{\left({\tau }_m\right),k}^{\left({\lambda }_m\right),\left({\beta }_m\right)}\left[p\left(t\right)\psi \left(t\right)\right]+{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[\varphi \left(t\right)p\left(t\right)\right]{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[r\left(t\right)\psi \left(t\right)\right]\right]\end{array}$$

(69)

provided $k>1,{\beta }_m;{\xi }_m\ge 0,{\tau }_m;{\delta }_m>0$, and ${\mu }_m>-1-{\displaystyle \frac{\vartheta }{{\delta }_m}};{\lambda }_m>-1-{\displaystyle \frac{\vartheta }{{\tau }_m}}$.

Proof. 

By making use of $u\left(t\right)=p\left(t\right),v\left(t\right)=q\left(t\right)$, in Theorem 7, we obtain the following:

$$\begin{array}{c}\hfill {\mathcal{I}}_{\left({\tau }_m\right),k}^{\left({\lambda }_m\right),\left({\beta }_m\right)}\left[q\left(t\right)\varphi \left(t\right)\psi \left(t\right)\right]{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[p\left(t\right)\right]+{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[\varphi \left(t\right)\psi \left(t\right)p\left(t\right)\right]{\mathcal{I}}_{\left({\tau }_m\right),k}^{\left({\lambda }_m\right),\left({\beta }_m\right)}\left[q\left(t\right)\right]\\ \hfill \ge {\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[p\left(t\right)\varphi \left(t\right)\right]{\mathcal{I}}_{\left({\tau }_m\right),k}^{\left({\lambda }_m\right),\left({\beta }_m\right)}\left[q\left(t\right)\psi \left(t\right)\right]+{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[p\left(t\right)\psi \left(t\right)\right]{\mathcal{I}}_{\left({\tau }_m\right),k}^{\left({\lambda }_m\right),\left({\beta }_m\right)}\left[q\left(t\right)\varphi \left(t\right)\right].\end{array}$$

(70)

Next, we multiply inequality (50) with the factor ${\mathcal{I}}_{\left({\tau }_m\right),k}^{\left({\lambda }_m\right),\left({\beta }_m\right)}\left[r\left(t\right)\right]$ to achieve the following

$$\begin{array}{c}\hfill {\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[r\left(t\right)\right]\left[{\mathcal{I}}_{\left({\tau }_m\right),k}^{\left({\lambda }_m\right),\left({\beta }_m\right)}\left[q\left(t\right)\varphi \left(t\right)\psi \left(t\right)\right]{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[p\left(t\right)\right]+{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[\varphi \left(t\right)\psi \left(t\right)p\left(t\right)\right]{\mathcal{I}}_{\left({\tau }_m\right),k}^{\left({\lambda }_m\right),\left({\beta }_m\right)}\left[q\left(t\right)\right]\right]\\ \hfill \ge {\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[r\left(t\right)\right]\left[{\mathcal{I}}_{\left({\tau }_m\right),k}^{\left({\lambda }_m\right),\left({\beta }_m\right)}\left[p\left(t\right)\varphi \left(t\right)\right]{\mathcal{I}}_{\left({\tau }_m\right),k}^{\left({\lambda }_m\right),\left({\beta }_m\right)}\left[q\left(t\right)\psi \left(t\right)\right]+{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[p\left(t\right)\psi \left(t\right)\right]{\mathcal{I}}_{\left({\tau }_m\right),k}^{\left({\lambda }_m\right),\left({\beta }_m\right)}\left[q\left(t\right)\varphi \left(t\right)\right]\right].\end{array}$$

(71)

Then again making use of $u\left(t\right)=p\left(t\right),v\left(t\right)=q\left(t\right)$, in Theorem 7, we obtain the following

$$\begin{array}{c}\hfill {\mathcal{I}}_{\left({\tau }_m\right),k}^{\left({\lambda }_m\right),\left({\beta }_m\right)}\left[q\left(t\right)\varphi \left(t\right)\psi \left(t\right)\right]{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[r\left(t\right)\right]+{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[\varphi \left(t\right)\psi \left(t\right)r\left(t\right)\right]{\mathcal{I}}_{\left({\tau }_m\right),k}^{\left({\lambda }_m\right),\left({\beta }_m\right)}\left[q\left(t\right)\right]\\ \hfill \ge {\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[r\left(t\right)\varphi \left(t\right)\right]{\mathcal{I}}_{\left({\tau }_m\right),k}^{\left({\lambda }_m\right),\left({\beta }_m\right)}\left[q\left(t\right)\psi \left(t\right)\right]+{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[r\left(t\right)\psi \left(t\right)\right]{\mathcal{I}}_{\left({\tau }_m\right),k}^{\left({\lambda }_m\right),\left({\beta }_m\right)}\left[q\left(t\right)\varphi \left(t\right)\right].\end{array}$$

(72)

We achieve the following inequality when multiplying (56) with the factor ${\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[p\left(x\right)\right]$

$$\begin{array}{c}\hfill {\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[p\left(x\right)\right]\left[{\mathcal{I}}_{\left({\tau }_m\right),k}^{\left({\lambda }_m\right),\left({\beta }_m\right)}\left[q\left(t\right)\varphi \left(t\right)\psi \left(t\right)\right]{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[r\left(t\right)\right]+{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[\varphi \left(t\right)\psi \left(t\right)r\left(t\right)\right]{\mathcal{I}}_{\left({\tau }_m\right),k}^{\left({\lambda }_m\right),\left({\beta }_m\right)}\left[q\left(t\right)\right]\right]\\ \hfill \ge {\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[p\left(x\right)\right]\left[{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[r\left(t\right)\varphi \left(t\right)\right]{\mathcal{I}}_{\left({\tau }_m\right),k}^{\left({\lambda }_m\right),\left({\beta }_m\right)}\left[q\left(t\right)\psi \left(t\right)\right]+{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[r\left(t\right)\psi \left(t\right)\right]{\mathcal{I}}_{\left({\tau }_m\right),k}^{\left({\lambda }_m\right),\left({\beta }_m\right)}\left[q\left(t\right)\varphi \left(t\right)\right]\right].\end{array}$$

(73)

Using the similar facts as used for the expressions (71) and (73) above, we can state that

$$\begin{array}{c}\hfill {\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[q\left(t\right)\right]\left[{\mathcal{I}}_{\left({\tau }_m\right),k}^{\left({\lambda }_m\right),\left({\beta }_m\right)}\left[p\left(t\right)\varphi \left(t\right)\psi \left(t\right)\right]{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[r\left(t\right)\right]+{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[\varphi \left(t\right)\psi \left(t\right)r\left(t\right)\right]{\mathcal{I}}_{\left({\tau }_m\right),k}^{\left({\lambda }_m\right),\left({\beta }_m\right)}\left[p\left(t\right)\right]\right]\\ \hfill \ge {\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[q\left(t\right)\right]\left[{\mathcal{I}}_{\left({\tau }_m\right),k}^{\left({\lambda }_m\right),\left({\beta }_m\right)}\left[p\left(t\right)\varphi \left(t\right)\right]{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[r\left(t\right)\psi \left(t\right)\right]+{\mathcal{I}}_{\left({\tau }_m\right),k}^{\left({\lambda }_m\right),\left({\beta }_m\right)}\left[p\left(t\right)\psi \left(t\right)\right]{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[r\left(t\right)\varphi \left(t\right)\right]\right.\end{array}$$

(74)

With the addition of the inequalities (71), (73), and (74), we establish the stated inequality (69) to complete the proof of Theorem 8. □

Corollary 8.

Consider two integrable as well as synchronous functions denoted as $\varphi \left(t\right)$ and $\psi \left(t\right)$ and defined on the interval $[0,\infty )$. Then, for $t>0$, prove that

$$\begin{array}{c}\hfill {\mathcal{G}}_{\left(\delta \right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[r\left(t\right)\right]\left[{\mathcal{G}}_{\left(\tau \right),k}^{\left({\lambda }_m\right),\left({\beta }_m\right)}\left[p\left(t\right)\varphi \left(t\right)\psi \left(t\right)\right]{\mathcal{G}}_{\left(\delta \right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[q\left(t\right)\right]+2{\mathcal{G}}_{\left(\tau \right),k}^{\left({\lambda }_m\right),\left({\beta }_m\right)}\left[\varphi \left(t\right)\psi \left(t\right)q\left(t\right)\right]{\mathcal{G}}_{\left(\delta \right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[p\left(t\right)\right]\right.\\ \hfill +\left.{\mathcal{G}}_{\left(\tau \right),k}^{\left({\lambda }_m\right),\left({\xi }_m\right)}\left[\varphi \left(t\right)\psi \left(t\right)p\left(t\right)\right]{\mathcal{G}}_{\left(\tau \right),k}^{\left({\lambda }_m\right),\left({\beta }_m\right)}\left[q\left(t\right)\right]\right]\\ \hfill +\left[{\mathcal{G}}_{\left(\delta \right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[p\left(t\right)\right]{\mathcal{G}}_{\left(\tau \right),k}^{\left({\lambda }_m\right),\left({\beta }_m\right)}\left[q\left(t\right)\right]+{\mathcal{G}}_{\left(\delta \right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[q\left(t\right)\right]{\mathcal{G}}_{\left(\tau \right),k}^{\left({\lambda }_m\right),\left({\beta }_m\right)}\left[p\left(t\right)\right]\right]{\mathcal{G}}_{\left(\delta \right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[rfg\right]\\ \hfill \ge {\mathcal{G}}_{\left(\delta \right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[r\left(t\right)\right]\left[{\mathcal{G}}_{\left(\delta \right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[p\left(t\right)\varphi \left(t\right)\right]{\mathcal{G}}_{\left(\tau \right),k}^{\left({\lambda }_m\right),\left({\beta }_m\right)}\left[q\left(t\right)\psi \left(t\right)\right]+{\mathcal{G}}_{\left(\tau \right),k}^{\left({\lambda }_m\right),\left({\beta }_m\right)}\left[\varphi \left(t\right)q\left(t\right)\right]{\mathcal{G}}_{\left(\tau \right),k}^{\left({\lambda }_m\right),\left({\beta }_m\right)}\left[p\left(t\right)\psi \left(t\right)\right]\right]\\ \hfill +{\mathcal{G}}_{\left(\delta \right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[p\left(t\right)\right]\left[{\mathcal{G}}_{\left(\delta \right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[r\left(t\right)\varphi \left(t\right)\right]{\mathcal{G}}_{\left(\tau \right),k}^{\left({\lambda }_m\right),\left({\xi }_m\right)}\left[q\left(t\right)\psi \left(t\right)\right]+{\mathcal{G}}_{\left(\tau \right),k}^{\left({\lambda }_m\right),\left({\beta }_m\right)}\left[\varphi \left(t\right)q\left(t\right)\right]{\mathcal{G}}_{\left(\delta \right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[r\left(t\right)\psi \left(t\right)\right]\right]\\ \hfill +{\mathcal{G}}_{\left(\delta \right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[q\left(t\right)\right]\left[{\mathcal{G}}_{\left(\delta \right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[r\left(t\right)\varphi \left(t\right)\right]{\mathcal{G}}_{\left(\tau \right),k}^{\left({\lambda }_m\right),\left({\beta }_m\right)}\left[p\left(t\right)\psi \left(t\right)\right]+{\mathcal{G}}_{\left(\delta \right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[\varphi \left(t\right)p\left(t\right)\right]{\mathcal{G}}_{\left(\delta \right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[r\left(t\right)\psi \left(t\right)\right]\right]\end{array}$$

(75)

provided $k>1,{\beta }_m;{\xi }_m\ge 0,\tau ;\delta >0$, and ${\mu }_m>-1-{\displaystyle \frac{\vartheta }{\delta }};{\lambda }_m>-1-{\displaystyle \frac{\vartheta }{\tau }}$.

Proof. 

This new inclusion involving the Meijer-G function can be proved by taking ${\delta }_m=\delta ,{\tau }_m=\tau$ in (69) and using the relation (15). □

Remark 9.

If the following criteria are met by a set of five functions $\varphi \left(t\right),\psi \left(t\right),r\left(t\right),p\left(t\right)$, and $q\left(t\right)$:

• Over the interval $[0,\infty )$, the functions $\varphi \left(t\right)$ and $\psi \left(t\right)$ are asynchronous.
• Over the interval $[0,\infty )$, three functions $r\left(t\right),p\left(t\right),q\left(t\right)$ are negative.
• One function on $[0,\infty )$ is negative, whereas the other two, $r\left(t\right),p\left(t\right),q\left(t\right)$, are positive.

In the light of the above, inequalities (66), (69), and (75) are inverted.

5. Conclusions and Future Directions

We use the m-E-K fractional integral operator to introduce new inequalities in the current paper. These inequalities are established for convex and synchronous functions and shed fresh insights on their applications in fractional calculus and statistics. We talk about a few implications based on the stated remarks. Compared to the conventional inequalities found in the literature, the inequalities produced in this study are more generic. A more general version of the numerous integral operators is the m-E-K operator. As a result, the inequalities found in this work will converge to those using Saigo and M-S-M fractional integral operators as discussed in Section 2. However, further generalizations are also obvious by making use of the operators defined in [42]. For functions that behave synchronously, Chebyshev’s inequality, a well-known inequality in the subject of inequality theory, produces amazing results. We establish generic forms for synchronous functions by using the m-E-K fractional integral operator, which has emerged as the most widely used idea in fractional analysis in recent years, to generate Chebyshev-type inequalities. The most remarkable conclusion is that some of the results may be simplified to the Chebyshev inequality for specific values of the fractional integral operator’s parameter. This study differs from others in the literature, in that it combines the widely accepted ideas of fractional analysis and synchronous functions, and the results produced using the m-E-K fractional integral operator have general forms. For example, using $u\left(t\right)=v\left(t\right)=1$ in Theorem 5, and then using the relation (14), we obtain the following new general form of the original Chebyshev inequality (3):

$${\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[\varphi \left(t\right)\psi \left(t\right)\right]\ge \prod _{m=1}^k{\displaystyle \frac{\Gamma ({\mu }_m+{\xi }_m+1)}{\Gamma ({\mu }_m+1)}}{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[\varphi \left(t\right)\right]{\mathcal{I}}_{\left({\delta }_m\right),k}^{\left({\mu }_m\right),\left({\xi }_m\right)}\left[\psi \left(t\right)\right].$$

(76)

This result generalizes many existing results in the literature; see, for example, [18]. Next, if we ignore the fractional parameters, then with the classical integration over $[a,b]$, we can obtain the original Chebyshev inequality (3). Similarly, a new set of further inequalities can be generated from Theorem 7. The results of this research could be more useful if combined with the new forms of the special functions obtained in [36,37,38] with the Fox-H function. Furthermore, we aim to enhance our research results in connection with Wu–Baleanu trajectories [43] and stochastic fractional differential equations [44].