Fractional Reverse Inequalities Involving Generic Interval-Valued Convex Functions and Applications

Abstract

The relation between fractional calculus and convexity significantly impacts the development of the theory of integral inequalities. In this paper, we explore the reverse of Minkowski and Hölder’s inequality, unified Jensen’s inequality, and Hermite–Hadamard ($\mathcal{H}$-$\mathcal{H}$)-like inequalities using fractional calculus and a generic class of interval-valued convexity. We introduce the concept of $\mathcal{I}.\mathcal{V}$-$(⋏,\hslash )$ generic class of convexity, which unifies several existing definitions of convexity. By utilizing Riemann–Liouville (R-L) fractional operators and $\mathcal{I}.\mathcal{V}$-$(⋏,\hslash )$ convexity to derive new improvements of the $\mathcal{H}$-$\mathcal{H}$- and Fejer and Pachpatte-like inequalities. Our results are quite unified; by substituting the different values of parameters, we obtain a blend of new and existing inequalities. These results are fruitful for establishing bounds for $\mathcal{I}.\mathcal{V}$ R-L integral operators. Furthermore, we discuss various implications of our findings, along with numerical examples and simulations to enhance the reliability of our results.

1. Introduction

Convex analysis based on the convex set and function whose epigraph is convex. It is an extensively investigated aspect due to some distinctive and unique properties regarding minima and maxima. As these concepts are purely geometric properties of sets and functions, they provide more information to enquire about numerous problems in linear programming and optimization. Using the idea of weighted means, different generalizations, refinements, and extensions have been created for different problems. These include weighted arithmetic means, harmonic means, geometric means, and other means. These have led to the classical convex set, the harmonic set, and the geometric set. Also, it is worth mentioning that non-convex sets and convex functions are effective tools to tackle non-convex sets, and functions introduced on these kinds of sets enjoy similar properties as convex functions.

The impact of convexity is unforgettable in various domains of mathematics, such as advanced analysis, topological spaces, particularly separation axioms, fixed point theory, and optimization. The convexity of the functions can be investigated through derivatives of the functions; a positive second-order derivative determines the concavity of functions.

The theory of inequalities covers many disciplines of mathematical analysis and other subjects due to its applicable characteristics. Among them, one of the applications of integral inequalities is to conclude some more refined upper bounds of error estimations of numerical quadrature rules like the trapezoidal rule, midpoint rule, Ostrowski’s rule, Simpson’s rules, etc. involving convex functions and their generalizations. Also, these bounds provide some relations between special means, special functions, probability theory, information theory, etc. By applying the notion of convexity, various fundamental inequalities can be extracted, like Jensen inequality, $\mathcal{H}$-$\mathcal{H}$ inequality, etc. From the following perspective, we revisit the familiar result due to Hermite and Hadamard separately: Let $\Phi :[{\mathfrak{c}}_3,{\mathfrak{c}}_4]\subset \mathbb{R}\to \mathbb{R}$ be a convex function, then:

$$\begin{array}{c}\hfill \Phi \left({\displaystyle \frac{{\mathfrak{c}}_3+{\mathfrak{c}}_4}{2}}\right)\le {\displaystyle \frac{1}{{\mathfrak{c}}_4-{\mathfrak{c}}_3}}{\int }_{{\mathfrak{c}}_3}^{{\mathfrak{c}}_4}\Phi \left(\varkappa \right)\mathrm{d}\varkappa \le {\displaystyle \frac{\Phi \left({\mathfrak{c}}_3\right)+\Phi \left({\mathfrak{c}}_4\right)}{2}}.\end{array}$$

This inequality can be viewed as another criterion of convex functions. For further detail, consult [1,2]. In 2019, Wu et al. [3] purported the unified form of convexity, which is described as follows.

 Definition 1 

([3]). Let $⋏:\mathfrak{M}\to \mathbb{R}$ be continuous monotonic function. Then, $\mathfrak{M}\subset \mathbb{R}$ is considered to be ⋏-convex set based on ⋏ if:

$$\begin{array}{c}\hfill {⋏}^{-1}((1-{‌}^1\ell )⋏\left(\varkappa \right)+{‌}^1\ell ⋏\left(y\right))\in \mathfrak{M}\forall \varkappa ,y\in \mathfrak{M}{‌}^1\ell \in [0,1].\end{array}$$

Now, we revisit the class ⋏-convex function, which is defined in [3].

 Definition 2 

([3]). A function $\Phi :\mathfrak{M}\to \mathbb{R}$ is said to be ⋏-convex function with respect to strictly monotonic continuous function ⋏ if:

$$\begin{array}{c}\hfill \Phi \left({⋏}^{-1}((1-{‌}^1\ell )⋏\left(\varkappa \right)+{‌}^1\ell ⋏\left(y\right))\right)\le (1-{‌}^1\ell )\Phi \left(\varkappa \right)+{‌}^1\ell \Phi \left(y\right)\forall \varkappa ,y\in \mathfrak{M}{‌}^1\ell \in [0,1].\end{array}$$

Another straightforward question in the realm of research is to formulate the set-valued counterparts of the known and new problems. An interval-valued function is a special type of set-valued function with co-domain as the collection of all real positive intervals. Moore was the first person to study the forgotten subject of interval analysis in a very systematic approach to derive the error estimate of finite machines. For more detail, see the monographs published by Moore [4]. The remarkable work of Moore is regarded as a re-emergence of this subject. After this, many authors have implemented $\mathcal{I}.\mathcal{V}$ concepts together with fractional calculus to study the various applicable fields of sciences. Working in the following directions, Breckner [5] delivered the conception of set-valued convex functions, which is demonstrated below.

 Definition 3. 

Let $\Phi :I=[{\mathfrak{c}}_3,{\mathfrak{c}}_4]\to {\mathbb{R}}_{+}$ is said to be $\mathcal{I}.\mathcal{V}$ convex function, if:

$$\begin{array}{c}\hfill \Phi ((1-{‌}^1\ell ){\mathfrak{c}}_3+{‌}^1\ell {\mathfrak{c}}_4)\supseteq (1-{‌}^1\ell )\Phi \left({\mathfrak{c}}_3\right)+{‌}^1\ell \Phi \left({\mathfrak{c}}_4\right),{‌}^1\ell \in [0,1].\end{array}$$

${\mathbb{R}}_I$ specifies the collection of all real intervals. For the first time, Sadowska [6] initiated the idea to extend the inequalities by set-valued functions and he investigated the $\mathcal{H}$-$\mathcal{H}$ inequality considering the set-valued convex function. Which is described below.

Let $\Phi :[{\mathfrak{c}}_3,{\mathfrak{c}}_4]\subseteq \mathbb{R}\to {\mathbb{R}}_I$ be an $\mathcal{I}.\mathcal{V}$ convex function, then:

$$\begin{array}{c}\hfill \Phi \left({\displaystyle \frac{{\mathfrak{c}}_3+{\mathfrak{c}}_4}{2}}\right)\supseteq {\displaystyle \frac{1}{{\mathfrak{c}}_4-{\mathfrak{c}}_3}}{\int }_{{\mathfrak{c}}_3}^{{\mathfrak{c}}_4}\Phi \left({‌}^1\ell \right)\mathrm{d}{‌}^1\ell \supseteq {\displaystyle \frac{\Phi \left({\mathfrak{c}}_3\right)+\Phi \left({\mathfrak{c}}_4\right)}{2}}.\end{array}$$

If $B\left([{\mathfrak{c}}_3,{\mathfrak{c}}_4]\right)$ is a collection of all divisions of $[{\mathfrak{c}}_3,{\mathfrak{c}}_4]$ and $B({\rho }_1,[{\mathfrak{c}}_3,{\mathfrak{c}}_4])$ be the collection of all divisions P such that mesh$\left(P\right)<{\rho }_1$, then $\Phi :[{\mathfrak{c}}_3,{\mathfrak{c}}_4]\to {\mathbb{R}}_{\mathbb{I}}$ is referred to as interval-valued Riemann integrable on $[{\mathfrak{c}}_3,{\mathfrak{c}}_4]$, if ∃${\mathfrak{M}}_1\in {\mathbb{R}}_{\mathbb{I}}$ and for each $ϵ>0$ there exit ${\rho }_1>0$ such that:

$$\begin{array}{c}\hfill d(S(\Phi ,P,{\rho }_1),{\mathfrak{M}}_1)<ϵ,\end{array}$$

(1)

where $S(\Phi ,P,{\rho }_1)$ specifies the Riemann sum of $Phi$ for any $PinB({\rho }_1,[{\mathfrak{c}}_3,{\rho }_4])$. The expression (1) represents that ${\mathfrak{M}}_1$ is the $\left(IR\right)$-integral of $\Phi$ such that:

$$\begin{array}{c}\hfill {\mathfrak{M}}_1=\left(IR\right){\int }_{{\mathfrak{c}}_3}^{{\mathfrak{c}}_4}\Phi \left({‌}^1\ell \right)\mathrm{d}{‌}^1\ell .\end{array}$$

For the sake of brevity, we specify the space of Riemann integrable functions and $\mathcal{I}.\mathcal{V}$ Riemann integration on $[{\mathfrak{c}}_3,{\mathfrak{c}}_4]$ and by $R_{[{\mathfrak{c}}_3,{\mathfrak{c}}_4]}$ and $I{R}_{[{\mathfrak{c}}_3,{\mathfrak{c}}_4]}$, respectively.

 Theorem 1 

([4]). Let $\Phi \left({‌}^1\ell \right):[{\mathfrak{c}}_3,{\mathfrak{c}}_4]\to \mathbb{R}$ be an $\mathcal{I}.\mathcal{V}$ function such that:

$$\begin{array}{c}\hfill \Phi \left({‌}^1\ell \right)=[{\Phi }_{*}\left({‌}^1\ell \right),{\Phi }^{*}\left({‌}^1\ell \right)],\Phi \left({‌}^1\ell \right)\in I{R}_{[{\mathfrak{c}}_3,{\mathfrak{c}}_4]}\iff {\Phi }_{*}\left({‌}^1\ell \right),{\Phi }^{*}\left({‌}^1\ell \right)\in R_{[{\mathfrak{c}}_3,{\mathfrak{c}}_4]}\end{array}$$

and

$$\begin{array}{c}\hfill \left(IR\right){\int }_{{\mathfrak{c}}_3}^{{\mathfrak{c}}_4}\Phi \left({‌}^1\ell \right)\mathrm{d}{‌}^1\ell =\left[\left(R\right){\int }_{{\mathfrak{c}}_3}^{{\mathfrak{c}}_4}{\Phi }_{*}\left({‌}^1\ell \right)\mathrm{d}{‌}^1\ell ,\left(R\right){\int }_{{\mathfrak{c}}_3}^{{\mathfrak{c}}_4}{\Phi }^{*}\left({‌}^1\ell \right)\mathrm{d}{‌}^1\ell \right].\end{array}$$

The space of Lebesgue integrable functions is defined by $L_1[{\mathfrak{c}}_3,{\mathfrak{c}}_4]$. Now, we recover the R-L integral operators, which are provided in [7].

 Definition 4. 

Let $\Phi \in L_1[{\mathfrak{c}}_3,{\mathfrak{c}}_4]$, then:

$$\begin{array}{cc}\hfill & J_{{\mathfrak{c}}_3^{+}}^{{\delta }_1}\Phi \left({\mathfrak{c}}_4\right)={\displaystyle \frac{1}{\Gamma \left({\delta }_1\right)}}{\int }_{{\mathfrak{c}}_3}^{{\mathfrak{c}}_4}\Phi \left(\varkappa \right){({\mathfrak{c}}_4-\varkappa )}^{{\delta }_1-1}\mathrm{d}\varkappa ,where{\mathfrak{c}}_4>{\mathfrak{c}}_3,{\delta }_1\ge 1.\hfill \end{array}$$

Analogously, the right side of the R-L operator is given as:

$$\begin{array}{c}\hfill J_{{\mathfrak{c}}_4^{-}}^{{\delta }_1}\Phi \left({\mathfrak{c}}_3\right)={\displaystyle \frac{1}{\Gamma \left({\delta }_1\right)}}{\int }_{{\mathfrak{c}}_3}^{{\mathfrak{c}}_4}\Phi \left(\varkappa \right){(\varkappa -{\mathfrak{c}}_3)}^{{\delta }_1-1}\mathrm{d}\varkappa ,where{\mathfrak{c}}_4>{\mathfrak{c}}_3,{\delta }_1\ge 1.\end{array}$$

where $\Gamma \left({\delta }_1\right)$ is the gamma function.

Now, we reproduce the $\mathcal{I}.\mathcal{V}$ R-L operators.

 Definition 5 

([8]). Let $\Phi \left(\varkappa \right)$ be an $\mathcal{I}.\mathcal{V}$ function such that $\underline{\Phi }\left(\varkappa \right),\overline{\Phi }\left(\varkappa \right)\in L_1[{\mathfrak{c}}_3,{\mathfrak{c}}_4]$, then:

$$\begin{array}{c}\hfill J_{{\varkappa }^{+}}^{{\delta }_1}\Phi \left({\mathfrak{c}}_4\right)={\displaystyle \frac{1}{\Gamma \left({\delta }_1\right)}}{\int }_{\varkappa }^{{\mathfrak{c}}_4}{({\mathfrak{c}}_4-{‌}^1\ell )}^{{\delta }_1-1}\Phi \left({‌}^1\ell \right)\mathrm{d}{‌}^1\ell ,{\mathfrak{c}}_4>\varkappa ,\end{array}$$

and

$$\begin{array}{c}\hfill J_{y^{-}}^{{\delta }_1}\Phi \left({\mathfrak{c}}_3\right)={\displaystyle \frac{1}{\Gamma \left({\delta }_1\right)}}{\int }_{{\mathfrak{c}}_3}^y{({‌}^1\ell -{\mathfrak{c}}_3)}^{{\delta }_1-1}\Phi \left({‌}^1\ell \right)\mathrm{d}{‌}^1\ell ,{\mathfrak{c}}_3<y,\end{array}$$

with ${\delta }_1\ge 0$. Obviously, we observe that:

$$\begin{array}{c}\hfill J_{{\varkappa }^{+}}^{{\delta }_1}\Phi \left({\mathfrak{c}}_4\right)=\left[J_{{\varkappa }^{+}}^{{\delta }_1}\left\{\underline{\Phi }\left({\mathfrak{c}}_4\right)\right\},J_{{\varkappa }^{+}}^{{\delta }_1}\left\{\overline{\Phi }\left({\mathfrak{c}}_4\right)\right\}\right],\end{array}$$

and

$$\begin{array}{c}\hfill J_{y^{-}}^{{\delta }_1}\Phi \left({\mathfrak{c}}_3\right)=\left[J_{y^{-}}^{{\delta }_1}\left\{\underline{\Phi }\left({\mathfrak{c}}_3\right)\right\},J_{y^{-}}^{{\delta }_1}\left\{\overline{\Phi }\left({\mathfrak{c}}_3\right)\right\}\right].\end{array}$$

Nowadays, several integral inequalities are generalized and rectified by implementing $\mathcal{I}.\mathcal{V}$ functions based on different partial and total ordered relations. Chalco-Cano et al. [9,10] analyzed the error inequality of the rectangular quadrature rule by making use of the generalized Hukuhara difference and interval-valued functions and also provided some useful applications, respectively.

Furthermore, Costa et al. [11] developed some interesting inequalities by considering the fuzzy real-valued functions. These papers were the initial attempts to make the inequalities more charming from the implementation aspect. Still, the above-mentioned studies are the gateway for further progress in the field of mathematical inequalities, especially those associated with set-valued functions. In the following scenario, Flores et al. [12] proposed the novel counterparts of renowned inequalities involving generalized functions. Sharma et al. [13] established the integral inequalities of trapezium like through non-convex $\mathcal{I}.\mathcal{V}$ functions defined over invex sets. In [14,15], Zhao and their coauthors investigated and explored the renowned Jensen’s and $\mathcal{H}$-$\mathcal{H}$-like containments by utilizing the $\mathcal{I}.\mathcal{V}$ unified convexity, which yields several notions of convexity through appropriate substitutions and Chebyshev-type inclusions, respectively. In [16], Abdeljawad et al. have derived some more general fractional containments of Hermite-Hadamar’s type inequalities involving p-$\mathcal{I}.\mathcal{V}$ convexity. Khan et al. [17] gave the conception of fuzzy convex functions for a coordinate system and applied it to construct fresh coordinated trapezium-type inclusions. Fractional calculus has valuable impacts on the rapid progression of several disciplines, including approximation theory and inequalities. For recent developments in fractional calculus and approximation theory, see [18,19,20,21,22,23].

Sarikaya et al. [24] started the use of non-integer order calculus to find fractional analogues of $\mathcal{H}$-$\mathcal{H}$ inequality considering R-L fractional operators. In [25], the authors explored the interval-valued unified fractional operator and concluded some fractional parametric integral inclusions.

Mohsin and their coauthors [26] explored two-dimensional $\mathcal{I}.\mathcal{V}$ harmonic convex functions and general family operators based on Raina’s function to develop new $\mathcal{H}$-$\mathcal{H}$-like inequalities. In [27], the authors formulated the fractional tempered trapzoidal-like inequalities and discussed the implications of primary findings. In [28], Akdemir and their coauthors calculated the Chebyshev-type formulas via a general family of fractional operators. Set et al. [29] explored the fractional versions of inequalities through Atangana-Baleanu fractional operators and the convexity property of the functions. In [8], Budak and their colleagues examined the fractional trapezium type inequalities through $\mathcal{I}.\mathcal{V}$ convex functions. In 2022, Du and Zhou [30] examined the coordinated set-valued variants of Hermite–Hadamard-like inequalities associated with fractional integral operators having non-singular kernels. Kara et al. [31] explored the two-dimensional interval-valued integral inequalities by means of unified fractional operators.

In [32], the authors implemented the technique of $\mathcal{I}.\mathcal{V}$ convex functions and post-quantum calculus to examine some fresh representations of already established findings. In [33], the authors investigated the non-convex functions based on $\beta$ connected sets in fuzzy domains and delivered some variants of trapezoid-type inequalities. Cortez et al. [34] gave the idea of a generic class of convexity and computed new counterparts of renowned Jensen’s and $\mathcal{H}$-$\mathcal{H}$ type inequalities along with applications.

Motivated by the above-mentioned studies, we aim to unify the existing notion of convexity in interval analysis. In the subsequent viewpoints, we introduced the idea of $\mathcal{I}.\mathcal{V}$ -$(⋏,\hslash )$ convex functions through the utilization of weighted arithmetic means involving a strictly monotone function ⋏ and non-negative function ℏ. In the setting of some suitable substitutions ⋏ and ℏ, it yields already-known notions and several new classes of convexity, which are discussed in the main section. Moreover, we provide the characterization of this class through inequalities. As applications of this class, we will derive some generic Jensen’s inequality, fractional variants of reverse Minkowski inequality, Hölders inequality, $\mathcal{H}$-$\mathcal{H}$ inequality, its weighted form known as Fejer-$\mathcal{H}$-$\mathcal{H}$ inequality, and some inequalities for the product of functions that are recognized as Pachpatee’s type inclusions. The novelty of the current proceeding is the generic class of convexity and its consequences in inequalities, because it is a larger space of functions containing convex and non-convex function classes. From our constructed results, one can characterize extensive function classes. Moreover, our results will be powerful tools for the computation of various bounds for $\mathcal{I},\mathcal{V}$ R-L fractional operators. To check the correctness of the proposed results, some simulations for numerical examples are listed. We hope this study helps curious readers to prove some other sort of inequalities and related problems in optimization.

2. Main Results

In this section, we present our main results.

2.1. Fractional Reverse Minkowski and Hölder’s Inequality

Now, we present the reverse fractional Minkowski’s integral inequality.

 Theorem 2. 

Let $\Phi ,\xi :[{\mathfrak{c}}_{\mathfrak{1}},{\mathfrak{c}}_{\mathfrak{2}}]\to {\mathbb{R}}_I^{+}$ be interval-valued functions such that $\Phi \left({‌}^1\ell \right)=[{\Phi }_{*},{\Phi }^{*}]$, $\xi \left({‌}^1\ell \right)=[{\xi }_{*},{\xi }^{*}]$, $J_{{{\mathfrak{c}}_{\mathfrak{1}}}^{+}}^{{\delta }_1}{\Phi }^p\left({‌}^1\ell \right)<\infty$ and $J_{{{\mathfrak{c}}_{\mathfrak{1}}}^{+}}^{{\delta }_1}{\xi }^p\left({‌}^1\ell \right)<\infty$, then:

$$\begin{array}{cc}\hfill & \left[{\displaystyle \frac{1+v(2+V)}{(1+v)(1+V)}},{\displaystyle \frac{1+V(2+v)}{(1+v)(1+V)}}\right]\left[{\left[J_{{{\mathfrak{c}}_{\mathfrak{1}}}^{+}}^{{\delta }_1}{(\Phi \left({‌}^1\ell \right)+\xi \left({‌}^1\ell \right))}^p\right]}^{{\displaystyle \frac{1}{p}}}\right]\hfill \\ \hfill & \supseteq {\left[J_{{{\mathfrak{c}}_{\mathfrak{1}}}^{+}}^{{\delta }_1}{\Phi }^p\left({‌}^1\ell \right)\right]}^{{\displaystyle \frac{1}{p}}}+{\left[J_{{{\mathfrak{c}}_{\mathfrak{1}}}^{+}}^{{\delta }_1}{\xi }^p\left({‌}^1\ell \right)\right]}^{{\displaystyle \frac{1}{p}}},\hfill \end{array}$$

where $0<v\le {\displaystyle \frac{{\Phi }_{*}\left({‌}^1\ell \right)}{{\xi }_{*}\left({‌}^1\ell \right)}}\le V$, $0<v\le {\displaystyle \frac{{\Phi }^{*}\left({‌}^1\ell \right)}{{\xi }^{*}\left({‌}^1\ell \right)}}\le V$, ${‌}^1\ell \in [{\mathfrak{c}}_{\mathfrak{1}},{\mathfrak{c}}_{\mathfrak{2}}]$ and $p\ge 1$ with ${\delta }_1>0$.

 Proof. 

Since ${\displaystyle \frac{{\Phi }^{*}\left({‌}^1\ell \right)}{{\xi }^{*}\left({‌}^1\ell \right)}}\le V$, then:

$$\begin{array}{c}\hfill {(V+1)}^p{\Phi }^{*p}\left({‌}^1\ell \right)\le V^p{({\Phi }^{*}\left({‌}^1\ell \right)+{\xi }^{*}\left({‌}^1\ell \right))}^p.\end{array}$$

(2)

Multiplying ${({\mathfrak{c}}_{\mathfrak{2}}-{‌}^1\ell )}^{{\delta }_1-1}$ on both sides of (2) and then taking the integration with respect to ${‌}^1\ell$ over $[{\mathfrak{c}}_{\mathfrak{1}},{\mathfrak{c}}_{\mathfrak{2}}]$, we have:

$$\begin{array}{c}\hfill {(V+1)}^p{\int }_{{\mathfrak{c}}_{\mathfrak{1}}}^{{\mathfrak{c}}_{\mathfrak{2}}}{({\mathfrak{c}}_{\mathfrak{2}}-{‌}^1\ell )}^{{\delta }_1-1}{\Phi }^{*p}\left({‌}^1\ell \right)\mathrm{d}{‌}^1\ell \le V^p{\int }_{{\mathfrak{c}}_{\mathfrak{1}}}^{{\mathfrak{c}}_{\mathfrak{2}}}{({\mathfrak{c}}_{\mathfrak{2}}-{‌}^1\ell )}^{{\delta }_1-1}{({\Phi }^{*}\left({‌}^1\ell \right)+{\xi }^{*}\left({‌}^1\ell \right))}^p\mathrm{d}{‌}^1\ell ,\end{array}$$

Now, by comparing the above expression with the R-L fractional operator, we obtain:

$$\begin{array}{c}\hfill {\left[J_{{{\mathfrak{c}}_{\mathfrak{1}}}^{+}}^{{\delta }_1}{\Phi }^{*p}\left({‌}^1\ell \right)\right]}^{{\displaystyle \frac{1}{p}}}\le {\displaystyle \frac{V}{1+V}}{\left[J_{{{\mathfrak{c}}_{\mathfrak{1}}}^{+}}^{{\delta }_1}{({\Phi }^{*}\left({‌}^1\ell \right)+{\xi }^{*}\left({‌}^1\ell \right))}^p\right]}^{{\displaystyle \frac{1}{p}}}.\end{array}$$

(3)

Also, $v\le {\displaystyle \frac{{\Phi }_{*}\left({‌}^1\ell \right)}{{\xi }_{*}\left({‌}^1\ell \right)}}$, then one can write:

$$\begin{array}{c}\hfill v^p{({\Phi }_{*}\left({‌}^1\ell \right)+{\xi }_{*}\left({‌}^1\ell \right))}^p\le {(1+v)}^p{f}_{*}^p\left({‌}^1\ell \right).\end{array}$$

(4)

Multiplying ${({\mathfrak{c}}_{\mathfrak{2}}-{‌}^1\ell )}^{{\delta }_1-1}$ on both sides of (4) and then taking the integration with respect to ${‌}^1\ell$ over $[{\mathfrak{c}}_{\mathfrak{1}},{\mathfrak{c}}_{\mathfrak{2}}]$, then:

$$\begin{array}{c}\hfill {\displaystyle \frac{v}{1+v}}{\left[J_{{{\mathfrak{c}}_{\mathfrak{1}}}^{+}}^{{\delta }_1}{({\Phi }_{*}^p\left({‌}^1\ell \right)+{\xi }_{*}^p\left({‌}^1\ell \right))}^p\right]}^{{\displaystyle \frac{1}{p}}}\le {\left[J_{{{\mathfrak{c}}_{\mathfrak{1}}}^{+}}^{{\delta }_1}{\Phi }_{*}^p\left({‌}^1\ell \right)\right]}^{{\displaystyle \frac{1}{p}}}.\end{array}$$

(5)

Furthermore, we note that:

$$\begin{array}{cc}\hfill & {\left[J_{{{\mathfrak{c}}_{\mathfrak{1}}}^{+}}^{{\delta }_1}{\Phi }^p\left({‌}^1\ell \right)\right]}^{{\displaystyle \frac{1}{p}}}\hfill \\ \hfill & =\left[{\left[J_{{{\mathfrak{c}}_{\mathfrak{1}}}^{+}}^{{\delta }_1}{\Phi }_{*}^p\left({‌}^1\ell \right)\right]}^{{\displaystyle \frac{1}{p}}},{\left[J_{{{\mathfrak{c}}_{\mathfrak{1}}}^{+}}^{{\delta }_1}{\Phi }^{*p}\left({‌}^1\ell \right)\right]}^{{\displaystyle \frac{1}{p}}}\right]\hfill \\ \hfill & \subseteq \left[{\displaystyle \frac{v}{1+v}}{\left[J_{{{\mathfrak{c}}_{\mathfrak{1}}}^{+}}^{{\delta }_1}{({\Phi }_{*}\left({‌}^1\ell \right)+{\xi }_{*}\left({‌}^1\ell \right))}^p\right]}^{{\displaystyle \frac{1}{p}}},{\displaystyle \frac{V}{1+V}}{\left[J_{{{\mathfrak{c}}_{\mathfrak{1}}}^{+}}^{{\delta }_1}{({\Phi }^{*}\left({‌}^1\ell \right)+{\xi }^{*}\left({‌}^1\ell \right))}^p\right]}^{{\displaystyle \frac{1}{p}}}\right]\hfill \\ \hfill & =\left[{\displaystyle \frac{v}{1+v}},{\displaystyle \frac{V}{1+V}}\right]{\left[J_{{{\mathfrak{c}}_{\mathfrak{1}}}^{+}}^{{\delta }_1}{(\Phi \left({‌}^1\ell \right)+\xi \left({‌}^1\ell \right))}^p\right]}^{{\displaystyle \frac{1}{p}}}.\hfill \end{array}$$

(6)

From $v\le {\displaystyle \frac{{\Phi }^{*}\left({‌}^1\ell \right)}{{\xi }^{*}\left({‌}^1\ell \right)}}$, we achieve:

$$\begin{array}{c}\hfill {(1+v)}^p{\xi }^p\left({‌}^1\ell \right)\le {({\Phi }^{*}\left({‌}^1\ell \right)+{\xi }^{*}\left({‌}^1\ell \right))}^p.\end{array}$$

(7)

Multiplying ${({\mathfrak{c}}_{\mathfrak{2}}-{‌}^1\ell )}^{{\delta }_1-1}$ on both sides of (7) and then taking the integration with respect to ${‌}^1\ell$ over $[{\mathfrak{c}}_{\mathfrak{1}},{\mathfrak{c}}_{\mathfrak{2}}]$, then:

$$\begin{array}{c}\hfill {\left[J_{{{\mathfrak{c}}_{\mathfrak{1}}}^{+}}^{{\delta }_1}{\xi }^{*p}\left({‌}^1\ell \right)\right]}^{{\displaystyle \frac{1}{p}}}\le {\displaystyle \frac{1}{1+v}}{\left[J_{{{\mathfrak{c}}_{\mathfrak{1}}}^{+}}^{{\delta }_1}{({\Phi }^{*}\left({‌}^1\ell \right)+{\xi }^{*}\left({‌}^1\ell \right))}^p\right]}^{{\displaystyle \frac{1}{p}}}.\end{array}$$

(8)

Now, from ${\displaystyle \frac{{\Phi }_{*}\left({‌}^1\ell \right)}{{\xi }_{*}\left({‌}^1\ell \right)}}\le V$, we obtain:

$$\begin{array}{c}\hfill {({\Phi }_{*}\left({‌}^1\ell \right)+{\xi }_{*}\left({‌}^1\ell \right))}^p\le {(1+V)}^p{g}_{*p}\left({‌}^1\ell \right).\end{array}$$

(9)

Multiplying ${({\mathfrak{c}}_{\mathfrak{2}}-{‌}^1\ell )}^{{\delta }_1-1}$ on both sides of (9) and taking the integration with respect to ${‌}^1\ell$ over $[{\mathfrak{c}}_{\mathfrak{1}},{\mathfrak{c}}_{\mathfrak{2}}]$, then:

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{1+V}}{\left[J_{{{\mathfrak{c}}_{\mathfrak{1}}}^{+}}^{{\delta }_1}{({\Phi }_{*}\left({‌}^1\ell \right)+{\xi }_{*}\left({‌}^1\ell \right))}^p\right]}^{{\displaystyle \frac{1}{p}}}\le {\left[J_{{{\mathfrak{c}}_{\mathfrak{1}}}^{+}}^{{\delta }_1}{\xi }_{*p}\left({‌}^1\ell \right)\right]}^{{\displaystyle \frac{1}{p}}}.\end{array}$$

(10)

From (8) and (10), we have:

$$\begin{array}{cc}\hfill & {\left[J_{{{\mathfrak{c}}_{\mathfrak{1}}}^{+}}^{{\delta }_1}{\xi }^p\left({‌}^1\ell \right)\right]}^{{\displaystyle \frac{1}{p}}}\hfill \\ \hfill & =\left[{\left[J_{{{\mathfrak{c}}_{\mathfrak{1}}}^{+}}^{{\delta }_1}{\xi }_{*}^p\left({‌}^1\ell \right)\right]}^{{\displaystyle \frac{1}{p}}},{\left[J_{{{\mathfrak{c}}_{\mathfrak{1}}}^{+}}^{{\delta }_1}{\xi }^{*p}\left({‌}^1\ell \right)\right]}^{{\displaystyle \frac{1}{p}}}\right]\hfill \\ \hfill & \subseteq \left[{\displaystyle \frac{v}{1+v}}{\left[J_{{{\mathfrak{c}}_{\mathfrak{1}}}^{+}}^{{\delta }_1}{({\Phi }_{*}\left({‌}^1\ell \right)+{\xi }_{*}\left({‌}^1\ell \right))}^p\right]}^{{\displaystyle \frac{1}{p}}},{\displaystyle \frac{V}{1+V}}{\left[J_{{{\mathfrak{c}}_{\mathfrak{1}}}^{+}}^{{\delta }_1}{({\Phi }^{*}\left({‌}^1\ell \right)+{\xi }^{*}\left({‌}^1\ell \right))}^p\right]}^{{\displaystyle \frac{1}{p}}}\right]\hfill \\ \hfill & =\left[{\displaystyle \frac{v}{1+v}},{\displaystyle \frac{V}{1+V}}\right]{[J_{{{\mathfrak{c}}_{\mathfrak{1}}}^{+}}^{{\delta }_1}{(\Phi \left({‌}^1\ell \right)+\xi \left({‌}^1\ell \right))}^p]}^{{\displaystyle \frac{1}{p}}}.\hfill \end{array}$$

(11)

Taking the sum of (6) and (11), we acquired our desired result. □

Now, we present the reverse fractional Hölder’s integral inequality.

 Theorem 3. 

Let $\Phi ,\xi :[{\mathfrak{c}}_{\mathfrak{1}},{\mathfrak{c}}_{\mathfrak{2}}]\to {\mathbb{R}}_I^{+}$ be interval-valued functions such that $\Phi \left({‌}^1\ell \right)=[{\Phi }_{*},{\Phi }^{*}]$, $\xi \left({‌}^1\ell \right)=[{\xi }_{*},{\xi }^{*}]$, $J_{{{\mathfrak{c}}_{\mathfrak{1}}}^{+}}^{{\delta }_1}{\Phi }^p\left({‌}^1\ell \right)<\infty$ and $J_{{{\mathfrak{c}}_{\mathfrak{1}}}^{+}}^{{\delta }_1}{\xi }^p\left({‌}^1\ell \right)<\infty$, then:

$$\begin{array}{c}\hfill \left[{\left({\displaystyle \frac{v}{V}}\right)}^{{\displaystyle \frac{1}{pq}}},{\left({\displaystyle \frac{V}{v}}\right)}^{{\displaystyle \frac{1}{pq}}}\right]J_{{{\mathfrak{c}}_{\mathfrak{1}}}^{+}}^{{\delta }_1}\left[{\Phi }^{{\displaystyle \frac{1}{p}}}\left({‌}^1\ell \right){\xi }^{{\displaystyle \frac{1}{q}}}\left({‌}^1\ell \right)\right]\supseteq {[J_{{{\mathfrak{c}}_{\mathfrak{1}}}^{+}}^{{\delta }_1}\Phi \left({‌}^1\ell \right)]}^{{\displaystyle \frac{1}{p}}}{\left[J_{{{\mathfrak{c}}_{\mathfrak{1}}}^{+}}^{{\delta }_1}\xi \left({‌}^1\ell \right)\right]}^{{\displaystyle \frac{1}{p}}},\end{array}$$

where $0<v\le {\displaystyle \frac{{\Phi }_{*}\left({‌}^1\ell \right)}{{\xi }_{*}\left({‌}^1\ell \right)}}\le V$, $0<v\le {\displaystyle \frac{{\Phi }^{*}\left({‌}^1\ell \right)}{{\xi }^{*}\left({‌}^1\ell \right)}}\le V$, ${‌}^1\ell \in [{\mathfrak{c}}_{\mathfrak{1}},{\mathfrak{c}}_{\mathfrak{2}}]$ and ${\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{q}}=1$, with $p>1$ and ${\delta }_1>0$.

 Proof. 

Since ${\displaystyle \frac{{\Phi }^{*}\left({‌}^1\ell \right)}{{\xi }^{*}\left({‌}^1\ell \right)}}\le V$, then:

$$\begin{array}{c}\hfill {\Phi }^{*}\left({‌}^1\ell \right)\le V^{{\displaystyle \frac{1}{q}}}{\Phi }^{*{\displaystyle \frac{1}{p}}}\left({‌}^1\ell \right){\Phi }^{*{\displaystyle \frac{1}{q}}}\left({‌}^1\ell \right).\end{array}$$

(12)

Multiplying ${({\mathfrak{c}}_{\mathfrak{2}}-{‌}^1\ell )}^{{\delta }_1-1}$ on both sides of (12) and then taking the integration with respect to ${‌}^1\ell$ over $[{\mathfrak{c}}_{\mathfrak{1}},{\mathfrak{c}}_{\mathfrak{2}}]$, then:

$$\begin{array}{c}\hfill {\left(J_{{{\mathfrak{c}}_{\mathfrak{1}}}^{+}}^{{\delta }_1}{\Phi }^{*}\left({‌}^1\ell \right)\right)}^{{\displaystyle \frac{1}{p}}}\le V^{{\displaystyle \frac{1}{pq}}}{J}_{{{\mathfrak{c}}_{\mathfrak{1}}}^{+}}^{{\delta }_1}{\left({\Phi }^{*{\displaystyle \frac{1}{p}}}\left({‌}^1\ell \right){\xi }^{*{\displaystyle \frac{1}{q}}}\left({‌}^1\ell \right)\right)}^{{\displaystyle \frac{1}{p}}}.\end{array}$$

(13)

Analogously, by the virtue of ${\displaystyle \frac{{\Phi }_{*}\left({‌}^1\ell \right)}{{\xi }_{*}\left({‌}^1\ell \right)}}\ge v$, we have:

$$\begin{array}{c}\hfill {\Phi }_{*}\left({‌}^1\ell \right)\ge v^{{\displaystyle \frac{1}{q}}}{\Phi }_{*{\displaystyle \frac{1}{p}}}\left({‌}^1\ell \right){\Phi }_{*{\displaystyle \frac{1}{q}}}\left({‌}^1\ell \right).\end{array}$$

(14)

Multiplying ${({\mathfrak{c}}_{\mathfrak{2}}-{‌}^1\ell )}^{{\delta }_1-1}$ on both sides of (14) and then taking the integration with respect to ${‌}^1\ell$ over $[{\mathfrak{c}}_{\mathfrak{1}},{\mathfrak{c}}_{\mathfrak{2}}]$, then:

$$\begin{array}{c}\hfill {\left(J_{{{\mathfrak{c}}_{\mathfrak{1}}}^{+}}^{{\delta }_1}{\Phi }_{*}\left({‌}^1\ell \right)\right)}^{{\displaystyle \frac{1}{p}}}\ge v^{{\displaystyle \frac{1}{pq}}}{J}_{{{\mathfrak{c}}_{\mathfrak{1}}}^{+}}^{{\delta }_1}{\left({\Phi }_{*}^{{\displaystyle \frac{1}{p}}}\left({‌}^1\ell \right){\xi }_{*}^{{\displaystyle \frac{1}{q}}}\left({‌}^1\ell \right)\right)}^{{\displaystyle \frac{1}{p}}}.\end{array}$$

(15)

Also, by combining (13) and (15), we have:

$$\begin{array}{cc}\hfill & {\left(J_{{{\mathfrak{c}}_{\mathfrak{1}}}^{+}}^{{\delta }_1}\Phi \left({‌}^1\ell \right)\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ \hfill & =\left[{\left(J_{{{\mathfrak{c}}_{\mathfrak{1}}}^{+}}^{{\delta }_1}{\Phi }_{*}\left({‌}^1\ell \right)\right)}^{{\displaystyle \frac{1}{p}}},{\left(J_{{{\mathfrak{c}}_{\mathfrak{1}}}^{+}}^{{\delta }_1}{\Phi }^{*}\left({‌}^1\ell \right)\right)}^{{\displaystyle \frac{1}{p}}}\right]\hfill \\ \hfill & \subseteq \left[v^{{\displaystyle \frac{1}{pq}}}{J}_{{{\mathfrak{c}}_{\mathfrak{1}}}^{+}}^{{\delta }_1}{\left({\Phi }_{*}^{{\displaystyle \frac{1}{p}}}\left({‌}^1\ell \right){\xi }_{*}^{{\displaystyle \frac{1}{q}}}\left({‌}^1\ell \right)\right)}^{{\displaystyle \frac{1}{p}}},V^{{\displaystyle \frac{1}{pq}}}{J}_{{{\mathfrak{c}}_{\mathfrak{1}}}^{+}}^{{\delta }_1}{\left({\Phi }^{*{\displaystyle \frac{1}{p}}}\left({‌}^1\ell \right){\xi }^{*{\displaystyle \frac{1}{q}}}\left({‌}^1\ell \right)\right)}^{{\displaystyle \frac{1}{p}}}\right]\hfill \\ \hfill & =\left[v^{{\displaystyle \frac{1}{pq}}},V^{{\displaystyle \frac{1}{pq}}}\right]J_{{{\mathfrak{c}}_{\mathfrak{1}}}^{+}}^{{\delta }_1}{\left({\Phi }^{{\displaystyle \frac{1}{p}}}\left({‌}^1\ell \right){\xi }^{{\displaystyle \frac{1}{q}}}\left({‌}^1\ell \right)\right)}^{{\displaystyle \frac{1}{p}}}.\hfill \end{array}$$

(16)

By similar proceedings, we acquire the following containment for ${\left(J_{{{\mathfrak{c}}_{\mathfrak{1}}}^{+}}^{{\delta }_1}\xi \left({‌}^1\ell \right)\right)}^{{\displaystyle \frac{1}{q}}}$, as:

$$\begin{array}{cc}\hfill & {\left(J_{a^{+}}^{{\delta }_1}\xi \left({‌}^1\ell \right)\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & =\left[{\left(J_{{{\mathfrak{c}}_{\mathfrak{1}}}^{+}}^{{\delta }_1}{\xi }_{*}\left({‌}^1\ell \right)\right)}^{{\displaystyle \frac{1}{q}}},{\left(J_{{{\mathfrak{c}}_{\mathfrak{1}}}^{+}}^{{\delta }_1}{\xi }^{*}\left({‌}^1\ell \right)\right)}^{{\displaystyle \frac{1}{q}}}\right]\hfill \\ \hfill & \subseteq \left[{\displaystyle \frac{1}{V^{\frac{1}{pq}}}}{J}_{{{\mathfrak{c}}_{\mathfrak{1}}}^{+}}^{{\delta }_1}{\left({\Phi }_{*}^{{\displaystyle \frac{1}{p}}}\left({‌}^1\ell \right){\xi }_{*}^{{\displaystyle \frac{1}{q}}}\left({‌}^1\ell \right)\right)}^{{\displaystyle \frac{1}{q}}},{\displaystyle \frac{1}{v^{\frac{1}{pq}}}}{J}_{{{\mathfrak{c}}_{\mathfrak{1}}}^{+}}^{{\delta }_1}{\left({\Phi }^{*{\displaystyle \frac{1}{p}}}\left({‌}^1\ell \right){\xi }^{*{\displaystyle \frac{1}{q}}}\left({‌}^1\ell \right)\right)}^{{\displaystyle \frac{1}{q}}}\right]\hfill \\ \hfill & =\left[{\displaystyle \frac{1}{V^{\frac{1}{pq}}}},{\displaystyle \frac{1}{v^{\frac{1}{pq}}}}\right]J_{{{\mathfrak{c}}_{\mathfrak{1}}}^{+}}^{{\delta }_1}{\left({\Phi }^{{\displaystyle \frac{1}{p}}}\left({‌}^1\ell \right){\xi }^{{\displaystyle \frac{1}{q}}}\left({‌}^1\ell \right)\right)}^{{\displaystyle \frac{1}{q}}}.\hfill \end{array}$$

(17)

This completes the proof. □

2.2. On $(⋏,\hslash )$ Interval-Valued Convex Functions and Related Results

Now, we initiate the concept of $\mathcal{I}.\mathcal{V}$-$(⋏,\hslash )$ convex functions, which shows that the several existing classes of convexity and various new generalizations of convexity can be obtained as special cases.

 Definition 6. 

Let $\Phi :[{\mathfrak{c}}_3,{\mathfrak{c}}_4]\to {\mathbb{R}}_I^{+}$ be an $\mathcal{I}.\mathcal{V}$-$(⋏,\hslash )$ function satisfying the condition $\Phi \left(\varkappa \right)=[{\Phi }_{*}\left(\varkappa \right),{\Phi }^{*}\left(\varkappa \right)]$ and $\hslash :[0,1]\to \mathbb{R}$ be a non-negative function. Then:

$$\begin{array}{c}\hfill \Phi \left({⋏}^{-1}((1-{‌}^1\ell )⋏\left({\mathfrak{c}}_3\right)+{‌}^1\ell ⋏\left({\mathfrak{c}}_4\right))\right)\supseteq \hslash (1-{‌}^1\ell )\Phi \left({\mathfrak{c}}_3\right)+\hslash \left({‌}^1\ell \right)\Phi \left({\mathfrak{c}}_4\right),\end{array}$$

(18)

$\forall \varkappa ,y\in [{\mathfrak{c}}_3,{\mathfrak{c}}_4]$ and ${‌}^1\ell \in [0,1]$.

Now, we report interesting outcomes concerning Definition 6.

• Choosing $\hslash \left({‌}^1\ell \right)={‌}^1\ell$, we obtain the ⋏-$\mathcal{I}.\mathcal{V}$ convex function:

  $$\begin{array}{c}\hfill \Phi \left({⋏}^{-1}((1-{‌}^1\ell )⋏\left({\mathfrak{c}}_3\right)+{‌}^1\ell ⋏\left({\mathfrak{c}}_4\right))\right)\supseteq (1-{‌}^1\ell )\Phi \left({\mathfrak{c}}_3\right)+{‌}^1\ell \Phi \left({\mathfrak{c}}_4\right).\end{array}$$
• Choosing $⋏\left(\varkappa \right)={\displaystyle \frac{1}{\varkappa }}$ and $\hslash \left({‌}^1\ell \right)={‌}^1\ell$. Then, we attain the $\mathcal{I}.\mathcal{V}$ harmonic convex function defined in [35]:

  $$\begin{array}{c}\hfill \Phi \left({\displaystyle \frac{{\mathfrak{c}}_3{\mathfrak{c}}_4}{{‌}^1\ell {\mathfrak{c}}_3+(1-{‌}^1\ell ){\mathfrak{c}}_4}}\right)\supseteq (1-{‌}^1\ell )\Phi \left({\mathfrak{c}}_3\right)+{‌}^1\ell \Phi \left({\mathfrak{c}}_4\right).\end{array}$$
• Choosing $⋏\left(\varkappa \right)={\varkappa }^p$ and $\hslash \left({‌}^1\ell \right)={‌}^1\ell$. Then, we attain the $\mathcal{I}.\mathcal{V}$-p convex functions, which are defined in [16]:

  $$\begin{array}{c}\hfill \Phi \left({((1-{‌}^1\ell ){{\mathfrak{c}}_3}^p+{‌}^1\ell {{\mathfrak{c}}_4}^p)}^{{\displaystyle \frac{1}{p}}}\right)\supseteq (1-{‌}^1\ell )\Phi \left({\mathfrak{c}}_3\right)+{‌}^1\ell \Phi \left({\mathfrak{c}}_4\right).\end{array}$$

 Remark 1. 

By fixing $⋏\left(\varkappa \right)=\varkappa$, we recover the definition of $\mathcal{I}.\mathcal{V}$ convex functions.

 Definition 7. 

By fixing $\hslash \left({‌}^1\ell \right)={{‌}^1\ell }^s$ in Definition 6, we acquire the $(⋏,s)$$\mathcal{I}.\mathcal{V}$ convex function:

$$\begin{array}{c}\hfill \Phi \left({⋏}^{-1}({‌}^1\ell ⋏\left({\mathfrak{c}}_3\right)+(1-{‌}^1\ell )⋏\left({\mathfrak{c}}_4\right))\right)\supseteq {(1-{‌}^1\ell )}^s\Phi \left({\mathfrak{c}}_3\right)+{{‌}^1\ell }^s\Phi \left({\mathfrak{c}}_4\right).\end{array}$$

Now, we report the primary deductions of Definition 7.

• By specifying $⋏\left(\varkappa \right)={\displaystyle \frac{1}{\varkappa }}$, we obtain $\mathcal{I}.\mathcal{V}$ harmonically s convex functions, which are as follows:

  $$\begin{array}{c}\hfill \Phi \left({\displaystyle \frac{{\mathfrak{c}}_3{\mathfrak{c}}_4}{{‌}^1\ell {\mathfrak{c}}_3+(1-{‌}^1\ell ){\mathfrak{c}}_4}}\right)\supseteq {(1-{‌}^1\ell )}^s\Phi \left({\mathfrak{c}}_3\right)+{{‌}^1\ell }^s\Phi \left({\mathfrak{c}}_4\right).\end{array}$$
• Setting $⋏\left(\varkappa \right)={\varkappa }^p$, $p\ge -1$, then we obtain $(p,s)$-$\mathcal{I}.\mathcal{V}$ convex functions, which are as follows:

  $$\begin{array}{c}\hfill \Phi \left({((1-{‌}^1\ell ){{\mathfrak{c}}_3}^p+{‌}^1\ell {{\mathfrak{c}}_4}^p)}^{{\displaystyle \frac{1}{p}}}\right)\supseteq {(1-{‌}^1\ell )}^s\Phi \left({\mathfrak{c}}_3\right)+{{‌}^1\ell }^s\Phi \left({\mathfrak{c}}_4\right).\end{array}$$

 Definition 8. 

Choosing $\hslash \left({‌}^1\ell \right)={{‌}^1\ell }^{-s}$ in Definition 6, we attain the $(⋏,s)$$\mathcal{I}.\mathcal{V}$ Godunova–Levin type convex function:

$$\begin{array}{c}\hfill \Phi \left({⋏}^{-1}({‌}^1\ell ⋏\left({\mathfrak{c}}_3\right)+(1-{‌}^1\ell )⋏\left({\mathfrak{c}}_4\right))\right)\supseteq {{‌}^1\ell }^{-s}\Phi \left({\mathfrak{c}}_3\right)+{(1-{‌}^1\ell )}^{-s}\Phi \left({\mathfrak{c}}_4\right).\end{array}$$

For the different choices of $⋏\left(\varkappa \right)$, we attain some interesting classes of convexity.

• Choosing $⋏\left(\varkappa \right)=\left(\varkappa \right)$, we achieve the $\mathcal{I}.\mathcal{V}$ Godunove–Levin s convex function:

  $$\begin{array}{c}\hfill \Phi \left({⋏}^{-1}((1-{‌}^1\ell )⋏\left({\mathfrak{c}}_3\right)+{‌}^1\ell ⋏\left({\mathfrak{c}}_4\right))\right)\supseteq {{‌}^1\ell }^{-s}\Phi \left({\mathfrak{c}}_3\right)+{(1-{‌}^1\ell )}^{-s}\Phi \left({\mathfrak{c}}_4\right).\end{array}$$
• Choosing $⋏\left(\varkappa \right)={\displaystyle \frac{1}{\varkappa }}$. Then, we attain the $\mathcal{I}.\mathcal{V}$ Godunova–Levin harmonic s convex functions:

  $$\begin{array}{c}\hfill \Phi \left({\displaystyle \frac{{\mathfrak{c}}_3{\mathfrak{c}}_4}{{‌}^1\ell {\mathfrak{c}}_3+(1-{‌}^1\ell ){\mathfrak{c}}_4}}\right)\supseteq {{‌}^1\ell }^{-s}\Phi \left({\mathfrak{c}}_3\right)+{(1-{‌}^1\ell )}^{-s}\Phi \left({\mathfrak{c}}_4\right).\end{array}$$
• Choosing $⋏\left(\varkappa \right)={\varkappa }^p$. Then, we attain the $\mathcal{I}.\mathcal{V}$ Godunova–Levin $(P,s)$ convex functions:

  $$\begin{array}{c}\hfill \Phi \left({((1-{‌}^1\ell ){{\mathfrak{c}}_3}^p+{‌}^1\ell {{\mathfrak{c}}_4}^p)}^{{\displaystyle \frac{1}{p}}}\right)\supseteq {{‌}^1\ell }^{-s}\Phi \left({\mathfrak{c}}_3\right)+{(1-{‌}^1\ell )}^{-s}\Phi \left({\mathfrak{c}}_4\right).\end{array}$$

 Definition 9. 

Choosing $\hslash \left({‌}^1\ell \right)={‌}^1\ell (1-{‌}^1\ell )$ in Definition 6, we attain $(⋏,s)$$\mathcal{I}.\mathcal{V}$-$tgs$ convex function:

$$\begin{array}{c}\hfill \Phi \left({⋏}^{-1}({‌}^1\ell ⋏\left({\mathfrak{c}}_3\right)+(1-{‌}^1\ell )⋏\left({\mathfrak{c}}_4\right))\right)\supseteq {‌}^1\ell (1-{‌}^1\ell )[\Phi \left({\mathfrak{c}}_3\right)+\Phi \left({\mathfrak{c}}_4\right)].\end{array}$$

For the different choices of $⋏\left(\varkappa \right)$, we attain some interesting classes of convexity.

For our own ease, the collection of $\mathcal{I}.\mathcal{V}$-$(⋏,\hslash )$ convex functions, $\mathcal{I}.\mathcal{V}$-$(⋏,\hslash )$ concave functions, $(⋏,\hslash )$ convex functions, and $(⋏,\hslash )$ concave functions are represented by $SIGX\left([{\mathfrak{c}}_3,{\mathfrak{c}}_4],{\mathbb{R}}_I^{+}\right)$, $SIGV\left([{\mathfrak{c}}_3,{\mathfrak{c}}_4],{\mathbb{R}}_I^{+}\right)$, $SGX\left([{\mathfrak{c}}_3,{\mathfrak{c}}_4],\mathbb{R}\right)$, and $SGV\left([{\mathfrak{c}}_3,{\mathfrak{c}}_4],\mathbb{R}\right)$, respectively.

 Theorem 4. 

Let $\Phi :[{\mathfrak{c}}_3,{\mathfrak{c}}_4]\to {\mathbb{R}}_I^{+}$ be an $\mathcal{I}.\mathcal{V}$ function such that $\Phi =[{\Phi }_{*},{\Phi }^{*}]$ with ${\Phi }_{*}\le {\Phi }^{*}$. Then, $\Phi \in SIGX\left([{\mathfrak{c}}_3,{\mathfrak{c}}_4],{\mathbb{R}}_I^{+}\right)\iff {\Phi }_{*}\in SGX\left([{\mathfrak{c}}_3,{\mathfrak{c}}_4],\mathbb{R}\right)\&{\Phi }^{*}\in SGV\left([{\mathfrak{c}}_3,{\mathfrak{c}}_4],\mathbb{R}\right)$.

 Proof. 

Assume that $\Phi \in SIGV\left([{\mathfrak{c}}_3,{\mathfrak{c}}_4],{\mathbb{R}}_I^{+}\right)$ and $\varkappa ,y\in [{\mathfrak{c}}_3,{\mathfrak{c}}_4]$ and ${‌}^1\ell \in [0,1]$, then:

$$\begin{array}{c}\hfill \Phi \left({⋏}^{-1}({‌}^1\ell ⋏\left(\varkappa \right)+(1-{‌}^1\ell )⋏\left(y\right))\right)\supseteq \hslash \left({‌}^1\ell \right)\Phi \left(\varkappa \right)+\hslash (1-{‌}^1\ell )\Phi \left(y\right).\end{array}$$

This implies that:

$$\begin{array}{cc}\hfill & \left[{\Phi }_{*}\left({⋏}^{-1}({‌}^1\ell ⋏\left(\varkappa \right)+(1-{‌}^1\ell )⋏\left(y\right))\right),{\Phi }^{*}\left({⋏}^{-1}({‌}^1\ell ⋏\left(\varkappa \right)+(1-{‌}^1\ell )⋏\left(y\right))\right)\right]\hfill \\ \hfill & \supseteq \left[\hslash \left({‌}^1\ell \right){\Phi }_{*}\left(\varkappa \right)+\hslash (1-{‌}^1\ell ){\Phi }_{*}\left(y\right),\hslash \left({‌}^1\ell \right){\Phi }^{*}\left(\varkappa \right)+\hslash (1-{‌}^1\ell ){\Phi }^{*}\left(y\right)\right].\hfill \end{array}$$

(19)

From (19), we have:

$$\begin{array}{c}\hfill {\Phi }_{*}\left({⋏}^{-1}({‌}^1\ell ⋏\left(\varkappa \right)+(1-{‌}^1\ell )⋏\left(y\right))\right)\le {{‌}^1\ell }^{{\delta }_1}{\Phi }_{*}\left(\varkappa \right)+{(1-{‌}^1\ell )}^{{\delta }_1}{\Phi }_{*}\left(y\right)\end{array}$$

(20)

and

$$\begin{array}{c}\hfill {\Phi }^{*}\left({⋏}^{-1}({‌}^1\ell ⋏\left(\varkappa \right)+(1-{‌}^1\ell )⋏\left(y\right))\right)\ge \hslash \left({‌}^1\ell \right){\Phi }^{*}\left(\varkappa \right)+\hslash (1-{‌}^1\ell ){\Phi }^{*}\left(y\right).\end{array}$$

(21)

Equations (20) and (21) indicate that ${\Phi }_{*}\in SGX\left([{\mathfrak{c}}_3,{\mathfrak{c}}_4],\mathbb{R}\right)$ and ${\Phi }^{*}\in SGV\left([{\mathfrak{c}}_3,{\mathfrak{c}}_4],\mathbb{R}\right)$. Conversely, suppose that ${\Phi }_{*}\in SGX\left([{\mathfrak{c}}_3,{\mathfrak{c}}_4],\mathbb{R}\right)$ and ${\Phi }^{*}\in SGV\left([{\mathfrak{c}}_3,{\mathfrak{c}}_4],\mathbb{R}\right)$, then:

$$\begin{array}{c}\hfill {\Phi }_{*}\left({⋏}^{-1}({‌}^1\ell ⋏\left(\varkappa \right)+(1-{‌}^1\ell )⋏\left(y\right))\right)\le \hslash \left({‌}^1\ell \right){\Phi }_{*}\left(\varkappa \right)+\hslash (1-{‌}^1\ell ){\Phi }_{*}\left(y\right)\end{array}$$

(22)

and

$$\begin{array}{c}\hfill {\Phi }^{*}\left({⋏}^{-1}({‌}^1\ell ⋏\left(\varkappa \right)+(1-{‌}^1\ell )⋏\left(y\right))\right)\ge \hslash \left({‌}^1\ell \right){\Phi }^{*}\left(\varkappa \right)+\hslash (1-{‌}^1\ell ){\Phi }^{*}\left(y\right).\end{array}$$

(23)

This implies:

$$\begin{array}{cc}\hfill & \left[{\Phi }_{*}\left({⋏}^{-1}({‌}^1\ell ⋏\left(\varkappa \right)+(1-{‌}^1\ell )⋏\left(y\right))\right),{\Phi }^{*}\left({⋏}^{-1}({‌}^1\ell ⋏\left(\varkappa \right)+(1-{‌}^1\ell )⋏\left(y\right))\right)\right]\hfill \\ \hfill & \supseteq \left[\hslash \left({‌}^1\ell \right){\Phi }_{*}\left(\varkappa \right)+\hslash (1-{‌}^1\ell ){\Phi }_{*}\left(y\right),\hslash \left({‌}^1\ell \right){‌}^1\ell {\Phi }^{*}\left(\varkappa \right)+\hslash (1-{‌}^1\ell ){\Phi }^{*}\left(y\right)\right].\hfill \end{array}$$

Hence, the result is achieved. □

Now, we establish Jensen’s inequality in a general sense.

 Theorem 5. 

Let $\Phi \in SIGX\left([{\mathfrak{c}}_3,{\mathfrak{c}}_4],{\mathbb{R}}_I^{+}\right)$, then:

$$\begin{array}{c}\hfill \Phi \left({⋏}^{-1}\left({\displaystyle \frac{1}{{\mathcal{W}}_n}}\sum _{i=1}^n{{‌}^1\ell }_i⋏\left({\varkappa }_i\right)\right)\right)\supseteq \sum _{i=1}^n{\hslash }_1\left({\displaystyle \frac{{{‌}^1\ell }_i}{{\mathcal{W}}_n}}\right)\Phi \left({\varkappa }_i\right),\end{array}$$

(24)

for ${\varkappa }_i\in [{\mathfrak{c}}_3,{\mathfrak{c}}_4]$ and ${\mathcal{W}}_n={\sum }_{i=1}^n{{‌}^1\ell }_i{\varkappa }_i$.

 Proof. 

To acquire our result, we utilize the technique of mathematical induction. Suppose $n=2$ in (24), then:

$$\begin{array}{c}\hfill \Phi \left({⋏}^{-1}\left({\displaystyle \frac{{{{‌}^1\ell }_i}_1⋏\left({\varkappa }_1\right)+{{{‌}^1\ell }_i}_2⋏\left({\varkappa }_2\right)}{{{{‌}^1\ell }_i}_1+{{{‌}^1\ell }_i}_2}}\right)\right)\supseteq {\hslash }_1\left({\displaystyle \frac{{{{‌}^1\ell }_i}_1}{{{{‌}^1\ell }_i}_1+{{{‌}^1\ell }_i}_2}}\right)\Phi \left({\varkappa }_1\right)+{\hslash }_1\left({\displaystyle \frac{{{{‌}^1\ell }_i}_2}{{{{‌}^1\ell }_i}_1+{{{‌}^1\ell }_i}_2}}\right)\Phi \left({\varkappa }_2\right).\end{array}$$

Presume that the inclusion (24) is satisfied for $n=z-1$ such that:

$$\begin{array}{c}\hfill \Phi \left({⋏}^{-1}\left({\displaystyle \frac{1}{{\mathcal{W}}_{z-1}}}\sum _{i=1}^{z-1}{{‌}^1\ell }_i⋏\left({\varkappa }_i\right)\right)\right)\supseteq \sum _{i=1}^{z-1}{\hslash }_1\left({\displaystyle \frac{{{‌}^1\ell }_i}{{\mathcal{W}}_{z-1}}}\right)\Phi \left({\varkappa }_i\right).\end{array}$$

(25)

Now, we show that it is true for $n=z$.

$$\begin{array}{cc}\hfill & \Phi \left({⋏}^{-1}\left({\displaystyle \frac{1}{{\mathcal{W}}_z}}\sum _{i=1}^z{{‌}^1\ell }_i⋏\left({\varkappa }_i\right)\right)\right)\hfill \\ \hfill & =\Phi \left({⋏}^{-1}\left({\displaystyle \frac{1}{{\mathcal{W}}_z}}\sum _{i=1}^{z-1}{{‌}^1\ell }_i⋏\left({\varkappa }_i\right)+{\displaystyle \frac{{{‌}^1\ell }_i⋏\left({\varkappa }_z\right)}{{\mathcal{W}}_z}}\right)\right)=\Phi \left({⋏}^{-1}\left({\displaystyle \frac{{\mathcal{W}}_{z-1}}{{\mathcal{W}}_z}}\sum _{i=1}^{z-1}{\displaystyle \frac{{{‌}^1\ell }_i⋏\left({\varkappa }_i\right)}{{\mathcal{W}}_{z-1}}}+{\displaystyle \frac{{{‌}^1\ell }_i⋏\left({\varkappa }_k\right)}{{\mathcal{W}}_z}}\right)\right)\hfill \\ \hfill & \supseteq {\hslash }_1\left({\displaystyle \frac{{{‌}^1\ell }_i}{{\mathcal{W}}_z}}\right)\Phi \left({\varkappa }_z\right)+{\hslash }_1\left({\displaystyle \frac{{\mathcal{W}}_{z-1}}{{\mathcal{W}}_z}}\right)\Phi \left({⋏}^{-1}\left(\sum _{i=1}^{z-1}{\displaystyle \frac{{{‌}^1\ell }_i⋏\left({\varkappa }_i\right)}{{\mathcal{W}}_{z-1}}}\right)\right)\hfill \\ \hfill & \supseteq {\hslash }_1\left({\displaystyle \frac{{{‌}^1\ell }_i}{{\mathcal{W}}_z}}\right)\Phi \left({\varkappa }_z\right)+\sum _{i=1}^{z-1}{\hslash }_1\left({\displaystyle \frac{{\mathcal{W}}_{z-1}}{{\mathcal{W}}_z}}\right)·{\hslash }_1\left({\displaystyle \frac{{{‌}^1\ell }_i}{{\mathcal{W}}_{z-1}}}\right)\Phi \left({\varkappa }_i\right)\hfill \\ \hfill & \supseteq {\hslash }_1\left({\displaystyle \frac{{{‌}^1\ell }_i}{{\mathcal{W}}_z}}\right)\Phi \left({\varkappa }_z\right)+\sum _{i=1}^{z-1}{\hslash }_1\left({\displaystyle \frac{{\mathcal{W}}_{z-1}}{{\mathcal{W}}_z}}·{\displaystyle \frac{{{‌}^1\ell }_i}{{\mathcal{W}}_{z-1}}}\right)\Phi \left({\varkappa }_i\right)\hfill \\ \hfill & =\sum _{i=1}^z{\hslash }_1\left({\displaystyle \frac{{{‌}^1\ell }_i}{{\mathcal{W}}_z}}\right)\Phi \left({\varkappa }_i\right).\hfill \end{array}$$

Hence, the result is achieved. □

Now, we report some consequences of Theorem 5.

• Substituting $⋏\left(\varkappa \right)=\varkappa$. Then, we obtain Jensen’s inequality for h-convex functions, which is derived in [14].
• Substituting $⋏\left(\varkappa \right)={\displaystyle \frac{1}{\varkappa }}$. Then, we obtain Jensen’s inequality for harmonically h-convex functions:

  $$\begin{array}{c}\hfill \Phi \left(\sum _{i=1}^n{\displaystyle \frac{1}{\frac{{{‌}^1\ell }_i}{{\mathcal{W}}_n{\varkappa }_i}}}\right)\supseteq \sum _{i=1}^n{\hslash }_1\left({\displaystyle \frac{{{‌}^1\ell }_i}{{\mathcal{W}}_n}}\right)\Phi \left({\varkappa }_i\right).\end{array}$$
• Substituting $⋏\left(\varkappa \right)={\varkappa }^p$, $p>-1$. Then, we obtain Jensen’s inequality for p-convex functions:

  $$\begin{array}{c}\hfill \Phi \left({\left({\displaystyle \frac{1}{{\mathcal{W}}_n}}\sum _{i=1}^n{{‌}^1\ell }_i{\varkappa }_i^p\right)}^{{\displaystyle \frac{1}{p}}}\right)\supseteq \sum _{i=1}^n{\hslash }_1\left({\displaystyle \frac{{{‌}^1\ell }_i}{{\mathcal{W}}_n}}\right)\Phi \left({\varkappa }_i\right).\end{array}$$
• Substituting $⋏\left(\varkappa \right)=\ln \left(\varkappa \right)$. Then, we obtain Jensen’s inequality for $GA$-h convex functions:

  $$\begin{array}{c}\hfill \Phi \left(\prod _{i=1}^n{\varkappa }_i^{{\displaystyle \frac{{{‌}^1\ell }_i}{{\mathcal{W}}_n}}}\right)\supseteq \sum _{i=1}^n{\hslash }_1\left({\displaystyle \frac{{{‌}^1\ell }_i}{{\mathcal{W}}_n}}\right)\Phi \left({\varkappa }_i\right).\end{array}$$

Now, we establish the unified $\mathcal{I}.\mathcal{V}$$\mathcal{H}$-$\mathcal{H}$ inequality.

 Theorem 6. 

Let $\Phi \in SIGX\left([{\mathfrak{c}}_3,{\mathfrak{c}}_4],{\mathbb{R}}_I^{+}\right)$, then:

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{\hslash \left(\frac{1}{2}\right)}}\Phi \left({⋏}^{-1}\left({\displaystyle \frac{⋏\left({\mathfrak{c}}_3\right)+⋏\left({\mathfrak{c}}_4\right)}{2}}\right)\right)\hfill \\ \hfill & \supseteq {\displaystyle \frac{\Gamma ({\delta }_1+1)}{{(⋏\left({\mathfrak{c}}_4\right)-⋏\left({\mathfrak{c}}_3\right))}^{{\delta }_1}}}\left[J_{⋏{\left({\mathfrak{c}}_3\right)}^{+}}^{{\delta }_1}\Phi \circ {⋏}^{-1}\left(⋏\left({\mathfrak{c}}_4\right)\right)+J_{{⋏\left({\mathfrak{c}}_4\right)}^{-}}^{{\delta }_1}\Phi \circ {⋏}^{-1}(⋏\left({\mathfrak{c}}_3\right))\right]\hfill \\ \hfill & \supseteq {\delta }_1[\Phi \left({\mathfrak{c}}_3\right)+\Phi \left({\mathfrak{c}}_4\right)]{\int }_0^1{{‌}^1\ell }^{{\delta }_1-1}[\hslash \left({‌}^1\ell \right)+\hslash (1-{‌}^1\ell )]\mathrm{d}{‌}^1\ell .\hfill \end{array}$$

 Proof. 

Since $\Phi$ is $\mathcal{I}.\mathcal{V}$-$(⋏,\hslash )$ convex function, then:

$$\begin{array}{c}\hfill \Phi \left({⋏}^{-1}\left({\displaystyle \frac{⋏\left(\varkappa \right)+⋏\left(y\right)}{2}}\right)\right)\supseteq \hslash \left({\displaystyle \frac{1}{2}}\right)[\Phi \left(\varkappa \right)+\Phi \left(y\right)].\end{array}$$

Substituting $\varkappa ={⋏}^{-1}({‌}^1\ell ⋏\left({\mathfrak{c}}_3\right)+(1-{‌}^1\ell )⋏\left({\mathfrak{c}}_4\right))$ and $y={⋏}^{-1}((1-{‌}^1\ell )⋏\left({\mathfrak{c}}_3\right)+{‌}^1\ell ⋏\left({\mathfrak{c}}_4\right))$ in the above inclusion and then taking the product on both sides by ${{‌}^1\ell }^{{\delta }_1-1}$ and then applying the integration with respect to ${‌}^1\ell$ on $[0,1]$:

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{\hslash \left(\frac{1}{2}\right)}}{\int }_0^1{{‌}^1\ell }^{{\delta }_1-1}\Phi \left({⋏}^{-1}\left({\displaystyle \frac{⋏\left({\mathfrak{c}}_3\right)+⋏\left({\mathfrak{c}}_4\right)}{2}}\right)\right)\mathrm{d}{‌}^1\ell \hfill \\ \hfill & \supseteq {\int }_0^1{{‌}^1\ell }^{{\delta }_1-1}\Phi \left({⋏}^{-1}({‌}^1\ell ⋏\left({\mathfrak{c}}_3\right)+(1-{‌}^1\ell )⋏\left({\mathfrak{c}}_4\right))\right)\mathrm{d}{‌}^1\ell \hfill \\ & +{\int }_0^1{{‌}^1\ell }^{{\delta }_1-1}\Phi \left({⋏}^{-1}((1-{‌}^1\ell )⋏\left({\mathfrak{c}}_3\right)+{‌}^1\ell ⋏\left({\mathfrak{c}}_4\right))\right)\mathrm{d}{‌}^1\ell .\hfill \end{array}$$

(26)

Now:

$$\begin{array}{cc}\hfill & \left(IR\right){\int }_0^1{{‌}^1\ell }^{{\delta }_1-1}\Phi \left({⋏}^{-1}\left({\displaystyle \frac{⋏\left({\mathfrak{c}}_3\right)+⋏\left({\mathfrak{c}}_4\right)}{2}}\right)\right)\mathrm{d}{‌}^1\ell \hfill \\ \hfill & =\left[\left(R\right){\int }_0^1{{‌}^1\ell }^{{\delta }_1-1}{\Phi }_{*}\left({⋏}^{-1}\left({\displaystyle \frac{⋏\left({\mathfrak{c}}_3\right)+⋏\left({\mathfrak{c}}_4\right)}{2}}\right)\right)\mathrm{d}{‌}^1\ell ,\left(R\right){\int }_0^1{{‌}^1\ell }^{{\delta }_1-1}{\Phi }^{*}\left({⋏}^{-1}\left({\displaystyle \frac{⋏\left({\mathfrak{c}}_3\right)+⋏\left({\mathfrak{c}}_4\right)}{2}}\right)\right)\mathrm{d}{‌}^1\ell \right]\hfill \\ \hfill & =\left[{\displaystyle \frac{1}{{\delta }_1}}{\Phi }_{*}\left({⋏}^{-1}\left({\displaystyle \frac{⋏\left({\mathfrak{c}}_3\right)+⋏\left({\mathfrak{c}}_4\right)}{2}}\right)\right)+{\displaystyle \frac{1}{{\delta }_1}}{\Phi }^{*}\left({⋏}^{-1}\left({\displaystyle \frac{⋏\left({\mathfrak{c}}_3\right)+⋏\left({\mathfrak{c}}_4\right)}{2}}\right)\right)\right]\hfill \\ \hfill & ={\displaystyle \frac{1}{{\delta }_1}}\Phi \left({⋏}^{-1}\left({\displaystyle \frac{2⋏\left({\mathfrak{c}}_3\right),⋏\left({\mathfrak{c}}_4\right)-⋏\left({\mathfrak{c}}_3\right)}{2}}\right)\right).\hfill \end{array}$$

(27)

Also:

$$\begin{array}{cc}\hfill & \left(IR\right){\int }_0^1{{‌}^1\ell }^{{\delta }_1-1}\Phi \left({⋏}^{-1}({‌}^1\ell ⋏\left({\mathfrak{c}}_3\right)+(1-{‌}^1\ell )⋏\left({\mathfrak{c}}_4\right))\right)\mathrm{d}{‌}^1\ell \hfill \\ & +\left(IR\right){\int }_0^1{{‌}^1\ell }^{{\delta }_1-1}\Phi \left({⋏}^{-1}((1-{‌}^1\ell )⋏\left({\mathfrak{c}}_3\right)+{‌}^1\ell ⋏\left({\mathfrak{c}}_4\right))\right)\mathrm{d}{‌}^1\ell \hfill \\ \hfill & =\left[\left(R\right){\int }_0^1{{‌}^1\ell }^{{\delta }_1-1}{\Phi }_{*}\left({⋏}^{-1}({‌}^1\ell ⋏\left({\mathfrak{c}}_3\right)+(1-{‌}^1\ell )⋏\left({\mathfrak{c}}_4\right))\right)\mathrm{d}{‌}^1\ell \right.\hfill \\ & +\left(R\right){\int }_0^1{{‌}^1\ell }^{{\delta }_1-1}{\Phi }_{*}\left({⋏}^{-1}((1-{‌}^1\ell )⋏\left({\mathfrak{c}}_3\right)+{‌}^1\ell ⋏\left({\mathfrak{c}}_4\right))\right)\mathrm{d}{‌}^1\ell ,\hfill \\ \hfill & \left(R\right){\int }_0^1{{‌}^1\ell }^{{\delta }_1-1}{\Phi }^{*}\left({⋏}^{-1}({‌}^1\ell ⋏\left({\mathfrak{c}}_3\right)+(1-{‌}^1\ell )⋏\left({\mathfrak{c}}_4\right))\right)\mathrm{d}{‌}^1\ell \hfill \\ & \left.+\left(R\right){\int }_0^1{{‌}^1\ell }^{{\delta }_1-1}{\Phi }^{*}\left({⋏}^{-1}((1-{‌}^1\ell )⋏\left({\mathfrak{c}}_3\right)+{‌}^1\ell ⋏\left({\mathfrak{c}}_4\right))\right)\mathrm{d}{‌}^1\ell \right]\hfill \\ \hfill & =\left[{\displaystyle \frac{1}{{(⋏\left({\mathfrak{c}}_4\right)-⋏\left({\mathfrak{c}}_3\right))}^{{\delta }_1}}}{\int }_{⋏\left({\mathfrak{c}}_3\right)}^{⋏\left({\mathfrak{c}}_4\right)}{(⋏\left({\mathfrak{c}}_4\right)-u)}^{{\delta }_1-1}{\Phi }_{*}\circ {⋏}^{-1}\left(u\right)\mathrm{d}u\right.\hfill \\ & +{\displaystyle \frac{1}{{(⋏\left({\mathfrak{c}}_4\right)-⋏\left({\mathfrak{c}}_3\right))}^{{\delta }_1}}}{\int }_{⋏\left({\mathfrak{c}}_3\right)}^{⋏\left({\mathfrak{c}}_4\right)}{(u-⋏\left({\mathfrak{c}}_3\right))}^{{\delta }_1-1}{\Phi }_{*}\circ {⋏}^{-1}\left(u\right)\mathrm{d}u,\hfill \\ \hfill & {\displaystyle \frac{1}{{(⋏\left({\mathfrak{c}}_4\right)-⋏\left({\mathfrak{c}}_3\right))}^{{\delta }_1}}}{\int }_{⋏\left({\mathfrak{c}}_3\right)}^{⋏\left({\mathfrak{c}}_4\right)}{(⋏\left({\mathfrak{c}}_4\right)-u)}^{{\delta }_1-1}{\Phi }^{*}\circ {⋏}^{-1}\left(u\right)\mathrm{d}u\hfill \\ & \left.+{\displaystyle \frac{1}{{(⋏\left({\mathfrak{c}}_4\right)-⋏\left({\mathfrak{c}}_3\right))}^{{\delta }_1}}}{\int }_{⋏\left({\mathfrak{c}}_3\right)}^{⋏\left({\mathfrak{c}}_4\right)}{(u-⋏\left({\mathfrak{c}}_3\right))}^{{\delta }_1-1}{\Phi }^{*}\circ {⋏}^{-1}\left(u\right)\mathrm{d}u\right].\hfill \end{array}$$

This implies:

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{\hslash \left(\frac{1}{2}\right){\delta }_1}}\Phi \left({⋏}^{-1}\left({\displaystyle \frac{⋏\left({\mathfrak{c}}_3\right)+⋏\left({\mathfrak{c}}_4\right)}{2}}\right)\right)\hfill \\ \hfill & \supseteq {\displaystyle \frac{1}{{(⋏\left({\mathfrak{c}}_4\right)-⋏\left({\mathfrak{c}}_3\right))}^{{\delta }_1}}}\left[{\int }_{⋏\left({\mathfrak{c}}_3\right)}^{⋏\left({\mathfrak{c}}_4\right)}{(⋏\left({\mathfrak{c}}_4\right)-u)}^{{\delta }_1-1}\Phi \circ {⋏}^{-1}\left(u\right)\mathrm{d}u\right.\hfill \\ \hfill & \left.+{\int }_{⋏\left({\mathfrak{c}}_3\right)}^{⋏\left({\mathfrak{c}}_4\right)}{(u-⋏\left({\mathfrak{c}}_3\right))}^{{\delta }_1-1}\Phi \circ {⋏}^{-1}\left(u\right)\mathrm{d}u\right].\hfill \end{array}$$

Combining (26) and (27), we obtain:

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{\hslash \left(\frac{1}{2}\right)}}\Phi \left({⋏}^{-1}\left({\displaystyle \frac{⋏\left({\mathfrak{c}}_3\right)+⋏\left({\mathfrak{c}}_4\right)}{2}}\right)\right)\hfill \\ \hfill & \supseteq {\displaystyle \frac{\Gamma ({\delta }_1+1)}{{(⋏\left({\mathfrak{c}}_4\right)-⋏\left({\mathfrak{c}}_3\right))}^{{\delta }_1}}}\left[J_{⋏{\left({\mathfrak{c}}_3\right)}^{+}}^{{\delta }_1}\Phi \circ {⋏}^{-1}\left(⋏\left({\mathfrak{c}}_4\right)\right)+J_{{⋏\left({\mathfrak{c}}_4\right)}^{-}}^{{\delta }_1}\Phi \circ {⋏}^{-1}(⋏\left({\mathfrak{c}}_3\right))\right].\hfill \end{array}$$

Again, employing $\mathcal{I}.\mathcal{V}$-$(⋏,\hslash )$ convexity of $\Phi$, we have

$$\begin{array}{cc}\hfill & \Phi \left({⋏}^{-1}({‌}^1\ell ⋏\left({\mathfrak{c}}_3\right)+(1-{‌}^1\ell )⋏\left({\mathfrak{c}}_4\right))\right)\supseteq \hslash \left({‌}^1\ell \right)\Phi \left({\mathfrak{c}}_3\right)+\hslash (1-{‌}^1\ell )\Phi \left({\mathfrak{c}}_4\right)\hfill \end{array}$$

(28)

and

$$\begin{array}{cc}\hfill & \Phi \left({⋏}^{-1}((1-{‌}^1\ell )⋏\left({\mathfrak{c}}_3\right)+{‌}^1\ell ⋏\left({\mathfrak{c}}_4\right))\right)\supseteq \hslash (1-{‌}^1\ell )\Phi \left({\mathfrak{c}}_3\right)+\hslash \left({‌}^1\ell \right)\Phi \left({\mathfrak{c}}_4\right).\hfill \end{array}$$

(29)

Adding (28) and (29), taking the product on both sides by ${{‌}^1\ell }^{{\delta }_1-1}$, and applying the integration with respect to ${‌}^1\ell$ on $[0,1]$, we acquired our required relation. □

Now, we provide novel results concerning with Theorem 6.

• Choosing $\hslash \left({‌}^1\ell \right)={‌}^1\ell$ in Theorem 6, we obtain:

  $$\begin{array}{cc}\hfill & \Phi \left({⋏}^{-1}\left({\displaystyle \frac{⋏\left({\mathfrak{c}}_3\right)+⋏\left({\mathfrak{c}}_4\right)}{2}}\right)\right)\hfill \\ \hfill & \supseteq {\displaystyle \frac{\Gamma ({\delta }_1+1)}{{(⋏\left({\mathfrak{c}}_4\right)-⋏\left({\mathfrak{c}}_3\right))}^{{\delta }_1}}}\left[J_{⋏{\left({\mathfrak{c}}_3\right)}^{+}}^{{\delta }_1}\Phi \circ {⋏}^{-1}\left(⋏\left({\mathfrak{c}}_4\right)\right)+J_{{⋏\left({\mathfrak{c}}_4\right)}^{-}}^{{\delta }_1}\Phi \circ (⋏\left({\mathfrak{c}}_3\right))\right]\hfill \\ \hfill & \supseteq {\displaystyle \frac{\Phi \left({\mathfrak{c}}_3\right)+\Phi \left({\mathfrak{c}}_4\right)}{2}}\hfill \end{array}$$
• Choosing $⋏\left(\varkappa \right)=\varkappa$$\hslash \left({‌}^1\ell \right)={‌}^1\ell$ in Theorem 6, we attain fractional $\mathcal{H}$-$\mathcal{H}$ inequality for $\mathcal{I}.\mathcal{V}$ convex functions, which is proved in [8]:

  $$\begin{array}{cc}\hfill \Phi \left({\displaystyle \frac{{\mathfrak{c}}_3+{\mathfrak{c}}_4}{2}}\right)& \supseteq {\displaystyle \frac{\Gamma ({\delta }_1+1)}{2{({\mathfrak{c}}_4-{\mathfrak{c}}_3)}^{{\delta }_1}}}\left[J_{{{\mathfrak{c}}_3}^{+}}^{{\delta }_1}\Phi \left({\mathfrak{c}}_4\right)+{J_{{{\mathfrak{c}}_4}^{-}}}^{{\delta }_1}\Phi \left({\mathfrak{c}}_3\right)\right]\supseteq {\displaystyle \frac{\Phi \left({\mathfrak{c}}_3\right)+\Phi \left({\mathfrak{c}}_4\right)}{2}}.\hfill \end{array}$$
• Choosing $⋏\left({‌}^1\ell \right)={{‌}^1\ell }^p$, $p\ge -1$ in Theorem 6, then:

  $$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{\hslash \left(\frac{1}{2}\right)}}\Phi \left({\left({\displaystyle \frac{{{\mathfrak{c}}_3}^p+{{\mathfrak{c}}_4}^p}{2}}\right)}^{{\displaystyle \frac{1}{p}}}\right)\hfill \\ \hfill & \supseteq {\displaystyle \frac{\Gamma ({\delta }_1+1)}{{({{\mathfrak{c}}_4}^p-{{\mathfrak{c}}_3}^p)}^{{\delta }_1}}}\left[J_{{{{\mathfrak{c}}_3}^p}^{+}}^{{\delta }_1}\Phi \circ k\left({{\mathfrak{c}}_4}^p\right)+J_{{{{\mathfrak{c}}_4}^p}^{-}}^{{\delta }_1}\Phi \circ k\left({{\mathfrak{c}}_3}^p\right)\right]\hfill \\ \hfill & \supseteq {\delta }_1[\Phi \left({\mathfrak{c}}_3\right)+\Phi \left({\mathfrak{c}}_4\right)]{\int }_0^1{{‌}^1\ell }^{{\delta }_1-1}[\hslash \left({‌}^1\ell \right)+\hslash (1-{‌}^1\ell )]\mathrm{d}{‌}^1\ell .\hfill \end{array}$$
• Choosing $⋏\left({‌}^1\ell \right)={{‌}^1\ell }^p$, $p\ge -1$ and $\hslash \left({‌}^1\ell \right)={‌}^1\ell$ in Theorem 6, we obtain:

  $$\begin{array}{cc}\hfill & \Phi \left({\left({\displaystyle \frac{{{\mathfrak{c}}_3}^p+{{\mathfrak{c}}_4}^p}{2}}\right)}^{{\displaystyle \frac{1}{p}}}\right)\hfill \\ \hfill & \supseteq {\displaystyle \frac{\Gamma ({\delta }_1+1)}{{({{\mathfrak{c}}_4}^p-{{\mathfrak{c}}_3}^p)}^{{\delta }_1}}}\left[J_{{{{\mathfrak{c}}_3}^p}^{+}}^{{\delta }_1}\Phi \circ k\left({{\mathfrak{c}}_4}^p\right)+J_{{{{\mathfrak{c}}_4}^p}^{-}}^{{\delta }_1}\Phi \circ k\left({{\mathfrak{c}}_3}^p\right)\right]\hfill \\ \hfill & \supseteq {\displaystyle \frac{[\Phi \left({\mathfrak{c}}_3\right)+\Phi \left({\mathfrak{c}}_4\right)]}{2}},\hfill \end{array}$$

  where $k\left(\varkappa \right)={\varkappa }^{{\displaystyle \frac{1}{p}}}$.

 Example 1. 

Presume each of the premises of Theorem 6 are met, we consider $\Phi \left({‌}^1\ell \right)=[4-\sqrt{\left({‌}^1\ell \right)},$$8+\sqrt{\left({‌}^1\ell \right)}]$ with $[{\mathfrak{c}}_3,{\mathfrak{c}}_4]=[0,2],⋏\left({‌}^1\ell \right)={‌}^1\ell$ and within ${\delta }_1=2$, then:

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{\Gamma ({\delta }_1+1)}}\Phi \left(1\right)=[1.5,4.5],\hfill \\ \hfill & {\displaystyle \frac{1}{2^{{\delta }_1+1}\Gamma \left({\delta }_1\right)}}\left[{\int }_0^2\left({(2-{‌}^1\ell )}^{{\delta }_1-1}+{{‌}^1\ell }^{{\delta }_1-1}\right)\Phi \left({‌}^1\ell \right)\mathrm{d}{‌}^1\ell \right]=[1.5286,4.4714],\hfill \\ \hfill & {\displaystyle \frac{(\Phi \left({\mathfrak{c}}_3\right)+\Phi \left({\mathfrak{c}}_4\right))}{2\Gamma ({\delta }_1+1)}}=[1.64645,4.3535].\hfill \end{array}$$

This example confirms the accuracy of the $\mathcal{H}$-$\mathcal{H}$-Fejer inequality for the product of $\mathcal{I}.\mathcal{V}$-$(⋏,\hslash )$ convex functions and symmetric functions. By considering these aforementioned computations, one can easily analyze bounds for $\mathcal{I}.\mathcal{V}$ R-L operator with different symmetric functions as a kernel.

For the pictorial demonstration, we vary $0\le {\delta }_1\le 3$.

Figure 1 reflects the comparison between the left, middle and right sides of Theorem 1. Furthermore red, blue and green color shows lower and upper functions of left, middle and right sides respectively.

Now, we present a unified fractional version of Fejer-Type $\mathcal{H}$-$\mathcal{H}$’s inequality.

 Theorem 7. 

Let $\Phi \in SIGX\left([{\mathfrak{c}}_3,{\mathfrak{c}}_4],{\mathbb{R}}_I^{+}\right)$, and $\Xi :[{\mathfrak{c}}_3,{\mathfrak{c}}_4]\to \mathbb{R}$ be a symmetric function with respect ${\displaystyle \frac{{\mathfrak{c}}_3+{\mathfrak{c}}_4}{2}}$, then:

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{2\hslash \left(\frac{1}{2}\right)}}\Phi \left({⋏}^{-1}\left({\displaystyle \frac{⋏\left({\mathfrak{c}}_3\right)+⋏\left({\mathfrak{c}}_4\right)}{2}}\right)\right)\left[J_{⋏{\left({\mathfrak{c}}_3\right)}^{+}}^{{\delta }_1}\Xi \circ {⋏}^{-1}(⋏\left({\mathfrak{c}}_4\right))+J_{⋏{(⋏\left({\mathfrak{c}}_4\right))}^{-}}^{{\delta }_1}\Xi \circ {⋏}^{-1}(⋏\left({\mathfrak{c}}_3\right))\right]\hfill \\ \hfill & \supseteq \left[J_{⋏{\left({\mathfrak{c}}_3\right)}^{+}}^{{\delta }_1}\Phi \Xi \circ {⋏}^{-1}(⋏\left({\mathfrak{c}}_4\right))+J_{{⋏\left({\mathfrak{c}}_4\right)}^{-}}^{{\delta }_1}\Phi \Xi \circ {⋏}^{-1}(⋏\left({\mathfrak{c}}_3\right))\right]\hfill \\ \hfill & \supseteq {\displaystyle \frac{{(⋏\left({\mathfrak{c}}_4\right)-⋏\left({\mathfrak{c}}_3\right))}^{{\delta }_1}}{\Gamma \left({\delta }_1\right)}}\left({\int }_0^1{{‌}^1\ell }^{{\delta }_1-1}[\hslash \left({‌}^1\ell \right)+\hslash (1-{‌}^1\ell )]\Xi \left({⋏}^{-1}({‌}^1\ell ⋏\left({\mathfrak{c}}_3\right)+(1-{‌}^1\ell )⋏\left({\mathfrak{c}}_4\right))\right)[\Phi \left({\mathfrak{c}}_3\right)+\Phi \left({\mathfrak{c}}_4\right)]\mathrm{d}{‌}^1\ell \right)\hfill \end{array}$$

${\delta }_1>0$ and $\forall \varkappa ,y\in [{\mathfrak{c}}_3,{\mathfrak{c}}_4]$.

 Proof. 

Since $\Phi$ is $\mathcal{I}.\mathcal{V}$-$(⋏,\hslash )$ convex function, for ${‌}^1\ell ={\displaystyle \frac{1}{2}}$, then we have:

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{\hslash \left(\frac{1}{2}\right)}}\Phi \left({⋏}^{-1}\left({\displaystyle \frac{⋏\left(\varkappa \right)+⋏\left(y\right)}{2}}\right)\right)\supseteq \Phi \left(\varkappa \right)+\Phi \left(y\right).\hfill \end{array}$$

Substituting $\varkappa ={⋏}^{-1}({‌}^1\ell ⋏\left({\mathfrak{c}}_3\right)+(1-{‌}^1\ell )⋏\left({\mathfrak{c}}_4\right))$ and $y={⋏}^{-1}((1-{‌}^1\ell )⋏\left({\mathfrak{c}}_3\right)+{‌}^1\ell ⋏\left({\mathfrak{c}}_4\right))$ in the above relation, then multiplying both sides by ${{‌}^1\ell }^{{\delta }_1-1}\Xi \left({⋏}^{-1}({‌}^1\ell ⋏\left({\mathfrak{c}}_3\right)+(1-{‌}^1\ell )⋏\left({\mathfrak{c}}_4\right))\right)$:

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{\hslash \left(\frac{1}{2}\right)}}{\int }_0^1{{‌}^1\ell }^{{\delta }_1-1}\Xi ({⋏}^{-1}({‌}^1\ell ⋏\left({\mathfrak{c}}_3\right)+(1-{‌}^1\ell )⋏\left({\mathfrak{c}}_4\right))\Phi \left({⋏}^{-1}\left({\displaystyle \frac{⋏\left({\mathfrak{c}}_3\right)+⋏\left({\mathfrak{c}}_4\right)}{2}}\right)\right)\mathrm{d}{‌}^1\ell \hfill \\ \hfill & \supseteq {\int }_0^1{{‌}^1\ell }^{{\delta }_1-1}\Phi \left({⋏}^{-1}({‌}^1\ell ⋏\left({\mathfrak{c}}_3\right)+(1-{‌}^1\ell )⋏\left({\mathfrak{c}}_4\right))\right)\Xi ({⋏}^{-1}({‌}^1\ell ⋏\left({\mathfrak{c}}_3\right)+(1-{‌}^1\ell )⋏\left({\mathfrak{c}}_4\right))\mathrm{d}{‌}^1\ell \hfill \\ \hfill & +{\int }_0^1{{‌}^1\ell }^{{\delta }_1-1}\Phi \left({⋏}^{-1}((1-{‌}^1\ell )⋏\left({\mathfrak{c}}_3\right)+{‌}^1\ell ⋏\left({\mathfrak{c}}_4\right))\right)\Xi ({⋏}^{-1}({‌}^1\ell ⋏\left({\mathfrak{c}}_3\right)+(1-{‌}^1\ell )⋏\left({\mathfrak{c}}_4\right))\mathrm{d}{‌}^1\ell .\hfill \end{array}$$

(30)

By taking the benefit of the fact $\Xi \left({⋏}^{-1}({‌}^1\ell ⋏\left({\mathfrak{c}}_3\right)+(1-{‌}^1\ell )⋏\left({\mathfrak{c}}_4\right))\right)=\Xi \left({⋏}^{-1}((1-{‌}^1\ell )⋏\left({\mathfrak{c}}_3\right)+{‌}^1\ell ⋏\left({\mathfrak{c}}_4\right))\right)$:

$$\begin{array}{cc}\hfill & {\int }_0^1{{‌}^1\ell }^{{\delta }_1-1}\Xi \left({⋏}^{-1}({‌}^1\ell ⋏\left({\mathfrak{c}}_3\right)+(1-{‌}^1\ell )⋏\left({\mathfrak{c}}_4\right))\right)\Phi \left({⋏}^{-1}\left({\displaystyle \frac{⋏\left({\mathfrak{c}}_3\right)+⋏\left({\mathfrak{c}}_4\right)}{2}}\right)\right)\mathrm{d}{‌}^1\ell \hfill \\ \hfill & ={\displaystyle \frac{1}{2}}\left[{\int }_0^1{{‌}^1\ell }^{{\delta }_1-1}\Xi \left({⋏}^{-1}({‌}^1\ell ⋏\left({\mathfrak{c}}_3\right)+(1-{‌}^1\ell )⋏\left({\mathfrak{c}}_4\right))\right)\Phi \left({⋏}^{-1}\left({\displaystyle \frac{⋏\left({\mathfrak{c}}_3\right)+⋏\left({\mathfrak{c}}_4\right)}{2}}\right)\right)\mathrm{d}{‌}^1\ell \right.\hfill \\ \hfill & \left.+{\int }_0^1{{‌}^1\ell }^{{\delta }_1-1}\Xi \left({⋏}^{-1}((1-{‌}^1\ell )⋏\left({\mathfrak{c}}_3\right)+{‌}^1\ell ⋏\left({\mathfrak{c}}_4\right))\right)\Phi \left({⋏}^{-1}\left({\displaystyle \frac{⋏\left({\mathfrak{c}}_3\right)+⋏\left({\mathfrak{c}}_4\right)}{2}}\right)\right)\mathrm{d}{‌}^1\ell \right]\hfill \\ \hfill & ={\displaystyle \frac{1}{2{(⋏\left({\mathfrak{c}}_4\right)-⋏\left({\mathfrak{c}}_3\right))}^{{\delta }_1}}}\left[{\int }_{⋏\left({\mathfrak{c}}_3\right)}^{⋏\left({\mathfrak{c}}_4\right)}{(⋏\left({\mathfrak{c}}_4\right)-u)}^{{\delta }_1-1}\Phi \left({⋏}^{-1}\left({\displaystyle \frac{⋏\left({\mathfrak{c}}_3\right)+⋏\left({\mathfrak{c}}_4\right)}{2}}\right)\right)\Xi \circ {⋏}^{-1}\left(u\right)\mathrm{d}u\right.\hfill \\ & \left.+{\int }_{⋏\left({\mathfrak{c}}_3\right)}^{⋏\left({\mathfrak{c}}_4\right)}{(u-⋏\left({\mathfrak{c}}_3\right))}^{{\delta }_1-1}\Phi \left({⋏}^{-1}\left({\displaystyle \frac{⋏\left({\mathfrak{c}}_3\right)+⋏\left({\mathfrak{c}}_4\right)}{2}}\right)\right)\Xi \circ {⋏}^{-1}\left(u\right)\mathrm{d}u\right]\hfill \\ \hfill & ={\displaystyle \frac{\Gamma \left({\delta }_1\right)}{2{(⋏\left({\mathfrak{c}}_4\right)-⋏\left({\mathfrak{c}}_3\right))}^{{\delta }_1}}}\Phi \left({⋏}^{-1}\left({\displaystyle \frac{⋏\left({\mathfrak{c}}_3\right)+⋏\left({\mathfrak{c}}_4\right)}{2}}\right)\right)\left[J_{⋏{\left({\mathfrak{c}}_3\right)}^{+}}^{{\delta }_1}\Xi \circ {⋏}^{-1}(⋏\left({\mathfrak{c}}_4\right))\right.\hfill \\ \hfill & \left.+J_{{⋏\left({\mathfrak{c}}_4\right)}^{-}}^{{\delta }_1}\Xi \circ {⋏}^{-1}(⋏\left({\mathfrak{c}}_3\right))\right].\hfill \end{array}$$

(31)

Now:

$$\begin{array}{cc}\hfill & {\int }_0^1{{‌}^1\ell }^{{\delta }_1-1}\Phi \left({⋏}^{-1}({‌}^1\ell ⋏\left({\mathfrak{c}}_3\right)+(1-{‌}^1\ell )⋏\left({\mathfrak{c}}_4\right))\right)\Xi ({⋏}^{-1}({‌}^1\ell ⋏\left({\mathfrak{c}}_3\right)+(1-{‌}^1\ell )⋏\left({\mathfrak{c}}_4\right))\mathrm{d}{‌}^1\ell \hfill \\ \hfill & +{\int }_0^1{{‌}^1\ell }^{{\delta }_1-1}\Phi \left({⋏}^{-1}((1-{‌}^1\ell )⋏\left({\mathfrak{c}}_3\right)+{‌}^1\ell ⋏\left({\mathfrak{c}}_4\right))\right)\Xi ({⋏}^{-1}({‌}^1\ell ⋏\left({\mathfrak{c}}_3\right)+(1-{‌}^1\ell )⋏\left({\mathfrak{c}}_4\right))\mathrm{d}{‌}^1\ell \hfill \\ \hfill & ={\displaystyle \frac{1}{{(⋏\left({\mathfrak{c}}_4\right)-⋏\left({\mathfrak{c}}_3\right))}^{{\delta }_1}}}\left[{\int }_{⋏\left({\mathfrak{c}}_3\right)}^{⋏\left({\mathfrak{c}}_4\right)}{(⋏\left({\mathfrak{c}}_4\right)-u)}^{{\delta }_1-1}\Phi \left({⋏}^{-1}\left(u\right)\right)\Xi \left({⋏}^{-1}\left(u\right)\right)\mathrm{d}u\right.\hfill \\ \hfill & \left.+{\int }_{⋏\left({\mathfrak{c}}_3\right)}^{⋏\left({\mathfrak{c}}_4\right)}{(u-⋏\left({\mathfrak{c}}_3\right))}^{{\delta }_1-1}\Phi \left({⋏}^{-1}\left(u\right)\right)\Xi \left({⋏}^{-1}\left(u\right)\right)\mathrm{d}u\right]\hfill \\ \hfill & ={\displaystyle \frac{\Gamma \left({\delta }_1\right)}{{(⋏\left({\mathfrak{c}}_4\right)-⋏\left({\mathfrak{c}}_3\right))}^{{\delta }_1}}}\left[J_{⋏{\left({\mathfrak{c}}_3\right)}^{+}}^{{\delta }_1}\Phi \Xi \circ {⋏}^{-1}(⋏\left({\mathfrak{c}}_4\right))+J_{{⋏\left({\mathfrak{c}}_4\right)}^{-}}^{{\delta }_1}\Phi \Xi \circ {⋏}^{-1}(⋏\left({\mathfrak{c}}_3\right))\right].\hfill \end{array}$$

(32)

In this way, we attain our desired inclusion.

To achieve our second inclusion, we take the sum of (28) and (29), then taking the product on both sides by ${{‌}^1\ell }^{{\delta }_1-1}\Xi ({⋏}^{-1}({‌}^1\ell ⋏\left({\mathfrak{c}}_3\right)+(1-{‌}^1\ell )⋏\left({\mathfrak{c}}_4\right))$ and integration with respect to ${‌}^1\ell$ on $[0,1]$, then:

$$\begin{array}{cc}\hfill & {\int }_0^1{{‌}^1\ell }^{{\delta }_1-1}\Phi \left({⋏}^{-1}({‌}^1\ell ⋏\left({\mathfrak{c}}_3\right)+(1-{‌}^1\ell )⋏\left({\mathfrak{c}}_4\right))\right)\Xi \left({⋏}^{-1}({‌}^1\ell ⋏\left({\mathfrak{c}}_3\right)+(1-{‌}^1\ell )⋏\left({\mathfrak{c}}_4\right))\right)\mathrm{d}{‌}^1\ell \hfill \\ \hfill & +{\int }_0^1{{‌}^1\ell }^{{\delta }_1-1}\Phi \left({⋏}^{-1}((1-{‌}^1\ell )⋏\left({\mathfrak{c}}_3\right)+{‌}^1\ell ⋏\left({\mathfrak{c}}_4\right))\right)\Xi ({⋏}^{-1}({‌}^1\ell ⋏\left({\mathfrak{c}}_3\right)+(1-{‌}^1\ell )⋏\left({\mathfrak{c}}_4\right))\mathrm{d}{‌}^1\ell \hfill \\ \hfill & \supseteq {\int }_0^1{{‌}^1\ell }^{{\delta }_1-1}[\hslash \left({‌}^1\ell \right)+\hslash (1-{‌}^1\ell )]\Xi ({⋏}^{-1}({‌}^1\ell ⋏\left({\mathfrak{c}}_3\right)+(1-{‌}^1\ell )⋏\left({\mathfrak{c}}_4\right))[\Phi \left({\mathfrak{c}}_3\right)+\Phi \left({\mathfrak{c}}_4\right)]\mathrm{d}{‌}^1\ell .\hfill \end{array}$$

Some simple calculations result in the desired inclusion. □

We extract important results concerning Theorem 7.

• Choosing $\hslash \left({‌}^1\ell \right)={‌}^1\ell$ in Theorem 7, we have:

  $$\begin{array}{cc}\hfill & \Phi \left({⋏}^{-1}\left({\displaystyle \frac{⋏\left({\mathfrak{c}}_3\right)+⋏\left({\mathfrak{c}}_4\right)}{2}}\right)\right)\left[J_{⋏{\left({\mathfrak{c}}_3\right)}^{+}}^{{\delta }_1}\Xi \circ {⋏}^{-1}(⋏\left({\mathfrak{c}}_4\right))+J_{⋏{(⋏\left({\mathfrak{c}}_4\right))}^{-}}^{{\delta }_1}\Xi \circ {⋏}^{-1}(⋏\left({\mathfrak{c}}_3\right))\right]\hfill \\ \hfill & \supseteq \left[J_{⋏{\left({\mathfrak{c}}_3\right)}^{+}}^{{\delta }_1}\Phi \Xi \circ {⋏}^{-1}(⋏\left({\mathfrak{c}}_4\right))+J_{{⋏\left({\mathfrak{c}}_4\right)}^{-}}^{{\delta }_1}\Phi \Xi \circ {⋏}^{-1}(⋏\left({\mathfrak{c}}_3\right))\right.\hfill \\ & \left.-{\displaystyle \frac{1-{\delta }_1}{{\mathfrak{c}}_4\left({\delta }_1\right)}}(\Phi \Xi \circ {⋏}^{-1}(⋏\left({\mathfrak{c}}_3\right))+\Phi \Xi \circ {⋏}^{-1}(⋏\left({\mathfrak{c}}_4\right)))\right]\hfill \\ \hfill & \supseteq {\displaystyle \frac{\Phi \left({\mathfrak{c}}_3\right)+\Phi \left({\mathfrak{c}}_4\right)}{2}}\left[J_{⋏{\left({\mathfrak{c}}_3\right)}^{+}}^{{\delta }_1}\Xi \circ {⋏}^{-1}(⋏\left({\mathfrak{c}}_4\right))+J_{⋏{(⋏\left({\mathfrak{c}}_4\right))}^{-}}^{{\delta }_1}\Xi \circ {⋏}^{-1}(⋏\left({\mathfrak{c}}_3\right))\right]\hfill \end{array}$$

  where ${\delta }_1>0$ and $\forall \varkappa ,y\in [{\mathfrak{c}}_3,{\mathfrak{c}}_4]$.
• Choosing $⋏\left(\varkappa \right)=\varkappa$ and $\hslash \left({‌}^1\ell \right)={‌}^1\ell$ in Theorem 7, then:

  $$\begin{array}{cc}\hfill & \Phi \left({\displaystyle \frac{{\mathfrak{c}}_3+{\mathfrak{c}}_4}{2}}\right)\left[J_{{{\mathfrak{c}}_3}^{+}}^{{\delta }_1}\Xi \left({\mathfrak{c}}_4\right)+J_{{{\mathfrak{c}}_4}^{-}}^{{\delta }_1}\Xi \left({\mathfrak{c}}_3\right)\right]\hfill \\ \hfill & \supseteq \left[J_{{{\mathfrak{c}}_3}^{+}}^{{\delta }_1}\Phi \Xi \left({\mathfrak{c}}_4\right)+J_{{{\mathfrak{c}}_4}^{-}}^{{\delta }_1}\Phi \Xi \left({\mathfrak{c}}_3\right)\right]\hfill \\ \hfill & \supseteq {\displaystyle \frac{\Phi \left({\mathfrak{c}}_3\right)+\Phi \left({\mathfrak{c}}_4\right)}{2}}\left[J_{{{\mathfrak{c}}_3}^{+}}^{{\delta }_1}\Xi \left({\mathfrak{c}}_4\right)+J_{{{\mathfrak{c}}_4}^{-}}^{{\delta }_1}\Xi \left({\mathfrak{c}}_3\right)\right]\hfill \end{array}$$

  where ${\delta }_1>0$ and $\forall \varkappa ,y\in [{\mathfrak{c}}_3,{\mathfrak{c}}_4]$.
• Choosing $⋏\left(\varkappa \right)={\varkappa }^p$ in Theorem 7, then:

  $$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{2\hslash \left(\frac{1}{2}\right)}}\Phi \left({\left({\displaystyle \frac{{{\mathfrak{c}}_3}^p+{{\mathfrak{c}}_4}^p}{2}}\right)}^{{\displaystyle \frac{1}{p}}}\right)\left[J_{{{{\mathfrak{c}}_3}^p}^{+}}^{{\delta }_1}\Xi \circ k\left({{\mathfrak{c}}_4}^p\right)+J_{{{{\mathfrak{c}}_4}^p}^{-}}^{{\delta }_1}\Xi \circ k\left({{\mathfrak{c}}_3}^p\right)\right]\hfill \\ \hfill & \supseteq \left[J_{{{{\mathfrak{c}}_3}^p}^{+}}^{{\delta }_1}\Phi \Xi \circ k\left({{\mathfrak{c}}_4}^p\right)+J_{{{{\mathfrak{c}}_4}^p}^{-}}^{{\delta }_1}\Phi \Xi \circ k\left({{\mathfrak{c}}_3}^p\right)\right]\hfill \\ \hfill & \supseteq {\displaystyle \frac{{(⋏\left({\mathfrak{c}}_4\right)-⋏\left({\mathfrak{c}}_3\right))}^{{\delta }_1}}{\Gamma \left({\delta }_1\right)}}\left({\int }_0^1{{‌}^1\ell }^{{\delta }_1-1}[\hslash \left({‌}^1\ell \right)+\hslash (1-{‌}^1\ell )]\Xi \circ k({‌}^1\ell {{\mathfrak{c}}_3}^p+(1-{‌}^1\ell ){{\mathfrak{c}}_4}^p)[\Phi \left({\mathfrak{c}}_3\right)+\Phi \left({\mathfrak{c}}_4\right)]\mathrm{d}{‌}^1\ell \right)\hfill \end{array}$$

  where ${\delta }_1>0$ and $\forall \varkappa ,y\in [{\mathfrak{c}}_3,{\mathfrak{c}}_4]$.
• Choosing $⋏\left(\varkappa \right)={\varkappa }^p$ and $\hslash \left({‌}^1\ell \right)={‌}^1\ell$ in Theorem 7, then:

  $$\begin{array}{cc}\hfill & \Phi \left({\left({\displaystyle \frac{{{\mathfrak{c}}_3}^p+{{\mathfrak{c}}_4}^p}{2}}\right)}^{{\displaystyle \frac{1}{p}}}\right)\left[J_{{{{\mathfrak{c}}_3}^p}^{+}}^{{\delta }_1}\Xi \circ k\left({{\mathfrak{c}}_4}^p\right)+J_{{{{\mathfrak{c}}_4}^p}^{-}}^{{\delta }_1}\Xi \circ k\left({{\mathfrak{c}}_3}^p\right)\right]\hfill \\ \hfill & \supseteq \left[J_{{{{\mathfrak{c}}_3}^p}^{+}}^{{\delta }_1}\Phi \Xi \circ k\left({{\mathfrak{c}}_4}^p\right)+J_{{{{\mathfrak{c}}_4}^p}^{-}}^{{\delta }_1}\Phi \Xi \circ k\left({{\mathfrak{c}}_3}^p\right)\right]\hfill \\ \hfill & \supseteq {\displaystyle \frac{\Phi \left({{\mathfrak{c}}_3}^p\right)+\Phi \left({{\mathfrak{c}}_4}^p\right)}{2}}\left[J_{{{{\mathfrak{c}}_3}^p}^{+}}^{{\delta }_1}\Xi \circ k\left({{\mathfrak{c}}_4}^p\right)+J_{{{{\mathfrak{c}}_4}^p}^{-}}^{{\delta }_1}\Xi \circ k\left({{\mathfrak{c}}_3}^p\right)\right]\hfill \end{array}$$

  where ${\delta }_1>0$ and $\forall \varkappa ,y\in [{\mathfrak{c}}_3,{\mathfrak{c}}_4]$.

 Example 2. 

Presume each of the premises of Theorem 7 are met, we consider $\Phi :[0,1]\to {\mathbb{R}}_I$, where $\Phi \left({‌}^1\ell \right)=[4-\sqrt{\left({‌}^1\ell \right)},8+\sqrt{\left({‌}^1\ell \right)}]$ with $⋏\left({‌}^1\ell \right)={‌}^1\ell$ and symmetric function $\Xi \left({‌}^1\ell \right)$ with respect to ${\displaystyle \frac{{\mathfrak{c}}_3+{\mathfrak{c}}_4}{2}}$ such that $\Xi ({\mathfrak{c}}_3+{\mathfrak{c}}_4-\varkappa )=\Xi \left(\varkappa \right),\varkappa \in [0,1]$ and $\Xi \left({‌}^1\ell \right)$ is defined as:

$$\begin{array}{c}\hfill \Xi \left({‌}^1\ell \right)=\left\{\begin{array}{c}{‌}^1\ell ,{‌}^1\ell \in [0,{\displaystyle \frac{1}{2}}],\hfill \\ 1-{‌}^1\ell ,{‌}^1\ell \in [{\displaystyle \frac{1}{2}},1].\hfill \end{array}\right.\end{array}$$

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\Phi \left(\frac{1}{2}\right)}{\Gamma \left({\delta }_1\right)}}\left[{\int }_0^1[{(1-{‌}^1\ell )}^{{\delta }_1-1}+{{‌}^1\ell }^{{\delta }_1-1}]\Xi \left({‌}^1\ell \right)\mathrm{d}{‌}^1\ell \right]=[1.64645,4.35355],\hfill \\ \hfill & {\displaystyle \frac{1}{\Gamma \left({\delta }_1\right)}}\left[{\int }_0^1[{(1-{‌}^1\ell )}^{{\delta }_1-1}+{{‌}^1\ell }^{{\delta }_1-1}]\Phi \Xi \left({‌}^1\ell \right)\mathrm{d}{‌}^1\ell \right]=[1.65523,4.34477],\hfill \\ \hfill & {\displaystyle \frac{(\Phi (0)+\Phi (1\left)\right)}{2\Gamma \left({\delta }_1\right)}}\left[{\int }_0^1[{(1-{‌}^1\ell )}^{{\delta }_1-1}+{{‌}^1\ell }^{{\delta }_1-1}]\Xi \left({‌}^1\ell \right)\mathrm{d}{‌}^1\ell \right]=[1.75,4.25].\hfill \end{array}$$

This example confirm the accuracy of $\mathcal{H}$-$\mathcal{H}$ inequality for the product of $\mathcal{I}.\mathcal{V}$-$(⋏,\hslash )$ convex functions. By considering these aforementioned computations, one can easily analyze bounds for $\mathcal{I}.\mathcal{V}$ R-L operator.

Figure 2 reflects the comparison between the left, middle and right sides of Theorem 7. Furthermore red, blue and green color shows lower and upper functions of left, middle and right sides respectively.

 Theorem 8. 

Let $\Phi ,\Xi \in SIGX\left([{\mathfrak{c}}_3,{\mathfrak{c}}_4],{\mathbb{R}}_I^{+}\right)$:

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\Gamma \left({\delta }_1\right)}{{(⋏\left({\mathfrak{c}}_4\right)-⋏\left({\mathfrak{c}}_3\right))}^{{\delta }_1}}}\left[J_{⋏{\left({\mathfrak{c}}_3\right)}^{+}}^{{\delta }_1}\Phi \Xi \circ {⋏}^{-1}(⋏\left({\mathfrak{c}}_4\right))+J_{{(⋏\left({\mathfrak{c}}_4\right))}^{-}}^{{\delta }_1}\Phi \Xi \circ {⋏}^{-1}(⋏\left({\mathfrak{c}}_3\right))\right]\hfill \\ \hfill & \supseteq \mathcal{P}({\mathfrak{c}}_3,{\mathfrak{c}}_4){\int }_0^1{{‌}^1\ell }^{{\delta }_1-1}[{\hslash }_1\left({‌}^1\ell \right){\hslash }_2\left({‌}^1\ell \right)+{\hslash }_1(1-{‌}^1\ell ){\hslash }_2(1-{‌}^1\ell )]\mathrm{d}{‌}^1\ell \hfill \\ \hfill & +\mathcal{Q}({\mathfrak{c}}_3,{\mathfrak{c}}_4){\int }_0^1{{‌}^1\ell }^{{\delta }_1-1}[{\hslash }_1\left({‌}^1\ell \right){\hslash }_2(1-{‌}^1\ell )+{\hslash }_1(1-{‌}^1\ell ){\hslash }_2\left({‌}^1\ell \right)]\mathrm{d}{‌}^1\ell ,\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill \mathcal{P}({\mathfrak{c}}_3,{\mathfrak{c}}_4)& =\Phi \left({\mathfrak{c}}_3\right)\Xi \left({\mathfrak{c}}_3\right)+\Phi \left({\mathfrak{c}}_4\right)\Xi \left({\mathfrak{c}}_4\right),\hfill \end{array}$$

(33)

$$\begin{array}{cc}\hfill \mathcal{Q}({\mathfrak{c}}_3,{\mathfrak{c}}_4)& =\Phi \left({\mathfrak{c}}_3\right)\Xi \left({\mathfrak{c}}_4\right)+\Phi \left({\mathfrak{c}}_4\right)\Xi \left({\mathfrak{c}}_3\right),\hfill \end{array}$$

(34)

${\delta }_1>0$ and $\forall \varkappa ,y\in [{\mathfrak{c}}_3,{\mathfrak{c}}_4]$.

 Proof. 

Since $\Phi ,\Xi \in SIGX\left([{\mathfrak{c}}_3,{\mathfrak{c}}_4],{\mathbb{R}}_I^{+}\right)$, then:

$$\begin{array}{cc}\hfill & \Phi \left({⋏}^{-1}({‌}^1\ell ⋏\left({\mathfrak{c}}_3\right)+(1-{‌}^1\ell )⋏\left({\mathfrak{c}}_4\right))\right)\supseteq {\hslash }_1\left({‌}^1\ell \right)\Phi \left({\mathfrak{c}}_3\right)+{\hslash }_1(1-{‌}^1\ell )\Phi \left({\mathfrak{c}}_4\right).\hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill & \Xi \left({⋏}^{-1}({‌}^1\ell ⋏\left({\mathfrak{c}}_3\right)+(1-{‌}^1\ell )⋏\left({\mathfrak{c}}_4\right))\right)\supseteq {\hslash }_2\left({‌}^1\ell \right)\Xi \left({\mathfrak{c}}_3\right)+{\hslash }_2(1-{‌}^1\ell )\Xi \left({\mathfrak{c}}_4\right).\hfill \end{array}$$

(36)

Multiplying (35) and (36), we have:

$$\begin{array}{cc}\hfill & \Phi \left({⋏}^{-1}({‌}^1\ell ⋏\left({\mathfrak{c}}_3\right)+(1-{‌}^1\ell )⋏\left({\mathfrak{c}}_4\right))\right)\Xi \left({⋏}^{-1}({‌}^1\ell ⋏\left({\mathfrak{c}}_3\right)+(1-{‌}^1\ell )⋏\left({\mathfrak{c}}_4\right))\right)\hfill \\ \hfill & \supseteq {\hslash }_1\left({‌}^1\ell \right){\hslash }_2\left({‌}^1\ell \right)\Phi \left({\mathfrak{c}}_3\right)\Xi \left({\mathfrak{c}}_3\right)+{\hslash }_1\left({‌}^1\ell \right){\hslash }_2(1-{‌}^1\ell )\Phi \left({\mathfrak{c}}_3\right)\Xi \left({\mathfrak{c}}_4\right)\hfill \\ \hfill & +{\hslash }_1(1-{‌}^1\ell ){\hslash }_2\left({‌}^1\ell \right)\Phi \left({\mathfrak{c}}_4\right)\Xi \left({\mathfrak{c}}_3\right)+{\hslash }_1(1-{‌}^1\ell ){\hslash }_2(1-{‌}^1\ell )\Phi \left({\mathfrak{c}}_4\right)\Xi \left({\mathfrak{c}}_4\right).\hfill \end{array}$$

(37)

Similarly, we have:

$$\begin{array}{cc}\hfill & \Phi \left({⋏}^{-1}((1-{‌}^1\ell )⋏\left({\mathfrak{c}}_3\right)+{‌}^1\ell ⋏\left({\mathfrak{c}}_4\right))\right)\Xi \left({⋏}^{-1}((1-{‌}^1\ell )⋏\left({\mathfrak{c}}_3\right)+{‌}^1\ell ⋏\left({\mathfrak{c}}_4\right))\right)\hfill \\ \hfill & \supseteq {\hslash }_1(1-{‌}^1\ell ){\hslash }_2(1-{‌}^1\ell )\Phi \left({\mathfrak{c}}_3\right)\Xi \left({\mathfrak{c}}_3\right)+{\hslash }_1(1-{‌}^1\ell ){\hslash }_2\left({‌}^1\ell \right)\Phi \left({\mathfrak{c}}_3\right)\Xi \left({\mathfrak{c}}_4\right)\hfill \\ \hfill & +{\hslash }_1\left({‌}^1\ell \right){\hslash }_2(1-{‌}^1\ell )\Phi \left({\mathfrak{c}}_4\right)\Xi \left({\mathfrak{c}}_3\right)+{\hslash }_1\left({‌}^1\ell \right){\hslash }_2\left({‌}^1\ell \right)\Phi \left({\mathfrak{c}}_4\right)\Xi \left({\mathfrak{c}}_4\right).\hfill \end{array}$$

(38)

By adding (37) and (38), taking the product of obtained result with ${{‌}^1\ell }^{{\delta }_1-1}$ and then applying the integration with respect to ${‌}^1\ell$ on $[0,1]$:

$$\begin{array}{cc}\hfill & {\int }_0^1{{‌}^1\ell }^{{\delta }_1-1}\Phi \left({⋏}^{-1}({‌}^1\ell ⋏\left({\mathfrak{c}}_3\right)+(1-{‌}^1\ell )⋏\left({\mathfrak{c}}_4\right))\right)\Xi \left({⋏}^{-1}({‌}^1\ell ⋏\left({\mathfrak{c}}_3\right)+(1-{‌}^1\ell )⋏\left({\mathfrak{c}}_4\right))\right)\mathrm{d}{‌}^1\ell \hfill \\ \hfill & +{\int }_0^1{{‌}^1\ell }^{{\delta }_1-1}\Phi \left({⋏}^{-1}((1-{‌}^1\ell )⋏\left({\mathfrak{c}}_3\right)+{‌}^1\ell ⋏\left({\mathfrak{c}}_4\right))\right)\Xi \left({⋏}^{-1}((1-{‌}^1\ell )⋏\left({\mathfrak{c}}_3\right)+{‌}^1\ell ⋏\left({\mathfrak{c}}_4\right))\right)\mathrm{d}{‌}^1\ell \hfill \\ \hfill & \supseteq {\int }_0^1{{‌}^1\ell }^{{\delta }_1-1}[{\hslash }_1\left({‌}^1\ell \right){\hslash }_2\left({‌}^1\ell \right)+{\hslash }_1(1-{‌}^1\ell ){\hslash }_2(1-{‌}^1\ell )][\Phi \left({\mathfrak{c}}_3\right)\Xi \left({\mathfrak{c}}_3\right)+\Phi \left({\mathfrak{c}}_4\right)\Xi \left({\mathfrak{c}}_4\right)]\mathrm{d}{‌}^1\ell \hfill \\ \hfill & +{\int }_0^1{{‌}^1\ell }^{{\delta }_1-1}[{\hslash }_1\left({‌}^1\ell \right){\hslash }_2(1-{‌}^1\ell )+{\hslash }_1(1-{‌}^1\ell ){\hslash }_2\left({‌}^1\ell \right)][\Phi \left({\mathfrak{c}}_3\right)\Xi \left({\mathfrak{c}}_4\right)+\Phi \left({\mathfrak{c}}_4\right)\Xi \left({\mathfrak{c}}_3\right)]\mathrm{d}{‌}^1\ell .\hfill \end{array}$$

(39)

Now, we utilize the R-L fractional operators, then:

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\Gamma \left({\delta }_1\right)}{{(⋏\left({\mathfrak{c}}_4\right)-⋏\left({\mathfrak{c}}_3\right))}^{{\delta }_1}}}\left[J_{⋏{\left({\mathfrak{c}}_3\right)}^{+}}^{{\delta }_1}\Phi \Xi \circ {⋏}^{-1}(⋏\left({\mathfrak{c}}_4\right))+J_{{(⋏\left({\mathfrak{c}}_4\right))}^{-}}^{{\delta }_1}\Phi \Xi \circ {⋏}^{-1}(⋏\left({\mathfrak{c}}_3\right))\right]\hfill \\ \hfill & \supseteq \mathcal{P}({\mathfrak{c}}_3,{\mathfrak{c}}_4){\int }_0^1{{‌}^1\ell }^{{\delta }_1-1}[{\hslash }_1\left({‌}^1\ell \right){\hslash }_2\left({‌}^1\ell \right)+{\hslash }_1(1-{‌}^1\ell ){\hslash }_2(1-{‌}^1\ell )]\mathrm{d}{‌}^1\ell \hfill \\ \hfill & +\mathcal{Q}({\mathfrak{c}}_3,{\mathfrak{c}}_4){\int }_0^1{{‌}^1\ell }^{{\delta }_1-1}[{\hslash }_1\left({‌}^1\ell \right){\hslash }_2(1-{‌}^1\ell )+{\hslash }_1(1-{‌}^1\ell ){\hslash }_2\left({‌}^1\ell \right)]\mathrm{d}{‌}^1\ell .\hfill \end{array}$$

This completes the proof. □

Now, we exhibit interesting results concerning Theorem 8.

• Choosing $\hslash \left({‌}^1\ell \right)={‌}^1\ell$ in Theorem 8, then:

  $$\begin{array}{cc}\hfill & {\displaystyle \frac{\Gamma \left({\delta }_1\right)}{2{(⋏\left({\mathfrak{c}}_4\right)-⋏\left({\mathfrak{c}}_3\right))}^{{\delta }_1}}}\left[J_{⋏{\left({\mathfrak{c}}_3\right)}^{+}}^{{\delta }_1}\Phi \Xi \circ {⋏}^{-1}(⋏\left({\mathfrak{c}}_4\right))+J_{{(⋏\left({\mathfrak{c}}_4\right))}^{-}}^{{\delta }_1}\Phi \Xi \circ {⋏}^{-1}(⋏\left({\mathfrak{c}}_3\right))\right]\hfill \\ \hfill & \supseteq \mathcal{P}({\mathfrak{c}}_3,{\mathfrak{c}}_4){\displaystyle \frac{{{\delta }_1}^2+{\delta }_1+2}{{\delta }_1({\delta }_1+2)({\delta }_1+1)}}+\mathcal{Q}({\mathfrak{c}}_3,{\mathfrak{c}}_4){\displaystyle \frac{2}{({\delta }_1+1)({\delta }_1+2)}}.\hfill \end{array}$$
• Choosing $⋏\left(\varkappa \right)=\varkappa$ and $\hslash \left({‌}^1\ell \right)={‌}^1\ell$ in Theorem 8, then:

  $$\begin{array}{cc}\hfill & {\displaystyle \frac{\Gamma \left({\delta }_1\right)}{2{({\mathfrak{c}}_4-{\mathfrak{c}}_3)}^{{\delta }_1}}}\left[J_{{{\mathfrak{c}}_3}^{+}}^{{\delta }_1}\Phi \Xi \left({\mathfrak{c}}_4\right)+J_{{\left({\mathfrak{c}}_4\right)}^{-}}^{{\delta }_1}\Phi \Xi \left({\mathfrak{c}}_3\right)\right]\hfill \\ \hfill & \supseteq \mathcal{P}({\mathfrak{c}}_3,{\mathfrak{c}}_4){\displaystyle \frac{{{\delta }_1}^2+{\delta }_1+2}{{\delta }_1({\delta }_1+2)({\delta }_1+1)}}+\mathcal{Q}({\mathfrak{c}}_3,{\mathfrak{c}}_4){\displaystyle \frac{2}{({\delta }_1+1)({\delta }_1+2)}}.\hfill \end{array}$$
• Choosing $⋏\left(\varkappa \right)={\varkappa }^p$ in Theorem 8, then:

  $$\begin{array}{cc}\hfill & {\displaystyle \frac{\Gamma \left({\delta }_1\right)}{2{({{\mathfrak{c}}_4}^p-{{\mathfrak{c}}_3}^p)}^{{\delta }_1}}}\left[J_{{{{\mathfrak{c}}_3}^p}^{+}}^{{\delta }_1}\Phi \Xi \circ k\left({{\mathfrak{c}}_4}^p\right)+J_{{\left({{\mathfrak{c}}_4}^p\right)}^{-}}^{{\delta }_1}\Phi \Xi \circ k\left({{\mathfrak{c}}_3}^p\right)\right]\hfill \\ \hfill & \supseteq \mathcal{P}({\mathfrak{c}}_3,{\mathfrak{c}}_4){\displaystyle \frac{{{\delta }_1}^2+{\delta }_1+2}{{\delta }_1({\delta }_1+2)({\delta }_1+1)}}+\mathcal{Q}({\mathfrak{c}}_3,{\mathfrak{c}}_4){\displaystyle \frac{2}{({\delta }_1+1)({\delta }_1+2)}}.\hfill \end{array}$$
• Choosing $⋏\left(\varkappa \right)={\varkappa }^p$ and $\hslash \left({‌}^1\ell \right)={‌}^1\ell$ in Theorem 8, then:

  $$\begin{array}{cc}\hfill & {\displaystyle \frac{\Gamma \left({\delta }_1\right)}{2{({{\mathfrak{c}}_4}^p-{{\mathfrak{c}}_3}^p)}^{{\delta }_1}}}\left[J_{{{{\mathfrak{c}}_3}^p}^{+}}^{{\delta }_1}\Phi \Xi \circ k\left({{\mathfrak{c}}_4}^p\right)+J_{{\left({{\mathfrak{c}}_4}^p\right)}^{-}}^{{\delta }_1}\Phi \Xi \circ k\left({{\mathfrak{c}}_3}^p\right)\right]\hfill \\ \hfill & \supseteq \mathcal{P}({\mathfrak{c}}_3,{\mathfrak{c}}_4){\displaystyle \frac{{{\delta }_1}^2+{\delta }_1+2}{{\delta }_1({\delta }_1+2)({\delta }_1+1)}}+\mathcal{Q}({\mathfrak{c}}_3,{\mathfrak{c}}_4){\displaystyle \frac{2}{({\delta }_1+1)({\delta }_1+2)}}.\hfill \end{array}$$

 Example 3. 

Presume each of the premises of Theorem 8 are met, we consider $\Phi ,\Xi :[0,1]\to {\mathbb{R}}_I$, where $\Phi \left({‌}^1\ell \right)=[{{‌}^1\ell }^2,2-{{‌}^1\ell }^2]$ and $\Xi \left({‌}^1\ell \right)=[4-\sqrt{\left({‌}^1\ell \right)},8+\sqrt{\left({‌}^1\ell \right)}]$ with $⋏\left({‌}^1\ell \right)={‌}^1\ell$, then:

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{\Gamma \left({\delta }_1\right)}}\left[{\int }_0^1\left({(1-{‌}^1\ell )}^{{\delta }_1-1}+{{‌}^1\ell }^{{\delta }_1-1}\right)\Phi \Xi \left({‌}^1\ell \right)\mathrm{d}{‌}^1\ell \right]=[1.04762,14.381],\hfill \\ \hfill & {\displaystyle \frac{1}{\Gamma \left({\delta }_1\right)}}\left[\mathcal{P}({\mathfrak{c}}_3,{\mathfrak{c}}_4){\displaystyle \frac{{{\delta }_1}^2+{\delta }_1+2}{{\delta }_1({\delta }_1+2)({\delta }_1+1)}}+\mathcal{Q}({\mathfrak{c}}_3,{\mathfrak{c}}_4){\displaystyle \frac{2}{({\delta }_1+1)({\delta }_1+2)}}\right]=[1.66667,12.6667].\hfill \end{array}$$

The aforementioned computations justify the accuracy of right Pachpatte’s type inequality for the product two $\mathcal{I}.\mathcal{V}$-$(⋏,\hslash )$ convex functions. By considering these calculations, one can easily evaluate the lower bounds for the $\mathcal{I}.\mathcal{V}$ R-L operator.

For pictorial depiction, we vary $0\le {\delta }_1\le 5$. Figure 3 reflects the comparison between the left and right sides of Theorem 8. Furthermore purple and green color shows lower and upper functions of left and right sides respectively.

 Theorem 9. 

Let $\Phi ,\Xi \in SIGX\left([{\mathfrak{c}}_3,{\mathfrak{c}}_4],{\mathbb{R}}_I^{+}\right)$, then:

$$\begin{array}{cc}\hfill & \Phi \left({⋏}^{-1}\left({\displaystyle \frac{⋏\left({\mathfrak{c}}_3\right)+⋏\left({\mathfrak{c}}_4\right)}{2}}\right)\right)\Xi \left({⋏}^{-1}\left({\displaystyle \frac{⋏\left({\mathfrak{c}}_3\right)+⋏\left({\mathfrak{c}}_4\right)}{2}}\right)\right)\hfill \\ \hfill & \supseteq {\displaystyle \frac{{\hslash }_1\left(\frac{1}{2}\right){\hslash }_2\left(\frac{1}{2}\right)\Gamma ({\delta }_1+1)}{{(⋏\left({\mathfrak{c}}_4\right)-⋏\left({\mathfrak{c}}_3\right))}^{{\delta }_1}}}\left[J_{{⋏\left({\mathfrak{c}}_4\right)}^{-}}^{{\delta }_1}\Phi \Xi \circ {⋏}^{-1}(⋏\left({\mathfrak{c}}_3\right))+J_{⋏{\left({\mathfrak{c}}_3\right)}^{+}}^{{\delta }_1}\Phi \Xi \circ (⋏\left({\mathfrak{c}}_4\right))\right]\hfill \\ \hfill & +{\delta }_1{\hslash }_1\left({\displaystyle \frac{1}{2}}\right){\hslash }_2\left({\displaystyle \frac{1}{2}}\right)\left[\mathcal{P}({\mathfrak{c}}_3,{\mathfrak{c}}_4){\int }_0^1{{‌}^1\ell }^{{\delta }_1-1}[{\hslash }_1\left({‌}^1\ell \right){\hslash }_2(1-{‌}^1\ell )+{\hslash }_1(1-{‌}^1\ell ){\hslash }_2\left({‌}^1\ell \right)]\right.\hfill \\ & \left.+\mathcal{Q}({\mathfrak{c}}_3,{\mathfrak{c}}_4){\int }_0^1{{‌}^1\ell }^{{\delta }_1-1}[{\hslash }_1\left({‌}^1\ell \right){\hslash }_2\left({‌}^1\ell \right)+{\hslash }_1(1-{‌}^1\ell ){\hslash }_2(1-{‌}^1\ell )]\mathrm{d}{‌}^1\ell \right],\hfill \end{array}$$

where $\mathcal{P}({\mathfrak{c}}_3,{\mathfrak{c}}_4)$ and $\mathcal{Q}({\mathfrak{c}}_3,{\mathfrak{c}}_4)$ are given by (33) and (34), respectively.

 Proof. 

Let $\Phi ,\Xi \in SIGX\left([{\mathfrak{c}}_3,{\mathfrak{c}}_4],{\mathbb{R}}_I^{+}\right)$, then:

$$\begin{array}{cc}\hfill & \Phi \left({⋏}^{-1}\left({\displaystyle \frac{⋏\left({\mathfrak{c}}_3\right)+⋏\left({\mathfrak{c}}_4\right)}{2}}\right)\right)\Xi \left({⋏}^{-1}\left({\displaystyle \frac{⋏\left({\mathfrak{c}}_3\right)+⋏\left({\mathfrak{c}}_4\right)}{2}}\right)\right)\hfill \\ \hfill & \supseteq {\hslash }_1\left({\displaystyle \frac{1}{2}}\right){\hslash }_2\left({\displaystyle \frac{1}{2}}\right)\left[\Phi \left({⋏}^{-1}((1-{‌}^1\ell )⋏\left({\mathfrak{c}}_3\right)+{‌}^1\ell ⋏\left({\mathfrak{c}}_4\right))\right)\Xi \left({⋏}^{-1}((1-{‌}^1\ell )⋏\left({\mathfrak{c}}_3\right)+{‌}^1\ell ⋏\left({\mathfrak{c}}_4\right))\right)\right.\hfill \\ \hfill & +\Phi \left({⋏}^{-1}((1-{‌}^1\ell )⋏\left({\mathfrak{c}}_3\right)+{‌}^1\ell ⋏\left({\mathfrak{c}}_4\right))\right)\Xi \left({⋏}^{-1}({‌}^1\ell ⋏\left({\mathfrak{c}}_3\right)+(1-{‌}^1\ell )⋏\left({\mathfrak{c}}_4\right))\right)\hfill \\ \hfill & +\Phi \left({⋏}^{-1}({‌}^1\ell ⋏\left({\mathfrak{c}}_3\right)+(1-{‌}^1\ell )⋏\left({\mathfrak{c}}_4\right))\right)\Xi \left({⋏}^{-1}((1-{‌}^1\ell )⋏\left({\mathfrak{c}}_3\right)+{‌}^1\ell ⋏\left({\mathfrak{c}}_4\right))\right)\hfill \\ \hfill & \left.+\Phi \left({⋏}^{-1}({‌}^1\ell ⋏\left({\mathfrak{c}}_3\right)+(1-{‌}^1\ell )⋏\left({\mathfrak{c}}_4\right))\right)\Xi \left({⋏}^{-1}({‌}^1\ell ⋏\left({\mathfrak{c}}_3\right)+(1-{‌}^1\ell )⋏\left({\mathfrak{c}}_4\right))\right)\right].\hfill \end{array}$$

Multiplying preceding inclusion by ${{‌}^1\ell }^{{\delta }_1-1}$ and applying integration with respect to ${‌}^1\ell$ on $[0,1]$, then:

$$\begin{array}{cc}\hfill & \left(IR\right){\int }_0^1{{‌}^1\ell }^{{\delta }_1-1}\Phi \left({⋏}^{-1}\left({\displaystyle \frac{⋏\left({\mathfrak{c}}_3\right)+⋏({{\mathfrak{c}}_4}^{)}}{2}}\right)\right)\Xi \left({⋏}^{-1}\left({\displaystyle \frac{⋏\left({\mathfrak{c}}_3\right)+⋏({{\mathfrak{c}}_4}^{)}}{2}}\right)\right)\mathrm{d}{‌}^1\ell \hfill \\ \hfill & \supseteq {\hslash }_1\left({\displaystyle \frac{1}{2}}\right){\hslash }_2\left({\displaystyle \frac{1}{2}}\right)\left[{\int }_0^1{{‌}^1\ell }^{{\delta }_1-1}\Phi \left({⋏}^{-1}((1-{‌}^1\ell )⋏\left({\mathfrak{c}}_3\right)+{‌}^1\ell ⋏\left({\mathfrak{c}}_4\right))\right)\Xi \left({⋏}^{-1}((1-{‌}^1\ell )⋏\left({\mathfrak{c}}_3\right)+{‌}^1\ell ⋏\left({\mathfrak{c}}_4\right))\right)\mathrm{d}{‌}^1\ell \right.\hfill \\ \hfill & +{\int }_0^1{{‌}^1\ell }^{{\delta }_1-1}\Phi \left({⋏}^{-1}((1-{‌}^1\ell )⋏\left({\mathfrak{c}}_3\right)+{‌}^1\ell ⋏\left({\mathfrak{c}}_4\right))\right)\Xi \left({⋏}^{-1}({‌}^1\ell ⋏\left({\mathfrak{c}}_3\right)+(1-{‌}^1\ell )⋏\left({\mathfrak{c}}_4\right))\right)\mathrm{d}{‌}^1\ell \hfill \\ \hfill & +{\int }_0^1{{‌}^1\ell }^{{\delta }_1-1}\Phi \left({⋏}^{-1}({‌}^1\ell ⋏\left({\mathfrak{c}}_3\right)+(1-{‌}^1\ell )⋏\left({\mathfrak{c}}_4\right))\right)\Xi \left({⋏}^{-1}((1-{‌}^1\ell )⋏\left({\mathfrak{c}}_3\right)+{‌}^1\ell ⋏\left({\mathfrak{c}}_4\right))\right)\mathrm{d}{‌}^1\ell \hfill \\ \hfill & \left.+{\int }_0^1{{‌}^1\ell }^{{\delta }_1-1}\Phi \left({⋏}^{-1}({‌}^1\ell ⋏\left({\mathfrak{c}}_3\right)+(1-{‌}^1\ell )⋏\left({\mathfrak{c}}_4\right))\right)\Xi \left({⋏}^{-1}({‌}^1\ell ⋏\left({\mathfrak{c}}_3\right)+(1-{‌}^1\ell )⋏\left({\mathfrak{c}}_4\right))\right)\mathrm{d}{‌}^1\ell \right].\hfill \end{array}$$

Employing the definition of $\mathcal{I}.\mathcal{V}$ convexity, we acquire:

$$\begin{array}{cc}\hfill & {\int }_0^1{{‌}^1\ell }^{{\delta }_1-1}\Phi \left({⋏}^{-1}\left({\displaystyle \frac{⋏\left({\mathfrak{c}}_3\right)+⋏({{\mathfrak{c}}_4}^{)}}{2}}\right)\right)\Xi \left({⋏}^{-1}\left({\displaystyle \frac{⋏\left({\mathfrak{c}}_3\right)+⋏({{\mathfrak{c}}_4}^{)}}{2}}\right)\mathrm{d}{‌}^1\ell \right)\hfill \\ \hfill & ={\displaystyle \frac{1}{{\delta }_1}}\Phi \left({⋏}^{-1}\left({\displaystyle \frac{⋏\left({\mathfrak{c}}_3\right)+⋏\left({\mathfrak{c}}_4\right)}{2}}\right)\right)\Xi \left({⋏}^{-1}\left({\displaystyle \frac{⋏\left({\mathfrak{c}}_3\right)+⋏\left({\mathfrak{c}}_4\right)}{2}}\right)\right)\hfill \\ \hfill & \supseteq {\hslash }_1\left({\displaystyle \frac{1}{2}}\right){\hslash }_2\left({\displaystyle \frac{1}{2}}\right)\left[{\int }_0^1{{‌}^1\ell }^{{\delta }_1-1}\Phi \left({⋏}^{-1}((1-{‌}^1\ell )⋏\left({\mathfrak{c}}_3\right)+{‌}^1\ell ⋏\left({\mathfrak{c}}_4\right))\right)\Xi \left({⋏}^{-1}((1-{‌}^1\ell )⋏\left({\mathfrak{c}}_3\right)+{‌}^1\ell ⋏\left({\mathfrak{c}}_4\right))\right)\mathrm{d}{‌}^1\ell \right.\hfill \\ \hfill & +{\int }_0^1{{‌}^1\ell }^{{\delta }_1-1}\Phi \left({⋏}^{-1}({‌}^1\ell ⋏\left({\mathfrak{c}}_3\right)+(1-{‌}^1\ell )⋏\left({\mathfrak{c}}_4\right))\right)\Xi \left({⋏}^{-1}({‌}^1\ell ⋏\left({\mathfrak{c}}_3\right)+(1-{‌}^1\ell )⋏\left({\mathfrak{c}}_4\right))\right)\mathrm{d}{‌}^1\ell \hfill \\ \hfill & +\mathcal{P}({\mathfrak{c}}_3,{\mathfrak{c}}_4){\int }_0^1{{‌}^1\ell }^{{\delta }_1-1}[{\hslash }_1\left({‌}^1\ell \right){\hslash }_2(1-{‌}^1\ell )+{\hslash }_1(1-{‌}^1\ell ){\hslash }_2\left({‌}^1\ell \right)]\mathrm{d}{‌}^1\ell \hfill \\ \hfill & \left.+\mathcal{Q}({\mathfrak{c}}_3,{\mathfrak{c}}_4){\int }_0^1{{‌}^1\ell }^{{\delta }_1-1}[{\hslash }_1\left({‌}^1\ell \right){\hslash }_2\left({‌}^1\ell \right)+{\hslash }_1(1-{‌}^1\ell ){\hslash }_2(1-{‌}^1\ell )]\mathrm{d}{‌}^1\ell \right].\hfill \end{array}$$

(40)

Also:

$$\begin{array}{cc}\hfill & \left(IR\right){\int }_0^1{{‌}^1\ell }^{{\delta }_1-1}\Phi \left({⋏}^{-1}((1-{‌}^1\ell )⋏\left({\mathfrak{c}}_3\right)+{‌}^1\ell ⋏\left({\mathfrak{c}}_4\right))\right)\Xi \left({⋏}^{-1}((1-{‌}^1\ell )⋏\left({\mathfrak{c}}_3\right)+{‌}^1\ell ⋏\left({\mathfrak{c}}_4\right))\right)\mathrm{d}{‌}^1\ell \hfill \\ \hfill & ={\displaystyle \frac{1}{{(⋏\left({\mathfrak{c}}_4\right)-⋏\left({\mathfrak{c}}_3\right))}^{{\delta }_1}}}{\int }_{⋏\left({\mathfrak{c}}_3\right)}^{⋏\left({\mathfrak{c}}_4\right)}{(u-⋏\left({\mathfrak{c}}_3\right))}^{{\delta }_1-1}\Phi \Xi \circ {⋏}^{-1}\left(u\right)\mathrm{d}u\hfill \\ \hfill & ={\displaystyle \frac{\Gamma \left({\delta }_1\right)}{{(⋏\left({\mathfrak{c}}_4\right)-⋏\left({\mathfrak{c}}_3\right))}^{{\delta }_1}}}{J}_{{⋏\left({\mathfrak{c}}_4\right)}^{-}}^{{\delta }_1}\Phi \Xi \circ {⋏}^{-1}(⋏\left({\mathfrak{c}}_3\right))\hfill \end{array}$$

(41)

and

$$\begin{array}{cc}\hfill & \left(IR\right){\int }_0^1{{‌}^1\ell }^{{\delta }_1-1}\Phi \left({⋏}^{-1}({‌}^1\ell ⋏\left({\mathfrak{c}}_3\right)+(1-{‌}^1\ell )⋏\left({\mathfrak{c}}_4\right))\right)\Xi \left({⋏}^{-1}({‌}^1\ell ⋏\left({\mathfrak{c}}_3\right)+(1-{‌}^1\ell )⋏\left({\mathfrak{c}}_4\right))\right)\mathrm{d}{‌}^1\ell \hfill \\ \hfill & ={\displaystyle \frac{\Gamma \left({\delta }_1\right)}{{(⋏\left({\mathfrak{c}}_4\right)-⋏\left({\mathfrak{c}}_3\right))}^{{\delta }_1}}}{J}_{⋏{\left({\mathfrak{c}}_3\right)}^{+}}^{{\delta }_1}\Phi \Xi \circ {⋏}^{-1}(⋏\left({\mathfrak{c}}_4\right)).\hfill \end{array}$$

(42)

From (40)–(42), we secured our result. □

Now, we explore some crucial results concerning with Theorem 9.

• Choosing $\hslash \left({‌}^1\ell \right)={‌}^1\ell$ in Theorem 9, then:

  $$\begin{array}{cc}\hfill & 2\Phi \left({⋏}^{-1}\left({\displaystyle \frac{⋏\left({\mathfrak{c}}_3\right)+⋏\left({\mathfrak{c}}_4\right)}{2}}\right)\right)\Xi \left({⋏}^{-1}\left({\displaystyle \frac{⋏\left({\mathfrak{c}}_3\right)+⋏\left({\mathfrak{c}}_4\right)}{2}}\right)\right)\hfill \\ \hfill & \supseteq {\displaystyle \frac{\Gamma ({\delta }_1+1)}{2{(⋏\left({\mathfrak{c}}_4\right)-⋏\left({\mathfrak{c}}_3\right))}^{{\delta }_1}}}\left[J_{{⋏\left({\mathfrak{c}}_4\right)}^{-}}^{{\delta }_1}\Phi \Xi \circ {⋏}^{-1}(⋏\left({\mathfrak{c}}_3\right))+J_{⋏{\left({\mathfrak{c}}_3\right)}^{+}}^{{\delta }_1}\Phi \Xi \circ (⋏\left({\mathfrak{c}}_4\right))\right]\hfill \\ \hfill & +\mathcal{P}({\mathfrak{c}}_3,{\mathfrak{c}}_4){\displaystyle \frac{{\delta }_1}{({\delta }_1+1)({\delta }_1+2)}}+\mathcal{Q}({\mathfrak{c}}_3,{\mathfrak{c}}_4){\displaystyle \frac{{{\delta }_1}^2+{\delta }_1+2}{2({\delta }_1+2)({\delta }_1+1)}},\hfill \end{array}$$

  where $\mathcal{P}({\mathfrak{c}}_3,{\mathfrak{c}}_4)$ and $\mathcal{Q}({\mathfrak{c}}_3,{\mathfrak{c}}_4)$ are given by (33) and (34), respectively.
• Choosing $⋏\left(\varkappa \right)=\varkappa$ and $\hslash \left({‌}^1\ell \right)={‌}^1\ell$ in Theorem 9, then:

  $$\begin{array}{cc}\hfill & 2\Phi \left({\displaystyle \frac{{\mathfrak{c}}_3+{\mathfrak{c}}_4}{2}}\right)\Xi \left({\displaystyle \frac{{\mathfrak{c}}_3+{\mathfrak{c}}_4}{2}}\right)\hfill \\ \hfill & \supseteq {\displaystyle \frac{\Gamma ({\delta }_1+1)}{2{({\mathfrak{c}}_4-{\mathfrak{c}}_3)}^{{\delta }_1}}}\left[J_{{{\mathfrak{c}}_4}^{-}}^{{\delta }_1}\Phi \Xi \left({\mathfrak{c}}_3\right)+J_{{{\mathfrak{c}}_3}^{+}}^{{\delta }_1}\Phi \Xi \left({\mathfrak{c}}_4\right)\right]\hfill \\ \hfill & +\mathcal{P}({\mathfrak{c}}_3,{\mathfrak{c}}_4){\displaystyle \frac{{\delta }_1}{({\delta }_1+1)({\delta }_1+2)}}+\mathcal{Q}({\mathfrak{c}}_3,{\mathfrak{c}}_4){\displaystyle \frac{{{\delta }_1}^2+{\delta }_1+2}{2({\delta }_1+2)({\delta }_1+1)}},\hfill \end{array}$$

  where $\mathcal{P}({\mathfrak{c}}_3,{\mathfrak{c}}_4)$ and $\mathcal{Q}({\mathfrak{c}}_3,{\mathfrak{c}}_4)$ are given by (33) and (34), respectively.
• Choosing $⋏\left(\varkappa \right)={\varkappa }^p$ and $\hslash \left({‌}^1\ell \right)={‌}^1\ell$ in Theorem 9, then:

  $$\begin{array}{cc}\hfill & 2\Phi \left({\left({\displaystyle \frac{{{\mathfrak{c}}_3}^p+{{\mathfrak{c}}_4}^p}{2}}\right)}^{{\displaystyle \frac{1}{p}}}\right)\Xi \left({\left({\displaystyle \frac{{{\mathfrak{c}}_3}^p+{{\mathfrak{c}}_4}^p}{2}}\right)}^{{\displaystyle \frac{1}{p}}}\right)\hfill \\ \hfill & \supseteq {\displaystyle \frac{\Gamma ({\delta }_1+1)}{2{({{\mathfrak{c}}_4}^p-{{\mathfrak{c}}_3}^p)}^{{\delta }_1}}}\left[J_{{{{\mathfrak{c}}_4}^p}^{-}}^{{\delta }_1}\Phi \Xi \circ k\left({{\mathfrak{c}}_3}^p\right)+J_{{{{\mathfrak{c}}_3}^p}^{+}}^{{\delta }_1}\Phi \Xi \circ k\left({{\mathfrak{c}}_4}^p\right)\right]\hfill \\ \hfill & +\mathcal{P}({\mathfrak{c}}_3,{\mathfrak{c}}_4){\displaystyle \frac{{\delta }_1}{({\delta }_1+1)({\delta }_1+2)}}+\mathcal{Q}({\mathfrak{c}}_3,{\mathfrak{c}}_4){\displaystyle \frac{{{\delta }_1}^2+{\delta }_1+2}{2({\delta }_1+2)({\delta }_1+1)}},\hfill \end{array}$$

  where $\mathcal{P}({\mathfrak{c}}_3,{\mathfrak{c}}_4)$ and $\mathcal{Q}({\mathfrak{c}}_3,{\mathfrak{c}}_4)$ are given by (33) and (34), respectively.

 Example 4. 

Presume each of the premises of Theorem 9 are met, we consider $\Phi ,\Xi :[0,1]\to {\mathbb{R}}_I$ where $\Phi \left({‌}^1\ell \right)=[{{‌}^1\ell }^2,2-{{‌}^1\ell }^2]$ and $\Xi \left({‌}^1\ell \right)=[4-\sqrt{{‌}^1\ell },\sqrt{{‌}^1\ell }+8]$ with $⋏\left({‌}^1\ell \right)={‌}^1\ell$, then:

$$\begin{array}{cc}\hfill & {\displaystyle \frac{2}{\Gamma ({\delta }_1+1)}}\Phi \left({\displaystyle \frac{1}{2}}\right)\Xi \left({\displaystyle \frac{1}{2}}\right)=[0.8232,15.2374],\hfill \\ \hfill & {\displaystyle \frac{1}{2\Gamma \left({\delta }_1\right)}}\left[{\int }_0^1\left({(1-{‌}^1\ell )}^{{\delta }_1-1}+{{‌}^1\ell }^{{\delta }_1-1}\right)\Phi \Xi \left({‌}^1\ell \right)\mathrm{d}{‌}^1\ell +\mathcal{P}({\mathfrak{c}}_3,{\mathfrak{c}}_4){\displaystyle \frac{2{\delta }_1}{{\delta }_1({\delta }_1+1)({\delta }_1+2)}}+\mathcal{Q}({\mathfrak{c}}_3,{\mathfrak{c}}_4){\displaystyle \frac{{{\delta }_1}^2+2{\delta }_1+4}{{\delta }_1({\delta }_1+2)({\delta }_1+1)}}\right]\hfill \\ \hfill & =[1.44048,13.6071].\hfill \end{array}$$

The aforementioned computations justify the accuracy of Theorem 9 for the product two $\mathcal{I}.\mathcal{V}$-$(⋏,\hslash )$ convex functions. Moreover, from these computations, one can easily calculate the upper bounds of the $\mathcal{I}.\mathcal{V}$ R-L operator.

For a pictorial depiction, we vary $0\le {\delta }_1\le 5$. Figure 4 reflects the comparison between both sides of Theorem 9. Furthermore purple and green color show lower and upper functions of left and right sides respectively.

3. Applications

Now, we demonstrate an interesting relation between means by taking the advantage of Hadamard’s inequality.

• The arithmetic mean:

  $$A({\mathfrak{c}}_3,{\mathfrak{c}}_4)={\displaystyle \frac{{\mathfrak{c}}_3+{\mathfrak{c}}_4}{2}},$$
• The generalized log-mean:

  $$L_r({\mathfrak{c}}_3,{\mathfrak{c}}_4)={\left[{\displaystyle \frac{{{\mathfrak{c}}_4}^{r+1}-{{\mathfrak{c}}_3}^{r+1}}{(r+1)({\mathfrak{c}}_4-{\mathfrak{c}}_3)}}\right]}^{{\displaystyle \frac{1}{r}}};r\in \mathbb{Z}\setminus \{-1,0\}.$$

 Proposition 1. 

For ${\mathfrak{c}}_3,{\mathfrak{c}}_4>0$, then:

$$\begin{array}{cc}\hfill \left[A^2({\mathfrak{c}}_3,{\mathfrak{c}}_4),-A^2({\mathfrak{c}}_3,{\mathfrak{c}}_4)+6\right]& \supseteq \left[{Ł}_2^2({\mathfrak{c}}_3,{\mathfrak{c}}_4),-{Ł}_2^2({\mathfrak{c}}_3,{\mathfrak{c}}_4)+6\right]\hfill \\ \hfill & \supseteq \left[A({{\mathfrak{c}}_3}^2,{{\mathfrak{c}}_4}^2),-A({{\mathfrak{c}}_3}^2,{{\mathfrak{c}}_4}^2)+6\right].\hfill \end{array}$$

 Proof. 

The assertion is followed directly by applying $\Phi \left({‌}^1\ell \right)=\left[{{‌}^1\ell }^2,-{{‌}^1\ell }^2+6\right]$ and $⋏\left({‌}^1\ell \right)={‌}^1\ell$ and $\hslash \left({‌}^1\ell \right)={‌}^1\ell$ in Theorem 6. □

4. Conclusions

In recent decades, the theory of inequalities has emerged as the main topic of research via fractional calculus and its various generalizations. Many well-known inequalities have been refined by applying different convexities and fractional concepts. We provided a very generic unification of classical concepts. Through this class of convexity, we obtained some new fractional counterparts of trapezium-type containment along with some interesting consequences and graphical explanations of key results through some examples. It is worth mentioning that we established the bounds of $\mathcal{I}.\mathcal{V}$ R-L operators for larger classes of functions. Because our proposed function class contains convex functions and several non-convex function classes. These results are useful for the development of bounds for special functions like modified Bessel functions, beta functions, etc. In the future, we will apply these notions to formulate some other types of Ostrowski’s type inequalities, $\mathcal{H}$-$\mathcal{H}$’s type inequalities in fuzzy settings, and quantum calculus. Also, by considering our proposed class, Jensen–Mercer-type inequalities can be obtained in different frameworks. By applying similar strategies, one can formulate bounds for double integral R-L operators, Hilfer fractional operators, and other fractional operators. We hope that the idea and technique considered in our work will articulate new insights for researchers.