New class of convex interval-valued functions and Riemann Liouville fractional integral inequalities

1.   Introduction

The study of fractional order integral and derivative functions over real and complex domains, as well as its applications, is now the focus of fractional calculus. In many issues, using arithmetic from classical analysis in fractional analysis is crucial for generating more realistic results. Differential equations of fractional order are capable of handling a wide range of mathematical models. Because fractional mathematical models are special instances of fractional order mathematical models, they have more broad and accurate conclusions than classical mathematical models. Integer orders aren't an effective model for nature in classical analysis. On the other side, fractional computing allows us to look at any number of orders and come up with far more quantifiable goals.

Wang et al. ^[1] looked into fractional integral identities for a differentiable mapping involving Riemann–Liouville fractional integrals and Hadamard fractional integrals, and came up with some inequalities using standard convex, $r$-convex, $m$-convex, $s$-convex, (s, m)-convex, $\left(\beta, m\right)$-convex, Işcan ^[10] also used fractional integrals for preinvex functions to get various Hermite–Hadamard type inequalities. See ^{[2,3,4,5,6,7]} for other applications of the Hermite-Hadamard inequality.

Moore ^[8] used interval arithmetic, interval-valued functions, and integrals of interval-valued functions to establish arbitrarily sharp upper and lower limits on accurate solutions to various problems in practical mathematics. Moore ^[8] shown that if a real-valued function $𝛶\left(\varpi \right)$ meets an ordinary Lipschitz condition in Y, $\left|𝛶\left(\varpi \right)-𝛶\left(y\right)\right|\le L\left|\varpi -y\right|$, for $\varpi, y\in Y$, then the united extension of is a Lipschitz interval extension in Y. Hilger ^[9] proposed a time scales theory that may be used to combine the study of discrete and continuous dynamical systems. The widespread use of dynamic equations and integral inequalities on time scales in domains as diverse as electrical engineering, quantum physics, heat transfer, neural networks, combinatorics, and population dynamics ^[9] has highlighted the need for this theory. Young's inequality, Hölder's inequality, Minkoswki's inequality, Jensen's inequality, Steffensen's inequality, Hermite–Hadamard inequality and Opial type inequality were all explored by Agarwal et al. ^[9]. Srivastava et al. ^[10] discovered some generic time scale weighted Opial type inequalities in 2010. Srivastava et al. ^[11] also proposed several time-based expansions and generalizations of Maroni's inequality. Under some favorable conditions, Wei et al. ^[12] created a local fractional integral counterpart of Anderson's inequality on fractal space and also demonstrated that the fractal space local fractional integral inequality is a novel extension of Anderson's inequality Tunç et al. ^[13] also constructed an identity for local fractional integrals and derived numerous modifications of the well-known Steffensen's inequality for fractional integrals. ^[10,14] and the references therein might be consulted for further information.

Bhurjee and Panda ^[15] identified the parametric form of an interval-valued function and devised a technique to investigate the existence of a generic interval optimization issue solution. Using the notion of the generalized Hukuhara difference, Lupulescu ^[16] developed differentiability and integrability for interval-valued functions on time scales. Cano et al. ^[17] developed a novel form of Ostrowski inequality for gH-differentiable interval-valued functions in 2015, and achieved an extension of the class of real functions that are not always differentiable. For gH-differentiable interval-valued functions, Cano et al. ^[17] found error limitations to quadrature rules. In addition, Roy and Panda ^[18] developed the idea of the -monotonic property of interval-valued functions in the higher dimension and used extended Hukuhara differentiability to obtain various conclusions.

The findings of Iscan ^[19] and Noor et al. ^[20] have largely influenced our research. The concept of harmonically h-convexity for interval-valued functions is introduced first. Then, given the introduced class of functions, we show certain new Hermite-Hadamard type inequalities. The conclusions from ^[19,20] have interval-valued analogues in our inequalities. We refer to ^{[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37]} and the references therein for further information on real-valued and interval-valued functions.

Furthermore, Khan et al. introduced the different classes of convex functions like $\left({{h}}_{1}, {{h}}_{2}\right)$-preinvex fuzzy IVFs ^[38], log-s-convex fuzzy IVFs in the second sense ^[39], harmonically convex fuzzy IVFs ^[40], $\left({{h}}_{1}, {{h}}_{2}\right)$-convex fuzzy IVFs ^[41], generalized p-convex fuzzy IVFs ^[42] and introduced Hermite-Hadamard type inequalities of these functions. For more information, see ^{[43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63]} and the references are therein.

The following is a breakdown of the paper's structure. Following the preliminaries in Section 2, Section 3 introduces the left and right harmonically $h$-convexity idea for interval-valued functions and proves new Hermite-Hadamard fractional integral type inequalities. Section 4 concludes with conclusions and further study.

2.   Preliminaries

First, we offer some background information on interval-valued functions, the theory of convexity, interval-valued integration, and interval-valued fractional integration, which will be utilized throughout the article.

We offer some fundamental arithmetic regarding interval analysis in this paragraph, which will be quite useful throughout the article.

Let ${\mathcal{K}}_{C}$, ${\mathcal{K}}_{C}^{+}$, ${\mathcal{K}}_{C}^{-}$ be the set of all closed intervals of $\mathbb{R}$, the set of all closed positive intervals of $\mathbb{R}$ and the set of all closed negative intervals of $\mathbb{R}$. Then, ${\mathcal{K}}_{C}$, ${\mathcal{K}}_{C}^{+}$, and ${\mathcal{K}}_{C}^{-}$ are defined as

For $\left[{\mathcal{Z}}_{{*}}, {\mathcal{Z}}^{{*}}\right], \left[{{{Q}}}_{*}, {{{Q}}}^{*}\right]\in {\mathcal{K}}_{C},$ the inclusion $"\subseteq "$ is defined by $\left[{\mathcal{Z}}_{{*}}, {\mathcal{Z}}^{{*}}\right]\subseteq \left[{{{Q}}}_{*}, {{{Q}}}^{*}\right]$, if and only if, ${{{Q}}}_{*}\le {\mathcal{Z}}_{{*}}$, ${\mathcal{Z}}^{{*}}\le {{{Q}}}^{*}.$

Remark 2.1. ^[36] The relation $"{\le }_{p}"$ defined on ${\mathcal{K}}_{C}$ by

for all $\left[{\mathcal{Q}}_{{*}}, {\mathcal{Q}}^{{*}}\right], \left[{\mathcal{Z}}_{{*}}, {\mathcal{Z}}^{{*}}\right]\in {\mathcal{K}}_{C},$ it is an pseudo-order relation.

Theorem 2.2. ^[9] If $𝛶:[\mu, \upsilon]\subset \mathbb{R}\to {\mathcal{K}}_{C}$ is an IV-F on such that $𝛶\left(\omega \right) = \left[{𝛶}_{*}\left(\omega \right), {𝛶}^{*}\left(\omega \right)\right]$, then $𝛶$ is Riemann integrable over $[\mu, \upsilon]$ if and only if, ${𝛶}_{*}\left(\omega \right)$ and ${𝛶}^{*}\left(\omega \right)$ are both Riemann integrable over $\left[\mu, \upsilon \right]$ such that

where, ${𝛶}_{*}, {𝛶}^{*}:[\mu, \upsilon]\to \mathbb{R}$.

The following interval Riemann-Liouville fractional integral operators were introduced by Budak et al. ^[26] and Lupulescu ^[37]:

Definition 2.3. ^[26,37] Let $\beta > 0$ and $L\left(\left[\mu, \upsilon \right], {\mathcal{K}}_{C}^{}\right)$ be the collection of all Lebesgue measurable IVFs on $[\mu, \upsilon]$. Then the interval left and right Riemann-Liouville fractional integral of $𝛶$ with order $\beta > 0$ are defined by

and

respectively, where $\mathit{\Gamma } \left(\varpi \right) = {\int }_{0}^{\infty }{\theta }^{\varpi -1}{e}^{-\theta }d\theta$ is the Euler gamma function. The interval left and right Riemann-Liouville fractional integral $\varpi$ based on left and right end point functions can be defined, that is

where

and

Similarly, the left and right end point functions can be used to define the right Riemann-Liouville fractional integral $𝛶$ of $\varpi$.

Definition 2.4. ^[19] A set $K = \left[\mu, \upsilon \right]\subset {\mathbb{R}}^{+} = \left(0, \infty \right)$ is said to be harmonically convex set, if, for all $\varpi, \mathcal{z}\in K, \theta \in \left[0, 1\right]$, we have

Definition 2.5. ^[19] The $𝛶:\left[\mu, \upsilon \right]\to {\mathbb{R}}^{+}$ is called harmonically convex function on $\left[\mu, \upsilon \right]$ if

for all $\varpi, \mathcal{z}\in \left[\mu, \upsilon \right], \theta \in \left[0, 1\right].$ If (13) is reversed then, $𝛶$ is called harmonically concave IVF on $\left[\mu, \upsilon \right]$.

Definition 2.6. ^[20] The positive real-valued function $𝛶:\left[\mu, \upsilon \right]\to {\mathbb{R}}^{+}$ is called $\mathcal{H}$−$h$-convex function on $\left[\mu, \upsilon \right]$ if

for all $\varpi, \mathcal{z}\in \left[\mu, \upsilon \right], \theta \in \left[0, 1\right],$ and ${{h}}:[0, 1]\subseteq [\mu, \upsilon]\to {\mathbb{R}}^{+}$ such that ${{h}}\not\equiv 0$. If (14) is reversed then, $𝛶$ is called $\mathcal{H}$−$h$-concave function on $\left[\mu, \upsilon \right]$. The set of all $\mathcal{H}$−$h$-convex ($\mathcal{H}$−$h$-concave) functions is denoted by

Definition 2.7. ^[35] The IVF $𝛶:\left[\mu, \upsilon \right]\to {\mathcal{K}}_{C}^{+}$ is called left and right ${{h}}$-convex IVF on $\left[\mu, \upsilon \right]$ if

for all $\varpi, \mathcal{z}\in \left[\mu, \upsilon \right], \theta \in \left[0, 1\right],$ and ${{h}}:[0, 1]\subseteq [\mu, \upsilon]\to {\mathbb{R}}^{+}$ such that ${{h}}\not\equiv 0$. If (15) is reversed then, $𝛶$ is called left and right ${{h}}$-concave IVF on $\left[\mu, \upsilon \right]$. The set of all left and right ${{h}}$-convex (left and right ${{h}}$-concave) IVF is denoted by

Definition 2.8. The IVF $𝛶:\left[\mu, \upsilon \right]\to {\mathcal{K}}_{C}^{+}$ is called left and right harmonically convex IVF (left and right $\mathcal{H}$-convex IVF) on $\left[\mu, \upsilon \right]$ if

for all $\varpi, \mathcal{z}\in \left[\mu, \upsilon \right], \theta \in \left[0, 1\right].$ If (16) is reversed then, $𝛶$ is called left and right harmonically concave IVF (left and right $\mathcal{H}$-concave IVF) on $\left[\mu, \upsilon \right]$.

Definition 2.9. The IVF $𝛶:\left[\mu, \upsilon \right]\to {\mathcal{K}}_{C}^{+}$ is called left and right harmonically $h$-convex (left and right $\mathcal{H}$−$h$-convex IVF) on $\left[\mu, \upsilon \right]$ if

for all $\varpi, \mathcal{z}\in \left[\mu, \upsilon \right], \theta \in \left[0, 1\right]$, for all $\varpi \in \left[\mu, \upsilon \right]$ and ${{h}}:[0, 1]\subseteq [\mu, \upsilon]\to {\mathbb{R}}^{+}$ such that ${{h}}\not\equiv 0$. If (17) is reversed then, $𝛶$ is called left and right $\mathcal{H}$−$h$-concave IVF on $\left[\mu, \upsilon \right]$. The set of all left and right $\mathcal{H}$−$h$-convex (left and right $\mathcal{H}$−$h$-concave) IVF is denoted by

Theorem 2.10. Let $\left[\mu, \upsilon \right]$ be harmonically convex set, and let $𝛶:\left[\mu, \upsilon \right]\to {\mathcal{K}}_{C}^{+}$ be an IVF such that

for all $\varpi \in \left[\mu, \upsilon \right]$. Then, $𝛶\in LRHSX\left(\left[\mu, \upsilon \right], {\mathcal{K}}_{C}^{+}, {{h}}\right),$ if and only if, ${𝛶}_{*}\left(\varpi \right)$, ${𝛶}^{*}\left(\varpi \right)\in HSX\left(\left[\mu, \upsilon \right], {\mathbb{R}}^{+}, {{h}}\right).$

Proof. Assume that ${𝛶}_{*}\left(\varpi \right)$, ${𝛶}^{*}\left(\varpi \right)\in HSX\left(\left[\mu, \upsilon \right], {\mathbb{R}}^{+}, {{h}}\right).$ Then, from (15), we have

And

Then by (18), (4) and (5), we obtain

that is

Hence, $𝛶\in LRHSX\left(\left[\mu, \upsilon \right], {\mathcal{K}}_{C}^{+}, {{h}}\right).$

Conversely, let $𝛶\in LRHSX\left(\left[\mu, \upsilon \right], {\mathcal{K}}_{C}^{+}, {{h}}\right).$ Then for all $\varpi, \mathcal{z}\in \left[\mu, \upsilon \right]$, $\theta \in \left[0, 1\right],$ we have

Therefore, from (18), left side of above inequality, we have

Again, from (18), we obtain

for all $\varpi, \mathcal{z}\in \left[\mu, \upsilon \right]$, $\theta \in \left[0, 1\right].$ Then by $\mathcal{H}$−$h$-convexity of $𝛶$, we have for all $\varpi, \mathcal{z}\in \left[\mu, \upsilon \right]$, $\theta \in \left[0, 1\right]$ such that

and

Hence, ${𝛶}_{*}\left(\varpi \right)$, ${𝛶}^{*}\left(\varpi \right)\in HSX\left(\left[\mu, \upsilon \right], {\mathbb{R}}^{+}, {{h}}\right)$.

Remark 2.11. On fixing ${𝛶}_{*}\left(\varpi \right) = {𝛶}^{*}\left(\varpi \right)$, then from Definition 2.9, we obtain Definition 2.6.

On fixing ${{h}}\left(\theta \right) = \theta$, then from Definition 2.9, we obtain Definition 2.8.

Example 2.12. We consider the IVFs $𝛶:[0, 2]\to {\mathcal{K}}_{C}^{+}$ defined by,

Since ${𝛶}_{*}\left(\varpi \right)$, ${𝛶}^{*}\left(\varpi \right)\in HSX\left(\left[\mu, \upsilon \right], {\mathbb{R}}^{+}, {{h}}\right)$ with ${{h}}\left(\theta \right) = \theta$. Hence, $𝛶\in LRHSX\left(\left[\mu, \upsilon \right], {\mathcal{K}}_{C}^{+}, {{h}}\right)$.

We shall develop a relationship between ${{h}}$-convex IVF and $\mathcal{H}-{{h}}$-convex IVF in the next finding.

Theorem 2.13. Let $𝛶:\left[\mu, \upsilon \right]\to {\mathcal{K}}_{C}^{+}$ be a IVF such that $𝛶\left(\varpi \right) = \left[{𝛶}_{*}\left(\varpi \right), {𝛶}^{*}\left(\varpi \right)\right],$ for all $\varpi \in \left[\mu, \upsilon \right]$. Then $𝛶\in LRHSX\left(\left[\mu, \upsilon \right], {\mathcal{K}}_{C}^{+}, {{h}}\right),$ if and only if, $𝛶\left(\frac{1}{\varpi }\right)\in LRSX\left(\left[\mu, \upsilon \right], {\mathcal{K}}_{C}^{+}, {{h}}\right).$

Proof. Since $𝛶\in LRHSX\left(\left[\mu, \upsilon \right], {\mathcal{K}}_{C}^{+}, {{h}}\right),$ then, for $\varpi, \mathcal{z}\in \left[\mu, \upsilon \right], \theta \in \left[0, 1\right]$, we have

Therefore, we have

Consider $\eta \left(\varpi \right) = 𝛶\left(\frac{1}{\varpi }\right)$. Taking $m = \frac{1}{\varpi }$ and $n = \frac{1}{\mathcal{z}}$ to replace $\varpi$ and $\mathcal{z}$, respectively. Then, by applying (20) we have

It follows that

that is

This concludes that $\eta \left(\varpi \right)\in LRSX\left(\left[\mu, \upsilon \right], {\mathcal{K}}_{C}^{+}, {{h}}\right),$.

Conversely, let $\eta \in LRSX\left(\left[\mu, \upsilon \right], {\mathcal{K}}_{C}^{+}, {{h}}\right), .$ Then, for all $\varpi, \mathcal{z}\in \left[\mu, \upsilon \right]$, $\theta \in \left[0, 1\right],$ we have

By using same steps as above, we have

that is

the proof the theorem has been completed.

Remark 2.14. If ${{h}}\left(\theta \right) = \theta$, and ${𝛶}_{*}\left(\varpi \right) = {𝛶}^{*}\left(\varpi \right)$, then from Theorem 2.14, we obtain Lemma 2.1 of ^[36].

3.   Hermite-Hadamard inequalities for harmonically ${{h}}$-convex fuzzy interval-valued functions

In this section, we will prove two types of inequalities. First one is $H$·$H$ and their variant forms, and the second one is $H$·$H$ Fejér inequalities for $\mathcal{H}$−$h$-convex IVFs where the integrands are IVFs. The family of Lebesgue measurable IVFs is denoted by $L\left(\left[\mu, \upsilon \right], {\mathcal{K}}_{C}^{+}\right)$ in the following.

Theorem 3.1. Let $𝛶:\left[\mu, \upsilon \right]\to {\mathcal{K}}_{C}^{+}$ be a IVF such that $𝛶\left(\varpi \right) = \left[{𝛶}_{*}\left(\varpi \right), {𝛶}^{*}\left(\varpi \right)\right],$ for all $\varpi \in \left[\mu, \upsilon \right].$ If $𝛶\in LRHSX\left(\left[\mu, \upsilon \right], {\mathcal{K}}_{C}^{+}, {{h}}\right)$ and $𝛶\in L\left(\left[\mu, \upsilon \right], {\mathcal{K}}_{C}^{+}\right)$, then

If $𝛶\left(\varpi \right)$ is concave IVF then

where $\delta \left(\varpi \right) = \frac{1}{\varpi }$.

Proof. Let $𝛶\in LRHSX\left(\left[\mu, \upsilon \right], {\mathcal{K}}_{C}^{+}, {{h}}\right),$. Then, by hypothesis, we have

Therefore, we have

Consider $\eta \left(\varpi \right) = 𝛶\left(\frac{1}{\varpi }\right).$ By Theorem 2.13, we have $\eta \left(\varpi \right)\in LRSX\left(\left[\mu, \upsilon \right], {\mathcal{K}}_{C}^{+}, {{h}}\right),$ then above inequality, we have

Multiplying both sides by ${\theta }^{\beta -1}$ and integrating the obtained result with respect to $\theta$ over $\left(\mathrm{0, 1}\right)$, we have

Let $\varpi = \frac{\left(1-\theta \right)\mu +\theta \upsilon }{\mu \upsilon }$ and $\mathcal{z} = \frac{\theta \mu +\left(1-\theta \right)\upsilon }{\mu \upsilon }.$ Then we have

It follows that

That is,

In a similar way as above, we have

Combining (23) and (24), we have

that is

Hence, the required result.

Remark 3.2. Followings results can be obtained through inequality (21):

On fixing ${{h}}\left(\theta \right) = \theta$, the following $H$·$H$ inequality is obtained, which is also new one;

On fixing ${{h}}\left(\theta \right) = \theta$ and $\beta = 1$, the following $H$·$H$ inequality is obtained, which is also new one;

On fixing ${{h}}\left(\theta \right) = \theta$ and ${𝛶}_{*}\left(\varpi \right) = {𝛶}^{*}\left(\varpi \right)$, the following $H$·$H$ inequality is obtained, see ^[36]:

On fixing ${{h}}\left(\theta \right) = \theta$ and ${𝛶}_{*}\left(\varpi \right) = {𝛶}^{*}\left(\varpi \right)$ with $\beta = 1$, the following $H$·$H$ inequality is obtained, see ^[19].

For the product of $\mathcal{H}$−$h$-convex IVFs, we now have some $H$·$H$ inequalities. These inequalities are modifications of previously published inequalities ^[38,34,43].

Theorem 3.3. Let $𝛶, \mathfrak{G}:\left[\mu, \upsilon \right]\to {\mathcal{K}}_{C}^{+}$ be a IVFs such that $𝛶\left(\varpi \right) = \left[{𝛶}_{*}\left(\varpi \right), {𝛶}^{*}\left(\varpi \right)\right]$ and $\mathfrak{G}\left(\varpi \right) = \left[{\mathfrak{G}}_{\mathfrak{*}}\left(\varpi \right), {\mathfrak{G}}^{\mathfrak{*}}\left(\varpi \right)\right]$ for all $\varpi \in \left[\mu, \upsilon \right]$, respectively. If $𝛶\in LRHSX\left(\left[\mu, \upsilon \right], {\mathcal{K}}_{C}^{+}, {{h}}_{1}\right)$, $\mathfrak{G}\in LRHSX\left(\left[\mu, \upsilon \right], {\mathcal{K}}_{C}^{+}, {{h}}_{2}\right)$, and $𝛶\times \mathfrak{G}\in L\left(\left[\mu, \upsilon \right], {\mathcal{K}}_{C}^{+}\right)$, then

Where $M\left(\mu, \upsilon \right) = 𝛶\left(\mu \right)\times \mathfrak{G}\left(\mu \right)+𝛶\left(\upsilon \right)\times \mathfrak{G}\left(\upsilon \right),$ $N\left(\mu, \upsilon \right) = 𝛶\left(\mu \right)\times \mathfrak{G}\left(\upsilon \right)+𝛶\left(\upsilon \right)\times \mathfrak{G}\left(\mu \right),$ and $M\left(\mu, \upsilon \right) = \left[{M}_{*}\left(\mu, \upsilon \right), {M}^{*}\left(\mu, \upsilon \right)\right]$ and $N\left(\mu, \upsilon \right) = \left[{N}_{*}\left(\mu, \upsilon \right), {N}^{*}\left(\mu, \upsilon \right)\right].$

Proof. Since $𝛶, \mathfrak{G}$ are $\mathcal{H}-{{h}}_{1}$ and $\mathcal{H}-{{h}}_{2}$-convex IVFs then, we have

and

From the definition of $\mathcal{H}$ −$h$-convex IVFs it follows that $0{\le }_{p}𝛶\left(\varpi \right)$ and $0{\le }_{p}\mathfrak{G}\left(\varpi \right)$, so

Analogously, we have

Adding (28) and (29), we have

Taking the result of multiplying (30) by ${\theta }^{\beta -1}$ and integrating it with respect to $\theta$ over (0, 1), we get

It follows that,

that is

Thus,

As a result, the theorem has been proven.

Theorem 3.4. Let $𝛶, \mathfrak{G}:\left[\mu, \upsilon \right]\to {\mathcal{K}}_{C}^{+}$ be a IVF such that $𝛶\left(\varpi \right) = \left[{𝛶}_{*}\left(\varpi \right), {𝛶}^{*}\left(\varpi \right)\right]$ and $\mathfrak{G}\left(\varpi \right) = \left[{\mathfrak{G}}_{\mathfrak{*}}\left(\varpi \right), {\mathfrak{G}}^{\mathfrak{*}}\left(\varpi \right)\right]$ for all $\varpi \in \left[\mu, \upsilon \right]$, respectively. If $𝛶\in LRHSX\left(\left[\mu, \upsilon \right], {\mathcal{K}}_{C}^{+}, {{h}}_{1}\right)$ and $\mathfrak{G}\in LRHSX\left(\left[\mu, \upsilon \right], {\mathcal{K}}_{C}^{+}, {{h}}_{2}\right)$ with ${{h}}_{1}\left(\frac{1}{2}\right){{h}}_{2}\left(\frac{1}{2}\right)\ne 0$, and $𝛶\times \mathfrak{G}\in L\left(\left[\mu, \upsilon \right], {\mathcal{K}}_{C}^{+}\right)$, then

Where $M\left(\mu, \upsilon \right) = 𝛶\left(\mu \right)\times \mathfrak{G}\left(\mu \right)+𝛶\left(\upsilon \right)\times \mathfrak{G}\left(\upsilon \right),$ $N\left(\mu, \upsilon \right) = 𝛶\left(\mu \right)\times \mathfrak{G}\left(\upsilon \right)+𝛶\left(\upsilon \right)\times \mathfrak{G}\left(\mu \right),$ and $M\left(\mu, \upsilon \right) = \left[{M}_{*}\left(\mu, \upsilon \right), {M}^{*}\left(\mu, \upsilon \right)\right]$ and $N\left(\mu, \upsilon \right) = \left[{N}_{*}\left(\mu, \upsilon \right), {N}^{*}\left(\mu, \upsilon \right)\right].$

Proof. Consider $𝛶, \mathfrak{G}:\left[\mu, \upsilon \right]\to {\mathcal{K}}_{C}^{+}$ are $\mathcal{H}-{{h}}_{1}$ and $\mathcal{H}-{{h}}_{2}$-convex IVFs. Then by hypothesis, we have

Inequality (31) is multiplied by ${\theta }^{\beta -1}$ and integrated over $(0, \mathrm{ }1),$

Taking $\varpi = \frac{\mu \upsilon }{\theta \mu +\left(1-\theta \right)\upsilon }$ and $\mathcal{z} = \frac{\mu \upsilon }{\left(1-\theta \right)\mu +\theta \upsilon }$, then we get

that is

As a result, the desired outcome has been achieved.

We now give $H$·$H$ Fejér inequalities for $\mathcal{H}$−$h$-convex IVFs. Firstly, we obtain the second $H$·$H$ Fejér inequality for $\mathcal{H}$−$h$-convex IVF.

Theorem 3.5. Let $𝛶:\left[\mu, \upsilon \right]\to {\mathcal{K}}_{C}^{+}$ be a IVF such that $𝛶\left(\varpi \right) = \left[{𝛶}_{*}\left(\varpi \right), {𝛶}^{*}\left(\varpi \right)\right]$ for all $\varpi \in \left[\mu, \upsilon \right]$, respectively. If $𝛶\in LRHSX\left(\left[\mu, \upsilon \right], {\mathcal{K}}_{C}^{+}, {{h}}\right)$, $𝛶\in L\left(\left[\mu, \upsilon \right], {\mathcal{K}}_{C}^{+}\right)$ and ${\mathit{\Omega}} :\left[\mu, \upsilon \right]\to \mathbb{R}, {\mathit{\Omega}} \left(\frac{1}{\frac{1}{\mu }+\frac{1}{\upsilon }-\frac{1}{\varpi }}\right) = {\mathit{\Omega}} \left(\varpi \right)\ge 0,$ then

If $𝛶\in LRHSV\left(\left[\mu, \upsilon \right], {\mathcal{K}}_{C}^{+}, {{h}}\right)$, then inequality (32) is reversed such that

Proof. Let $𝛶\in LRHSX\left(\left[\mu, \upsilon \right], {\mathcal{K}}_{C}^{+}, {{h}}\right)$ and ${\theta }^{\beta -1}{\mathit{\Omega}} \left(\frac{\mu \upsilon }{\theta \mu +\left(1-\theta \right)\upsilon }\right)\ge 0$. Then, we have

And

After adding (33) and (34), and integrating over $\left[0, 1\right],$ we get

that is

As a result, the desired result has been achieved.

Following result obtain the first interval fractional $H\cdot H$ Fejér inequality.

Theorem 3.6. Let $𝛶:\left[\mu, \upsilon \right]\to {\mathcal{K}}_{C}^{+}$ be a IVF such that $𝛶\left(\varpi \right) = \left[{𝛶}_{*}\left(\varpi \right), {𝛶}^{*}\left(\varpi \right)\right]$ for all $\varpi \in \left[\mu, \upsilon \right]$, respectively. If $𝛶\in LRHSX\left(\left[\mu, \upsilon \right], {\mathcal{K}}_{C}^{+}, {{h}}\right)$, $𝛶\in L\left(\left[\mu, \upsilon \right], {\mathcal{K}}_{C}^{+}\right)$ and ${\mathit{\Omega}} :\left[\mu, \upsilon \right]\to \mathbb{R}, {\mathit{\Omega}} \left(\frac{1}{\frac{1}{\mu }+\frac{1}{\upsilon }-\frac{1}{\varpi }}\right) = {\mathit{\Omega}} \left(\varpi \right)\ge 0,$ then

If $𝛶\in LRHSV\left(\left[\mu, \upsilon \right], {\mathcal{K}}_{C}^{+}, {{h}}\right)$, then inequality (36) is reversed such that

Proof. Since $𝛶\in LRHSX\left(\left[\mu, \upsilon \right], {\mathcal{K}}_{C}^{+}, {{h}}\right)$, then we have

Multiplying both sides by (37) by ${\theta }^{\beta -1}{\mathit{\Omega}} \left(\frac{\mu \upsilon }{\left(1-\theta \right)\mu +\theta \upsilon }\right)$ and then integrating the resultant with respect to $\theta$ over $\left[0, 1\right],$ we obtain

Let $\varpi = \frac{\mu \upsilon }{\theta \mu +\left(1-\theta \right)\upsilon }$. Then, we have

From (39), we have

that is

The theorem has been proved.

Remark 3.7. From Theorem 3.5 and Theorem 3.6, following result can be obtained:

On fixing ${{h}}\left(\theta \right) = \theta$ and $\beta = 1$ then following $H$·$H$ inequality is obtained, which is also new one:

On fixing ${{h}}\left(\theta \right) = \theta$ and ${\mathit{\Omega}} \left(\varpi \right) = 1$, the inequality (21) is obtained.

On fixing ${{h}}\left(\theta \right) = \theta$, ${\mathit{\Omega}} \left(\varpi \right) = 1$ and $\beta = 1$, the following $H$·$H$ inequality is obtained:

4.   Conclusion and future plan

The concept of left and right $\mathcal{H}$−$h$-convex IVFs has been discussed. We've also demonstrated that this type of convexity includes a few other types of classical convexity. The left and right $\mathcal{H}$−$h$-convex IVF is a harmonically convex function with distinct approaches that has been suggested. Furthermore, we have developed several novel classical and fractional integral inequalities of the Hermite-Hadamard and related Hermite-Hadamard type inequalities using the concept of left and right $\mathcal{H}$-$h$-convex IVF. The findings in this article can be used to identify different types of inequality. Furthermore, these results are universal and may be used to generate other, potentially useful, and fascinating inequalities using various fractional operators. In near future, we will generalize the class of left and right $\mathcal{H}$−$h$-convex IVFs and with the help of this class, we will try to obtain some new version of different type inequalities.

Conflict of interests

The authors declare that they have no competing interests.