Generalized Interval-Valued Convexity in Fractal Geometry

Abstract

The main goal of this study is to explain the idea of generalized interval-valued ($\mathbb{I}.\mathbb{V}$) convexity on a fractal set. We first define the basic operations for a generalized interval of ${\mathbb{R}}^s$ with $0<s\le 1$. Then, we expand the idea of ($\mathbb{I}.\mathbb{V}$) Riemann integration to ($\mathbb{I}.\mathbb{V}$) local fractal integration, which sets the stage for further research. This is followed by the proof of new Jensen, Hermite, Hadamard, Pachpatte, and Fejer inequalities that are ($\mathbb{I}.\mathbb{V}$) and have to do with the generalized class of ($\mathbb{I}.\mathbb{V}$) convexity defined over the fractal domain. We furnish validation through visual and comparative approaches. Our outcomes are the refinement of many existing results, indicating that they are fruitful. In fractal settings, this is the first paper to work on ($\mathbb{I}.\mathbb{V}$) convexity and some set-valued versions of Hermite–Hadamard-type containments.

1. Introduction and Preliminaries

The theory of convexity has made significant contributions in several fields of mathematical and applied sciences. Researchers have frequently used innovative and novel concepts to investigate and refine convexity and its related terms in recent years. It has appeared as an important subject of analysis due to its geometrical and analytical aspects. The study of convexity has progressed in many areas, but the theory of convex mappings is one of the most studied sub-domain because it has so many precise and unique uses. These kinds of mappings provide us unique minima and enjoy similar properties even though the dimension of space is not finite. Over the years, a wide range of generalizations of convexity have been used to refine the various existing inequalities.

The theory of inequalities has flourished exponentially due to its potential applications through the applications of convexity. The ongoing research and applicability of this theory motivated the researchers to work further on inequalities and convex mappings. The idea of convex maps and their generalizations can be used to produce the various fundamental inequalities.

Let $\mathsf{\Theta }:[\sigma ,\varsigma ]\to \mathbb{R}$ be a convex mapping. Then the Hadmard inequality is given as

$$\begin{array}{c}\hfill \mathsf{\Theta }\left({\displaystyle \frac{\sigma +\varsigma }{2}}\right)\le {\displaystyle \frac{1}{\varsigma -\sigma }}{\int }_{\sigma }^{\varsigma }\mathsf{\Theta }\left(\varrho \right)\mathrm{d}\varrho \le {\displaystyle \frac{\mathsf{\Theta }\left(\sigma \right)+\mathsf{\Theta }\left(\varsigma \right)}{2}}.\end{array}$$

This result is considered as a necessary and sufficient condition for a convex mapping. Also, it provides the error estimation of mid-point and trapezium-type quadrature rules. The literature is evident that Moore’s work [1] started the use of ($\mathbb{I}.\mathbb{V}$) calculus in numerical analysis, especially in automated error analysis. This work has remained a useful resource for researchers studying ($\mathbb{I}.\mathbb{V}$) techniques, which replace single-valued mappings, as well as for those dealing with uncertainty issues. On the other side, interval analysis has also gained the researcher’s attention because of its applications in error analysis.

In 2012, Chalco-Cano et al. [2] opened a new venue to conduct more research in mathematical inequalities via ($\mathbb{I}.\mathbb{V}$) mappings, especially utilizing the generalized Hukuhara differences. In 2017, Costa et al. [3] used the fuzzy valued mappings to construct novel forms of classical inequalities. While [4] introduces and studies coordinated log-h h-convexity for interval-valued functions, to make fractional inclusion relations, the authors of [5] used the idea of ($\mathbb{I}.\mathbb{V}$) p-convex mapping. After these changes, Budak et al. [6] were able to successfully explore the idea of fractional ($\mathbb{I}.\mathbb{V}$) variations oin the Hermite–Hadamard type inequality. In [7], authors developed a fresh generic class of convexity using the quasi-weighted mean and containment ordering relation. They have derived some unified ($\mathbb{I}.\mathbb{V}$) inequalities utilizing the notion of generalized convexity. In 2021, Kara et al. [8] developed some two-dimensional containments utilizing coordinated ($\mathbb{I}.\mathbb{V}$) convex mappings. Khan et al. [9] used generalized ($\mathbb{I}.\mathbb{V}$) fuzzy convex mappings to come up with some fuzzy fractional versions of Jensen [10] and Hermite-Hadamard [11,12] inequalities. The authors of [13] talked about the concept of left-right ($\mathbb{I}.\mathbb{V}$) fuzzy bi-convex mapping and looked at some fractional integral containments that are connected to the Atangana-Baleanu fractional operator. For more detail, see [14,15,16].

Many significant refinements of calculus have emerged in recent eras to overcome the limitations of classical relations and to visualize real-world problems. Fractional calculus addresses the problem of fractional order derivatives and antiderivatives and generalizes the already known concepts. In 2012, Yang [17] initiated the idea of local fractional calculus based on a fractal set to seek out the problems concerned with Riemann and Caputo fractional calculus. From these operators, one can recover the classical results. The concept of fractal sets was presented and elaborated by Mandelbrot [18] (pp. 25–74). Fractals have utility in photography creation, soil mechanic, and scattering theory, etc. These days, fractal calculus is very useful and complete for studying math problems with mappings that are not differentiable and appearing in nature sciences. Inspired by the properties above, Mo et al. [19] set up the idea of extended convexity over ${\mathbb{R}}^s$ and examined specific algebraic characteristics associated with that novel class. Additionally, she created the new fractal counterparts of inequalities.

In [20], Sarikaya and Budak created the Ostrowski type of integral inequalities in 2017. They did this by using the idea of extended convexity and a local fractional procedure. The idea of generalized harmonic convex mappings to formulate Hermite-Hadamard and Hermite-Hadamard Fejer inequalities was proved by Noor et al. [21] in 2018. They also examined various specific situations to supplement their findings with existing research. Sun et al. [22] used general harmonic convex mapping to prove new trapezium-type inequalities with local fractional identity in 2020. Also, the Hermite-Hadamard type inequality for general harmonic convex mappings over fractal sets employing local fractional calculus was discussed in [23]. For comprehensive investigation, see [24,25,26,27,28,29].

2. Preliminaries

To further explore the Yang [17] calculus, we recollect some well-known s-sets.

1.

${\mathbb{Z}}^s:=\{\pm 0^s,\pm 1^s,\pm 2^s,\dots \}$

2.

${\mathbb{Q}}^s:=\{v^s={\left({\displaystyle \frac{p_1}{q_1}}\right)}^s={\displaystyle \frac{p_1^s}{q_1^s}}:p_1,q_1\in \mathbb{Z},q_1\ne 0\}$

3.

${\left({\mathbb{Q}}^s\right)}^c:=\{v^s\ne \left({\displaystyle \frac{p_1^s}{q_1^s}}\right):p_1,q_1\in \mathbb{Z},q_1\ne 0\}$

4.

${\mathbb{R}}^s:={\mathbb{Q}}^s\cup {\left({\mathbb{Q}}^s\right)}^c$.

The operations addition ′ + ′, and multiplication ′ * ′ over s-like subsets ${\mathbb{R}}^s$ are defined as follows:

$$\begin{array}{c}\hfill {c_1}^s+{d_1}^s:={(c_1+d_1)}^s,\end{array}$$

$$\begin{array}{c}\hfill {c_1}^s\ast {d_1}^s={c_1}^s{d_1}^s:={\left(c_1{d}_1\right)}^s,\end{array}$$

and both ${c_1}^s+{d_1}^s,{c_1}^s{d_1}^s\in {\mathbb{R}}^s$.

Observe that $({\mathbb{R}}^s,+)$ is a commutative group. Let ${c_1}^s,{d_1}^s{e_1}^s\in {\mathbb{R}}^s$ be arbitrary, and then,

1.

${c_1}^s+{d_1}^s={d_1}^s+{c_1}^s$

2.

$({c_1}^s+{d_1}^s)+{e_1}^s={c_1}^s+({d_1}^s+{e_1}^s)$

3.

$0^s$ is the additive identity of ${\mathbb{R}}^s$ such that $0^s+{c_1}^s={c_1}^s+0^s$, $\forall {c_1}^s\in {\mathbb{R}}^s$

4.

For any ${c_1}^s$ then there exists ${(-c_1)}^s\in {\mathbb{R}}^s$ such that ${c_1}^s+{(-c_1)}^s=0^s$.

Also $({\mathbb{R}}^s,\ast )\setminus \left\{0\right\}$ is a commutative group. For any ${c_1}^s,{d_1}^s,{e_1}^s\in {\mathbb{R}}^s$ be arbitrary, and then:

1.

${c_1}^s{d_1}^s={d_1}^s{c_1}^s$

2.

$\left({c_1}^s{d_1}^s\right){e_1}^s={c_1}^s\left({d_1}^s{e_1}^s\right)$

3.

$1^s\in {\mathbb{R}}^s$ then for each ${c_1}^s\in {\mathbb{R}}^s$ we have $1^s{c_1}^s={c_1}^s{1}^s={c_1}^s$

4.

For each ${c_1}^s\in {\mathbb{R}}^s\setminus \left\{0\right\}$ there exists ${\left({\displaystyle \frac{1}{c_1}}\right)}^s$ such that ${c_1}^s{\left({\displaystyle \frac{1}{c_1}}\right)}^s={\left(c_1{\displaystyle \frac{1}{c_1}}\right)}^s=1^s$.

Remark 1. 

• $({\mathbb{R}}^s,+,\ast )$ is a field.
• The order relation ≤ on ${\mathbb{R}}^s$ is defined as follows ${c_1}^s\le {d_1}^s$⇔$c_1\le d_1$ in $\mathbb{R}$. Then, $({\mathbb{R}}^s,+,\ast )$ is an ordered field.

Definition 1. 

A mapping $\mathsf{\Theta }:\mathbb{R}\to {\mathbb{R}}^s$ is called local fractional continuous at $v_0$, if for all $ϵ>0$ there exists $\delta >0$ such that

$$\begin{array}{c}\hfill \left|\mathsf{\Theta }\left(v\right)-\mathsf{\Theta }\left(v_0\right)\right|<{ϵ}^s,\mathit{for}\mathit{all}v\mathit{satisfying}|v-v_0|<\delta .\end{array}$$

If $\mathsf{\Theta }\left(v\right)$ is local fractional continuous at $v\in (\sigma ,\varsigma )$, then we can write $\mathsf{\Theta }\left(v\right)\in C_s(\sigma ,\varsigma )$.

Now, we deliver the local differentiability.

Definition 2. 

The local fractional derivative of Θ defined over s-type sets is defined as

$$\begin{array}{c}\hfill D_v^s\mathsf{\Theta }\left(v\right)={\mathsf{\Theta }}^s\left(v\right)={\left.{\displaystyle \frac{{\mathrm{d}}^s\mathsf{\Theta }\left(v\right)}{{\left(\mathrm{d}v\right)}^s}}\right|}_{v=v_0}:=\underset{v\to v_0}{lim}{\displaystyle \frac{{▵}^s(\mathsf{\Theta }\left(v\right)-\mathsf{\Theta }\left(v_0\right))}{{(v-v_0)}^s}},\end{array}$$

provided that limit exists and using the notation ${▵}^s(\mathsf{\Theta }\left(v\right)-\mathsf{\Theta }\left(v_0\right))=\mathsf{\Gamma }(1+s)(\mathsf{\Theta }\left(v\right)-\mathsf{\Theta }\left(v_0\right))$.

Moreover, we iterate the above definition and write ${\mathsf{\Theta }}^{\left(k\right)s}\left(v\right)=\stackrel{\left(k\right)times}{\overbrace{D_v^s\mathsf{\Theta }\left(v\right)·D_v^s\mathsf{\Theta }\left(v\right)\dots D_v^s\mathsf{\Theta }\left(v\right)}}$ for any $v\in [\sigma ,\varsigma ]$; when the corresponding iterated limit exists we call this expression the k-th order derivative of s-type. If ${\mathsf{\Theta }}^{\left(k\right)s}$ exists at each $v\in (\sigma ,\varsigma )$, then we write $\mathsf{\Theta }\in D_{\left(k\right)s}$, where the space of k-order local differentiable mappings is represented by $D_{(k+1)s}$ and $k\in \mathbb{N}$.

Analogously, the integral operators is defined as

Definition 3. 

Let $▵=\{{\varkappa }_0,{\varkappa }_1,{\varkappa }_2,\dots ,{\varkappa }_n\}$, $n\in \mathbb{N}$, be a division of $[\sigma ,\varsigma ]$ such that ${\varkappa }_0<{\varkappa }_1<{\varkappa }_2<\dots <{\varkappa }_n$. Then, the fractal integral of Θ on $[\sigma ,\varsigma ]$ is defined as follows:

$$\begin{array}{c}\hfill {}_{\sigma }I_{\varsigma }^s\mathsf{\Theta }\left(\varrho \right)={\displaystyle \frac{1}{\mathsf{\Gamma }(1+s)}}{\int }_{\sigma }^{\varsigma }\mathsf{\Theta }\left(\varrho \right){\left(\mathrm{d}\varrho \right)}^s={\displaystyle \frac{1}{\mathsf{\Gamma }(1+s)}}\underset{n\to \infty }{lim}\sum _{i=1}^n\mathsf{\Theta }\left({\varrho }_i\right){(▵{\varrho }_i)}^s,\end{array}$$

where $▵{\varrho }_i={\varkappa }_i-{\varkappa }_{i-1}$ and ${\varrho }_i\in [{\varkappa }_{i-1},{\varkappa }_i]$ for $i=1,2,3\dots n$. If the limit exists, then we say that Θ is local fractional integrable.

Lemma 1. 

The following equalities hold:

1.

If $\mathsf{\Theta }\left(\varrho \right)=r^s\left(\varrho \right)\in C_s[\sigma ,\varsigma ]$, then

$$\begin{array}{c}\hfill {}_{\sigma }I_{\varsigma }^s\mathsf{\Theta }\left(\varrho \right)=r\left(\varsigma \right)-r\left(\sigma \right).\end{array}$$

2.

If $\mathsf{\Theta }\left(\varrho \right)={\varrho }^{ks}\in C_s[\sigma ,\varsigma ]$, then

$$\begin{array}{c}\hfill {\displaystyle \frac{{\mathrm{d}}^s{\varrho }^{ks}}{{\left(\mathrm{d}\varrho \right)}^s}}={\displaystyle \frac{\mathsf{\Gamma }(1+ks)}{\mathsf{\Gamma }(1+(k-1\left)s\right)}}{\varrho }^{(k-1)s}.\end{array}$$

3.

If $\mathsf{\Theta }\left(\varrho \right)={\varrho }^{ks}\in C_s[\sigma ,\varsigma ]$, then

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{\mathsf{\Gamma }(1+s)}}{\int }_{\sigma }^{\varsigma }{\varrho }^{ks}{\left(\mathrm{d}\varrho \right)}^s={\displaystyle \frac{\mathsf{\Gamma }(1+ks)}{\mathsf{\Gamma }(1+(k+1\left)s\right)}}({\varsigma }^{(k+1)s}-{\sigma }^{(k+1)s}).\end{array}$$

In 2014, Mo et al. [19] studied the concept of fractal convexity in the context of local calculus.

Definition 4. 

A function $\mathsf{\Theta }:[\sigma ,\varsigma ]\to {\mathbb{R}}^s$ is called generalized convex if

$$\begin{array}{c}\hfill \mathsf{\Theta }(\varkappa \sigma +(1-\varkappa )\varsigma )\le {\varkappa }^s\mathsf{\Theta }\left(\sigma \right)+{(1-\varkappa )}^s\mathsf{\Theta }\left(\varsigma \right),\end{array}$$

(1)

$\varkappa \in [0,1]$ and $0<s\le 1.$

Additionally, Mo and Sui [19] formulated the Jensen inequality via generalized convexity.

Theorem 1. 

Let $\mathsf{\Theta }:[\sigma ,\varsigma ]\to {\mathbb{R}}^s$ be a generalized convex mapping, then

$$\begin{array}{c}\hfill \mathsf{\Theta }\left(\sum _{i=1}^n{\varkappa }_i{\varrho }_i\right)\le \sum _{i=1}^n{\varkappa }_i^s\mathsf{\Theta }\left({\varrho }_i\right).\end{array}$$

In the next section, we provide our main outcomes.

3. Main Results

Here, ${\mathbb{R}}_I^s$ denote the space of all intervals of the space ${\mathbb{R}}^s$ with $0<s\le 1$. If $\mathcal{J}\in {\mathbb{R}}^s$, then

$$\mathcal{J}=[{\sigma }^s,{\varsigma }^s]=\{\varrho \in {\mathbb{R}}^s|{\sigma }^s\le \varrho \le {\varsigma }^s{\sigma }^s,{\varsigma }^s\in {\mathbb{R}}^s\}.$$

If ${\sigma }^s={\varsigma }^s$, then the interval is said to be degenerate. If $\mathcal{J}$ is positive, then ${\sigma }^s>0$ and similarly $\mathcal{J}$ is negative if ${\varsigma }^s<0$. From now on, let ${\mathbb{R}}_I^{s,+}$ denote the set of positive and ${\mathbb{R}}_I^{s,-}$ the set of negative fractals intervals of ${\mathbb{R}}^s$, respectively.

Now, we define the scalar multiplication of any fractal interval in the following way:

$$\begin{array}{c}\hfill {\lambda }^s\mathcal{J}=\left\{\begin{array}{c}\left[{\lambda }^s{\sigma }^s,{\lambda }^s{\varsigma }^s\right]=\left[{\left(\lambda \sigma \right)}^s,{\left(\lambda \varsigma \right)}^s\right],{\lambda }^s>0\hfill \\ \left[{\lambda }^s{\varsigma }^s,{\lambda }^s{\sigma }^s\right]=\left[{\left(\lambda \varsigma \right)}^s,{\left(\lambda \sigma \right)}^s\right],{\lambda }^s<0.\hfill \end{array}\right.\end{array}$$

The inclusion between two fractal intervals ${\mathcal{J}}_1=[{\sigma }^s,{\varsigma }^s]$ and ${\mathcal{J}}_2=[{c_1}^s,{d_1}^s]$ is defined as follows:

$$\begin{array}{c}\hfill {\mathcal{J}}_1\subseteq {\mathcal{J}}_2\iff {c_1}^s\le {\sigma }^s\&{\varsigma }^s\le {d_1}^s.\end{array}$$

(2)

The operations of Minkowski addition and difference for fractal intervals are given as follows: Let ${\mathcal{J}}_1=\left[{\sigma }^s,{\varsigma }^s\right]$ and ${\mathcal{J}}_2=\left[{c_1}^s,{d_1}^s\right]$; then,

$$\begin{array}{cc}\hfill & {\mathcal{J}}_1+{\mathcal{J}}_2=\left[{\sigma }^s,{\varsigma }^s\right]+\left[{c_1}^s,{d_1}^s\right]=\left[{\sigma }^s+{c_1}^s,{\varsigma }^s+{d_1}^s\right]=\left[{(\sigma +c_1)}^s,{(\varsigma +d_1)}^s\right].\hfill \\ \hfill & {\mathcal{J}}_1-{\mathcal{J}}_2=\left[{\sigma }^s,{\varsigma }^s\right]-\left[{c_1}^s,{d_1}^s\right]=\left[{\sigma }^s-{c_1}^s,{\varsigma }^s-{d_1}^s\right]=\left[{(\sigma -c_1)}^s,{(\varsigma -d_1)}^s\right].\hfill \end{array}$$

It is worth noting that the previous differences only hold when $l\left({\mathcal{J}}_1\right)>l\left({\mathcal{J}}_2\right)$, where l denotes the length of an interval which is defined as $l\left(\mathcal{J}\right)={\varsigma }^s-{\sigma }^s={(\varsigma -\sigma )}^s$.

Now, we define the notion of the Hausdorff-Pompeiu distance between two fractal intervals ${\mathcal{J}}_1=[{\sigma }^s,{\varsigma }^s]$ and ${\mathcal{J}}_2=[{c_1}^s,{d_1}^s]$ as follows:

$$\begin{array}{c}\hfill d({\mathcal{J}}_1,{\mathcal{J}}_2)=max\left\{\right|{\sigma }^s-{c_1}^s|,|{\varsigma }^s-{d_1}^s\left|\right\}=max\left\{\right|{(\sigma -c_1)}^s|,|{(\varsigma -d_1)}^s\left|\right\}\end{array}$$

In the following sequel, we define the product of ${\mathcal{J}}_1$ and ${\mathcal{J}}_2$ as follows:

$$\begin{array}{c}\hfill {\mathcal{J}}_1{\mathcal{J}}_2=\left[min\{{\left({\sigma }^s{c}_1^s\right)}^s,{\left({\sigma }^s{d}_1^s\right)}^s,{\left({\varsigma }^s{c}_1^s\right)}^s,{\left({\varsigma }^s{d}_1^s\right)}^s\},max\{{\left({\sigma }^s{c}_1^s\right)}^s,{\left({\sigma }^s{d}_1^s\right)}^s,{\left({\varsigma }^s{c}_1^s\right)}^s,{\left({\varsigma }^s{d}_1^s\right)}^s\}\right].\end{array}$$

Now, we present some algebraic properties of fractal intervals of ${\mathbb{R}}^s$.

1.

Commutativity under addition and multiplications:

$$\begin{array}{cc}\hfill & \left(i\right){\mathcal{J}}_1+{\mathcal{J}}_2={\mathcal{J}}_2+{\mathcal{J}}_1,\forall {\mathcal{J}}_1,{\mathcal{J}}_2\in {\mathbb{R}}_I^s.\hfill \\ \hfill & \left(ii\right){\mathcal{J}}_1{\mathcal{J}}_2={\mathcal{J}}_2{\mathcal{J}}_1,\forall {\mathcal{J}}_1,{\mathcal{J}}_2\in {\mathbb{R}}_I^s.\hfill \end{array}$$

2.

Associativity under addition and multiplications:

$$\begin{array}{cc}\hfill & \left(i\right)({\mathcal{J}}_1+{\mathcal{J}}_2)+{\mathcal{J}}_3={\mathcal{J}}_1+({\mathcal{J}}_2+{\mathcal{J}}_3),\forall {\mathcal{J}}_1,{\mathcal{J}}_2,{\mathcal{J}}_3\in {\mathbb{R}}_I^s.\hfill \\ \hfill & \left(ii\right)\left({\mathcal{J}}_1{\mathcal{J}}_2\right){\mathcal{J}}_3={\mathcal{J}}_1\left({\mathcal{J}}_2{\mathcal{J}}_3\right),\forall {\mathcal{J}}_1,{\mathcal{J}}_2,{\mathcal{J}}_3\in {\mathbb{R}}_I^s.\hfill \end{array}$$

3.

Existence of both identities:

$$\begin{array}{cc}\hfill & 0^s+\mathcal{J}=[0^s,0^s]+[{\sigma }^s,{\varsigma }^s]=\mathcal{J}+0^s,\forall \mathcal{J}\in {\mathbb{R}}_I^s.\hfill \\ \hfill & 1^s\mathcal{J}=[1^s,1^s].[{\sigma }^s,{\varsigma }^s]=\mathcal{J}{1}^s,\forall \mathcal{J}\in {\mathbb{R}}_I^s.\hfill \end{array}$$

4.

Associativity:

$$\begin{array}{c}\hfill u^s\left(v^s\mathcal{J}\right)={\left(uv\right)}^s\mathcal{J},u^s,v^s\in {\mathbb{R}}_s\&\mathcal{J}\in {\mathbb{R}}_I^s.\end{array}$$

5.

First distributive law:

$$\begin{array}{c}\hfill u^s({\mathcal{J}}_1+{\mathcal{J}}_2)=u^s{\mathcal{J}}_1+u^s{\mathcal{J}}_2,u^s\in {\mathbb{R}}^s\&{\mathcal{J}}_1,{\mathcal{J}}_2\in {\mathbb{R}}_I^s.\end{array}$$

6.

Second distributive law:

$$\begin{array}{c}\hfill (u^s+v^s)\mathcal{J}={(u+v)}^s=u^s\mathcal{J}+v^s\mathcal{J},u^s,v^s\in {\mathbb{R}}^s\&\mathcal{J}\in {\mathbb{R}}_I^s.\end{array}$$

7.

In general, distributive law does not hold. One can easily check that for ${\mathcal{J}}_1=[2^s,3^s]$, ${\mathcal{J}}_2=[1^s,2^s]$ and ${\mathcal{J}}_3=[{(-2)}^s,{(-1)}^s]$, distributive property does not hold.

8.

Inverse does not exist. One can verify that for $[2^s,5^s]$, the inverse does not exist.

Local Integration

Let $\mathsf{\Theta }:[\sigma ,\varsigma ]\to {\mathcal{R}}_I^s$ be a generalized $(\mathbb{I},\mathbb{V})$ mapping. Let $\mathcal{P}=\{{\varkappa }_0,{\varkappa }_1,\dots ,{\varkappa }_{n-1},{\varkappa }_n\}$ be a partition of $[\sigma ,\varsigma ]$, where $[{\varkappa }_{i-1},{\varkappa }_i],i=1,2,3,\dots n$ is a general subinterval of $[\sigma ,\varsigma ]$. Let $\mathsf{\Psi }\left([\sigma ,\varsigma ]\right)$ be the set of all partitions of $[\sigma ,\varsigma ]$ and $\mathsf{\Psi }\left(\rho ,[\sigma ,\varsigma ]\right)$ be the collection of all points such that the mesh of any partition $\mathcal{P}$ is $\parallel \mathcal{P}\parallel <\rho$. Moreover, the norm or mesh is the length of the longest generalized interval. Let ${\varrho }_i\in [{\varkappa }_{i-1},{\varkappa }_i]$, then the local sum is defined as

$$\begin{array}{c}\hfill S(\mathsf{\Theta },\mathcal{P},\rho )={\displaystyle \frac{1}{\mathsf{\Gamma }(1+s)}}\sum _{i=1}^n\mathsf{\Theta }\left({\varrho }_i\right){({\varkappa }_i-{\varkappa }_{i-1})}^s.\end{array}$$

Any $\mathsf{\Theta }:[\sigma ,\varsigma ]\to {\mathcal{R}}_I^s$ is called $(\mathbb{I},\mathbb{V})$ local fractional integrable on $[\sigma ,\varsigma ]$; if there exists $\mathcal{C}\in {\mathbb{R}}_I^s$, the following holds: For each $ϵ>0$ there exists $\rho >o$, such that

$$\begin{array}{c}\hfill d\left(S(\mathsf{\Theta },\mathcal{P},\rho ),\mathcal{C}\right)<{ϵ}^s\end{array}$$

for every local sum S of $\mathsf{\Theta }$ related to each $\mathcal{P}\in \mathsf{\Psi }\left(\rho ,[\sigma ,\varsigma ]\right)$. Here, $\mathcal{C}$ is called the ($\mathbb{I}.\mathbb{V}$) local fractional integral of $\mathsf{\Theta }$ on $[\sigma ,\varsigma ]$ and is given as follows:

If $\parallel \mathcal{P}\parallel \to 0$, then $\underset{n\to \infty }{lim}{\displaystyle \frac{1}{\mathsf{\Gamma }(1+s)}}{\sum }_{i=1}^n\mathsf{\Theta }\left({\varrho }_i\right){({\varkappa }_i-{\varkappa }_{i-1})}^s={\displaystyle \frac{1}{\mathsf{\Gamma }(1+s)}}{\int }_{\sigma }^{\varsigma }\mathsf{\Theta }\left(\chi \right){\left(\mathrm{d}\chi \right)}^s$. Also, its end point representation is given as

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{\mathsf{\Gamma }(1+s)}}{\int }_{\sigma }^{\varsigma }\mathsf{\Theta }\left(\chi \right){\left(\mathrm{d}\chi \right)}^s=\left[{\displaystyle \frac{1}{\mathsf{\Gamma }(1+s)}}{\int }_{\sigma }^{\varsigma }{\mathsf{\Theta }}_{\ast }\left(\chi \right){\left(\mathrm{d}\chi \right)}^s,{\displaystyle \frac{1}{\mathsf{\Gamma }(1+s)}}{\int }_{\sigma }^{\varsigma }{\mathsf{\Theta }}^{\ast }\left(\chi \right){\left(\mathrm{d}\chi \right)}^s\right].\end{array}$$

Theorem 2. 

Let $\mathsf{\Theta }:[\sigma ,\varsigma ]\to {\mathbb{R}}_I^s$ be an ($\mathbb{I}.\mathbb{V}$) mapping, such that $\mathsf{\Theta }=[{\mathsf{\Theta }}_{\ast },{\mathsf{\Theta }}^{\ast }]$ with ${\mathsf{\Theta }}_{\ast }\le {\mathsf{\Theta }}^{\ast }$. Then, Θ is ($\mathbb{I}.\mathbb{V}$) local fractional integrable if and only if ${\mathsf{\Theta }}_{\ast }$ and ${\mathsf{\Theta }}^{\ast }$ are local fractional integrable on $[\sigma ,\varsigma ]$.

Proof. 

Let $\mathcal{P}$ be a partition and let Θ be a ($\mathbb{I}.\mathbb{V}$) local fractional integrable mapping. Then, for each ${ϵ}^s\ge 0^s$, there exists $C\in {\mathbb{R}}_I^{s,+}$ such that

$$\begin{array}{c}\hfill d\left(S(\mathsf{\Theta },\mathcal{P},\rho ),\mathcal{C}\right)<{ϵ}^s\end{array}$$

This can be transformed as

$$\begin{array}{c}\hfill \left|S({\mathsf{\Theta }}_{\ast },\mathcal{P},\rho )-{\mathcal{C}}_{\ast }\right|<{ϵ}^s,\end{array}$$

(3)

and

$$\begin{array}{c}\hfill \left|S({\mathsf{\Theta }}^{\ast },\mathcal{P},\rho )-{\mathcal{C}}^{\ast }\right|<{ϵ}^s.\end{array}$$

(4)

From (3) and (4), it is clear that both ${\mathsf{\Theta }}_{\ast }$ and ${\mathsf{\Theta }}^{\ast }$ are local integrable mappings. To prove the converse, we assume that ${\mathsf{\Theta }}_{\ast }$ and ${\mathsf{\Theta }}^{\ast }$ are local integrable mappings and ${\mathsf{\Theta }}_{\ast }\le {\mathsf{\Theta }}^{\ast }$. Then, it can be written as ${\displaystyle \frac{1}{\mathsf{\Gamma }(1+s)}}{\int }_{\sigma }^{\varsigma }{\mathsf{\Theta }}_{\ast }\left(\varkappa \right){\left(\mathrm{d}\varkappa \right)}^s={\mathcal{C}}_{\ast }$ and ${\displaystyle \frac{1}{\mathsf{\Gamma }(1+s)}}{\int }_{\sigma }^{\varsigma }{\mathsf{\Theta }}^{\ast }\left(\varkappa \right){\left(\mathrm{d}\varkappa \right)}^s=C^{\ast }$ for some $C_{\ast },C^{\ast }\in {\mathbb{R}}^s$. Then for each $ϵ>0$ there exist ${\rho }_{\ast }$ and ${\rho }^{\ast }$ such that

$$\begin{array}{c}\hfill \left|S({\mathsf{\Theta }}_{\ast },\mathcal{P},{\rho }_{\ast })-{\mathcal{C}}_{\ast }\right|<{ϵ}^s,\end{array}$$

(5)

and

$$\begin{array}{c}\hfill \left|S({\mathsf{\Theta }}_{\ast },\mathcal{P},{\rho }^{\ast })-{\mathcal{C}}^{\ast }\right|<{ϵ}^s,\end{array}$$

(6)

By selecting $\rho =min\{{\rho }_{\ast },{\rho }^{\ast }\}$ and combining (5) and (6), we obtain the desired definition and this concludes the proof. □

4. Interval-Valued Generalized Convex Functions

Now, we introduce the concept of ($\mathbb{I}.\mathbb{V}$) convex mappings over fractal sets, which is defined as follows.

Definition 5. 

Suppose $\mathsf{\Theta }:[\sigma ,\varsigma ]\to {\mathbb{R}}_I^s$ is an ($\mathbb{I}.\mathbb{V}$) mapping such that $\mathsf{\Theta }=[{\mathsf{\Theta }}_{\ast },{\mathsf{\Theta }}^{\ast }]$ with ${\mathsf{\Theta }}_{\ast }\le {\mathsf{\Theta }}^{\ast }$, then Θ is called a generalized ($\mathbb{I}.\mathbb{V}$) convex mapping if

$$\begin{array}{c}\hfill \mathsf{\Theta }(\varkappa \sigma +(1-\varkappa )\varsigma )\supseteq {\varkappa }^s\mathsf{\Theta }\left(\sigma \right)+{(1-\varkappa )}^s\mathsf{\Theta }\left(\varsigma \right),\end{array}$$

where $\varkappa \in [0,1],0<s\le 1$.

If the inclusion holds in opposite direction, then Θ is said to be a generalized $(\mathbb{I},\mathbb{V})$ concave mapping.

For the convenience the collection of generalized ($\mathbb{I}.\mathbb{V}$) convex mappings, generalized ($\mathbb{I}.\mathbb{V}$) concave mappings, generalized convex mappings and generalized concave mappings are denoted by $SIGX\left([\sigma ,\varsigma ],{\mathbb{R}}_I^s\right)$, $SIGV\left([\sigma ,\varsigma ],{\mathbb{R}}_I^s\right)$, $SGX\left([\sigma ,\varsigma ],{\mathbb{R}}^s\right)$ and $SGV\left([\sigma ,\varsigma ],{\mathbb{R}}^s\right)$, respectively.

Theorem 3. 

Suppose $\mathsf{\Theta }:[\sigma ,\varsigma ]\to {\mathbb{R}}_I^s$ is an ($\mathbb{I}.\mathbb{V}$) mapping, such that $\mathsf{\Theta }=[{\mathsf{\Theta }}_{\ast },{\mathsf{\Theta }}^{\ast }]$ with ${\mathsf{\Theta }}_{\ast }\le {\mathsf{\Theta }}^{\ast }$. Then, $\mathsf{\Theta }\in SIGX\left([\sigma ,\varsigma ],{\mathbb{R}}_I^s\right)\iff {\mathsf{\Theta }}_{\ast }\in SGX\left([\sigma ,\varsigma ],{\mathbb{R}}^s\right)$ & ${\mathsf{\Theta }}^{\ast }\in SGV\left([\sigma ,\varsigma ],{\mathbb{R}}^s\right)$.

Proof. 

Assume that $\mathsf{\Theta }\in SIGV\left([\sigma ,\varsigma ],{\mathbb{R}}_I^s\right)$ and $\varrho ,y\in [\sigma ,\varsigma ]$ and $\varkappa \in [0,1]$; then,

$$\begin{array}{c}\hfill \mathsf{\Theta }(\varkappa \varrho +(1-\varkappa )y)\supseteq {\varkappa }^s\mathsf{\Theta }\left(\varrho \right)+{(1-\varkappa )}^s\mathsf{\Theta }\left(y\right).\end{array}$$

This suggests that

$$\begin{array}{cc}\hfill & \left[{\mathsf{\Theta }}_{\ast }(\varkappa \varrho +(1-\varkappa )y),{\mathsf{\Theta }}^{\ast }(\varkappa \varrho +(1-\varkappa )y)\right]\hfill \\ \hfill & \supseteq \left[{\varkappa }^s{\mathsf{\Theta }}_{\ast }\left(\varrho \right)+{(1-\varkappa )}^s{\mathsf{\Theta }}_{\ast }\left(y\right),{\varkappa }^s{\mathsf{\Theta }}^{\ast }\left(\varrho \right)+{(1-\varkappa )}^s{\mathsf{\Theta }}^{\ast }\left(y\right)\right].\hfill \end{array}$$

From (2), we have

$$\begin{array}{c}\hfill {\mathsf{\Theta }}_{\ast }(\varkappa \varrho +(1-\varkappa )y)\le {\varkappa }^s{\mathsf{\Theta }}_{\ast }\left(\varrho \right)+{(1-\varkappa )}^s{\mathsf{\Theta }}_{\ast }\left(y\right),\end{array}$$

(7)

and

$$\begin{array}{c}\hfill {\mathsf{\Theta }}^{\ast }(\varkappa \varrho +(1-\varkappa )y)\ge {\varkappa }^s{\mathsf{\Theta }}^{\ast }\left(\varrho \right)+{(1-\varkappa )}^s{\mathsf{\Theta }}^{\ast }\left(y\right).\end{array}$$

(8)

Equations (7) and (8) indicate that ${\mathsf{\Theta }}_{\ast }\in SGX\left([\sigma ,\varsigma ],{\mathbb{R}}^s\right)$ and ${\mathsf{\Theta }}^{\ast }\in SGV\left([\sigma ,\varsigma ],{\mathbb{R}}^s\right)$.

• Conversely, suppose that ${\mathsf{\Theta }}_{\ast }\in SGX\left([\sigma ,\varsigma ],{\mathbb{R}}^s\right)$ and ${\mathsf{\Theta }}^{\ast }\in SGV\left([\sigma ,\varsigma ],{\mathbb{R}}^s\right)$; then,

$$\begin{array}{c}\hfill {\mathsf{\Theta }}_{\ast }(\varkappa \varrho +(1-\varkappa )y)\le {\varkappa }^s{\mathsf{\Theta }}_{\ast }\left(\varrho \right)+{(1-\varkappa )}^s{\mathsf{\Theta }}_{\ast }\left(y\right),\end{array}$$

(9)

and

$$\begin{array}{c}\hfill {\mathsf{\Theta }}^{\ast }(\varkappa \varrho +(1-\varkappa )y)\ge {\varkappa }^s{\mathsf{\Theta }}^{\ast }\left(\varrho \right)+{(1-\varkappa )}^s{\mathsf{\Theta }}^{\ast }\left(y\right).\end{array}$$

(10)

Again from (2) and the fact ${\mathsf{\Theta }}_{\ast }\le {\mathsf{\Theta }}^{\ast }$,

$$\begin{array}{cc}\hfill & \left[{\mathsf{\Theta }}_{\ast }(\varkappa \varrho +(1-\varkappa )y),{\mathsf{\Theta }}^{\ast }(\varkappa \varrho +(1-\varkappa )y)\right]\hfill \\ \hfill & \supseteq \left[{\varkappa }^s{\mathsf{\Theta }}_{\ast }\left(\varrho \right)+{(1-\varkappa )}^s{\mathsf{\Theta }}_{\ast }\left(y\right),{\varkappa }^s{\mathsf{\Theta }}^{\ast }\left(\varrho \right)+{(1-\varkappa )}^s{\mathsf{\Theta }}^{\ast }\left(y\right)\right]\hfill \end{array}$$

□

Theorem 4. 

Suppose $\mathsf{\Theta }:[\sigma ,\varsigma ]\to {\mathbb{R}}_I^s$ is an ($\mathbb{I}.\mathbb{V}$) mapping such that $\mathsf{\Theta }=[{\mathsf{\Theta }}_{\ast },{\mathsf{\Theta }}^{\ast }]$ with ${\mathsf{\Theta }}_{\ast }\le {\mathsf{\Theta }}^{\ast }$. Then $\mathsf{\Theta }\in SIGV\left([\sigma ,\varsigma ],{\mathbb{R}}_I^s\right)\iff {\mathsf{\Theta }}_{\ast }\in SGV\left([\sigma ,\varsigma ],{\mathbb{R}}^s\right)\&{\mathsf{\Theta }}^{\ast }\in SGX\left([\sigma ,\varsigma ],{\mathbb{R}}^s\right)$.

Proof. 

The proof is similar to Theorem 3. □

Example 1. 

Let $\mathsf{\Theta }:[1,2]\to {\mathbb{R}}_I^s$ be an $(\mathbb{I},\mathbb{V})$ mapping and $\mathsf{\Theta }\left(\varrho \right)=\left[{\varrho }^{2s},-{\varrho }^{2s}+5^s\right]$.

With the help of Theorem 3, one can clearly visualize that $\mathsf{\Theta }\in SIGX\left([\sigma ,\varsigma ],{\mathbb{R}}_I^s\right)$.

Now we establish the Jensen’s inequality for ($\mathbb{I}.\mathbb{V}$) generalized convex mappings.

Theorem 5. 

Let $\mathsf{\Theta }\in SIGX\left([\sigma ,\varsigma ],{\mathbb{R}}_I^s\right)$, then we have

$$\begin{array}{c}\hfill \mathsf{\Theta }\left(\sum _{i=1}^n{\varkappa }_i{\varrho }_i\right)\supseteq \sum _{i=1}^n{\varkappa }_i^s\mathsf{\Theta }\left({\varrho }_i\right),\end{array}$$

(11)

where ${\varrho }_i\in [\sigma ,\varsigma ],{\sum }_{i=1}^n{\varkappa }_i=1$ and $0<s\le 1$, ${\varkappa }_i\in [0,1]$.

If $\mathsf{\Theta }\in SIGV\left([\sigma ,\varsigma ],{\mathbb{R}}_I^s\right)$, then (11) holds in reverse direction.

Proof. 

Since $\mathsf{\Theta }\in SIGX\left([\sigma ,\varsigma ],{\mathbb{R}}_I^s\right)$, for $n=2$ by choosing ${\varkappa }_1=\varkappa$ & ${\varkappa }_2=(1-\varkappa )$ and by using the notion of ($\mathbb{I}.\mathbb{V}$) generalized convexity, the result holds.

Assume that the result holds true for $n=k-1$; then, for any ${\varrho }_i\in [\sigma ,\varsigma ]$ for $i=1,2,3\dots ,k-1$ and ${\sum }_{i=1}^{k-1}{\sigma }_i=1$, we have

$$\begin{array}{c}\hfill \mathsf{\Theta }\left(\sum _{i=1}^{k-1}{\sigma }_i{\varrho }_i\right)\supseteq \sum _{i=1}^{k-1}{\sigma }_i^s\mathsf{\Theta }\left({\varrho }_i\right).\end{array}$$

If ${\varrho }_1,{\varrho }_2,{\varrho }_3,\dots {\varrho }_k\in [\sigma ,\varsigma ]$, ${\sum }_{i=1}^k{\mu }_i=1$, ${\mu }_i\in [0,1]$ and ${\mu }_k\ne 1$, then set ${\sigma }_i:={\displaystyle \frac{{\mu }_i}{1-{\mu }_k}}$, i = 1, 2, 3…$k-$1 and so ${\sum }_{i=1}^{k-1}{\sigma }_i=1$.

Thus,

$$\begin{array}{cc}\hfill & \mathsf{\Theta }\left({\mu }_1{\varrho }_1+{\mu }_2{\varrho }_2+{\mu }_3{\varrho }_3+\dots +{\mu }_{k-1}{\varrho }_{k-1}+{\mu }_k{\varrho }_k\right)\hfill \\ \hfill & =\mathsf{\Theta }\left(\left(1-{\mu }_k\right){\displaystyle \frac{{\mu }_1{\varrho }_1+{\mu }_2{\varrho }_2+{\mu }_3{\varrho }_3+...+{\mu }_{k-1}{\varrho }_{k-1}}{1-{\mu }_k}}+{\mu }_k{\varrho }_k\right)\hfill \\ \hfill & \supseteq {(1-{\mu }_k)}^s\mathsf{\Theta }\left({\displaystyle \frac{{\mu }_1}{1-{\mu }_k}}{\varrho }_1+{\displaystyle \frac{{\mu }_2}{1-{\mu }_k}}{\varrho }_2+\dots +{\displaystyle \frac{{\mu }_{k-1}}{1-{\mu }_k}}{\varrho }_{k-1}\right)+{\mu }_k^s\mathsf{\Theta }\left({\varrho }_k\right)\hfill \\ & ={(1-{\mu }_k)}^s\mathsf{\Theta }\left({\sigma }_1{\varrho }_1+{\sigma }_2{\varrho }_2+{\sigma }_3{\varrho }_3+\dots +{\sigma }_{k-1}{\varrho }_{k-1}\right)+{\mu }_k^s\mathsf{\Theta }\left({\varrho }_k\right)\hfill \\ \hfill & \supseteq {(1-{\mu }_k)}^s\left({\sigma }_1^s\mathsf{\Theta }\left({\varrho }_1\right)+{\sigma }_2^s\mathsf{\Theta }\left({\varrho }_2\right)+\dots +{\sigma }_{k-1}^s\mathsf{\Theta }\left({\varrho }_{k-1}\right)\right)+{\mu }_k^s\mathsf{\Theta }\left({\varrho }_k\right)\hfill \\ \hfill & ={(1-{\mu }_k)}^s\left({\left({\displaystyle \frac{{\mu }_1}{1-{\mu }_k}}\right)}^s\mathsf{\Theta }\left({\varrho }_1\right)+{\left({\displaystyle \frac{{\mu }_2}{1-{\mu }_k}}\right)}^s\mathsf{\Theta }\left({\varrho }_2\right)+\dots +{\left({\displaystyle \frac{{\mu }_{k-1}}{1-{\mu }_k}}\right)}^s\mathsf{\Theta }\left({\varrho }_{k-1}\right)\right)+{\mu }_k^s\mathsf{\Theta }\left({\varrho }_k\right)\hfill \\ \hfill & =\sum _{i=1}^k{\mu }_i^s\mathsf{\Theta }\left({\varrho }_i\right).\hfill \end{array}$$

□

Our next result is Hermite–Hadamard inclusion.

Theorem 6. 

Let $\mathsf{\Theta }:[\sigma ,\varsigma ]\to {\mathbb{R}}_I^s$ be a local ($\mathbb{I}.\mathbb{V}$) mapping, such that $\mathsf{\Theta }=[{\mathsf{\Theta }}_{\ast },{\mathsf{\Theta }}^{\ast }]$ and $\mathsf{\Theta }\in SIGX\left([\sigma ,\varsigma ],{\mathbb{R}}_I^s\right)$; then,

$$\begin{array}{c}\hfill \mathsf{\Theta }\left({\displaystyle \frac{\sigma +\varsigma }{2}}\right)\supseteq {\displaystyle \frac{\mathsf{\Gamma }(1+s)}{{(\varsigma -\sigma )}^s}}{}_{\sigma }I_{\varsigma }^s\mathsf{\Theta }\left(\varrho \right)\supseteq {\displaystyle \frac{\mathsf{\Theta }\left(\sigma \right)+\mathsf{\Theta }\left(\varsigma \right)}{2^s}},\end{array}$$

(12)

where $0<s\le 1$.

If $\mathsf{\Theta }\in SIGV\left([\sigma ,\varsigma ],{\mathbb{R}}_I^s\right)$, then the following inequality holds:

$$\begin{array}{c}\hfill \mathsf{\Theta }\left({\displaystyle \frac{\sigma +\varsigma }{2}}\right)\subseteq {\displaystyle \frac{\mathsf{\Gamma }(1+s)}{{(\varsigma -\sigma )}^s}}{}_{\sigma }I_{\varsigma }^s\mathsf{\Theta }\left(\varrho \right)\subseteq {\displaystyle \frac{\mathsf{\Theta }\left(\sigma \right)+\mathsf{\Theta }\left(\varsigma \right)}{2^s}}.\end{array}$$

(13)

Proof. 

Let $\varrho ,y\in [\sigma ,\varsigma ]$ and $\varkappa \in [0,1]$, then

$$\begin{array}{c}\hfill \mathsf{\Theta }(\varkappa \varrho +(1-\varkappa )y)\supseteq {\varkappa }^s\mathsf{\Theta }\left(\varrho \right)+{(1-\varkappa )}^s\mathsf{\Theta }\left(y\right)\end{array}$$

By taking $\varkappa ={\displaystyle \frac{1}{2}}$ in above inequality, we obtain

$$\begin{array}{c}\hfill \mathsf{\Theta }\left({\displaystyle \frac{\varrho +y}{2}}\right)\supseteq {\displaystyle \frac{1}{2^s}}\left[\mathsf{\Theta }\left(\varrho \right)+\mathsf{\Theta }\left(y\right)\right].\end{array}$$

By substituting $\varrho =\varkappa \sigma +(1-\varkappa )\varsigma$ and $y=(1-\varkappa )\sigma +\varkappa \varsigma$, we obtain

$$\begin{array}{c}\hfill \mathsf{\Theta }\left({\displaystyle \frac{\sigma +\varsigma }{2}}\right)\supseteq {\displaystyle \frac{1}{2^s}}\left[\mathsf{\Theta }(\varkappa \sigma +(1-\varkappa \left)\varsigma \right)+\mathsf{\Theta }\left(\right(1-\varkappa )\sigma +\varkappa \varsigma )\right].\end{array}$$

Now taking the local fractional integration with respect to ′ ϰ ′ over $[0,1]$,

$$\begin{array}{cc}\hfill & \mathsf{\Theta }\left({\displaystyle \frac{\sigma +\varsigma }{2}}\right){\displaystyle \frac{1}{\mathsf{\Gamma }(1+s)}}{\int }_0^1{\left(\mathrm{d}\varkappa \right)}^s\hfill \\ \hfill & \supseteq {\displaystyle \frac{1}{2^s\mathsf{\Gamma }(1+s)}}{\int }_0^1\left[\mathsf{\Theta }(\varkappa \sigma +(1-\varkappa \left)\varsigma \right)+\mathsf{\Theta }\left(\right(1-\varkappa )\sigma +\varkappa \varsigma )\right]{\left(\mathrm{d}\varkappa \right)}^s.\hfill \end{array}$$

(14)

Also

$$\begin{array}{cc}\hfill \mathsf{\Theta }\left({\displaystyle \frac{\sigma +\varsigma }{2}}\right){\displaystyle \frac{1}{\mathsf{\Gamma }(1+s)}}{\int }_0^1{\left(\mathrm{d}\varkappa \right)}^s& =\left[{\mathsf{\Theta }}_{\ast }\left({\displaystyle \frac{\sigma +\varsigma }{2}}\right){\displaystyle \frac{1}{\mathsf{\Gamma }(1+s)}}{\int }_0^1{\left(\mathrm{d}\varkappa \right)}^s,{\mathsf{\Theta }}^{\ast }\left({\displaystyle \frac{\sigma +\varsigma }{2}}\right){\displaystyle \frac{1}{\mathsf{\Gamma }(1+s)}}{\int }_0^1{\left(\mathrm{d}\varkappa \right)}^s\right]\hfill \\ \hfill & =\left[{\displaystyle \frac{1}{\mathsf{\Gamma }(1+s)}}{\mathsf{\Theta }}_{\ast }\left({\displaystyle \frac{\sigma +\varsigma }{2}}\right),{\displaystyle \frac{1}{\mathsf{\Gamma }(1+s)}}{\mathsf{\Theta }}^{\ast }\left({\displaystyle \frac{\sigma +\varsigma }{2}}\right)\right]\hfill \\ \hfill & ={\displaystyle \frac{1}{\mathsf{\Gamma }(1+s)}}\mathsf{\Theta }\left({\displaystyle \frac{\sigma +\varsigma }{2}}\right),\hfill \end{array}$$

(15)

and

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{\mathsf{\Gamma }(1+s)}}{\int }_0^1\mathsf{\Theta }(\varkappa \sigma +(1-\varkappa )\varsigma ){\left(\mathrm{d}\varkappa \right)}^s\hfill \\ \hfill & =\left[{\displaystyle \frac{1}{\mathsf{\Gamma }(1+s)}}{\int }_0^1{\mathsf{\Theta }}_{\ast }(\varkappa \sigma +(1-\varkappa )\varsigma ){\left(\mathrm{d}\varkappa \right)}^s,{\displaystyle \frac{1}{\mathsf{\Gamma }(1+s)}}{\int }_0^1{\mathsf{\Theta }}^{\ast }(\varkappa \sigma +(1-\varkappa )\varsigma ){\left(\mathrm{d}\varkappa \right)}^s\right]\hfill \\ \hfill & =\left[{\displaystyle \frac{1}{{(\varsigma -\sigma )}^s\mathsf{\Gamma }(1+s)}}{\int }_{\sigma }^{\varsigma }{\mathsf{\Theta }}_{\ast }\left(\varrho \right){\left(\mathrm{d}\varrho \right)}^s,{\displaystyle \frac{1}{{(\varsigma -\sigma )}^s\mathsf{\Gamma }(1+s)}}{\int }_{\sigma }^{\varsigma }{\mathsf{\Theta }}^{\ast }\left(\varrho \right){\left(\mathrm{d}\varrho \right)}^s\right]\hfill \\ \hfill & ={\displaystyle \frac{1}{{(\varsigma -\sigma )}^s}}{}_{\sigma }I_{\varsigma }^s\mathsf{\Theta }\left(\varrho \right).\hfill \end{array}$$

(16)

Similarly,

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{\mathsf{\Gamma }(1+s)}}{\int }_0^1\mathsf{\Theta }((1-\varkappa )\sigma +\varkappa \varsigma ){\left(\mathrm{d}\varkappa \right)}^s={\displaystyle \frac{1}{{(\varsigma -\sigma )}^s}}{}_{\sigma }I_{\varsigma }^s\mathsf{\Theta }\left(\varrho \right).\end{array}$$

(17)

We combine (14)–(17) and achieve the left inequality of (13).

To prove second inclusion, we utilize the ($\mathbb{I}.\mathbb{V}$) generalized convexity,

$$\begin{array}{cc}\hfill & \mathsf{\Theta }(\varkappa \sigma +(1-\varkappa )\varsigma )\supseteq {\varkappa }^s\mathsf{\Theta }\left(\sigma \right)+{(1-\varkappa )}^s\mathsf{\Theta }\left(\varsigma \right).\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill & \mathsf{\Theta }((1-\varkappa )\sigma +\varkappa \varsigma )\supseteq {(1-\varkappa )}^s\mathsf{\Theta }\left(\sigma \right)+{\varkappa }^s\mathsf{\Theta }\left(\varsigma \right).\hfill \end{array}$$

(19)

Adding (18) and (19) and then taking the local fractional integration with respect to ′ ϰ ′ over $[0,1]$ gives

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{\mathsf{\Gamma }(1+s)}}{\int }_0^1\left[\mathsf{\Theta }(\varkappa \sigma +(1-\varkappa \left)\varsigma \right)+\mathsf{\Theta }\left(\right(1-\varkappa )\sigma +\varkappa \varsigma )\right]{\left(\mathrm{d}\varkappa \right)}^s\hfill \\ & \supseteq (\mathsf{\Theta }\left(\sigma \right)+\mathsf{\Theta }\left(\varsigma \right)){\displaystyle \frac{1}{\mathsf{\Gamma }(1+s)}}{\int }_0^1{\left(\mathrm{d}\varkappa \right)}^s.\hfill \end{array}$$

(20)

Again, by using (16), (17) and (20), we obtain the right Hermite–Hadamard’s inequality. □

Now, we provide the general Hermite–Hadamard–Fejer inequality by the means of symmetric mappings.

Theorem 7. 

[Right-sided Fejer Inequality]

Let $\mathsf{\Theta }:[\sigma ,\varsigma ]\to {\mathbb{R}}_I^s$ be a local ($\mathbb{I}.\mathbb{V}$) mapping such that $\mathsf{\Theta }=[{\mathsf{\Theta }}_{\ast },{\mathsf{\Theta }}^{\ast }]$. If $\mathsf{\Theta }\in SIGX\left([\sigma ,\varsigma ],{\mathbb{R}}_I^s\right)$ and g is a symmetric mapping with respect to ${\displaystyle \frac{\sigma +\varsigma }{2}}$, then

$$\begin{array}{c}\hfill {}_{\sigma }I_{\varsigma }^s\mathsf{\Theta }\left(\varrho \right)g\left(\varrho \right)\supseteq {\displaystyle \frac{\mathsf{\Theta }\left(\sigma \right)+\mathsf{\Theta }\left(\varsigma \right)}{2^s}}{}_{\sigma }I_{\varsigma }^{s}g\left(\varrho \right).\end{array}$$

If $\mathsf{\Theta }\in SIGV\left([\sigma ,\varsigma ],{\mathbb{R}}_I^s\right)$ and g is a symmetric mapping with respect to ${\displaystyle \frac{\sigma +\varsigma }{2}}$, then

$$\begin{array}{c}\hfill {}_{\sigma }I_{\varsigma }^s\mathsf{\Theta }\left(\varrho \right)g\left(\varrho \right)\subseteq {\displaystyle \frac{\mathsf{\Theta }\left(\sigma \right)+\mathsf{\Theta }\left(\varsigma \right)}{2^s}}{}_{\sigma }I_{\varsigma }^s\mathsf{\Theta }\left(\varrho \right)g\left(\varrho \right).\end{array}$$

Proof. 

We multiply (18) by $g(\varkappa \sigma +(1-\varkappa \left)\varsigma \right)$ and (19) by $g\left(\right(1-\varkappa )\sigma +\varkappa \varsigma )$, and so we obtain

$$\begin{array}{cc}\hfill & \mathsf{\Theta }(\varkappa \sigma +(1-\varkappa )\varsigma )g(\varkappa \sigma +(1-\varkappa )\varsigma )\supseteq \left({\varkappa }^s\mathsf{\Theta }\left(\sigma \right)+{(1-\varkappa )}^s\mathsf{\Theta }\left(\varsigma \right)\right)g(\varkappa \sigma +(1-\varkappa )\varsigma ).\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill & \mathsf{\Theta }((1-\varkappa )\sigma +\varkappa \varsigma )g((1-\varkappa )\sigma +\varkappa \varsigma )\supseteq \left({(1-\varkappa )}^s\mathsf{\Theta }\left(\sigma \right)+{\varkappa }^s\mathsf{\Theta }\left(\varsigma \right)\right)g((1-\varkappa )\sigma +\varkappa \varsigma ).\hfill \end{array}$$

(22)

Adding (21) and (22) and using the symmetry of g gives

• $g\left(\right(1-\varkappa )\sigma +\varkappa \varsigma )=g(\varkappa \sigma +(1-\varkappa \left)\varsigma \right)$, then

$$\begin{array}{cc}\hfill & \mathsf{\Theta }(\varkappa \sigma +(1-\varkappa \left)\varsigma \right)g(\varkappa \sigma +(1-\varkappa \left)\varsigma \right)+\mathsf{\Theta }\left(\right(1-\varkappa )\sigma +\varkappa \varsigma )g\left(\right(1-\varkappa )\sigma +\varkappa \varsigma )\hfill \\ \hfill & \supseteq (\mathsf{\Theta }\left(\sigma \right)+\mathsf{\Theta }\left(\varsigma \right))({\varkappa }^s+{(1-\varkappa )}^s)g(\varkappa \sigma +(1-\varkappa )\varsigma )\hfill \\ \hfill & =\left(\mathsf{\Theta }\right(\sigma )+\mathsf{\Theta }(\varsigma \left)\right)g(\varkappa \sigma +(1-\varkappa \left)\varsigma \right).\hfill \end{array}$$

(23)

Now, take the local fractional integration of (23) with respect to ′ ϰ ′ over $[0,1]$, then

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{\mathsf{\Gamma }(1+s)}}{\int }_0^1\left[\mathsf{\Theta }(\varkappa \sigma +(1-\varkappa \left)\varsigma \right)g(\varkappa \sigma +(1-\varkappa \left)\varsigma \right)\right.\hfill \\ \hfill & \left.+\mathsf{\Theta }\left(\right(1-\varkappa )\sigma +\varkappa \varsigma )g\left(\right(1-\varkappa )\sigma +\varkappa \varsigma )\right]{\left(\mathrm{d}\varkappa \right)}^s\hfill \\ \hfill & \supseteq {\displaystyle \frac{\left(\mathsf{\Theta }\right(\sigma )+\mathsf{\Theta }(\varsigma \left)\right)}{\mathsf{\Gamma }(1+s)}}{\int }_0^{1}g(\varkappa \sigma +(1-\varkappa )\varsigma ){\left(\mathrm{d}\varkappa \right)}^s,\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{\mathsf{\Gamma }(1+s)}}{\int }_0^1\mathsf{\Theta }(\varkappa \sigma +(1-\varkappa )\varsigma )g(\varkappa \sigma +(1-\varkappa )\varsigma ){\left(\mathrm{d}\varkappa \right)}^s\hfill \\ \hfill & =\left[{\displaystyle \frac{1}{\mathsf{\Gamma }(1+s)}}{\int }_0^1{\mathsf{\Theta }}_{\ast }(\varkappa \sigma +(1-\varkappa )\varsigma )g_{\ast }(\varkappa \sigma +(1-\varkappa )\varsigma ){\left(\mathrm{d}\varkappa \right)}^s,\right.\hfill \\ \hfill & \left.{\displaystyle \frac{1}{\mathsf{\Gamma }(1+s)}}{\int }_0^1{\mathsf{\Theta }}^{\ast }(\varkappa \sigma +(1-\varkappa )\varsigma )g^{\ast }(\varkappa \sigma +(1-\varkappa )\varsigma ){\left(\mathrm{d}\varkappa \right)}^s\right]\hfill \\ \hfill & =\left[{\displaystyle \frac{1}{{(\varsigma -\sigma )}^s\mathsf{\Gamma }(1+s)}}{\int }_{\sigma }^{\varsigma }{\mathsf{\Theta }}_{\ast }\left(\varrho \right)g_{\ast }\left(\varrho \right){\left(\mathrm{d}\varrho \right)}^s,{\displaystyle \frac{1}{{(\varsigma -\sigma )}^s\mathsf{\Gamma }(1+s)}}{\int }_{\sigma }^{\varsigma }{\mathsf{\Theta }}^{\ast }\left(\varrho \right)g^{\ast }\left(\varrho \right){\left(\mathrm{d}\varrho \right)}^s\right]\hfill \\ \hfill & ={\displaystyle \frac{1}{{(\varsigma -\sigma )}^s}}{}_{\sigma }I_{\varsigma }^s\mathsf{\Theta }\left(\varrho \right)g\left(\varrho \right).\hfill \end{array}$$

(24)

Similarly,

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{\mathsf{\Gamma }(1+s)}}{\int }_0^1\mathsf{\Theta }((1-\varkappa )\sigma +\varkappa \varsigma )g((1-\varkappa )\sigma +\varkappa \varsigma ){\left(\mathrm{d}\varkappa \right)}^s\hfill \\ \hfill & ={\displaystyle \frac{1}{{(\varsigma -\sigma )}^s}}{}_{\sigma }I_{\varsigma }^s\mathsf{\Theta }\left(\varrho \right)g\left(\varrho \right),\hfill \end{array}$$

(25)

and

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{\mathsf{\Gamma }(1+s)}}{\int }_0^{1}g(\varkappa \sigma +(1-\varkappa )\varsigma ){\left(\mathrm{d}\varkappa \right)}^s={\displaystyle \frac{1}{{(\varsigma -\sigma )}^s}}{}_{\sigma }I_{\varsigma }^{s}g\left(\varrho \right).\end{array}$$

(26)

By combining (23)–(26), we obtain

$$\begin{array}{c}\hfill 2^s{}_{\sigma }I_{\varsigma }^s\mathsf{\Theta }\left(\varrho \right)g\left(\varrho \right)\supseteq \left[\mathsf{\Theta }\left(\sigma \right)+\mathsf{\Theta }\left(\varsigma \right)\right]{}_{\sigma }I_{\varsigma }^{s}g\left(\varrho \right).\end{array}$$

□

Theorem 8 

(Left sided Fejer Inequality). Let $\mathsf{\Theta }:[\sigma ,\varsigma ]\to {\mathbb{R}}_I^s$ be a local ($\mathbb{I}.\mathbb{V}$) mapping such that $\mathsf{\Theta }=[{\mathsf{\Theta }}_{\ast },{\mathsf{\Theta }}^{\ast }]$. If $\mathsf{\Theta }\in SIGX\left([\sigma ,\varsigma ],{\mathbb{R}}_I^s\right)$ and g is a symmetric mapping with respect to ${\displaystyle \frac{\sigma +\varsigma }{2}}$; then,

$$\begin{array}{c}\hfill \mathsf{\Theta }\left({\displaystyle \frac{\sigma +\varsigma }{2}}\right){}_{\sigma }I_{\varsigma }^{s}g\left(\varrho \right)\supseteq {}_{\sigma }I_{\varsigma }^s\mathsf{\Theta }\left(\varrho \right)g\left(\varrho \right).\end{array}$$

If $\mathsf{\Theta }\in SIGV\left([\sigma ,\varsigma ],{\mathbb{R}}_I^s\right)$ and g is a symmetric mapping with respect to ${\displaystyle \frac{\sigma +\varsigma }{2}}$, then

$$\begin{array}{c}\hfill \mathsf{\Theta }\left({\displaystyle \frac{\sigma +\varsigma }{2}}\right){}_{\sigma }I_{\varsigma }^{s}g\left(\varrho \right)\subseteq {}_{\sigma }I_{\varsigma }^s\mathsf{\Theta }\left(\varrho \right)g\left(\varrho \right).\end{array}$$

Proof. 

Since $\mathsf{\Theta }\in SIGX\left([\sigma ,\varsigma ]\right)$, then

$$\begin{array}{c}\hfill \mathsf{\Theta }\left({\displaystyle \frac{\sigma +\varsigma }{2}}\right)\supseteq {\displaystyle \frac{1}{2^s}}\left[\mathsf{\Theta }(\varkappa \sigma +(1-\varkappa \left)\varsigma \right)+\mathsf{\Theta }\left(\right(1-\varkappa )\sigma +\varkappa \varsigma )\right].\end{array}$$

(27)

Multiplying both sides of (27) by $g(\varkappa \sigma +(1-\varkappa \left)\varsigma \right)$ and taking the integration with respect to ′ ϰ ′ over $[0,1]$, then

$$\begin{array}{cc}\hfill & \mathsf{\Theta }\left({\displaystyle \frac{\sigma +\varsigma }{2}}\right){\displaystyle \frac{1}{\mathsf{\Gamma }(1+s)}}{\int }_0^{1}g(\varkappa \sigma +(1-\varkappa )\varsigma ){\left(\mathrm{d}\varkappa \right)}^s\hfill \\ \hfill & \supseteq {\displaystyle \frac{1}{2^s\mathsf{\Gamma }(1+s)}}{\int }_0^1\left[\mathsf{\Theta }(\varkappa \sigma +(1-\varkappa \left)\varsigma \right)+\mathsf{\Theta }\left(\right(1-\varkappa )\sigma +\varkappa \varsigma )\right]g(\varkappa \sigma +(1-\varkappa )\varsigma ){\left(\mathrm{d}\varkappa \right)}^s.\hfill \end{array}$$

Now, utilizing the symmetry of g such that $g(\varkappa \sigma +(1-\varkappa \left)\varsigma \right)=g\left(\right(1-\varkappa )\sigma +\varkappa \varsigma )$, we obtain

$$\begin{array}{cc}\hfill & \mathsf{\Theta }\left({\displaystyle \frac{\sigma +\varsigma }{2}}\right){\displaystyle \frac{1}{\mathsf{\Gamma }(1+s)}}{\int }_0^{1}g(\varkappa \sigma +(1-\varkappa )\varsigma ){\left(\mathrm{d}\varkappa \right)}^s\hfill \\ \hfill & \supseteq {\displaystyle \frac{1}{2^s\mathsf{\Gamma }(1+s)}}{\int }_0^1\left[\mathsf{\Theta }(\varkappa \sigma +(1-\varkappa \left)\varsigma \right)g(\varkappa \sigma +(1-\varkappa \left)\varsigma \right)\right.\hfill \\ \hfill & \left.+\mathsf{\Theta }\left(\right(1-\varkappa )\sigma +\varkappa \varsigma )g\left(\right(1-\varkappa )\sigma +\varkappa \varsigma )\right]{\left(\mathrm{d}\varkappa \right)}^s.\hfill \end{array}$$

Simply by using the substitution principle, and adding the similar integrals, we obtain the intended inequality. □

Now, we give some product trapezium-like inequalities, which are known as Pachpatte-type inequalities.

Theorem 9. 

Let $\mathsf{\Theta },g:[\sigma ,\varsigma ]\to {\mathbb{R}}_I^s$ be two $(\mathbb{I},\mathbb{V})$ mappings, such that $\mathsf{\Theta }=[{\mathsf{\Theta }}_{\ast },{\mathsf{\Theta }}^{\ast }]$ and $g=[g_{\ast },g^{\ast }]$. If $\mathsf{\Theta },g\in SIGX\left([\sigma ,\varsigma ],{\mathbb{R}}_I^s\right)$, then

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{{(\varsigma -\sigma )}^s}}{}_{\sigma }I_{\varsigma }^s\mathsf{\Theta }\left(\varrho \right)g\left(\varrho \right)\supseteq {\displaystyle \frac{\mathsf{\Gamma }(1+2s)}{\mathsf{\Gamma }(1+3s)}}{\mathcal{Q}}_1(\sigma ,\varsigma )+\left({\displaystyle \frac{\mathsf{\Gamma }(1+s)}{\mathsf{\Gamma }(1+2s)}}-{\displaystyle \frac{\mathsf{\Gamma }(1+2s)}{\mathsf{\Gamma }(1+3s)}}\right){\mathcal{Q}}_2(\sigma ,\varsigma ),\end{array}$$

where

$$\begin{array}{cc}\hfill & {\mathcal{Q}}_1(\sigma ,\varsigma )=\mathsf{\Theta }\left(\sigma \right)g\left(\sigma \right)+\mathsf{\Theta }\left(\varsigma \right)g\left(\varsigma \right)\hfill \\ \hfill & {\mathcal{Q}}_2(\sigma ,\varsigma )=\mathsf{\Theta }\left(\sigma \right)g\left(\varsigma \right)+\mathsf{\Theta }\left(\varsigma \right)g\left(\sigma \right).\hfill \end{array}$$

If $\mathsf{\Theta },g\in SIGV\left([\sigma ,\varsigma ],{\mathbb{R}}_I^s\right)$, then

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{{(\varsigma -\sigma )}^s}}{}_{\sigma }I_{\varsigma }^s\mathsf{\Theta }\left(\varrho \right)g\left(\varrho \right)\subseteq {\displaystyle \frac{\mathsf{\Gamma }(1+2s)}{\mathsf{\Gamma }(1+3s)}}{\mathcal{Q}}_1(\sigma ,\varsigma )+\left({\displaystyle \frac{\mathsf{\Gamma }(1+s)}{\mathsf{\Gamma }(1+2s)}}-{\displaystyle \frac{\mathsf{\Gamma }(1+2s)}{\mathsf{\Gamma }(1+3s)}}\right){\mathcal{Q}}_2(\sigma ,\varsigma ).\end{array}$$

Proof. 

Since $\mathsf{\Theta },g\in SIGX\left([\sigma ,\varsigma ],{\mathbb{R}}_I^s\right)$, then

$$\begin{array}{cc}\hfill & \mathsf{\Theta }(\varkappa \sigma +(1-\varkappa )\varsigma )\supseteq {\varkappa }^s\mathsf{\Theta }\left(\sigma \right)+{(1-\varkappa )}^s\mathsf{\Theta }\left(\varsigma \right)\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill & g(\varkappa \sigma +(1-\varkappa )\varsigma )\supseteq {\varkappa }^{s}g\left(\sigma \right)+{(1-\varkappa )}^{s}g\left(\varsigma \right).\hfill \end{array}$$

(29)

When multiplying (28) and (29), then

$$\begin{array}{cc}\hfill & \mathsf{\Theta }(\varkappa \sigma +(1-\varkappa \left)\varsigma \right)g(\varkappa \sigma +(1-\varkappa \left)\varsigma \right)\hfill \\ \hfill & \supseteq {\varkappa }^{2s}\mathsf{\Theta }\left(\sigma \right)g\left(\sigma \right)+{\varkappa }^s{(1-\varkappa )}^s[\mathsf{\Theta }\left(\sigma \right)g\left(\varsigma \right)+\mathsf{\Theta }\left(\varsigma \right)g\left(\sigma \right)]+{(1-\varkappa )}^{2s}\mathsf{\Theta }\left(\varsigma \right)g\left(\varsigma \right).\hfill \end{array}$$

Now taking the local fractional integration with respect to ′ ϰ ′ over $[0,1]$,

$$\begin{array}{cc}& {\displaystyle \frac{1}{\mathsf{\Gamma }(1+s)}}{\int }_0^1\mathsf{\Theta }(\varkappa \sigma +(1-\varkappa )\varsigma )g(\varkappa \sigma +(1-\varkappa )\varsigma ){\left(\mathrm{d}\varkappa \right)}^s\hfill \\ \hfill & \supseteq {\displaystyle \frac{1}{\mathsf{\Gamma }(1+s)}}{\int }_0^1\left[{\varkappa }^{2s}\mathsf{\Theta }\left(\sigma \right)g\left(\sigma \right)+{\varkappa }^s{(1-\varkappa )}^s[\mathsf{\Theta }\left(\sigma \right)g\left(\varsigma \right)+\mathsf{\Theta }\left(\varsigma \right)g\left(\sigma \right)]+{(1-\varkappa )}^{2s}\mathsf{\Theta }\left(\varsigma \right)g\left(\varsigma \right)\right]{\left(\mathrm{d}\varkappa \right)}^s\hfill \\ \hfill & ={\displaystyle \frac{\mathsf{\Gamma }(1+2s)}{\mathsf{\Gamma }(1+3s)}}{\mathcal{Q}}_1(\sigma ,\varsigma )+\left({\displaystyle \frac{\mathsf{\Gamma }(1+s)}{\mathsf{\Gamma }(1+2s)}}-{\displaystyle \frac{\mathsf{\Gamma }(1+2s)}{\mathsf{\Gamma }(1+3s)}}\right){\mathcal{Q}}_2(\sigma ,\varsigma ).\hfill \end{array}$$

□

Theorem 10. 

Let $\mathsf{\Theta },g:[\sigma ,\varsigma ]\to {\mathbb{R}}_I^s$ be two $(\mathbb{I},\mathbb{V})$ mappings such that $\mathsf{\Theta }=[{\mathsf{\Theta }}_{\ast },{\mathsf{\Theta }}^{\ast }]$ and $g=[g_{\ast },g^{\ast }]$. If $\mathsf{\Theta },g\in SIGX\left([\sigma ,\varsigma ],{\mathbb{R}}_I^s\right)$, then

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{\mathsf{\Gamma }(1+s)}}\mathsf{\Theta }\left({\displaystyle \frac{\sigma +\varsigma }{2}}\right)g\left({\displaystyle \frac{\sigma +\varsigma }{2}}\right)\hfill \\ \hfill & \supseteq {\displaystyle \frac{1}{2^s{(\varsigma -\sigma )}^s}}{}_{\sigma }I_{\varsigma }^s\mathsf{\Theta }\left(\varrho \right)g\left(\varrho \right)+{\displaystyle \frac{{\mathcal{Q}}_1(\sigma ,\varsigma )}{2^s}}\left[{\displaystyle \frac{\mathsf{\Gamma }(1+s)}{\mathsf{\Gamma }(1+2s)}}-{\displaystyle \frac{\mathsf{\Gamma }(1+2s)}{\mathsf{\Gamma }(1+3s)}}\right]+{\mathcal{Q}}_2(\sigma ,\varsigma ){\displaystyle \frac{\mathsf{\Gamma }(1+2s)}{\mathsf{\Gamma }(1+3s)}},\hfill \end{array}$$

where ${\mathcal{Q}}_1(\sigma ,\varsigma )$ and ${\mathcal{Q}}_2(\sigma ,\varsigma )$ are already defined in Theorem 9. The above inclusion holds in the reverse direction for interval-valued generalized concave mappings.

Proof. 

Since $\mathsf{\Theta },g\in SIGX\left([\sigma ,\varsigma ],{\mathbb{R}}_I^s\right)$, then

$$\begin{array}{cc}\hfill & \mathsf{\Theta }\left({\displaystyle \frac{\sigma +\varsigma }{2}}\right)g\left({\displaystyle \frac{\sigma +\varsigma }{2}}\right)\hfill \\ \hfill & \supseteq {\displaystyle \frac{1}{4^s}}\left[\mathsf{\Theta }(\varkappa \sigma +(1-\varkappa \left)\varsigma \right)+\mathsf{\Theta }\left(\right(1-\varkappa )\sigma +\varkappa \varsigma )\right]\left[g(\varkappa \sigma +(1-\varkappa \left)\varsigma \right)+g\left(\right(1-\varkappa )\sigma +\varkappa \varsigma )\right]\hfill \\ \hfill & \supseteq {\displaystyle \frac{1}{4^s}}\left[\mathsf{\Theta }(\varkappa \sigma +(1-\varkappa \left)\varsigma \right)g(\varkappa \sigma +(1-\varkappa \left)\varsigma \right)+\mathsf{\Theta }\left(\right(1-\varkappa )\sigma +\varkappa \varsigma )g\left(\right(1-\varkappa )\sigma +\varkappa \varsigma )\right]\hfill \\ \hfill & +{\displaystyle \frac{1}{4^s}}\left[\mathsf{\Theta }\left(\right(1-\varkappa )\sigma +\varkappa \varsigma )g(\varkappa \sigma +(1-\varkappa \left)\varsigma \right)+\mathsf{\Theta }(\varkappa \sigma +(1-\varkappa \left)\varsigma \right)g\left(\right(1-\varkappa )\sigma +\varkappa \varsigma )\right]\hfill \\ \hfill & \supseteq {\displaystyle \frac{1}{4^s}}\left[\mathsf{\Theta }(\varkappa \sigma +(1-\varkappa \left)\varsigma \right)g(\varkappa \sigma +(1-\varkappa \left)\varsigma \right)+\mathsf{\Theta }\left(\right(1-\varkappa )\sigma +\varkappa \varsigma )g\left(\right(1-\varkappa )\sigma +\varkappa \varsigma )\right]\hfill \\ \hfill & +{\displaystyle \frac{1}{4^s}}\left[\{{(1-\varkappa )}^s\mathsf{\Theta }\left(\sigma \right)+{\varkappa }^s\mathsf{\Theta }\left(\varsigma \right)\}\{{\varkappa }^{s}g\left(\sigma \right)+{(1-\varkappa )}^{s}g\left(\varsigma \right)\}\right.\hfill \\ \hfill & \left.+\{{\varkappa }^s\mathsf{\Theta }\left(\sigma \right)+{(1-\varkappa )}^s\mathsf{\Theta }\left(\varsigma \right)\}\{{(1-\varkappa )}^{s}g\left(\sigma \right)+{\varkappa }^{s}g\left(\varsigma \right)\}\right]\hfill \\ \hfill & \supseteq {\displaystyle \frac{1}{4^s}}\left[\mathsf{\Theta }(\varkappa \sigma +(1-\varkappa \left)\varsigma \right)g(\varkappa \sigma +(1-\varkappa \left)\varsigma \right)+\mathsf{\Theta }\left(\right(1-\varkappa )\sigma +\varkappa \varsigma )g\left(\right(1-\varkappa )\sigma +\varkappa \varsigma )\right]\hfill \\ \hfill & +{\displaystyle \frac{1}{4^s}}\left[2^s{\varkappa }^s{(1-\varkappa )}^s[\mathsf{\Theta }\left(\sigma \right)g\left(\sigma \right)+\mathsf{\Theta }\left(\varsigma \right)g\left(\varsigma \right)]+2^s[{\varkappa }^{2s}+{(1-\varkappa )}^{2s}][\mathsf{\Theta }\left(\sigma \right)g\left(\varsigma \right)+\mathsf{\Theta }\left(\varsigma \right)g\left(\sigma \right)]\right]\hfill \\ \hfill & \supseteq {\displaystyle \frac{1}{4^s}}\left[\mathsf{\Theta }(\varkappa \sigma +(1-\varkappa \left)\varsigma \right)g(\varkappa \sigma +(1-\varkappa \left)\varsigma \right)+\mathsf{\Theta }\left(\right(1-\varkappa )\sigma +\varkappa \varsigma )g\left(\right(1-\varkappa )\sigma +\varkappa \varsigma )\right]\hfill \\ \hfill & +{\displaystyle \frac{1}{2^s}}\left[{\varkappa }^s{(1-\varkappa )}^s{\mathcal{Q}}_1+{\mathcal{Q}}_2[{\varkappa }^{2s}+{(1-\varkappa )}^{2s}]\right]\hfill \\ \hfill & \supseteq {\displaystyle \frac{1}{4^s}}\left[\mathsf{\Theta }(\varkappa \sigma +(1-\varkappa \left)\varsigma \right)g(\varkappa \sigma +(1-\varkappa \left)\varsigma \right)+\mathsf{\Theta }\left(\right(1-\varkappa )\sigma +\varkappa \varsigma )g\left(\right(1-\varkappa )\sigma +\varkappa \varsigma )\right]\hfill \\ \hfill & +{\displaystyle \frac{1}{2^s}}\left[({\varkappa }^s-{\varkappa }^{2s}){\mathcal{Q}}_1+{\mathcal{Q}}_2[{\varkappa }^{2s}+{(1-\varkappa )}^{2s}]\right].\hfill \end{array}$$

(30)

Applying local fractional integration on (30) with respect to ′ ϰ′ over $[0,1]$, then

$$\begin{array}{cc}\hfill & \mathsf{\Theta }\left({\displaystyle \frac{\sigma +\varsigma }{2}}\right)g\left({\displaystyle \frac{\sigma +\varsigma }{2}}\right){\displaystyle \frac{1}{\mathsf{\Gamma }(1+s)}}{\int }_0^1{\left(\mathrm{d}\varkappa \right)}^s\hfill \\ \hfill & \supseteq {\displaystyle \frac{1}{4^s\mathsf{\Gamma }(1+s)}}{\int }_0^1\left[\mathsf{\Theta }(\varkappa \sigma +(1-\varkappa \left)\varsigma \right)g(\varkappa \sigma +(1-\varkappa \left)\varsigma \right)\right.\hfill \\ \hfill & \left.+\mathsf{\Theta }\left(\right(1-\varkappa )\sigma +\varkappa \varsigma )g\left(\right(1-\varkappa )\sigma +\varkappa \varsigma )\right]{\left(\mathrm{d}\varkappa \right)}^s\hfill \\ \hfill & +{\displaystyle \frac{1}{2^s\mathsf{\Gamma }(1+s)}}{\int }_0^1\left[({\varkappa }^s-{\varkappa }^{2s}){\mathcal{Q}}_1+{\mathcal{Q}}_2[{\varkappa }^{2s}+{(1-\varkappa )}^{2s}]{\left(\mathrm{d}\varkappa \right)}^s\right].\hfill \end{array}$$

After performing simple integrations by using power formula and substitution techniques, we obtain

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{\mathsf{\Gamma }(1+s)}}\mathsf{\Theta }\left({\displaystyle \frac{\sigma +\varsigma }{2}}\right)g\left({\displaystyle \frac{\sigma +\varsigma }{2}}\right)\hfill \\ \hfill & \supseteq {\displaystyle \frac{{}_{\sigma }I_{\varsigma }^s\mathsf{\Theta }\left(\varrho \right)g\left(\varrho \right)+{}_{\sigma }I_{\varsigma }^s\mathsf{\Theta }\left(\varrho \right)g\left(\varrho \right)}{4^s{(\varsigma -\sigma )}^s}}+{\displaystyle \frac{{\mathcal{Q}}_1(\sigma ,\varsigma )}{2^s}}\left[{\displaystyle \frac{\mathsf{\Gamma }(1+s)}{\mathsf{\Gamma }(1+2s)}}-{\displaystyle \frac{\mathsf{\Gamma }(1+2s)}{\mathsf{\Gamma }(1+3s)}}\right]+{\mathcal{Q}}_2(\sigma ,\varsigma ){\displaystyle \frac{\mathsf{\Gamma }(1+2s)}{\mathsf{\Gamma }(1+3s)}}.\hfill \end{array}$$

It ends the proof. □

5. Visual Analysis

In this section, we present numerical and graphical validation of our main results.

Example 2. 

We consider that $\mathsf{\Theta }\left(\varrho \right)=\left[{\varrho }^2,10-{\varrho }^2\right]$. If $s=1$, $\sigma =0$ and $\varsigma \in [0.5,1]$, then we have

$$\begin{array}{cc}\hfill & \mathsf{\Theta }\left({\displaystyle \frac{\sigma +\varsigma }{2}}\right)=\left[{\left({\displaystyle \frac{\varsigma }{2}}\right)}^2,10-{\left({\displaystyle \frac{\varsigma }{2}}\right)}^2\right].\hfill \end{array}$$

In addition, the middle part of Theorem 6 becomes (see Figure 1)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\Gamma (s+1)}{{(\varsigma -\sigma )}^s}}{}_{\sigma }I_{\varsigma }^s\mathsf{\Theta }\left(\varrho \right)=\left[{\displaystyle \frac{{\varsigma }^2}{3}},10\varsigma -{\displaystyle \frac{{\varsigma }^2}{3}}\right].\hfill \end{array}$$

The right-hand side of Theorem 6 becomes (see Figure 1)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\mathsf{\Theta }\left(\sigma \right)+\mathsf{\Theta }\left(\varsigma \right)}{2^s}}=\left[{\displaystyle \frac{{\varsigma }^2}{2}},{\displaystyle \frac{10-{\varsigma }^2}{2}}\right].\hfill \end{array}$$

Figure 1. Graphical representation of Theorem 6 based on the values from Table 1 and Table 2, for $\varsigma \in [0.5,1]$.

Table 1. Numerical validation of Example 2 for ${\mathsf{\Theta }}_{\ast }$ based on Theorem 6 for $\varsigma \in [0.5,1]$.

Values of $\mathit{\varsigma }$	Left Term	Middle Term	Right Term
0.5	0.0625	0.08333	0.1250
0.6	0.0900	0.1200	0.1800
0.7	0.1225	0.1633	0.2450
0.8	0.1600	0.2133	0.3200
0.9	0.2025	0.2700	0.4050
1	0.2500	0.3333	0.5000

Table 1. Numerical validation of Example 2 for ${\mathsf{\Theta }}_{\ast }$ based on Theorem 6 for $\varsigma \in [0.5,1]$.

Values of $\mathit{\varsigma }$	Left Term	Middle Term	Right Term
0.5	0.0625	0.08333	0.1250
0.6	0.0900	0.1200	0.1800
0.7	0.1225	0.1633	0.2450
0.8	0.1600	0.2133	0.3200
0.9	0.2025	0.2700	0.4050
1	0.2500	0.3333	0.5000

Table 2. Numerical validation of Example 2 for ${\mathsf{\Theta }}^{\ast }$ based on Theorem 6 for $\varsigma \in [0.5,1]$.

Values of $\mathit{\varsigma }$	Left Term	Middle Term	Right Term
0.5	9.9375	4.9167	4.8750
0.6	9.9100	5.8800	4.8200
0.7	9.8775	6.8367	4.7550
0.8	9.8400	7.7867	4.6800
0.9	9.7975	8.7300	4.5950
1	9.7500	9.6667	4.5000

Table 2. Numerical validation of Example 2 for ${\mathsf{\Theta }}^{\ast }$ based on Theorem 6 for $\varsigma \in [0.5,1]$.

Values of $\mathit{\varsigma }$	Left Term	Middle Term	Right Term
0.5	9.9375	4.9167	4.8750
0.6	9.9100	5.8800	4.8200
0.7	9.8775	6.8367	4.7550
0.8	9.8400	7.7867	4.6800
0.9	9.7975	8.7300	4.5950
1	9.7500	9.6667	4.5000

Example 3. 

We consider that $\mathsf{\Theta }\left(\varrho \right)=\left[{\varrho }^2,10-{\varrho }^2\right]$, $g\left(\varrho \right)={\left(\varrho -{\displaystyle \frac{1}{2}}\right)}^2$. If $s=1$, $\sigma =0$ and $\varsigma \in [0.2,1]$, then the left side of Theorem 7 can be written as (see Figure 2)

$$\begin{array}{cc}\hfill & {}_{\sigma }I_{\varsigma }^s\mathsf{\Theta }\left(\varrho \right)g\left(\varrho \right)\hfill \\ \hfill & =[{\displaystyle \frac{{\varsigma }^5}{5}}-{\displaystyle \frac{{\varsigma }^4}{4}}+{\displaystyle \frac{{\varsigma }^3}{12}},{\displaystyle \frac{{\varsigma }^5}{5}}-{\displaystyle \frac{21{\varsigma }^4}{4}}+{\displaystyle \frac{481{\varsigma }^3}{12}}-{\displaystyle \frac{105{\varsigma }^2}{2}}+25\varsigma ].\hfill \end{array}$$

In addition, the right-hand side of Theorem 7 becomes (see Figure 2)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\mathsf{\Theta }\left(\sigma \right)+\mathsf{\Theta }\left(\varsigma \right)}{2^s}}{}_{\sigma }I_{\varsigma }^{s}g\left(\varrho \right)\hfill \\ \hfill & =\left[{\displaystyle \frac{{\varsigma }^2}{2}}\left({\displaystyle \frac{{\varsigma }^3}{3}}-{\displaystyle \frac{{\varsigma }^2}{2}}+{\displaystyle \frac{\varsigma }{4}}\right),{\displaystyle \frac{20-{\varsigma }^2}{2}}\left({\displaystyle \frac{{\varsigma }^3}{3}}-{\displaystyle \frac{{\varsigma }^2}{2}}+{\displaystyle \frac{\varsigma }{4}}\right)\right].\hfill \end{array}$$

Figure 2. Graphical representation of Theorem 7 based on the values from Table 3 and Table 4, for $\varsigma \in \left[0.2,1\right]$.

Table 3. Numerical validation of Example 3 for ${\mathsf{\Theta }}_{\ast }$ based on Theorem 7 for $\varsigma \in [0.2,1]$.

Values of $\mathit{\varsigma }$	Left Term	Right Term
0.2	0.000331	0.000653
0.3	0.000711	0.001755
0.4	0.000981	0.003307
0.5	0.001042	0.005208
0.6	0.001152	0.007560
0.7	0.002172	0.010862
0.8	0.005803	0.016213
0.9	0.014823	0.025515

Table 3. Numerical validation of Example 3 for ${\mathsf{\Theta }}_{\ast }$ based on Theorem 7 for $\varsigma \in [0.2,1]$.

Values of $\mathit{\varsigma }$	Left Term	Right Term
0.2	0.000331	0.000653
0.3	0.000711	0.001755
0.4	0.000981	0.003307
0.5	0.001042	0.005208
0.6	0.001152	0.007560
0.7	0.002172	0.010862
0.8	0.005803	0.016213
0.9	0.014823	0.025515

Table 4. Numerical validation of Example 3 for ${\mathsf{\Theta }}^{\ast }$ based on Theorem 7 for $\varsigma \in [0.2,1]$.

Values of $\mathit{\varsigma }$	Left Term	Right Term
0.2	0.000331	0.000653
0.3	0.000711	0.001755
0.4	0.000981	0.003307
0.5	0.001042	0.005208
0.6	0.001152	0.007560
0.7	0.002172	0.010862
0.8	0.005803	0.016213
0.9	0.014823	0.025515

Table 4. Numerical validation of Example 3 for ${\mathsf{\Theta }}^{\ast }$ based on Theorem 7 for $\varsigma \in [0.2,1]$.

Values of $\mathit{\varsigma }$	Left Term	Right Term
0.2	0.000331	0.000653
0.3	0.000711	0.001755
0.4	0.000981	0.003307
0.5	0.001042	0.005208
0.6	0.001152	0.007560
0.7	0.002172	0.010862
0.8	0.005803	0.016213
0.9	0.014823	0.025515

Example 4. 

We consider that $\mathsf{\Theta }\left(\varrho \right)=[{\varrho }^2,10-{\varrho }^2]$, $g\left(\varrho \right)=[\varrho ,5-\varrho ]$. If $s=1$, $\sigma =0$ and $\varsigma \in [1,2]$, then we have (see Figure 3)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{{(\varsigma -\sigma )}^s}}{}_{\sigma }I_{\varsigma }^s\mathsf{\Theta }\left(\varrho \right)g\left(\varrho \right)=\left[{\displaystyle \frac{{\varsigma }^3}{4}},50+{\displaystyle \frac{{\varsigma }^3}{4}}-{\displaystyle \frac{5{\varsigma }^2}{3}}-5\varsigma \right].\hfill \end{array}$$

In addition, the right-hand side Theorem 9 becomes (see Figure 3)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\Gamma (1+2s)}{\Gamma (1+3s)}}{Q}_1(\sigma ,\varsigma )+\left({\displaystyle \frac{\Gamma (1+s)}{\Gamma (1+2s)}}-{\displaystyle \frac{\Gamma (1+2s)}{\Gamma (1+3s)}}\right)Q_2(\sigma ,\varsigma )\hfill \\ \hfill & =\left[{\displaystyle \frac{{\varsigma }^3}{3}},50+{\displaystyle \frac{{\varsigma }^3}{3}}-{\displaystyle \frac{5{\varsigma }^2}{2}}-5\varsigma \right].\hfill \end{array}$$

Figure 3. Graphical representation of Theorem 9 based on the values from Table 5 and Table 6, for $\varsigma \in \left[1,2\right]$.

Table 5. Numerical validation of Example 4 for ${\mathsf{\Theta }}_{\ast }$ based on Theorem 9 for $\varsigma \in [1,2]$.

Values of $\mathit{\varsigma }$	Left Term	Right Term
1.1	0.3328	0.4437
1.2	0.4320	0.5760
1.3	0.5493	0.7323
1.4	0.6860	0.9147
1.5	0.8438	1.1250
1.6	1.0240	1.3653
1.7	1.2283	1.6377
1.8	1.4580	1.9440
1.9	1.7148	2.2863

Table 5. Numerical validation of Example 4 for ${\mathsf{\Theta }}_{\ast }$ based on Theorem 9 for $\varsigma \in [1,2]$.

Values of $\mathit{\varsigma }$	Left Term	Right Term
1.1	0.3328	0.4437
1.2	0.4320	0.5760
1.3	0.5493	0.7323
1.4	0.6860	0.9147
1.5	0.8438	1.1250
1.6	1.0240	1.3653
1.7	1.2283	1.6377
1.8	1.4580	1.9440
1.9	1.7148	2.2863

Table 6. Numerical validation of Example 4 for ${\mathsf{\Theta }}^{\ast }$ based on Theorem 9 for $\varsigma \in [1,2]$.

Values of $\mathit{\varsigma }$	Left Term	Right Term
1.1	42.8161	41.9187
1.2	42.0320	40.9760
1.3	41.2326	40.0073
1.4	40.4193	39.0147
1.5	39.5938	38.0000
1.6	38.7573	36.9653
1.7	37.9116	35.9127
1.8	37.0580	34.8440
1.9	36.1981	33.7613

Table 6. Numerical validation of Example 4 for ${\mathsf{\Theta }}^{\ast }$ based on Theorem 9 for $\varsigma \in [1,2]$.

Values of $\mathit{\varsigma }$	Left Term	Right Term
1.1	42.8161	41.9187
1.2	42.0320	40.9760
1.3	41.2326	40.0073
1.4	40.4193	39.0147
1.5	39.5938	38.0000
1.6	38.7573	36.9653
1.7	37.9116	35.9127
1.8	37.0580	34.8440
1.9	36.1981	33.7613

6. Conclusions

In recent years, fractional calculus approaches have often been utilized to analyze many scientific models. It has been modified and refined by various techniques. Yang [17] has successfully introduced local fractal calculus to tackle non-differentiable scientific problems. Due to the effective range of applications, researchers have tried formulating the fractal version of classical inequalities. The subject of fractal inequalities is a widely explored site of research. We have defined some operations on fractal intervals, such as local fractional integration of fractal intervals and order relations to compare the fractal intervals. These were influenced by research that is still going on. With the help of these developments, we have constructed some well-known integral inequalities. The idea and technique of the paper will create more opportunities to derive new results in various domains of mathematics.