Certain Novel Fractional Integral Inequalities via Fuzzy Interval Valued Functions

Abstract

Fuzzy-interval valued functions (FIVFs) are the generalization of interval valued and real valued functions, which have a great contribution to resolve the problems arising in the theory of interval analysis. In this article, we elaborate the convexities and pre-invexities in aspects of FIVFs and investigate the existence of fuzzy fractional integral operators (FFIOs) having a generalized Bessel–Maitland function as their kernel. Using the class of convexities and pre-invexities FIVFs, we prove some Hermite–Hadamard (H-H) and trapezoid-type inequalities by the implementation of FFIOs. Additionally, we obtain other well known inequalities having significant behavior in the field of fuzzy interval analysis.

1. Introduction

The theory of convexity is a dynamic, addictive, and significant area of study that has made major contributions to other fields of research, such as mathematical analysis, optimization problems, control theories, economics, finance, and game theory. By the theory of convexity, researchers have created unified numerical structures that can be used to resolve the wide range of problems, which have arisen in pure and applied mathematics. Convexity has been through significant advancements, generalizations, and extensions in a few decades. The study of fractional analysis has increased the demand of fractional operators in different areas of mathematics. To fulfil this requirement, many researchers have worked to develop the fractional operators by utilizing the non-singular special functions as their kernel and obtained modified versions of inequalities. Generalized fractional operators are one of the techniques used to improve the fractional inequalities for different convexities and pre-invexities, which have significant applications in the field of analysis.

Fractional calculus is the generalization of classical calculus, which plays an important role in pure, applied, and computation fields of mathematics. The research work in the field of fractional analysis has made a great contribution in various directions, such as signal-image processing, biology, physics, control operator theory, computer structure optimizations, and fluid dynamics [1,2,3]. During the last few decades, most of the scientists have worked to obtain the generalized versions of well-known inequalities and discussed a huge number of applications in the fields of analysis and discrete optimization. Many authors have worked extensively [4,5,6,7,8,9,10,11,12] and discussed the refinements and extensions in different areas of mathematics. The advanced analysis of inequalities is possible for the development of fractional operators by means of their kernel in multi-dimension functions, which play an ideal role to create new horizons to study the behavior of inequalities in multi-discipline branches of mathematics.

There are many significant applications of fuzzy set theory, which deals with the problems incorporating ambiguous, vague, and imprecise information, and makes decisions for individual or group collaboration. This initiative developed the idea of interval analysis [13] by Moore Ramon in 1966. Many scientists are attracted towards this field because of to its decision-making evaluation. The investigation on interval analysis proved to be beneficial in global optimization and constraint solution algorithms for decision makers and in the last few decades, it has become very popular among experts. It has provided effective and valid results, minimized the errors, and improved the accuracy. Due to this motivation, several researchers started their research in inequalities to get the desired results.

Zhao et al. [14] were the first who introduced the interval-valued-function (IVF). Many mathematicians introduced a strong relation between inequalities and IVFs by the implementation of different integral operators [15,16,17,18,19]. Many scholars have elaborated the applications of fuzzy differential equations and fuzzy interval analysis, which deal with many mathematical or computer modules [20,21,22,23,24]. In different research articles, many people illustrated several inequalities, such as Hermite–Hadamard inequality, Jensen inequality, Mercer inequality, Schur inequality, and trapezoid-type inequalities with the help of a fuzzy interval valued function [25,26].

Motivation

Convexity and generalized convexity are important concepts in optimization under the fuzzy domain, which provide fuzzy variational inequalities as a result of characterizing the optimality condition of convexity. Variational inequality and fuzzy set theory have generated strong techniques, which resolved many mathematical problems related to minimization theory and interval analysis. Fuzzy mappings are also termed as fuzzy-IVFs. There are many fractional integrals that contain fuzzy-IVFs for lower and upper cases. The behavior of well known inequalities can be checked by the successful implementation of such kinds of fractional integrals. These investigations demonstrate that this strategy is important from both a theoretical and a practical standpoint to transform the actual integral inequalities to the fuzzy integral inequalities.

2. Preliminaries

In this section, we will discuss the basic definitions and results, which help to understand the concepts of our new results.

Definition 1. 

[27] Let $k\in \mathbb{R}$ be convex set, then the convex function for $\mathsf{ϝ}:K\to \mathbb{R}$ is defined as follows:

$$\begin{array}{c}\hfill \mathsf{ϝ}\left[\sigma u+(1-\sigma )v\right]\le \sigma \mathsf{ϝ}\left(u\right)+(1-\sigma )\mathsf{ϝ}\left(v\right).\end{array}$$

(1)

for $\sigma \in [0,1]$, $\forall ð,v\in K$.

The concave function for ϝ is to have the reversed inequality, defined in Equation (1).

Definition 2. 

The Hermite–Hadamard type inequality [28,29,30,31] is as follows:

$$\begin{array}{c}\hfill \mathsf{ϝ}\left(\frac{\mathsf{ȷ}+\aleph }{2}\right)\le \frac{1}{\aleph -\mathsf{ȷ}}{\int }_{\mathsf{ȷ}}^{\aleph }\mathsf{ϝ}\left(z\right)dz\le \frac{\mathsf{ϝ}\left(\mathsf{ȷ}\right)+\mathsf{ϝ}(\aleph )}{2}.\end{array}$$

(2)

where $\mathsf{ϝ}:K\to R$ is the convex function, and $[\mathsf{ȷ},\aleph ]\in K\subseteq \mathbb{R},\mathsf{ȷ}<\aleph ,\mathsf{ȷ},\aleph \in R$.

The modified form of fractional H-H inequality for the convex function is as follows:

$$\begin{array}{c}\hfill \mathsf{ϝ}\left(\frac{\mathsf{ȷ}+\aleph }{2}\right)\le \frac{1}{2\varsigma {(v,ð)}^{{\nu }^{{}^{\prime }}+1}}\left[{\mathcal{I}}_{v,v^{\prime }}^{{ð}^{+}}(\mathsf{\Omega },\mathsf{ϝ})+{\mathcal{I}}_{ð,v^{\prime }}^{v^{-}}(\mathsf{\Omega },\mathsf{ϝ})\right]\le \frac{\mathsf{ϝ}\left(\mathsf{ȷ}\right)+\mathsf{ϝ}(\aleph )}{2}\end{array}$$

(3)

Definition 3. 

[25] Let $\mathsf{ϝ}:[ð,v]\subset \mathbb{R}\to {\mathsf{\Omega }}_c^{+}$ (a space of all positive closed and bounded intervals of $\mathbb{R}$) is a convex IVF given by $\mathsf{ϝ}\left(\alpha \right)=[\mathsf{ϝ}{\left(\alpha \right)}_{*},\mathsf{ϝ}{\left(\alpha \right)}^{*}]$ for all $\alpha \in [ð,v]$ where $\mathsf{ϝ}{\left(\alpha \right)}_{*}$ is convex function and $\mathsf{ϝ}{\left(\alpha \right)}^{*}$ is concave function. If ϝ is Riemann integrable function, then:

$$\begin{array}{c}\hfill \mathsf{ϝ}\left(\frac{\alpha +\mathsf{\Lambda }}{2}\right)\supseteq \left(IR\right)\frac{1}{\mathsf{\Lambda }-\alpha }{\int }_{\alpha }^{\mathsf{\Lambda }}\mathsf{ϝ}\left(z\right)dz\supseteq \frac{\mathsf{ϝ}\left(\alpha \right)+\mathsf{ϝ}\left(\mathsf{\Lambda }\right)}{2}\end{array}$$

(4)

The Hermite–Hadamard inequality for inclusion relation is as follows:

$$\begin{array}{c}\hfill \mathsf{ϝ}\left(\frac{\alpha +\mathsf{\Lambda }}{2}\right)\supseteq \frac{1}{2\varsigma {(v,ð)}^{{\nu }^{{}^{\prime }}+1}}\left[{\mathcal{I}}_{v,v^{\prime }}^{{ð}^{+}}(\mathsf{\Omega },\mathsf{ϝ})+{\mathcal{I}}_{ð,v^{\prime }}^{v^{-}}(\mathsf{\Omega },\mathsf{ϝ})\right]\supseteq \frac{\mathsf{ϝ}\left(\alpha \right)+\mathsf{ϝ}\left(\mathsf{\Lambda }\right)}{2}\end{array}$$

(5)

Proposition 1. 

[16] If $\mathsf{ϝ},\gimel \in {\mathbb{F}}_0$, then for ${\mathbb{F}}_0$, relation $“\preccurlyeq ”$ is defined as:

$$\begin{array}{c}\hfill \mathsf{ϝ}\preccurlyeq \gimel \mathit{if}\mathit{and}\mathit{only}\mathit{if}{\left[\mathsf{ϝ}\right]}^{\mathsf{ı}}{\le }_I{[\gimel ]}^{\mathsf{ı}}\mathit{for}\mathit{all}\mathsf{ı}\in [0,1],\end{array}$$

(6)

the relation given above is also called partial order relation.

For $\mathsf{ϝ},\gimel \in {\mathbb{F}}_0$, their scalar and vector properties are defined for $\mathsf{ı}\in [0,1]$, as follows:

$$\begin{array}{ccc}\hfill {(\mathsf{ϝ}\widetilde{+}\gimel )}^{\mathsf{ı}}& =& {\left(\mathsf{ϝ}\right)}^{\mathsf{ı}}+{(\gimel )}^{\mathsf{ı}}\hfill \\ \hfill {(\mathsf{ϝ}\widetilde{+}\gimel )}^{\mathsf{ı}}& =& \mathsf{ϝ}+{(\gimel )}^{\mathsf{ı}}\hfill \\ \hfill {(\wp .\mathsf{ϝ})}^{\mathsf{ı}}& =& \wp .{\left(\mathsf{ϝ}\right)}^{\mathsf{ı}}\hfill \\ \hfill {(\mathsf{ϝ}\tilde{\times }\gimel )}^{\mathsf{ı}}& =& {\left(\mathsf{ϝ}\right)}^{\mathsf{ı}}\times {(\gimel )}^{\mathsf{ı}}\hfill \end{array}$$

The Hukuhara difference of ϝ and ℷ, for $\beth \in {\mathbb{F}}_0$ and $\mathsf{ϝ}=\gimel \widetilde{+}\beth$, then ℶ is the Hukuhara difference of ϝ and ℷ, which is defined as follows:

$$\begin{array}{c}\hfill {(\beth )}^{*}\left(\mathsf{ı}\right)={(\mathsf{ϝ}\widetilde{-}\gimel )}^{*}\left(\mathsf{ı}\right)={\mathsf{ϝ}}^{*}\left(\mathsf{ı}\right)-{\gimel }^{*}\left(\mathsf{ı}\right),\\ \hfill {(\beth )}_{*}\left(\mathsf{ı}\right)={(\mathsf{ϝ}\widetilde{-}\gimel )}_{*}\left(\mathsf{ı}\right)={\mathsf{ϝ}}_{*}\left(\mathsf{ı}\right)-{\gimel }_{*}\left(\mathsf{ı}\right)\end{array}$$

Definition 4. 

[11] Let P partition be the partition on the closed interval $[ð,v]$ as the form:

$$\begin{array}{c}\hfill P=ð={\mathsf{ȷ}}_1<{\mathsf{ȷ}}_2<{\mathsf{ȷ}}_3<{\mathsf{ȷ}}_4<{\mathsf{ȷ}}_5<....{\mathsf{ȷ}}_k=v.\end{array}$$

The subintervals containing point P have a maximum length, which is called the mesh of a partition and defined as follows:

$$\begin{array}{c}\hfill mesh\left(P\right)=max({\mathsf{ȷ}}_j-{\mathsf{ȷ}}_{j-1}:j=1,2,3...k)\end{array}$$

(7)

Let $P(\sigma ,[ð,v\left]\right)$ be the set of all $p\in P(\sigma ,[ð,v\left]\right)$ and mesh$\left(p\right)<\sigma$. By taking arbitrary point ${\gimel }_j$ on each subinterval $[{\mathsf{ȷ}}_{j-1},{\mathsf{ȷ}}_j]$, where $1\le j\le k$, such that:

$$S(\mathsf{ϝ},p,\sigma )=\sum _{j=1}^k\mathsf{ϝ}\left({\gimel }_j\right)({\mathsf{ȷ}}_j-{\mathsf{ȷ}}_{j-1}),$$

where $\mathsf{ϝ}:[ð,v]\to {\mathbb{R}}_I$ is the real valued function and $S(\mathsf{ϝ},p,\sigma )$ is called the Riemann sum of ϝ corresponding to $p\in P(\sigma ,[ð,v\left]\right)$.

Definition 5. 

[32] A function $\mathsf{ϝ}:[ð,v]\to {\mathbb{R}}_I$ is called the Reimann integrable (IR-integrable) on the closed interval $[ð,v]$ if there exists $B\in {\mathbb{R}}_I$ and for each ϵ and $\sigma >0$, such that:

$$\begin{array}{c}\hfill d\left(S\right(\mathsf{ϝ},P,\sigma ),B)<ϵ.\end{array}$$

for every Riemann sum of ϝ corresponding to the partition $P\in p(\sigma ,[ð,v\left]\right)$, and arbitrary choice of ${\gimel }_j\in [{\mathsf{ȷ}}_{j-1},{\mathsf{ȷ}}_j]$ for $1\le j\le k$; then, we have B be the IR-integral of ϝ on $[ð,v]$, which is denoted by $B=\left(IR\right){\int }_{ð}^v\mathsf{ϝ}\left(t\right)dt$.

Theorem 1. 

[33] The real interval valued function $\mathsf{ϝ}:[ð,v]\subseteq \mathbb{R}\to {\mathbb{R}}_I$ and $\mathsf{ϝ}\left(\mathsf{ȷ}\right)=[{\mathsf{ϝ}}_{*},{\mathsf{ϝ}}^{*}]$; then, ϝ is the integrable function on $[ð,v]$ if and only if ${\mathsf{ϝ}}_{*}$ and ${\mathsf{ϝ}}^{*}$ are both integrable functions over $[ð,v]$, if:

$$\begin{array}{c}\hfill \left(IR\right){\int }_{ð}^v\mathsf{ϝ}\left(\mathsf{ȷ}\right)dx=\left[\left(R\right){\int }_{ð}^v{\mathsf{ϝ}}_{*}\left(\mathsf{ȷ}\right),\left(R\right){\int }_{ð}^v{\mathsf{ϝ}}^{*}\left(\mathsf{ȷ}\right)dx\right].\end{array}$$

(8)

The representation of integrable function real valued functions and generalized integrable interval valued functions are ${\mathit{R}}_{[c,d]},{\mathit{IR}}_{[ð,v]}$, respectively.

Definition 6. 

[34] If for each $\mathsf{ı}\in [0,1]$ and let $\mathsf{ϝ}:k\subseteq R\to {\mathbb{F}}_0$ be fuzzy IVF, the ı-levels define on IVF $\mathsf{ϝ}:k\subseteq R\to {\mathsf{\Omega }}_c$ are given by ${\mathsf{ϝ}}_{\mathsf{ı}}\left(\mathsf{ȷ}\right)=\left[{\mathsf{ϝ}}_{*}(\mathsf{ȷ},\mathsf{ı}),{\mathsf{ϝ}}^{*}(\mathsf{ȷ},\mathsf{ı})\right]$, $\forall \mathsf{ȷ}\in \mathsf{\Omega }$. Both the real-valued functions ${\mathsf{ϝ}}_{*}(\mathsf{ȷ},\mathsf{ı}),{\mathsf{ϝ}}^{*}(\mathsf{ȷ},\mathsf{ı}):\mathsf{\Omega }\to \mathbb{R}$, called the upper and lower functions of ϝ.

Remark 1. 

Let $\mathsf{ϝ}:k\subseteq R\to {\mathbb{F}}_0$ be the FIV function, and for each $\mathsf{ı}\in [0,1]$, then $\mathsf{ϝ}\left(\mathsf{ȷ}\right)$ is called the continuous function at $\mathsf{ȷ}\in \mathsf{\Omega }$, and both the left and right real valued functions ${\mathsf{ϝ}}_{*}(\mathsf{ȷ},\mathsf{ı}),{\mathsf{ϝ}}^{*}(\mathsf{ȷ},\mathsf{ı})$ are continuous at $\mathsf{ȷ}\in K$.

Definition 7. 

[11] Let all closed and bounded intervals ${\mathsf{\Omega }}_C$ of $\mathbb{R}$ and $\tau \in {\mathsf{\Omega }}_C$, described as follows:

$$\begin{array}{c}\hfill \tau =[{\tau }_{*},{\tau }^{*}]=\{\alpha \in \mathbb{R}|{\tau }_{*}\preccurlyeq \alpha \preccurlyeq {\tau }^{*}\},({\tau }_{*},{\tau }^{*}\in \mathbb{R}).\end{array}$$

• If ${\tau }_{*}={\tau }^{*}$, then we say that $\tau$ is degenerate.
• If ${\tau }_{*}\ge 0$, then $[{\tau }_{*},{\tau }^{*}]$ is said to be a positive interval. All positive intervals are denoted by ${\mathsf{\Omega }}_C^{+}$ and defined as:

  $$\begin{array}{c}\hfill {\mathsf{\Omega }}_C^{+}=\{[{\tau }_{*},{\tau }^{*}]:[{\tau }_{*},{\tau }^{*}]\in {\mathsf{\Omega }}_C and {\tau }_{*}\ge 0\}.\end{array}$$

Definition 8. 

[11] Let $\sigma \in \mathbb{R}$ and $\sigma \tau$ be defined as:

$$\begin{array}{c}\hfill \sigma \tau =[\sigma {\tau }_{*},\sigma {\tau }^{*}]\mathit{if}\sigma \ge 0,[\sigma {\tau }^{*},\sigma {\tau }_{*}]if\sigma \preccurlyeq 0.\end{array}$$

(9)

Algebraic Minkowski properties are defined for $\tau ,\varsigma \in {\mathsf{\Omega }}_C$ as follows:

$$\begin{array}{c}\hfill [{\varsigma }_{*},{\varsigma }^{*}]-[{\tau }_{*},{\tau }^{*}]=[{\varsigma }_{*}-{\tau }_{*},{\varsigma }^{*}-{\tau }^{*}]\\ \hfill ,[{\varsigma }_{*},{\varsigma }^{*}]+[{\tau }_{*},{\tau }^{*}]=[{\varsigma }_{*}+{\tau }_{*},{\varsigma }^{*}+{\tau }^{*}]\end{array}$$

The inclusion $“\subseteq ”$ property is defined as:

$$\varsigma \subseteq \tau ,\mathit{then}[{\varsigma }_{*},{\varsigma }^{*}]\subseteq [{\tau }_{*},{\tau }^{*}]\mathit{implies}\mathit{that}{\varsigma }_{*}\preccurlyeq {\tau }_{*},{\varsigma }^{*}\preccurlyeq {\tau }^{*}.$$

Remark 2. 

[35] The relation $“{\preccurlyeq }_1”$ is defined on ${\mathsf{\Omega }}_C$ as follows:

$$\begin{array}{c}\hfill [{\sigma }_{*},{\sigma }^{*}]{\preccurlyeq }_1[{\tau }_{*},{\tau }^{*}]\mathrm{if}\mathrm{and}\mathrm{only}\mathrm{if}{\sigma }_{*}\preccurlyeq {\tau }_{*},{\sigma }^{*}\preccurlyeq {\tau }^{*}\end{array}$$

for all $[{\sigma }_{*},{\sigma }^{*}],[{\tau }_{*},{\tau }^{*}]\in {\mathsf{\Omega }}_C$, is an order relation. We have $[{\sigma }_{*},{\sigma }^{*}],[{\tau }_{*},{\tau }^{*}]\in {\mathsf{\Omega }}_C$, then $[{\sigma }_{*},{\sigma }^{*}]{\preccurlyeq }_1[{\tau }_{*},{\tau }^{*}]$ if and only if ${\sigma }_{*}\preccurlyeq {\tau }_{*}$ or ${\sigma }^{*}\preccurlyeq {\tau }^{*}$, $\sigma <\tau .$

Definition 9. 

[11] For $ð,v\in K$, $\lambda \in [0,1]$, if $\mathsf{ϝ}:K\times K\to \mathbb{R}$ is the bi-function, then the invex set $K\subseteq \mathbb{R}$ is defined as follows:

$$\begin{array}{c}\hfill v+\lambda \mathsf{ϝ}(ð,v)\in K.\end{array}$$

Definition 10. 

[11] Let K be an invex set with respect to ς and $\mathsf{ϝ}:K\to \mathbb{R}$ is the pre-invex function, defined for $\mathsf{ȷ},\aleph \in K$ as follows:

$$\begin{array}{c}\hfill \mathsf{ϝ}(\aleph +\lambda \varsigma (\mathsf{ȷ},\aleph \left)\right)\le \lambda \mathsf{ϝ}\left(\mathsf{ȷ}\right)+(1-\lambda )\mathsf{ϝ}(\aleph ),\end{array}$$

(10)

where $\lambda \in [0,1]$.

Definition 11. 

[22] If $\mathsf{ϝ}:K\to \mathbb{R}$, then the convex fuzzy interval valued function is defined as follows:

$$\begin{array}{c}\hfill \mathsf{ϝ}\left[\sigma ð+(1-\sigma )v\right]\preccurlyeq \sigma \mathsf{ϝ}\left(ð\right)\widetilde{+}(1-\sigma )\mathsf{ϝ}\left(v\right).\end{array}$$

(11)

for $\sigma \in [0,1]$, $\forall ð,v\in K$, and then we call ϝ concave if inequality (11) is reversed.

Remark 3. 

If ${\mathsf{ϝ}}_{*}(\mathsf{ȷ},\mathsf{ı})={\mathsf{ϝ}}^{*}(\mathsf{ȷ},\mathsf{ı})$ and $\mathsf{ı}=1$, then we obtain the inequality (1).

Definition 12. 

[23] The pre-invex FIV function $\mathsf{ϝ}:K\to \mathbb{R}$ is defined for $\mathsf{ȷ},\aleph \in K$ and $\lambda \in [0,1]$ as follows:

$$\begin{array}{c}\hfill \mathsf{ϝ}(\aleph +\lambda \varsigma (\mathsf{ȷ},\aleph ))\preccurlyeq \lambda \mathsf{ϝ}\left(\mathsf{ȷ}\right)\widetilde{+}(1-\lambda )\mathsf{ϝ}(\aleph ),\end{array}$$

(12)

where K is an open invex set with respect to ς if we reversed the inequality (12), then named as pre-incave FIV functions.

Definition 13. 

[36] The gamma function is defined in an integral form as follows:

$$\begin{array}{c}\hfill \mathsf{\Gamma }\left(\sigma \right)={\int }_0^{\infty }{z}^{\sigma -1}{e}^{-z}dz,\end{array}$$

for $\Re \left(t\right)>0$.

Definition 14. 

[36] The Pochammer’s symbol is defined as follows:

$$\begin{array}{c}\hfill {(\wp )}_{\sigma }=\left\{\begin{array}{cc}1,\hfill & \mathit{for}\sigma =0,\wp \ne 0\hfill \\ \wp (\wp +1)\cdots (\wp +\sigma -1),\hfill & \mathit{for}\sigma \ge 1,\hfill \end{array}\right\}\end{array}$$

For $\sigma \in \mathbb{N}$ and $\wp \in \mathbb{C}$:

$$\begin{array}{ccc}\hfill {(\wp )}_n& =& \frac{\mathsf{\Gamma }(\wp +n)}{\mathsf{\Gamma }(\wp )}\hfill \\ \hfill {(\wp )}_{kn}& =& \frac{\mathsf{\Gamma }(\wp +kn)}{\mathsf{\Gamma }(\wp )}\hfill \end{array}$$

where Γ is the gamma function.

Definition 15. 

[37] The beta function for $\Re \left(m\right)>0$ and $\Re \left(n\right)>0$ is defined as follows:

$$\begin{array}{cc}\hfill \mathsf{\Lambda }(g,h)=& {\int }_0^1{\sigma }^{g-1}{(1-\sigma )}^{h-1}d\sigma \hfill \\ & =\frac{\mathsf{\Gamma }\left(g\right)\mathsf{\Gamma }\left(h\right)}{\mathsf{\Gamma }(g+h)}.\hfill \end{array}$$

Definition 16. 

[38] The extended version of beta functions is defined for $\Re \left(g\right)>0$, $\Re \left(h\right)>0$, $\Re \left(p\right)>0$ as follows:

$$\begin{array}{c}\hfill {\mathsf{\Lambda }}_p(g,h)={\int }_0^1{z}^{g-1}{(1-z)}^{h-1}exp\left(\frac{-p}{z(1-z)}\right)dz.\end{array}$$

If we replace $p=1$, then the extended beta function is going to replace the classical beta function.

Definition 17. 

[39,40] The generalized Bessel–Maitland (eight-parameter) function is defined as follows:

$$\begin{array}{c}\hfill {\mathrm{J}}_{\xi ,\hslash ,m,\sigma }^{\beth ,\mathsf{\Theta },\sigma ,\vartheta }(\aleph )=\sum _{p=0}^{\infty }\frac{{\left(\mathsf{\Theta }\right)}_{\hslash p}{\left(\vartheta \right)}_{\sigma p}{(-\aleph )}^p}{\wp (\xi p+\beth +1){\left(\sigma \right)}_{mp}},\end{array}$$

(13)

where $\mathsf{ϝ},\beth ,\sigma ,\mathsf{ϝ},\vartheta \in \mathbb{C}$, $\Re \left(\mathsf{ϝ}\right)>0$, $\Re \left(\mathsf{ϝ}\right)>0$, $\Re \left(\nu \right)\ge -1$, $\Re \left(\sigma \right)>0$, $\Re \left(\vartheta \right)>0$; $m,\hslash ,\sigma \ge 0$ and $\hslash >\Re \left(\mathsf{ϝ}\right)+\sigma ,m$.

Definition 18. 

[40] The extended version of the Bessel–Maitland function is defined for $\mathsf{\Xi },\gimel ,\nu ,\wp ,\rho ,c\in \mathbb{C}$, $\Re \left(\mathsf{\Xi }\right)>0$, $\Re \left(\nu \right)\ge -1$, $\Re (\gimel )>0$, $\Re \left(\rho \right)>0$, $\Re (\wp )>0$; $\hslash ,\sigma ,m\ge 0$ and $\hslash ,m>\Re \left(\mathsf{\Xi }\right)+\sigma$ as follows:

$$\begin{array}{c}\hfill {\mathrm{J}}_{v,\gimel ,\rho ,\wp }^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega };p)=\sum _{n=0}^{\infty }\frac{{\mathsf{\Lambda }}_p(\gimel +\hslash n,c-\gimel ){\left(c\right)}_{\hslash n}{(\wp )}_{\sigma n}}{\mathsf{\Lambda }(\gimel ,c-\gimel )\wp (\mathsf{\Xi }n+v+1){\left(\rho \right)}_{mn}}{(-\mathsf{\Omega })}^n.\end{array}$$

(14)

Definition 19. 

Let $\mathsf{ϝ}:K\subseteq \mathbb{R}\to \mathbb{R}$ be the positive real valued function, then the Godunova–Levin FIV function is defined as follows:

$$\mathsf{ϝ}(\sigma ð+(1-\sigma )v)\preccurlyeq \frac{\mathsf{ϝ}\left(ð\right)}{\sigma }\widetilde{+}\frac{\mathsf{ϝ}\left(v\right)}{1-\sigma },$$

where $ð,v\in K$, $\sigma \in (0,1)$

Definition 20. 

Let $h:(0,1)\to \mathbb{R}$ and $\mathsf{ϝ}:K\to \mathbb{R}$ be the non-negative real valued function, the h-Godunova–Levin FIV function is defined for $\sigma \in (0,1)$, $ð,v\in K$ as follows:

$$\mathsf{ϝ}(\sigma ð+(1-\sigma )v)\preccurlyeq \frac{\mathsf{ϝ}\left(ð\right)}{h\left(\sigma \right)}\widetilde{+}\frac{\mathsf{ϝ}\left(v\right)}{h(1-\sigma )}.$$

Definition 21. 

A real valued function $\mathsf{ϝ}:K\to \mathbb{R}$ is said to be h-Godunova–Levin pre-invex FIV function with respect to ς if for all $ð,v\in K$, $\mathsf{\Phi }\in (0,1)$ as follows:

$$\mathsf{ϝ}(ð+\mathsf{\Phi }\varsigma (v,ð))\preccurlyeq \frac{\mathsf{ϝ}\left(ð\right)}{h(1-\mathsf{\Phi })}\widetilde{+}\frac{\mathsf{ϝ}\left(v\right)}{h\left(\mathsf{\Phi }\right)}$$

holds.

Definition 22. 

[40] The generalized fractional integral operators having an extended version of the Bessel–Maitland function as their kernel are defined for $\mathsf{\Xi },\nu ,\gimel ,\rho ,\wp ,c\in \mathbb{C}$, $\Re \left(\mathsf{\Xi }\right)>0$, $\Re \left(\nu \right)\ge -1$, $\Re \left(\rho \right)>0$, $\Re (\gimel )>0$, $\Re (\wp )>0$; $\hslash ,\sigma ,m\ge 0$ and $\hslash ,m>\Re \left(\mathsf{\Xi }\right)+\sigma$ as follows:

$$\begin{array}{cc}\hfill & \left({\mathfrak{T}}_{v,\gimel ,\rho ,\wp ;p^{+}}^{\mathsf{\Xi },\hslash ,m,\sigma ,c}f\right)(\mathsf{ȷ},r)={\int }_p^{\mathsf{ȷ}}{(\mathsf{ȷ}-t)}^v{\mathrm{J}}_{v,\gimel ,\rho ,\wp }^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega }{(\mathsf{ȷ}-t)}^{\mathsf{\Xi }};r)f\left(ð\right)dð,(\mathsf{ȷ}>p)\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & \left({\mathfrak{T}}_{v,\gimel ,\rho ,\wp ;q^{-}}^{\mathsf{\Xi },\hslash ,m,\sigma ,c}f\right)(\mathsf{ȷ},r)={\int }_{\mathsf{ȷ}}^q{(t-\mathsf{ȷ})}^v{\mathrm{J}}_{v,\gimel ,\rho ,\wp }^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega }{(t-\mathsf{ȷ})}^{\mathsf{\Xi }};r)f\left(ð\right)dð,(\mathsf{ȷ}<q).\hfill \end{array}$$

Remark 4. 

The generalized fractional operators are represented in short notations [41] as follows:

$$\begin{array}{cc}& \left({\mathcal{I}}_{v,v^{\prime }}^{{ð}^{+}}\right)(\mathsf{\Omega },\mathsf{ϝ})=\left({\mathfrak{T}}_{v^{\prime },\gimel ,\rho ,\wp ;{ð}^{+}}^{\mathsf{\Xi },\hslash ,m,\sigma ,c}\mathsf{ϝ}\right)(v,p)\\ & \left({\mathcal{I}}_{ð,v^{\prime }}^{v^{-}}\right)(\mathsf{\Omega },\mathsf{ϝ})=\left({\mathfrak{T}}_{v^{\prime },\gimel ,\rho ,\wp ;v^{-}}^{\mathsf{\Xi },\hslash ,m,\sigma ,c}\mathsf{ϝ}\right)(ð,p).\end{array}$$

and

$$\begin{array}{c}\hfill {\int }_v^u{(v-x)}^{v^{\prime }}{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\mathsf{ı}}^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega }{\left(\frac{v-x}{\varsigma (v,ð)}\right)}^{\mathsf{\Xi }};p)dx=\left({\mathfrak{I}}_{ð,v^{\prime }}^{v^{-}}\right)({\mathsf{\Omega }}^{\prime },1)\end{array}$$

3. Hermite–Hadamard (H-H) Integral Inequalities via Convex FIVF

Here, we define the left- and right-sided generalized fuzzy fractional integral operators, discuss the existence of H-H type inequality for the h-Godnova–Levin convex FIVF, and deduce some corollaries from our main results.

Definition 23. 

The left- and right-sided generalized fuzzy fractional integral operators based on the left and right end point function for $\mathsf{\Xi },\nu ,\gimel ,\rho ,\wp ,c\in \mathbb{C}$, $\Re \left(\mathsf{\Xi }\right)>0$, $\Re \left(\nu \right)\ge -1$, $\Re (\gimel )>0$, $\Re \left(\rho \right)>0$, $\Re (\wp )>0$; $\hslash ,m,\sigma \ge 0$ and $m,\hslash >\Re \left(\mathsf{\Xi }\right)+\sigma$ are defined as follows:

$$\begin{array}{cc}\hfill & {\left[\left({\mathfrak{T}}_{v,\gimel ,\rho ,\wp ;p^{+}}^{\mathsf{\Xi },\hslash ,m,\sigma ,c}f\right)(\mathsf{ȷ},r)\right]}^{\mathsf{ı}}={\int }_p^{\mathsf{ȷ}}{(\mathsf{ȷ}-t)}^v{\mathrm{J}}_{v,\gimel ,\rho ,\wp }^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega }{(\mathsf{ȷ}-t)}^{\mathsf{\Xi }};r)f_{\mathsf{ı}}\left(ð\right)dð\hfill \end{array}$$

$$\begin{array}{cc}\hfill & ={\int }_p^{\mathsf{ȷ}}{(\mathsf{ȷ}-t)}^v{\mathrm{J}}_{v,\gimel ,\rho ,\wp }^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega }{(\mathsf{ȷ}-t)}^{\mathsf{\Xi }};r)[f_{*}(ð,\mathsf{ı}),f^{*}(ð,\mathsf{ı})]dð,(\mathsf{ȷ}>p)\hfill \end{array}$$

where

$$\begin{array}{cc}& \left({\mathfrak{T}}_{v,\gimel ,\rho ,\wp ;p^{+}}^{\mathsf{\Xi },\hslash ,m,\sigma ,c}{f}_{*}\right)\left((\mathsf{ȷ},r),\mathsf{ı}\right)={\int }_p^{\mathsf{ȷ}}{(\mathsf{ȷ}-t)}^v{\mathrm{J}}_{v,\gimel ,\rho ,\wp }^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega }{(\mathsf{ȷ}-t)}^{\mathsf{\Xi }};r)f_{*}(ð,\mathsf{ı})dð,(\mathsf{ȷ}>p)\end{array}$$

$$\begin{array}{cc}& \left({\mathfrak{T}}_{v,\gimel ,\rho ,\wp ;p^{+}}^{\mathsf{\Xi },m,\hslash ,\sigma ,c}{f}^{*}\right)\left((\mathsf{ȷ},r),\mathsf{ı}\right)={\int }_p^{\mathsf{ȷ}}{(\mathsf{ȷ}-t)}^v{\mathrm{J}}_{v,\gimel ,\rho ,\wp }^{\mathsf{\Xi },m,\hslash ,\sigma ,c}(\mathsf{\Omega }{(\mathsf{ȷ}-t)}^{\mathsf{\Xi }};r)f^{*}(ð,\mathsf{ı})dð,(\mathsf{ȷ}>p)\end{array}$$

On the same pattern, we can easily define the right-sided generalized fractional integral operator based on left and right end point functions.

Remark 5. 

The generalized fuzzy fractional integral operators are represented in short notations as follows:

$$\begin{array}{ccc}\hfill & & {\int }_u^v{(x-u)}^{v^{\prime }}{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\mathsf{ı}}^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega }{(\frac{x-u}{\varsigma (v,ð)})}^{\mathsf{\Xi }};p){\mathsf{ϝ}}_{*}(x,\mathsf{ı})dx=\left({\mathfrak{I}}_{v,v^{\prime }}^{{ð}^{+}}\right)({\mathsf{\Omega }}^{\prime };{\mathsf{ϝ}}_{*})\hfill \\ & & {\int }_v^u{(v-x)}^{v^{\prime }}{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\mathsf{ı}}^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega }{(\frac{v-x}{\varsigma (v,ð)})}^{\mathsf{\Xi }};p){\mathsf{ϝ}}_{*}(x,\mathsf{ı})dx=\left({\mathfrak{I}}_{ð,v^{\prime }}^{v^{-}}\right)({\mathsf{\Omega }}^{\prime };{\mathsf{ϝ}}_{*})\hfill \end{array}$$

(15)

Theorem 2. 

Let $h:(0,1)\to \mathbb{R}$ be a positive function, $h\left(\sigma \right)\ne 0$ and h-Godunova Levin convex fuzzy interval valued function $\mathsf{ϝ}:[ð,v]\to \mathbb{R}$ with $\mathsf{ϝ}\in L_1[ð,v]$ and $0<ð<v$, then with the generalized fractional integral described in (22), we have;

$$\begin{array}{cc}\hfill & \frac{h\left(\frac{1}{2}\right)}{2}\mathsf{ϝ}\left(\frac{ð+v}{2}\right)\left({\mathfrak{I}}_{ð,v^{\prime }}^{v^{-}}\right)({\mathsf{\Omega }}^{\prime },1)\preccurlyeq \frac{1}{2}\left[\left({\mathfrak{I}}_{ð,v^{\prime }}^{v^{-}}\right)({\mathsf{\Omega }}^{\prime },\mathsf{ϝ})\widetilde{+}\left({\mathfrak{I}}_{v,v^{\prime }}^{{ð}^{+}}\right)({\mathsf{\Omega }}^{\prime },\mathsf{ϝ})\right]\hfill \\ & \preccurlyeq \frac{\mathsf{ϝ}\left(ð\right)\widetilde{+}\mathsf{ϝ}\left(v\right)}{2}{\int }_0^1\left[\frac{1}{h\left(\sigma \right)}+\frac{1}{h(1-\sigma )}\right]{\sigma }^{v^{\prime }}{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\mathsf{ı}}^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega }{\sigma }^{\mathsf{\Xi }};p)d\sigma ,\hfill \end{array}$$

(16)

where ${\mathsf{\Omega }}^{\prime }=\frac{\mathsf{\Omega }}{\varsigma {(v,ð)}^{\mathsf{\Xi }}}$

Proof. 

Consider $\mathsf{ϝ}$ to be the h-Godunova–Levin convex function for $\mathsf{ȷ},\aleph \in [ð,v]$, we have:

$$\mathsf{ϝ}(\hslash \mathsf{ȷ}+(1-\hslash )\aleph )\le \frac{\mathsf{ϝ}\left(\mathsf{ȷ}\right)}{h(\hslash )}+\frac{\mathsf{ϝ}(\aleph )}{h(1-\hslash )}$$

Replacing the values $\mathsf{ȷ}=\sigma ð+(1-\sigma )v,\aleph =(1-\sigma )ð+\sigma v$ and $\hslash =\frac{1}{2}$, we have:

$$\begin{array}{c}\hfill \mathsf{ϝ}\left(\frac{ð+v}{2}\right)\le \frac{1}{h\left(\frac{1}{2}\right)}\left[\mathsf{ϝ}(\sigma ð+(1-\sigma \left)v\right)+\mathsf{ϝ}\left(\right(1-\sigma )ð+\sigma v)\right]\\ \hfill h\left(\frac{1}{2}\right)\mathsf{ϝ}\left(\frac{ð+v}{2}\right)\le \mathsf{ϝ}(\sigma ð+(1-\sigma )v)+\mathsf{ϝ}((1-\sigma )ð+\sigma v)\end{array}$$

If we take for every $\mathsf{ı}\in [0,1]$, then we have:

$$\begin{array}{c}\hfill \mathsf{ϝ}(\frac{ð+v}{2},\mathsf{ı})\le \frac{1}{h\left(\frac{1}{2}\right)}\left[{\mathsf{ϝ}}_{*}(\sigma ð+(1-\sigma )v,\mathsf{ı})+{\mathsf{ϝ}}_{*}((1-\sigma )ð+\sigma v),\mathsf{ı}\right]\end{array}$$

(17)

Multiplying by ${\sigma }^{v^{\prime }}{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\mathsf{ı}}^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega }{\sigma }^{\mathsf{\Xi }};p)$ in Equation (17) and then integrating with respect to $\sigma$ over the interval $[0,1]$, we have:

$$\begin{array}{ccc}& & h\left(\frac{1}{2}\right){\mathsf{ϝ}}_{*}(\frac{ð+v}{2},\mathsf{ı}){\int }_0^1{\sigma }^{v^{\prime }}{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\mathsf{ı}}^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega }{\sigma }^{\mathsf{\Xi }};p)d\sigma \le {\int }_0^1{\sigma }^{v^{\prime }}{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\mathsf{ı}}^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega }{\sigma }^{\mathsf{\Xi }};p){\mathsf{ϝ}}_{*}(\sigma ð+(1-\sigma )v,\mathsf{ı})d\sigma \hfill \\ & & +{\int }_0^1{\sigma }^{v^{\prime }}{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\mathsf{ı}}^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega }{\sigma }^{\mathsf{\Xi }};p){\mathsf{ϝ}}_{*}((1-\sigma )ð+\sigma v,\mathsf{ı})d\sigma \hfill \end{array}$$

$$\begin{array}{cc}\hfill & h\left(\frac{1}{2}\right){\mathsf{ϝ}}_{*}(\frac{ð+v}{2},\mathsf{ı})\sum _{n=0}^{\infty }\frac{{\mathsf{\Lambda }}_p(\gimel +\hslash n,c-\gimel ){\left(c\right)}_{\hslash n}{\left(\mathsf{ı}\right)}_{\sigma n}}{\mathsf{\Lambda }(\gimel ,c-\gimel )\mathsf{ı}(\mathsf{\Xi }n+v^{\prime }+1){\left(\rho \right)}_{mn}}{(-\mathsf{\Omega })}^n{\int }_0^1{\sigma }^{v^{\prime }+\mathsf{\Xi }n}d\sigma \hfill \\ & \le \sum _{n=0}^{\infty }\frac{{\mathsf{\Lambda }}_p(\gimel +\hslash n,c-\gimel ){\left(c\right)}_{\hslash n}{\left(\mathsf{ı}\right)}_{\sigma n}}{\mathsf{\Lambda }(\gimel ,c-\gimel )\mathsf{ı}(\mathsf{\Xi }n+v^{\prime }+1){\left(\rho \right)}_{mn}}{(-\mathsf{\Omega })}^n[{\int }_0^1{\sigma }^{v^{\prime }+\mathsf{\Xi }n}{\mathsf{ϝ}}_{*}(\sigma ð+(1-\sigma )v,\mathsf{ı})d\sigma \hfill \\ & +{\int }_0^1{\sigma }^{v^{\prime }+\mathsf{\Xi }n}{\mathsf{ϝ}}_{*}((1-\sigma )ð+\sigma v,\mathsf{ı})d\sigma ].\hfill \end{array}$$

(18)

After simplification of the integral inequality (18) by using (15), we obtain:

$$\begin{array}{cc}\hfill & \frac{h\left(\frac{1}{2}\right)}{2}{\mathsf{ϝ}}_{*}(\frac{ð+v}{2},\mathsf{ı})\left({\mathfrak{I}}_{ð,v^{\prime }}^{v^{-}}\right)({\mathsf{\Omega }}^{\prime },1)\le \frac{1}{2}\left[\left({\mathfrak{I}}_{v,v^{\prime }}^{{ð}^{+}}\right)({\mathsf{\Omega }}^{\prime };{\mathsf{ϝ}}_{*})+\left({\mathfrak{I}}_{ð,v^{\prime }}^{v^{-}}\right)({\mathsf{\Omega }}^{\prime };{\mathsf{ϝ}}_{*})\right].\hfill \end{array}$$

(19)

Now, again consider the h-Godunova–Levin convex on $\mathsf{ϝ}$:

$$\begin{array}{c}\hfill \mathsf{ϝ}(\sigma ð+(1-\sigma )v)\le \frac{\mathsf{ϝ}\left(ð\right)}{h\left(\sigma \right)}+\frac{\mathsf{ϝ}\left(v\right)}{h(1-\sigma )}\end{array}$$

(20)

Re-writing Equation (20), we have:

$$\begin{array}{c}\hfill \mathsf{ϝ}((1-\sigma )ð+\sigma v)\le \frac{\mathsf{ϝ}\left(ð\right)}{h(1-\sigma )}+\frac{\mathsf{ϝ}\left(v\right)}{h\left(\sigma \right)}.\end{array}$$

(21)

Adding Equations (20) and (21) gives the following inequality:

$$\begin{array}{c}\hfill \mathsf{ϝ}(\sigma ð+(1-\sigma )v)+\mathsf{ϝ}((1-\sigma )ð+\sigma v)\le (\mathsf{ϝ}\left(ð\right)+\mathsf{ϝ}\left(v\right))\left[\frac{1}{h\left(\sigma \right)}+\frac{1}{h(1-\sigma )}\right].\end{array}$$

If we take for every $\mathsf{ı}\in [0,1]$, then we are given the following inequality:

$$\begin{array}{c}\hfill {\mathsf{ϝ}}_{*}(\sigma ð+(1-\sigma )v,\mathsf{ı})+{\mathsf{ϝ}}_{*}((1-\sigma )ð+\sigma v,\mathsf{ı})\le ({\mathsf{ϝ}}_{*}(ð,\mathsf{ı})+{\mathsf{ϝ}}_{*}(v,\mathsf{ı}))\left[\frac{1}{h\left(\sigma \right)}+\frac{1}{h(1-\sigma )}\right]\end{array}$$

(22)

Multiplying by ${\sigma }^{v^{\prime }}{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\mathsf{ı}}^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\omega {\sigma }^{\mathsf{\Xi }};p)$ in Equation (22), then integrating with respect to $\sigma$ over the interval $[0,1]$, we get:

$$\begin{array}{c}\hfill {\int }_0^1{\sigma }^{v^{\prime }}{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\mathsf{ı}}^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega }{\sigma }^{\mathsf{\Xi }};p){\mathsf{ϝ}}_{*}(\sigma ð+(1-\sigma )v,\mathsf{ı})d\sigma +{\int }_0^1{\sigma }^{v^{\prime }}{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\mathsf{ı}}^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega }{\sigma }^{\mathsf{\Xi }};p){\mathsf{ϝ}}_{*}((1-\sigma )ð+\sigma v,\mathsf{ı})d\sigma \\ \hfill \le ({\mathsf{ϝ}}_{*}(ð,\mathsf{ı})+{\mathsf{ϝ}}_{*}(v,\mathsf{ı})){\int }_0^1[\frac{1}{h\left(\sigma \right)}+\frac{1}{h(1-\sigma )}]{\sigma }^{v^{\prime }}{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\mathsf{ı}}^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega }{\sigma }^{\mathsf{\Xi }};p)d\sigma .\end{array}$$

(23)

After simplification of the inequality (23), we have:

$$\begin{array}{c}\hfill \frac{1}{2}\left[\left({\mathfrak{I}}_{v,v^{\prime }}^{{ð}^{+}}\right)({\mathsf{\Omega }}^{\prime };{\mathsf{ϝ}}_{*})+\left({\mathfrak{I}}_{ð,v^{\prime }}^{v^{-}}\right)({\mathsf{\Omega }}^{\prime };{\mathsf{ϝ}}_{*})\right]\le \frac{{\mathsf{ϝ}}_{*}(ð,\mathsf{ı})+{\mathsf{ϝ}}_{*}(v,\mathsf{ı})}{2}{\int }_0^1\left[\frac{1}{h\left(\sigma \right)}+\frac{1}{h(1-\sigma )}\right]{\sigma }^{v^{\prime }}{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\mathsf{ı}}^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega }{\sigma }^{\mathsf{\Xi }};p)d\sigma .\end{array}$$

(24)

Combining (19) and (24), we get inequality.

$$\begin{array}{c}\hfill \frac{h\left(\frac{1}{2}\right)}{2}{\mathsf{ϝ}}_{*}(\frac{ð+v}{2},\mathsf{ı})({\mathfrak{I}}_{ð,v^{\prime }}^{v^{-}})({\mathsf{\Omega }}^{\prime },1)\le \frac{1}{2}\left[\left({\mathfrak{I}}_{v,v^{\prime }}^{{ð}^{+}}\right)({\mathsf{\Omega }}^{\prime };{\mathsf{ϝ}}_{*})+\left({\mathfrak{I}}_{ð,v^{\prime }}^{v^{-}}\right)({\mathsf{\Omega }}^{\prime };{\mathsf{ϝ}}_{*})\right]\\ \hfill \le \frac{{\mathsf{ϝ}}_{*}(ð,\mathsf{ı})+{\mathsf{ϝ}}_{*}(v,\mathsf{ı})}{2}{\int }_0^1[\frac{1}{h\left(\sigma \right)}+\frac{1}{h(1-\sigma )}]{\sigma }^{v^{\prime }}{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\mathsf{ı}}^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega }{\sigma }^{\mathsf{\Xi }};p)d\sigma .\end{array}$$

(25)

Similarly for ${\mathsf{ϝ}}^{*}$:

$$\begin{array}{c}\hfill \frac{h\left(\frac{1}{2}\right)}{2}{\mathsf{ϝ}}^{*}(\frac{ð+v}{2},\mathsf{ı})\left({\mathfrak{I}}_{ð,v^{\prime }}^{v^{-}}\right)({\mathsf{\Omega }}^{\prime },1)\le \frac{1}{2}\left[\left({\mathfrak{I}}_{v,v^{\prime }}^{{ð}^{+}}\right)({\mathsf{\Omega }}^{\prime };{\mathsf{ϝ}}^{*})+\left({\mathfrak{I}}_{ð,v^{\prime }}^{v^{-}}\right)({\mathsf{\Omega }}^{\prime };{\mathsf{ϝ}}^{*})\right]\\ \hfill \le \frac{{\mathsf{ϝ}}^{*}(ð,\mathsf{ı})+{\mathsf{ϝ}}^{*}(v,\mathsf{ı})}{2}{\int }_0^1[\frac{1}{h\left(\sigma \right)}+\frac{1}{h(1-\sigma )}]{\sigma }^{v^{\prime }}{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\mathsf{ı}}^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega }{\sigma }^{\mathsf{\Xi }};p)d\sigma .\end{array}$$

(26)

From Equations (25) and (26):

$$\begin{array}{cc}\hfill & \frac{h\left(\frac{1}{2}\right)}{2}[{\mathsf{ϝ}}^{*}(\frac{ð+v}{2},\mathsf{ı}),{\mathsf{ϝ}}_{*}(\frac{ð+v}{2},\mathsf{ı})]({\mathfrak{I}}_{ð,v^{\prime }}^{v^{-}})({\mathsf{\Omega }}^{\prime },1)\hfill \\ \hfill & {\le }_I\left[\left({\mathfrak{I}}_{v,v^{\prime }}^{{ð}^{+}}\right)({\mathsf{\Omega }}^{\prime };{\mathsf{ϝ}}_{*})+\left({\mathfrak{I}}_{ð,v^{\prime }}^{v^{-}}\right)({\mathsf{\Omega }}^{\prime };{\mathsf{ϝ}}_{*})\right],\left[\left({\mathfrak{I}}_{v,v^{\prime }}^{{ð}^{+}}\right)({\mathsf{\Omega }}^{\prime };{\mathsf{ϝ}}^{*})+\left({\mathfrak{I}}_{ð,v^{\prime }}^{v^{-}}\right)({\mathsf{\Omega }}^{\prime };{\mathsf{ϝ}}^{*})\right]\hfill \\ \hfill & {\le }_I\left[\{{\mathsf{ϝ}}^{*}(ð,\mathsf{ı})+{\mathsf{ϝ}}^{*}(v,\mathsf{ı})\},\{{\mathsf{ϝ}}_{*}(ð,\mathsf{ı})+{\mathsf{ϝ}}_{*}(v,\mathsf{ı})\}\right]{\int }_0^1[\frac{1}{h\left(\sigma \right)}+\frac{1}{h(1-\sigma )}]{\sigma }^{v^{\prime }}{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\mathsf{ı}}^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega }{\sigma }^{\mathsf{\Xi }};p)d\sigma .\hfill \end{array}$$

After simplification:

$$\begin{array}{cc}\hfill & \frac{h\left(\frac{1}{2}\right)}{2}\mathsf{ϝ}\left(\frac{ð+v}{2}\right)\left({\mathfrak{I}}_{ð,v^{\prime }}^{v^{-}}\right)({\mathsf{\Omega }}^{\prime },1)\preccurlyeq \frac{1}{2}\left[\left({\mathfrak{I}}_{ð,v^{\prime }}^{v^{-}}\right)({\mathsf{\Omega }}^{\prime },\mathsf{ϝ})\widetilde{+}\left({\mathfrak{I}}_{v,v^{\prime }}^{{ð}^{+}}\right)({\mathsf{\Omega }}^{\prime },\mathsf{ϝ})\right]\preccurlyeq \frac{\mathsf{ϝ}\left(ð\right)\widetilde{+}\mathsf{ϝ}\left(v\right)}{2}\hfill \\ & {\int }_0^1\left[\frac{1}{h\left(\sigma \right)}+\frac{1}{h(1-\sigma )}\right]{\sigma }^{v^{\prime }}{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\mathsf{ı}}^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega }{\sigma }^{\mathsf{\Xi }};p)d\sigma ,\hfill \end{array}$$

□

Corollary 1. 

We obtain H-H-type inequality for the p-type FIV function if we substitute $h\left(\sigma \right)=1$ in Theorem 2:

$$\begin{array}{cc}\hfill & \frac{1}{2}\mathsf{ϝ}\left(\frac{ð+v}{2}\right)\left({\mathcal{I}}_{ð,v^{\prime }}^{v^{-}}\right)({\mathsf{\Omega }}^{\prime },1)\preccurlyeq \frac{1}{2}\left[\left({\mathcal{I}}_{ð,v^{\prime }}^{v^{-}}\right)({\mathsf{\Omega }}^{\prime },\mathsf{ϝ})\widetilde{+}\left({\mathcal{I}}_{v,v^{\prime }}^{{ð}^{+}}\right)({\mathsf{\Omega }}^{\prime },\mathsf{ϝ})\right]\hfill \\ & \preccurlyeq \left(\mathsf{ϝ}\left(ð\right)\widetilde{+}\mathsf{ϝ}\left(v\right)\right)\left({\mathcal{I}}_{v,v^{\prime }}^{{ð}^{+}}\right)({\mathsf{\Omega }}^{\prime },1)\hfill \end{array}$$

Corollary 2. 

We obtained H-H-type inequality for the s-Godunova–Levin FIV function after replacing $h\left(\sigma \right)={\sigma }^s$ in Theorem 2.

$$\begin{array}{cc}\hfill & \frac{{\left(\frac{1}{2}\right)}^s}{2}\mathsf{ϝ}\left(\frac{ð+v}{2}\right)\left({\mathcal{I}}_{ð,v^{\prime }}^{v^{-}}\right)({\mathsf{\Omega }}^{\prime },‘1)\preccurlyeq \frac{1}{2}\left[\left({\mathcal{I}}_{ð,v^{\prime }}^{v^{-}}\right)({\mathsf{\Omega }}^{\prime },\mathsf{ϝ})\widetilde{+}\left({\mathcal{I}}_{v,v^{\prime }}^{{ð}^{+}}\right)({\mathsf{\Omega }}^{\prime },\mathsf{ϝ})\right]\preccurlyeq \frac{\mathsf{ϝ}\left(ð\right)\widetilde{+}\mathsf{ϝ}\left(v\right)}{2}\hfill \\ & {\int }_0^1\left[\frac{1}{{\sigma }^s}+\frac{1}{{(1-\sigma )}^s}\right]{\sigma }^{v^{\prime }}{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\wp }^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega }{\sigma }^{\mathsf{\Xi }};p)d\sigma .\hfill \end{array}$$

Corollary 3. 

Putting the value $h\left(\sigma \right)=\frac{1}{\sigma }$ in Theorem 2, we obtain Hermite–Hadamard-type inequality for the FIV convex function.

$$\begin{array}{cc}\hfill & \mathsf{ϝ}\left(\frac{ð+v}{2}\right)\left({\mathcal{I}}_{ð,v^{\prime }}^{v^{-}}\right)({\mathsf{\Omega }}^{\prime },1)\preccurlyeq \frac{1}{2}\left[\left({\mathcal{I}}_{ð,v^{\prime }}^{v^{-}}\right)({\mathsf{\Omega }}^{\prime },\mathsf{ϝ})\widetilde{+}\left({\mathcal{I}}_{v,v^{\prime }}^{{ð}^{+}}\right)({\mathsf{\Omega }}^{\prime },\mathsf{ϝ})\right]\hfill \\ & \preccurlyeq \frac{\mathsf{ϝ}\left(ð\right)\widetilde{+}\mathsf{ϝ}\left(v\right)}{2}\left({\mathcal{I}}_{v,v^{\prime }}^{{ð}^{+}}\right)({\mathsf{\Omega }}^{\prime },1)\hfill \end{array}$$

Corollary 4. 

If we take $h\left(\sigma \right)=\sigma$ in Theorem 2, we obtain Hermite–Hadamard-type inequality for the Godunova–Levin FIV function.

$$\begin{array}{cc}\hfill & \frac{1}{4}\mathsf{ϝ}\left(\frac{ð+v}{2}\right)\left({\mathcal{I}}_{ð,v^{\prime }}^{v^{-}}\right)({\mathsf{\Omega }}^{\prime },1)\preccurlyeq \frac{1}{2}\left[\left({\mathcal{I}}_{ð,v^{\prime }}^{v^{-}}\right)({\mathsf{\Omega }}^{\prime },\mathsf{ϝ})\widetilde{+}\left({\mathcal{I}}_{v,v^{\prime }}^{{ð}^{+}}\right)({\mathsf{\Omega }}^{\prime },\mathsf{ϝ})\right]\preccurlyeq \frac{\mathsf{ϝ}\left(ð\right)\widetilde{+}\mathsf{ϝ}\left(v\right)}{2}\hfill \\ & {\int }_0^1\left[\frac{{\sigma }^{v^{\prime }-1}}{1-\sigma }\right]{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\wp }^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega }{\sigma }^{\mathsf{\Xi }};p)d\sigma .\hfill \end{array}$$

Corollary 5. 

If we choose $h\left(\sigma \right)=\frac{1}{{\sigma }^s}$ in Theorem 2, we obtain Hermite–Hadamard-type inequality for the s-convex FIV function.

$$\begin{array}{cc}\hfill & 2^{s-1}\mathsf{ϝ}\left(\frac{ð+v}{2}\right)\left({\mathcal{I}}_{ð,v^{\prime }}^{v^{-}}\right)({\mathsf{\Omega }}^{\prime },1)\preccurlyeq \frac{1}{2}\left[\left({\mathcal{I}}_{ð,v^{\prime }}^{v^{-}}\right)({\mathsf{\Omega }}^{\prime },\mathsf{ϝ})\widetilde{+}\left({\mathcal{I}}_{v,v^{\prime }}^{{ð}^{+}}\right)({\mathsf{\Omega }}^{\prime },\mathsf{ϝ})\right]\hfill \\ & \preccurlyeq \frac{\mathsf{ϝ}\left(ð\right)\widetilde{+}\mathsf{ϝ}\left(v\right)}{2}{\int }_0^1\left[{\sigma }^s+{(1-\sigma )}^s\right]{\sigma }^{v^{\prime }}{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\wp }^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega }{\sigma }^{\mathsf{\Xi }};p)d\sigma .\hfill \end{array}$$

4. Applications of Trapezoid Type Inequalities via Pre-Invex Fuzzy Interval Valued Function (FIV)

In this section, we discuss the important result in the form of lemma, which are used to develop our main results related to the trapezoid-type inequalities by the implementation of generalized fractional operators for the h-Godunova–Levin pre-invex fuzzy interval valued function.

Lemma 1. 

Let J be an open invex set with respect to $\varsigma :J\times J\to \mathbb{R}$, $\varsigma (v,ð)>0$ for $ð,v\in J$, $\mathsf{ϝ}\in L_1[ð,ð+\varsigma (v,ð)]$ be a differentiable function and $\mathsf{ϝ}:J=[ð,ð+\varsigma (v,ð\left)\right]\to \mathbb{R}$ with $ð,v\in \mathbb{R}$, then the following result holds:

$$\begin{array}{cc}& \frac{\mathsf{ϝ}\left(ð\right)\widetilde{+}\mathsf{ϝ}(ð+\varsigma (v,ð))}{2}{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\wp }^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega };p)-\frac{1}{2\varsigma {(v,ð)}^{v^{\prime }}}[\left({\mathcal{I}}_{ð+\varsigma (v,ð),v^{\prime }-1}^{{ð}^{+}}\right)({\mathsf{\Omega }}^{\prime };\mathsf{ϝ})\widetilde{+}\\ & \left({\mathcal{I}}_{ð,v^{\prime }-1}^{{(ð+{\gimel }_1(v,ð))}^{-}}\right)({\mathsf{\Omega }}^{\prime };\mathsf{ϝ})]=\frac{\varsigma (v,ð)}{2}J\end{array}$$

(27)

where ${\mathsf{\Omega }}^{\prime }=\frac{\mathsf{\Omega }}{\varsigma {(v,ð)}^{\mathsf{\Xi }}}$ and $J={\int }_0^1{\sigma }^{v^{\prime }}{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\wp }^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega }{\left(\sigma \right)}^{\mathsf{\Xi }};p){\mathsf{ϝ}}^{\prime }(ð+\sigma \varsigma (v,ð))d\sigma +{\int }_0^1-{(1-\sigma )}^{v^{\prime }}$${\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\wp }^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega }{(1-\sigma )}^{\mathsf{\Xi }};p){\mathsf{ϝ}}^{\prime }(ð+\sigma \varsigma (v,ð))d\sigma ,$

Proof. 

Consider the integral:

$$\begin{array}{c}\hfill J={\int }_0^1{\sigma }^{v^{\prime }}{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\wp }^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega }{\left(\sigma \right)}^{\mathsf{\Xi }};p){\mathsf{ϝ}}^{\prime }(ð+\sigma \varsigma (v,ð))d\sigma +{\int }_0^1-{(1-\sigma )}^{v^{\prime }}\\ \hfill {\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\wp }^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega }{(1-\sigma )}^{\mathsf{\Xi }};p){\mathsf{ϝ}}^{\prime }(ð+\sigma \varsigma (v,ð))d\sigma .\end{array}$$

Therefore, by taking the value of $\mathsf{ı}\in [0,1]$, we have:

$$\begin{array}{c}\hfill J={\int }_0^1{\sigma }^{v^{\prime }}{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\wp }^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega }{\left(\sigma \right)}^{\mathsf{\Xi }};p){\mathsf{ϝ}}_{*}^{{}^{\prime }}(ð+\sigma \varsigma (v,ð),\mathsf{ı})d\sigma +{\int }_0^1-{(1-\sigma )}^{v^{\prime }}\\ \hfill {\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\wp }^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega }{(1-\sigma )}^{\mathsf{\Xi }};p){\mathsf{ϝ}}_{*}^{{}^{\prime }}(ð+\sigma \varsigma (v,ð),\mathsf{ı})d\sigma .\end{array}$$

(28)

Now, we take the integrals:

$$\begin{array}{ccc}& & J_1={\int }_0^1{\sigma }^{v^{\prime }}{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\wp }^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega }{\left(\sigma \right)}^{\mathsf{\Xi }};p){\mathsf{ϝ}}_{*}^{{}^{\prime }}(ð+\sigma \varsigma (v,ð),\mathsf{ı})d\sigma \hfill \\ & & J_1=\sum _{n=0}^{\infty }\frac{{\mathsf{\Lambda }}_p(\gimel +\hslash n,c-\gimel ){\left(c\right)}_{\hslash n}{(\wp )}_{\sigma n}}{\mathsf{\Lambda }(\gimel ,c-\gimel )\wp (\mathsf{\Xi }n+v^{\prime }+1){\left(\rho \right)}_{mn}}{(-\mathsf{\Omega })}^n{\int }_0^1{\sigma }^{v^{\prime }+\mathsf{\Xi }n}{\mathsf{ϝ}}_{*}^{{}^{\prime }}(ð+\sigma \varsigma (v,ð),\mathsf{ı})d\sigma .\hfill \\ & & J_2={\int }_0^1-{(1-\sigma )}^{v^{\prime }}{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\wp }^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega }{(1-\sigma )}^{\mathsf{\Xi }};p){\mathsf{ϝ}}_{*}^{{}^{\prime }}(ð+\sigma \varsigma (v,ð),\mathsf{ı})d\sigma .\hfill \end{array}$$

$$\begin{array}{cc}\hfill & J_2=\sum _{n=0}^{\infty }\frac{{\mathsf{\Lambda }}_p(\gimel +\hslash n,c-\gimel ){\left(c\right)}_{\hslash n}{(\wp )}_{\sigma n}}{\mathsf{\Lambda }(\gimel ,c-\gimel )\wp (\mathsf{\Xi }n+v^{\prime }+1){\left(\rho \right)}_{mn}}{(-\mathsf{\Omega })}^n{\int }_0^1-{(1-\sigma )}^{v^{\prime }+\mathsf{\Xi }n}{\mathsf{ϝ}}_{*}^{{}^{\prime }}(ð+\sigma \varsigma (v,ð),\mathsf{ı})d\sigma .\hfill \end{array}$$

For solving the integral $J_1$:

$$\begin{array}{cc}\hfill & J_1=\sum _{n=0}^{\infty }\frac{{\mathsf{\Lambda }}_p(\gimel +\hslash n,c-\gimel ){\left(c\right)}_{\hslash n}{(\wp )}_{\sigma n}}{\mathsf{\Lambda }(\gimel ,c-\gimel )\wp (\mathsf{\Xi }n+v^{\prime }+1){\left(\rho \right)}_{mn}}{(-\mathsf{\Omega })}^n[{\sigma }^{v^{\prime }+\mathsf{\Xi }n}\frac{{\mathsf{ϝ}}_{*}(ð+\sigma \varsigma (v,ð),\mathsf{ı})}{\varsigma (v,ð)}{|}_0^1\hfill \\ & -\frac{v^{\prime }+\mathsf{\Xi }n}{\varsigma (v,ð)}{\int }_0^1{\sigma }^{v^{\prime }+\mathsf{\Xi }n-1}{\mathsf{ϝ}}_{*}(ð+\sigma \varsigma (v,ð),\mathsf{ı})d\sigma ]\hfill \\ & J_1=\sum _{n=0}^{\infty }\frac{{\mathsf{\Lambda }}_p(\gimel +\hslash n,c-\gimel ){\left(c\right)}_{\hslash n}{(\wp )}_{\sigma n}}{\mathsf{\Lambda }(\gimel ,c-\gimel )\wp (\mathsf{\Xi }n+v^{\prime }+1){\left(\rho \right)}_{mn}}{(-\mathsf{\Omega })}^n[\frac{{\mathsf{ϝ}}_{*}(ð+\varsigma (v,ð),\mathsf{ı})}{\varsigma (v,ð)}\hfill \\ & -\frac{v^{\prime }+\mathsf{\Xi }n}{\varsigma (v,ð)}{\int }_0^1{\sigma }^{v^{\prime }+\mathsf{\Xi }n-1}\mathsf{ϝ}(ð+\sigma \varsigma (v,ð))d\sigma ]\hfill \\ \hfill & J_1=\frac{{\mathsf{ϝ}}_{*}(ð+\varsigma (v,ð),\mathsf{ı})}{\varsigma (v,ð)}{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\wp }^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega };p)-\frac{1}{{\left(\varsigma (v,ð)\right)}^{v^{\prime }+1}}\left({\mathcal{I}}_{ð,v^{\prime }-1}^{{(ð+\varsigma (v,ð),\mathsf{ı})}^{-}}\right)({\mathsf{\Omega }}^{\prime };{\mathsf{ϝ}}_{*})\hfill \end{array}$$

By applying the same procedure for lower FIVF of the integral $J_2$, we have:

$$\begin{array}{cc}\hfill & J_2=\frac{{\mathsf{ϝ}}_{*}(ð,\mathsf{ı})}{\varsigma (v,ð)}{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\wp }^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega };p)-\frac{1}{{\left(\varsigma (v,ð)\right)}^{v^{\prime }+1}}\left({\mathcal{I}}_{(ð+\varsigma (v,ð),\mathsf{ı}),v^{\prime }-1}^{{ð}^{+}}\right)({\mathsf{\Omega }}^{\prime },{\mathsf{ϝ}}_{*})\hfill \\ \end{array}$$

(29)

Substituting the values of $J_1$ and $J_2$ in (28), we have:

$$\begin{array}{ccc}\hfill & & J=\frac{{\mathsf{ϝ}}_{*}(ð,\mathsf{ı})+{\mathsf{ϝ}}_{*}(ð+\varsigma (v,ð),\mathsf{ı})}{\varsigma (v,ð)}{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\wp }^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega };p)-\frac{1}{{\left(\varsigma (v,ð)\right)}^{v^{\prime }+1}}\hfill \\ & & \left[\left({\mathcal{I}}_{ð,v^{\prime }-1}^{{(ð+\varsigma (v,ð),\mathsf{ı})}^{-}}\right)({\mathsf{\Omega }}^{\prime },{\mathsf{ϝ}}_{*})+\left({\mathcal{I}}_{ð+\varsigma (v,ð),v^{\prime }-1}^{{(ð,\mathsf{ı})}^{+}}\right)({\mathsf{\Omega }}^{\prime },{\mathsf{ϝ}}_{*})\right].\hfill \end{array}$$

(30)

Similarly, solving the expression of J for upper FIVF ${\mathsf{ϝ}}^{*}$, we obtain:

$$\begin{array}{cc}\hfill & J=\frac{{\mathsf{ϝ}}^{*}(ð,\mathsf{ı})+{\mathsf{ϝ}}^{*}(ð+\varsigma (v,ð),\mathsf{ı})}{\varsigma (v,ð)}{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\wp }^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega };p)-\frac{1}{{\left(\varsigma (v,ð)\right)}^{v^{\prime }+1}}\times \hfill \\ & \left[\left({\mathcal{I}}_{ð,v^{\prime }-1}^{{(ð+\varsigma (v,ð),\mathsf{ı})}^{-}}\right)({\mathsf{\Omega }}^{\prime },{\mathsf{ϝ}}^{*})+\left({\mathcal{I}}_{ð+\varsigma (v,ð),v^{\prime }-1}^{{(ð,\mathsf{ı})}^{+}}\right)({\mathsf{\Omega }}^{\prime },{\mathsf{ϝ}}^{*})\right].\hfill \end{array}$$

(31)

By combining (30) and (31), we get:

$$\begin{array}{cc}\hfill & J=\frac{{\mathsf{ϝ}}^{*}(ð,\mathsf{ı}),{\mathsf{ϝ}}_{*}(ð,\mathsf{ı})+{\mathsf{ϝ}}^{*}(ð+\varsigma (v,ð),\mathsf{ı}),{\mathsf{ϝ}}_{*}(ð+\varsigma (v,ð),\mathsf{ı})}{\varsigma (v,ð)}{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\wp }^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega };p)-\frac{1}{{\left(\varsigma (v,ð)\right)}^{v^{\prime }+1}}\times \hfill \\ & \left[\left({\mathcal{I}}_{ð,v^{\prime }-1}^{{(ð+\varsigma (v,ð),\mathsf{ı})}^{-}}\right)({\mathsf{\Omega }}^{\prime },{\mathsf{ϝ}}^{*})+\left({\mathcal{I}}_{ð+\varsigma (v,ð),v^{\prime }-1}^{{(ð,\mathsf{ı})}^{+}}\right)({\mathsf{\Omega }}^{\prime },{\mathsf{ϝ}}^{*}),\left({\mathcal{I}}_{ð,v^{\prime }-1}^{{(ð+\varsigma (v,ð),\mathsf{ı})}^{-}}\right)({\omega }^{\prime },{\mathsf{ϝ}}^{*})+\left({\mathcal{I}}_{ð+\varsigma (v,ð),v^{\prime }-1}^{{(ð,\mathsf{ı})}^{+}}\right)({\mathsf{\Omega }}^{\prime },{\mathsf{ϝ}}^{*})\right].\hfill \\ \hfill & J=\frac{\mathsf{ϝ}\left(ð\right)\widetilde{+}\mathsf{ϝ}(ð+\varsigma (v,ð))}{\varsigma (v,ð)}{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\wp }^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega };p)-\frac{1}{{\left(\varsigma (v,ð)\right)}^{v^{\prime }+1}}[\left({\mathcal{I}}_{ð,v^{\prime }-1}^{{(ð+\varsigma (v,ð))}^{-}}\right)({\mathsf{\Omega }}^{\prime },\mathsf{ϝ})\hfill \\ & \widetilde{+}\left({\mathcal{I}}_{(ð+\varsigma (v,ð)),v^{\prime }-1}^{{ð}^{+}}\right)({\mathsf{\Omega }}^{\prime },\mathsf{ϝ})].\hfill \end{array}$$

(32)

Multiplying by $\frac{\varsigma (v,ð)}{2}$, we get the required result. □

Theorem 3. 

Let $J\in \mathbb{R}$ be a differentiable function on J with a function $\mathsf{ϝ}:J=[ð,ð+\varsigma (v,ð\left)\right]\to (0,\infty )$ for the generalized the fractional integral defined in (22) with the restricted extended generalized Bessel–Maitland function to a real valued function and suppose that $|{\mathsf{ϝ}}^{\prime }|$ is a h-Godunova–Levin pre-invex FIV function on J, then we have:

$$\begin{array}{cc}\hfill & |\frac{\mathsf{ϝ}\left(ð\right)\widetilde{+}\mathsf{ϝ}(ð+\varsigma (v,ð))}{2}{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\wp }^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega };p)-\frac{1}{2\varsigma {(v,ð)}^{v^{\prime }}}[\left({\mathcal{I}}_{ð+\varsigma (v,ð),v^{\prime }-1}^{{ð}^{+}}\right)({\mathsf{\Omega }}^{\prime },\mathsf{ϝ})\widetilde{+}\hfill \\ & \left({\mathcal{I}}_{ð,v^{\prime }-1}^{{(ð+\varsigma (v,ð))}^{-}}\right)({\mathsf{\Omega }}^{\prime },\mathsf{ϝ})\left]\right|\hfill \\ & \preccurlyeq \frac{\varsigma (v,ð)}{2}\left(|{\mathsf{ϝ}}^{\prime }\left(ð\right)|\widetilde{+}\left|{\mathsf{ϝ}}^{\prime }\left(v\right)\right|\right){\int }_0^1\sum _{n=0}^{\infty }\left|\frac{{\mathsf{\Lambda }}_p(\gimel +\hslash n,c-\gimel ){\left(c\right)}_{\hslash n}{(\wp )}_{\sigma n}}{\mathsf{\Lambda }(\gimel ,c-\gimel )\wp (\mathsf{\Xi }n+v^{\prime }+1){\left(\rho \right)}_{mn}}{(-\mathsf{\Omega })}^n\right|\hfill \\ & \left|\frac{{\sigma }^{v^{\prime }+\mathsf{\Xi }n}-{(1-\sigma )}^{v^{\prime }+\mathsf{\Xi }n}}{h\left(\sigma \right)}\right|d\sigma .\hfill \end{array}$$

Proof. 

By considering previous lemma and taking mod on both sides:

$$\begin{array}{cc}\hfill & |\frac{\mathsf{ϝ}\left(ð\right)\widetilde{+}\mathsf{ϝ}(ð+\varsigma (v,ð))}{2}{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\wp }^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega };p)-\frac{1}{2\varsigma {(v,ð)}^{v^{\prime }}}[\left({\mathcal{I}}_{ð+\varsigma (v,ð),v^{\prime }-1}^{{ð}^{+}}\right)({\mathsf{\Omega }}^{\prime },\mathsf{ϝ})\widetilde{+}\hfill \\ & \left({\mathcal{I}}_{ð,v^{\prime }-1}^{{(ð+\varsigma (v,ð))}^{-}}\right)({\mathsf{\Omega }}^{\prime },\mathsf{ϝ})\left]\right|=\left|\frac{\varsigma (v,ð)}{2}J\right|\hfill \end{array}$$

by taking the value $\mathsf{ı}\in [0,1]$, we have:

$$\begin{array}{c}\hfill =\left|\frac{\varsigma (v,ð)}{2}J\right|\end{array}$$

$$\begin{array}{cc}\hfill & \le \frac{\varsigma (v,ð)}{2}\sum _{n=0}^{\infty }|\frac{{\mathsf{\Lambda }}_p(\gimel +\hslash n,c-\gimel ){\left(c\right)}_{\hslash n}{(\wp )}_{\sigma n}}{\mathsf{\Lambda }(\gimel ,c-\gimel )\wp (\mathsf{\Xi }n+v^{\prime }+1){\left(\rho \right)}_{mn}}{(-\mathsf{\Omega })}^n|{\int }_0^1|{\sigma }^{v^{\prime }+\mathsf{\Xi }n}-{(1-\sigma )}^{v^{\prime }+\mathsf{\Xi }n}|\hfill \\ \hfill & |{\mathsf{ϝ}}_{*}^{{}^{\prime }}(ð+\sigma \varsigma (v,ð),\mathsf{ı})|d\sigma \hfill \\ \hfill & \le \frac{\varsigma (v,ð)}{2}\sum _{n=0}^{\infty }|\frac{{\mathsf{\Lambda }}_p(\gimel +\hslash n,c-\gimel ){\left(c\right)}_{\hslash n}{(\wp )}_{\sigma n}}{\mathsf{\Lambda }(\gimel ,c-\gimel )\wp (\mathsf{\Xi }n+v^{\prime }+1){\left(\rho \right)}_{mn}}{(-\mathsf{\Omega })}^n|{\int }_0^1|{\sigma }^{v^{\prime }+\mathsf{\Xi }n}-{(1-\sigma )}^{v^{\prime }+\mathsf{\Xi }n}|\hfill \\ & |\frac{{\mathsf{ϝ}}_{*}^{{}^{\prime }}]\left(ð\right)}{h\left(\sigma \right)}+\frac{{\mathsf{ϝ}}_{*}^{{}^{\prime }}\left(v\right)}{h(1-\sigma )}|d\sigma .\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \le \frac{\varsigma (v,ð)}{2}\sum _{n=0}^{\infty }|\frac{{\mathsf{\Lambda }}_p(\gimel +\hslash n,c-\gimel ){\left(c\right)}_{\hslash n}{(\wp )}_{\sigma n}}{\mathsf{\Lambda }(\gimel ,c-\gimel )\wp (\mathsf{\Xi }n+v^{\prime }+1){\left(\rho \right)}_{mn}}{(-\mathsf{\Omega })}^n|\hfill \\ & \left[|{\mathsf{ϝ}}_{*}^{{}^{\prime }}\left(ð\right)|{\int }_0^1|{\sigma }^{v^{\prime }+\mathsf{\Xi }n}-{(1-\sigma )}^{v^{\prime }+\mathsf{\Xi }n}||\frac{1}{h\left(\sigma \right)}|d\sigma +|{\mathsf{ϝ}}_{*}^{{}^{\prime }}\left(v\right)|{\int }_0^1|{\sigma }^{v^{\prime }+\mathsf{\Xi }n}-{(1-\sigma )}^{v^{\prime }+\mathsf{\Xi }n}||\frac{1}{h(1-\sigma )}|d\sigma \right]\hfill \\ & =\frac{\varsigma (v,ð)}{2}\left(|{\mathsf{ϝ}}_{*}^{{}^{\prime }}(ð,\mathsf{ı})|+|{\mathsf{ϝ}}_{*}^{{}^{\prime }}(v,\mathsf{ı})|\right){\int }_0^1\sum _{n=0}^{\infty }|\frac{{\mathsf{\Lambda }}_p(\gimel +\hslash n,c-\gimel ){\left(c\right)}_{\hslash n}{(\wp )}_{\sigma n}}{\mathsf{\Lambda }(\gimel ,c-\gimel )\wp (\mathsf{\Xi }n+v^{\prime }+1){\left(\rho \right)}_{mn}}{(-\mathsf{\Omega })}^n|\hfill \\ & |\frac{{\sigma }^{v^{\prime }+\mathsf{\Xi }n}-{(1-\sigma )}^{v^{\prime }+\mathsf{\Xi }n}}{h\left(\sigma \right)}|d\sigma .\hfill \end{array}$$

(33)

Similarly, if we solve for upper FIV function ${\mathsf{ϝ}}^{*}$, we have:

$$\begin{array}{cc}\hfill & \le \frac{\varsigma (v,ð)}{2}\left(|{\mathsf{ϝ}}^{{}^{\prime }*}(ð,\mathsf{ı})|+|{\mathsf{ϝ}}^{{}^{\prime }*}(v,\mathsf{ı})|\right){\int }_0^1\sum _{n=0}^{\infty }|\frac{{\mathsf{\Lambda }}_p(\gimel +\hslash n,c-\gimel ){\left(c\right)}_{\hslash n}{(\wp )}_{\sigma n}}{\mathsf{\Lambda }(\gimel ,c-\gimel )\wp (\mathsf{\Xi }n+v^{\prime }+1){\left(\rho \right)}_{mn}}{(-\mathsf{\Omega })}^n|\times \hfill \\ & |\frac{{\sigma }^{v^{\prime }+\mathsf{\Xi }n}-{(1-\sigma )}^{v^{\prime }+\mathsf{\Xi }n}}{h\left(\sigma \right)}|d\sigma .\hfill \end{array}$$

(34)

By combining Equations (33) and (34), we have the required result:

$$\begin{array}{ccc}\hfill & & \le \frac{\varsigma (v,ð)}{2}\left(|{\mathsf{ϝ}}_{*}^{\prime }(ð,\mathsf{ı})|+|{\mathsf{ϝ}}_{*}^{\prime }(v,\mathsf{ı})|,|{\mathsf{ϝ}}^{{}^{\prime }*}(ð,\mathsf{ı})|+|{\mathsf{ϝ}}^{{}^{\prime }*}(v,\mathsf{ı})|\right)\hfill \\ & & {\int }_0^1\sum _{n=0}^{\infty }|\frac{{\mathsf{\Lambda }}_p(\gimel +\hslash n,c-\gimel ){\left(c\right)}_{\hslash n}{(\wp )}_{\sigma n}}{\mathsf{\Lambda }(\gimel ,c-\gimel )\wp (\mathsf{\Xi }n+v^{\prime }+1){\left(\rho \right)}_{mn}}{(-\mathsf{\Omega })}^n||\frac{{\sigma }^{v^{\prime }+\mathsf{\Xi }n}-{(1-\sigma )}^{v^{\prime }+\mathsf{\Xi }n}}{h\left(\sigma \right)}|d\sigma .\hfill \\ & & \preccurlyeq \frac{\varsigma (v,ð)}{2}\left(|{\mathsf{ϝ}}^{\prime }\left(ð\right)|\widetilde{+}|{\mathsf{ϝ}}^{\prime }\left(v\right)|\right){\int }_0^1\sum _{n=0}^{\infty }|\frac{{\mathsf{\Lambda }}_p(\gimel +\hslash n,c-\gimel ){\left(c\right)}_{\hslash n}{(\wp )}_{\sigma n}}{\mathsf{\Lambda }(\gimel ,c-\gimel )\wp (\mathsf{\Xi }n+v^{\prime }+1){\left(\rho \right)}_{mn}}{(-\mathsf{\Omega })}^n|\hfill \\ & & |\frac{{\sigma }^{v^{\prime }+\mathsf{\Xi }n}-{(1-\sigma )}^{v^{\prime }+\mathsf{\Xi }n}}{h\left(\sigma \right)}|d\sigma .\hfill \end{array}$$

□

Corollary 6. 

Taking $\varsigma (v,ð)=v-ð$ in Theorem 3, we obtain the following inequality:

$$\begin{array}{cc}\hfill & |\frac{\mathsf{ϝ}\left(ð\right)\widetilde{+}\mathsf{ϝ}\left(v\right)}{2}{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\wp }^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega };p)-\frac{1}{2{(v-ð)}^{v^{\prime }}}\left[\left({\mathcal{I}}_{v,v^{\prime }-1}^{{ð}^{+}}\right)({\mathsf{\Omega }}^{\prime },\mathsf{ϝ})\widetilde{+}\left({\mathcal{I}}_{ð,v^{\prime }-1}^{v^{-}}\right)({\mathsf{\Omega }}^{\prime },\mathsf{ϝ})\right]|\hfill \\ & \preccurlyeq \frac{v-ð}{2}\left(|{\mathsf{ϝ}}^{\prime }\left(ð\right)|\widetilde{+}|{\mathsf{ϝ}}^{\prime }\left(v\right)|\right){\int }_0^1\sum _{n=0}^{\infty }|\frac{{\mathsf{\Lambda }}_p(\gimel +\hslash n,c-\gimel ){\left(c\right)}_{\hslash n}{(\wp )}_{\sigma n}}{\mathsf{\Lambda }(\gimel ,c-\gimel )\wp (\mathsf{\Xi }n+v^{\prime }+1){\left(\rho \right)}_{mn}}{(-\mathsf{\Omega })}^n|\hfill \\ & |\frac{{\sigma }^{v^{\prime }+\mathsf{\Xi }n}-{(1-\sigma )}^{v^{\prime }+\mathsf{\Xi }n}}{h\left(\sigma \right)}|d\sigma .\hfill \end{array}$$

Theorem 4. 

Let $\mathsf{ϝ}:J=[ð,ð+\varsigma (v,ð\left)\right]\to (0,\infty )$ be a differentiable function on $J\in \mathbb{R}$, and suppose that $|{\mathsf{ϝ}}^{\prime }{|}^q$ is a h-Godunova–Levin pre-invex FIV function on J with $p>1$ and $q=\left(p\right){(p-1)}^{-1}$, then for the generalized fractional integral defined in (22) with the restricted extended generalized Bessel–Maitland function to a real valued function, we have:

$$\begin{array}{cc}\hfill & |\frac{\mathsf{ϝ}\left(ð\right)\widetilde{+}\mathsf{ϝ}(ð+\varsigma (v,ð))}{2}{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\wp }^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega };p)-\frac{1}{2\varsigma {(v,ð)}^{v^{\prime }}}\left[\left({\mathcal{I}}_{ð+\varsigma (v,ð),v^{\prime }-1}^{{ð}^{+}}\right)({\mathsf{\Omega }}^{\prime },\mathsf{ϝ})\widetilde{+}\left({\mathcal{I}}_{ð,v^{\prime }-1}^{{(ð+\varsigma (v,ð))}^{-}}\right)({\mathsf{\Omega }}^{\prime },\mathsf{ϝ})\right]|\hfill \\ & \preccurlyeq \frac{\varsigma (v,ð)}{2}{\left(|{\mathsf{ϝ}}^{\prime }{\left(ð\right)|}^q\widetilde{+}{\left|{\mathsf{ϝ}}^{\prime }\left(v\right)\right|}^q\right)}^{\frac{1}{q}}\hfill \\ & {\left({\int }_0^1|{\sigma }^{v^{\prime }}{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\wp }^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega }{\sigma }^{\mathsf{\Xi }};p)-{(1-\sigma )}^{v^{\prime }}{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\wp }^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega }{(1-\sigma )}^{\mathsf{\Xi }};p){|}^{p}d\sigma \right)}^{\frac{1}{p}}{\left({\int }_0^1\frac{1}{h\left(\sigma \right)}d\sigma \right)}^{\frac{1}{q}}.\hfill \end{array}$$

Proof. 

By using Lemma 1, we have:

$$\begin{array}{cc}\hfill & |\frac{\mathsf{ϝ}\left(ð\right)\widetilde{+}\mathsf{ϝ}(ð+\varsigma (v,ð))}{2}{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\wp }^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega };p)-\frac{1}{2\varsigma {(v,ð)}^{v^{\prime }}}\left[\left({\mathcal{I}}_{ð+\varsigma (v,ð),v^{\prime }-1}^{{ð}^{+}}\right)({\mathsf{\Omega }}^{\prime },\mathsf{ϝ})\widetilde{+}\left({\mathcal{I}}_{ð,v^{\prime }-1}^{{(ð+\varsigma (v,ð))}^{-}}\right)({\mathsf{\Omega }}^{\prime },\mathsf{ϝ})\right]|\hfill \\ & =\left|\frac{\varsigma (v,ð)}{2}J\right|\hfill \\ \end{array}$$

by taking the value $\mathsf{ı}\in [0,1]$, we have:

$$\begin{array}{c}\hfill =\left|\frac{\varsigma (v,ð)}{2}J\right|\end{array}$$

$$\begin{array}{c}\hfill \le \frac{\varsigma (v,ð)}{2}{\int }_0^1|{\sigma }^{v^{\prime }}{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\wp }^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega }{\sigma }^{\mathsf{\Xi }};p)-{(1-\sigma )}^{v^{\prime }}{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\wp }^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega }{(1-\sigma )}^{\mathsf{\Xi }};p)|\left|{\mathsf{ϝ}}_{*}^{{}^{\prime }}(ð+\sigma \varsigma (v,ð),\mathsf{ı})\right|d\sigma \end{array}$$

Using Hölder’s integral inequality, we have:

$$\begin{array}{cc}\hfill & \le \frac{\varsigma (v,ð)}{2}{\left({\int }_0^1|{\sigma }^{v^{\prime }}{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\wp }^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega }{\sigma }^{\mathsf{\Xi }};p)-{(1-\sigma )}^{v^{\prime }}{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\wp }^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega }{(1-\sigma )}^{\mathsf{\Xi }};p){|}^{p}d\sigma \right)}^{\frac{1}{p}}\times \hfill \\ \hfill & {\left({\int }_0^1|{\mathsf{ϝ}}_{*}^{{}^{\prime }}(ð+\sigma \varsigma (v,ð),\mathsf{ı}){|}^{q}d\sigma \right)}^{\frac{1}{q}}\hfill \end{array}$$

(35)

where $p^{-1}+q^{-1}=1$.

Considering the h-Godunova–Levin pre-invex FIV function $|{\mathsf{ϝ}}_{*}^{{}^{\prime }}{|}^q$, we have:

$$\begin{array}{cc}\hfill & {\int }_0^1{\left|{\mathsf{ϝ}}_{*}^{{}^{\prime }}(ð+\sigma \varsigma (v,ð))\right|}^{q}d\sigma \le {\int }_0^1\left(\frac{|{\mathsf{ϝ}}^{\prime }{\left(ð\right)|}^q}{h\left(\sigma \right)}+\frac{|{\mathsf{ϝ}}^{\prime }{\left(v\right)|}^q}{h(1-\sigma )}\right)d\sigma \hfill \\ & \le \left(|{\mathsf{ϝ}}_{*}^{{}^{\prime }}{(ð,\mathsf{ı})|}^q+{\left|{\mathsf{ϝ}}_{*}^{{}^{\prime }}(v,\mathsf{ı})\right|}^q\right){\int }_0^1\frac{1}{h\left(\sigma \right)}d\sigma .\hfill \end{array}$$

(36)

By using Equation (36) in Equation (35), we obtain:

$$\begin{array}{cc}\hfill & \le \frac{\varsigma (v,ð)}{2}{\left(|{\mathsf{ϝ}}_{*}^{{}^{\prime }}{(ð,\mathsf{ı})|}^q+{\left|{\mathsf{ϝ}}_{*}^{{}^{\prime }}(v,\mathsf{ı})\right|}^q\right)}^{\frac{1}{q}}\hfill \\ & {\left({\int }_0^1|{\sigma }^{v^{\prime }}{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\wp }^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega }{\sigma }^{\mathsf{\Xi }};p)-{(1-\sigma )}^{v^{\prime }}{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\wp }^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega }{(1-\sigma )}^{\mathsf{\Xi }};p){|}^{p}d\sigma \right)}^{\frac{1}{p}}{\left({\int }_0^1\frac{1}{h\left(\sigma \right)}d\sigma \right)}^{\frac{1}{q}}.\hfill \end{array}$$

(37)

Similarly, if we solve for upper FIV function for $|{\mathsf{ϝ}}^{{}^{\prime }*}{|}^q$, we are given the following inequality:

$$\begin{array}{cc}\hfill & \le \frac{\varsigma (v,ð)}{2}{\left(|{\mathsf{ϝ}}^{{}^{\prime }*}{(ð,\mathsf{ı})|}^q+{\left|{\mathsf{ϝ}}^{{}^{\prime }*}(v,\mathsf{ı})\right|}^q\right)}^{\frac{1}{q}}\hfill \\ & {\left({\int }_0^1|{\sigma }^{v^{\prime }}{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\wp }^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega }{\sigma }^{\mathsf{\Xi }};p)-{(1-\sigma )}^{v^{\prime }}{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\wp }^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega }{(1-\sigma )}^{\mathsf{\Xi }};p){|}^{p}d\sigma \right)}^{\frac{1}{p}}{\left({\int }_0^1\frac{1}{h\left(\sigma \right)}d\sigma \right)}^{\frac{1}{q}}.\hfill \end{array}$$

(38)

By combining Equations (37) and (38), we have:

$$\begin{array}{cc}\hfill & {\le }_I\frac{\varsigma (v,ð)}{2}{\left(|{\mathsf{ϝ}}_{*}^{\prime }{(ð,\mathsf{ı})|}^q+|{\mathsf{ϝ}}_{*}^{\prime }{(v,\mathsf{ı})|}^q,|{\mathsf{ϝ}}^{{}^{\prime }*}{(ð,\mathsf{ı})|}^q+{\left|{\mathsf{ϝ}}^{{}^{\prime }*}(v,\mathsf{ı})\right|}^q\right)}^{\frac{1}{q}}\hfill \\ & {\left({\int }_0^1|{\sigma }^{v^{\prime }}{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\wp }^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega }{\sigma }^{\mathsf{\Xi }};p)-{(1-\sigma )}^{v^{\prime }}{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\wp }^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega }{(1-\sigma )}^{\mathsf{\Xi }};p){|}^{p}d\sigma \right)}^{\frac{1}{p}}{\left({\int }_0^1\frac{1}{h\left(\sigma \right)}d\sigma \right)}^{\frac{1}{q}}.\hfill \end{array}$$

(39)

After replacing the value $|{\mathsf{ϝ}}_{*}^{{}^{\prime }}{(ð,\mathsf{ı})|}^q+|{\mathsf{ϝ}}_{*}^{{}^{\prime }}{(v,\mathsf{ı})|}^q,|{\mathsf{ϝ}}^{{}^{\prime }*}{(ð,\mathsf{ı})|}^q+|{\mathsf{ϝ}}^{{}^{\prime }*}{(v,\mathsf{ı})|}^q\preccurlyeq {\left|{\mathsf{ϝ}}^{\prime }\left(ð\right)\right|}^q\widetilde{+}$$|{\mathsf{ϝ}}^{\prime }{\left(v\right)|}^q$, we have the required result. □

Theorem 5. 

With the assumptions of Theorem 4, we get the inequality related to Hermite–Hadamard inequality as follows:

$$\begin{array}{cc}\hfill & |\frac{\mathsf{ϝ}\left(ð\right)\widetilde{+}\mathsf{ϝ}(ð+\varsigma (v,ð))}{2}{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\wp }^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega };p)-\frac{1}{2\varsigma {(v,ð)}^{v^{\prime }}}\left[\left({\mathcal{I}}_{ð+\varsigma (v,ð),v^{\prime }-1}^{{ð}^{+}}\right)({\mathsf{\Omega }}^{\prime },\mathsf{ϝ})\widetilde{+}\left({\mathcal{I}}_{ð,v^{\prime }-1}^{{(ð+\varsigma (v,ð))}^{-}}\right)({\mathsf{\Omega }}^{\prime },\mathsf{ϝ})\right]|\hfill \\ & \preccurlyeq \frac{\varsigma (v,ð)}{2^{\frac{1}{q}}}{\left(|{\mathsf{ϝ}}^{\prime }{\left(ð\right)|}^q\widetilde{+}{\left|{\mathsf{ϝ}}^{\prime }\left(v\right)\right|}^q\right)}^{\frac{1}{q}}{\left[{\mathcal{J}}_{v^{\prime }+1,\gimel ,\rho ,\wp }^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega };p)-{\left(\frac{1}{2}\right)}^{v^{\prime }}{\mathcal{J}}_{v^{\prime }+1,\gimel ,\rho ,\wp }^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega }{\left(\frac{1}{2}\right)}^{\mathsf{\Xi }};p)\right]}^{1-\frac{1}{q}}\hfill \\ & {\left[{\int }_0^1\frac{|{\sigma }^{v^{\prime }}{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\wp }^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega }{\sigma }^{\mathsf{\Xi }};p)-{(1-\sigma )}^{v^{\prime }}{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\wp }^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega }{(1-\sigma )}^{\mathsf{\Xi }};p)|}{h\left(\sigma \right)}d\sigma \right]}^{\frac{1}{q}},\hfill \end{array}$$

where $v^{\prime },\mathsf{\Xi }\in {\mathbb{R}}^{+}$.

Proof. 

Considering Lemma 1, we have:

$$\begin{array}{cc}& |\frac{\mathsf{ϝ}\left(ð\right)\widetilde{+}\mathsf{ϝ}(ð+\varsigma (v,ð))}{2}{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\wp }^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega };p)-\frac{1}{2\varsigma {(v,ð)}^{v^{\prime }}}\left[\left({\mathcal{I}}_{ð+\varsigma (v,ð),v^{\prime }-1}^{{ð}^{+}}\right)({\mathsf{\Omega }}^{\prime },\mathsf{ϝ})\widetilde{+}\left({\mathcal{I}}_{ð,v^{\prime }-1}^{{(ð+\varsigma (v,ð))}^{-}}\right)({\mathsf{\Omega }}^{\prime },\mathsf{ϝ})\right]|=\left|\frac{\varsigma (v,ð)}{2}J\right|\end{array}$$

by taking the value $\mathsf{ı}\in [0,1]$, we have:

$$\begin{array}{c}\hfill =\left|\frac{\varsigma (v,ð)}{2}J\right|\end{array}$$

$$\begin{array}{c}\hfill \le \frac{\varsigma (v,ð)}{2}{\int }_0^1|{\sigma }^{v^{\prime }}{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\wp }^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega }{\sigma }^{\mathsf{\Xi }};p)-{(1-\sigma )}^{v^{\prime }}{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\wp }^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega }{(1-\sigma )}^{\mathsf{\Xi }};p)|\left|{\mathsf{ϝ}}_{*}^{{}^{\prime }}(ð+\sigma \varsigma (v,ð),\mathsf{ı})\right|d\sigma .\end{array}$$

By applying the power mean inequality, we get:

$$\begin{array}{cc}\hfill & |\frac{\mathsf{ϝ}\left(ð\right)+\mathsf{ϝ}(ð+\varsigma (v,ð\left)\right)}{2}{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\wp }^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega };p)-\frac{1}{2\varsigma {(v,ð)}^{v^{\prime }}}\left[\left({\mathcal{I}}_{ð+\varsigma (v,ð),v^{\prime }-1}^{{ð}^{+}}\right)({\mathsf{\Omega }}^{\prime },\mathsf{ϝ})+\left({\mathcal{I}}_{ð,v^{\prime }-1}^{{(ð+\varsigma (v,ð))}^{-}}\right)({\mathsf{\Omega }}^{\prime },\mathsf{ϝ})\right]|\\ & \le \frac{\varsigma (v,ð)}{2}{\left({\int }_0^1|{\sigma }^{v^{\prime }}{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\wp }^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega }{\sigma }^{\mathsf{\Xi }};p)-{(1-\sigma )}^{v^{\prime }}{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\wp }^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega }{(1-\sigma )}^{\mathsf{\Xi }};p)|d\sigma \right)}^{1-\frac{1}{q}}\times \\ & {\left({\int }_0^1|{\sigma }^{v^{\prime }}{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\wp }^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega }{\sigma }^{\mathsf{\Xi }};p)-{(1-\sigma )}^{v^{\prime }}{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\wp }^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega }{(1-\sigma )}^{\mathsf{\Xi }};p)||{\mathsf{ϝ}}_{*}^{{}^{\prime }}(ð+\sigma \varsigma (v,ð),\mathsf{ı}){|}^{q}d\sigma \right)}^{\frac{1}{q}}.\end{array}$$

(40)

Let

$$\begin{array}{c}\hfill I={\int }_0^1|{\sigma }^{v^{\prime }}{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\wp }^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega }{\sigma }^{\mathsf{\Xi }};p)-{(1-\sigma )}^{v^{\prime }}{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\wp }^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega }{(1-\sigma )}^{\mathsf{\Xi }};p)||{\mathsf{ϝ}}_{*}^{{}^{\prime }}(ð+\sigma \varsigma (v,ð),\mathsf{ı}){|}^{q}d\sigma \end{array}$$

By applying the definition of the h-Godunova–Levin pre-invex function FIV function on $|{\mathsf{ϝ}}_{*}^{{}^{\prime }}(ð+\sigma \varsigma (v,ð),\mathsf{ı}){|}^q$, we have:

$$\begin{array}{ccc}\hfill & & \le {\int }_0^1\frac{|{\sigma }^{v^{\prime }}{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\wp }^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega }{\sigma }^{\mathsf{\Xi }};p)-{(1-\sigma )}^{v^{\prime }}{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\wp }^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega }{(1-\sigma )}^{\mathsf{\Xi }};p)|}{h\left(\sigma \right)}|{\mathsf{ϝ}}_{*}^{{}^{\prime }}(ð,\mathsf{ı}){|}^q\hfill \\ & & +\frac{|{\sigma }^{v^{\prime }}{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\wp }^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega }{\sigma }^{\mathsf{\Xi }};p)-{(1-\sigma )}^{v^{\prime }}{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\wp }^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega }{(1-\sigma )}^{\mathsf{\Xi }};p)|}{h(1-\sigma )}|{\mathsf{ϝ}}_{*}^{{}^{\prime }}(v,\mathsf{ı}){|}^q)d\sigma \hfill \\ & & =\left(|{\mathsf{ϝ}}_{*}^{{}^{\prime }}{(ð,\mathsf{ı})|}^q+{|{\mathsf{ϝ}}_{*}^{{}^{\prime }}(v,\mathsf{ı})|}^q\right)\hfill \\ & & {\int }_0^1\frac{|{\sigma }^{v^{\prime }}{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\wp }^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega }{\sigma }^{\mathsf{\Xi }};p)-{(1-\sigma )}^{v^{\prime }}{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\wp }^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega }{(1-\sigma )}^{\mathsf{\Xi }};p)|}{h\left(\sigma \right)}.\hfill \end{array}$$

(41)

Similarly, we solve for the upper FIV function $|{\mathsf{ϝ}}^{{}^{\prime }*}{|}^q$:

$$\begin{array}{cc}\hfill & I\le {\int }_0^1\frac{|{\sigma }^{v^{\prime }}{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\wp }^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega }{\sigma }^{\mathsf{\Xi }};p)-{(1-\sigma )}^{v^{\prime }}{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\wp }^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega }{(1-\sigma )}^{\mathsf{\Xi }};p)|}{h\left(\sigma \right)}|{\mathsf{ϝ}}^{{}^{\prime }*}(ð,\mathsf{ı}){|}^q\hfill \\ & +\frac{|{\sigma }^{v^{\prime }}{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\wp }^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega }{\sigma }^{\mathsf{\Xi }};p)-{(1-\sigma )}^{v^{\prime }}{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\wp }^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega }{(1-\sigma )}^{\mathsf{\Xi }};p)|}{h(1-\sigma )}|{\mathsf{ϝ}}^{{}^{\prime }*}(v,\mathsf{ı}){|}^q)d\sigma \hfill \\ & =\left(|{\mathsf{ϝ}}^{{}^{\prime }*}{(ð,\mathsf{ı})|}^q+{|{\mathsf{ϝ}}^{{}^{\prime }*}(v,\mathsf{ı})|}^q\right){\int }_0^1\frac{|{\sigma }^{v^{\prime }}{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\wp }^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega }{\sigma }^{\mathsf{\Xi }};p)-{(1-\sigma )}^{v^{\prime }}{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\wp }^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega }{(1-\sigma )}^{\mathsf{\Xi }};p)|}{h\left(\sigma \right)}.\hfill \end{array}$$

(42)

Combining the upper and lower value of FIV function for I, we have:

$$\begin{array}{cc}\hfill & I{\le }_I\left(|{\mathsf{ϝ}}_{*}^{{}^{\prime }}{(ð,\mathsf{ı})|}^q+|{\mathsf{ϝ}}_{*}^{{}^{\prime }}{(v,\mathsf{ı})|}^q,|{\mathsf{ϝ}}^{{}^{\prime }*}{(ð,\mathsf{ı})|}^q+{|{\mathsf{ϝ}}^{{}^{\prime }*}(v,\mathsf{ı})|}^q\right)\hfill \\ \hfill & {\int }_0^1\frac{|{\sigma }^{v^{\prime }}{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\wp }^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega }{\sigma }^{\mathsf{\Xi }};p)-{(1-\sigma )}^{v^{\prime }}{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\wp }^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega }{(1-\sigma )}^{\mathsf{\Xi }};p)|}{h\left(\sigma \right)}.\hfill \\ \hfill & \preccurlyeq \left(|{\mathsf{ϝ}}^{{}^{\prime }}{\left(ð\right)|}^q\widetilde{+}{|{\mathsf{ϝ}}^{{}^{\prime }}\left(v\right)|}^q\right){\int }_0^1\frac{|{\sigma }^{v^{\prime }}{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\wp }^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega }{\sigma }^{\mathsf{\Xi }};p)-{(1-\sigma )}^{v^{\prime }}{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\wp }^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega }{(1-\sigma )}^{\mathsf{\Xi }};p)|}{h\left(\sigma \right)}.\hfill \end{array}$$

(43)

Now, consider the integral:

$$\begin{array}{ccc}\hfill & & {\int }_0^1|{\sigma }^{v^{\prime }}{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\wp }^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega }{\sigma }^{\mathsf{\Xi }};p)-{(1-\sigma )}^{v^{\prime }}{\mathcal{J}}_{v^{\prime },\gimel ,\rho ,\wp }^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega }{(1-\sigma )}^{\mathsf{\Xi }};p)|\hfill \\ & & =\sum _{n=0}^{\infty }|\frac{{\mathsf{\Lambda }}_p(\gimel +\hslash n,c-\gimel ){\left(c\right)}_{\hslash n}{(\wp )}_{\sigma n}}{\mathsf{\Lambda }(\gimel ,c-\gimel )\wp (\mathsf{\Xi }n+v^{\prime }+1){\left(\rho \right)}_{mn}}{(-\mathsf{\Omega })}^n|{\int }_0^1|{\sigma }^{v^{\prime }+\mathsf{\Xi }n}-{(1-\sigma )}^{v^{\prime }+\mathsf{\Xi }n}|d\sigma \hfill \\ & & =\sum _{n=0}^{\infty }|\frac{{\mathsf{\Lambda }}_p(\gimel +\hslash n,c-\gimel ){\left(c\right)}_{\hslash n}{(\wp )}_{\sigma n}}{\mathsf{\Lambda }(\gimel ,c-\gimel )\wp (\mathsf{\Xi }n+v^{\prime }+1){\left(\rho \right)}_{mn}}{(-\mathsf{\Omega })}^n|[{\int }_0^{\frac{1}{2}}\left({(1-\sigma )}^{v^{\prime }+\mathsf{\Xi }n}-{\sigma }^{v^{\prime }+\mathsf{\Xi }n}\right)d\sigma +\hfill \\ & & {\int }_{\frac{1}{2}}^1\left({\sigma }^{v^{\prime }+\mathsf{\Xi }n}-{(1-\sigma )}^{v^{\prime }+\mathsf{\Xi }n}\right)]\hfill \\ & & =2\left[{\mathcal{J}}_{v^{\prime }+1,\gimel ,\rho ,\wp }^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega };p)-{\left(\frac{1}{2}\right)}^{v^{\prime }}{\mathcal{J}}_{v^{\prime }+1,\gimel ,\rho ,\wp }^{\mathsf{\Xi },\hslash ,m,\sigma ,c}(\mathsf{\Omega }{\left(\frac{1}{2}\right)}^{\mathsf{\Xi }};p)\right].\hfill \end{array}$$

(44)

Putting the values (43) and (44) in (40), we have the required result. □

5. Conclusions

In research work, we investigated the existence of inequalities such as Hermite–Hadamard-type inequalities and trapezoid-type inequalities for h-Godunova–Levin convex and pre-invex fuzzy interval valued functions by the implementation of FFIPs. The obtained inequalities are the generalizations and extensions by means of fuzzy interval valued functions. A lot of work could be conducted in the field of analysis by improving the convex and non-convex functions in FIVFs. In future work, we will attempt to investigate this idea for generalized convex fuzzy-IVFs and various applications in fuzzy-interval nonlinear programming. The new area of research in convex analysis and optimization theory can be found by applying this idea.