Fractional Calculus for Convex Functions in Interval-Valued Settings and Inequalities

Abstract

In this paper, we discuss the Riemann–Liouville fractional integral operator for left and right convex interval-valued functions (left and right convex I∙V-F), as well as various related notions and concepts. First, the authors used the Riemann–Liouville fractional integral to prove Hermite–Hadamard type (𝓗–𝓗 type) inequality. Furthermore, 𝓗–𝓗 type inequalities for the product of two left and right convex I∙V-Fs have been established. Finally, for left and right convex I∙V-Fs, we found the Riemann–Liouville fractional integral Hermite–Hadamard type inequality (𝓗–𝓗 Fejér type inequality). The findings of this research show that this methodology may be applied directly and is computationally simple and precise.

1. Introduction

Mathematical inequality, finance, engineering, statistics, and probability all use convex functions in some way. Convex and symmetric convex functions have strong relationships with inequalities. Because of their intriguing features in the mathematical sciences, there are expansive properties and strong links between the symmetric function and different fields of convexity, including convex functions, probability theory, and convex geometry on convex sets. Convex functions have a long and illustrious history in science, and they have been a hot focus of study for more than a century. Several researchers have proposed different convex function guesses, expansions, and variants. Many inequalities or equalities, such as the Ostrowski-type inequality, Hardy-type inequality, Opial-type inequality, Simpson inequality, Fejér-type inequality, and Cebysev-type inequalities, have been established using convex functions. Among these inequalities, the 𝓗–𝓗 inequality [1,2], on which many publications have been published, is likely the one that attracts the most attention from scholars. 𝓗–𝓗 inequality has been regarded as the most useful inequality in mathematical analysis since its discovery in 1883. It is also known as the conventional 𝓗–𝓗 Inequality equation. The expansions and generalizations of the 𝓗–𝓗 inequality have piqued the curiosity of a number of mathematicians. For various classes of convex functions and mappings, a number of mathematicians in the fields of pure and applied mathematics have worked to expand, generalize, counterpart, and enhance the 𝓗–𝓗 inequality (references [3,4,5,6,7,8,9,10,11,12,13] are a good place to start for interested readers).

Historically, Leibnitz and L’Hospital (1695) are credited with the invention of fractional calculus; however, Riemann, Liouville, and Grunwald–Letnikov, among others, made significant contributions to the field later on. The way that fractional operator speculation deciphers nature’s existence in a grand and intentional fashion [14,15,16,17,18,19] has piqued the curiosity of researchers. By offering an enhanced form of an integral representation for the Appell k-series, Mubeen and Iqbal [20] have contributed to the present research.

Moreover, Khan et al. [21] exploited fuzzy order relations to introduce a new class of convex fuzzy-interval-valued functions (convex F-I∙V-Fs), known as ($h_1,h_2$)-convex F-I∙V-Fs, as well as a novel version of the 𝓗–𝓗 type inequality for ($h_1,h_2$)-convex F-I∙V-Fs that incorporates the fuzzy interval Riemann integral. Khan et al. went a step further by providing new convex and extended convex I∙V-F classes, as well as new fractional 𝓗–𝓗 type and 𝓗–𝓗 type inequalities for left and right ($h_1,h_2$)-preinvex I∙V-F [22], left and right p-convex I∙V-Fs [23], left and right log-h-convex I∙V-Fs [24], and the references therein. For further analysis of the literature on the applications and properties of fuzzy Riemannian integrals, inequalities, and generalized convex fuzzy mappings, we refer the readers to cited works [25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56] and the references therein.

Motivated and inspired by the fascinating features of symmetry, convexity, and the fractional operator, we study the new 𝓗–𝓗 and related 𝓗–𝓗 type inequalities for left and right convex I∙V-Fs, based upon the pseudo order relation and the Riemann–Liouville fractional integral operator.

2. Preliminaries

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

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

$${\rm Y}=\left[{{\rm Y}}_{*},{{\rm Y}}^{*}\right],Q=\left[Q_{*},Q^{*}\right]\hspace{1em}\left({{\rm Y}}_{*}\le \omega \le {{\rm Y}}^{*} and Q_{*}\le z\le Q^{*} \omega , z\in ℝ\right)$$

$$\begin{gathered} {\rm Y}+Q=\left[{{\rm Y}}_{*},{{\rm Y}}^{*}\right]+\left[Q_{*},Q^{*}\right]=\left[{{\rm Y}}_{*}+Q_{*},{{\rm Y}}^{*}+Q^{*}\right], \\ {\rm Y}-Q=\left[{{\rm Y}}_{*},{{\rm Y}}^{*}\right]-\left[Q_{*},Q^{*}\right]=\left[{{\rm Y}}_{*}-Q_{*},{{\rm Y}}^{*}-Q^{*}\right], \\ {\rm Y}\times Q=\left[{{\rm Y}}_{*},{{\rm Y}}^{*}\right]\times \left[Q_{*},Q^{*}\right]=\left[\min \mathcal{K},\max \mathcal{K}\right] \\ \min \mathcal{K}=\min \left\{{{\rm Y}}_{*}{Q}_{*},{{\rm Y}}^{*}{Q}_{*},{{\rm Y}}_{*}{Q}^{*},{{\rm Y}}^{*}{Q}^{*}\right\},\max \mathcal{K}=\max \left\{{{\rm Y}}_{*}{Q}_{*},{{\rm Y}}^{*}{Q}_{*},{{\rm Y}}_{*}{Q}^{*},{{\rm Y}}^{*}{Q}^{*}\right\} \\ \nu .\left[{{\rm Y}}_{*},{{\rm Y}}^{*}\right]=\left\{\begin{array}{c}\left[\nu {{\rm Y}}_{*},\nu {{\rm Y}}^{*}\right] \mathrm{if} \nu >0,\\ \left\{0\right\} \mathrm{if} \nu =0,\\ \left[\nu {{\rm Y}}^{*},\nu {{\rm Y}}_{*}\right] \mathrm{if} \nu <0. \end{array}\right. \end{gathered}$$

Let ${\mathcal{X}}_I$, ${\mathcal{X}}_I^{+}$, ${\mathcal{X}}_I^{-}$ be the set of all closed intervals of $ℝ$, the set of all closed positive intervals of $ℝ$, and the set of all closed negative intervals of $ℝ$, respectively.

For $\left[{{\rm Y}}_{*},{{\rm Y}}^{*}\right],\left[Q_{*},Q^{*}\right]\in {\mathcal{X}}_I,$ the inclusion $\text{“}\subseteq \text{”}$ is defined by $\left[{{\rm Y}}_{*},{{\rm Y}}^{*}\right]\subseteq \left[Q_{*},Q^{*}\right]$, if and only if, $Q_{*}\le {{\rm Y}}_{*}$, ${{\rm Y}}^{*}\le Q^{*}.$

Remark 1.

[21] The left and right relation$“{\le }_p”$, defined on${\mathcal{X}}_I$by$\left[{{\rm Y}}_{*},{{\rm Y}}^{*}\right]{\le }_p\left[Q_{*},Q^{*}\right]$. , if and only if,${{\rm Y}}_{*}\le Q_{*}, {{\rm Y}}^{*}\le Q^{*},$for all$\left[{{\rm Y}}_{*}, {{\rm Y}}^{*}\right], \left[Q_{*}, Q^{*}\right]\in {\mathcal{X}}_I,$it is a pseudo order relation. For a given$\left[{{\rm Y}}_{*}, {{\rm Y}}^{*}\right], \left[Q_{*}, Q^{*}\right]\in {\mathcal{X}}_I,$we say that$\left[{{\rm Y}}_{*}, {{\rm Y}}^{*}\right]{\le }_p\left[Q_{*}, Q^{*}\right]$, if and only if,${{\rm Y}}_{*}\le Q_{*}, {{\rm Y}}^{*}\le Q^{*}$or${{\rm Y}}_{*}\le Q_{*}, {{\rm Y}}^{*}<Q^{*}$.

Theorem 1.

[33] If${\rm Y}:\left[t,s\right]\subset ℝ\to {\mathcal{X}}_I$is an I∙V-F on such that${\rm Y}\left(\omega \right)=\left[{{\rm Y}}_{*}\left(\omega \right), {{\rm Y}}^{*}\left(\omega \right)\right],$then${\rm Y}$is Riemann integrable over $\left[t,s\right]$, if and only if, ${{\rm Y}}_{*}$and ${{\rm Y}}^{*}$ are both Riemann integrable over $\left[t,s\right]$, such that

$$\left(IR\right){{\displaystyle \int }}_t^s{\rm Y}\left(\omega \right)d\omega =\left[\left(R\right){{\displaystyle \int }}_t^s{{\rm Y}}_{*}\left(\omega \right)d\omega , \left(R\right){{\displaystyle \int }}_t^s{{\rm Y}}^{*}\left(\omega \right)d\omega \right].$$

Definition 1.

[28,30] Let ${\rm Y}\in$ $L\left(\left[t,s\right],{\mathcal{X}}_I^{+}\right)$. Then, interval fractional integrals, ${\mathcal{I}}_{t^{+}}^{\mathfrak{a}}$ and ${\mathcal{I}}_{s^{-}}^{\mathfrak{a}}$, of order $\mathfrak{a}>0$ are defined by

$${\mathcal{I}}_{t^{+}}^{\mathfrak{a}} {\rm Y}\left(\omega \right)=\frac{1}{\Gamma \left(\mathfrak{a}\right)}{{\displaystyle \int }}_t^{\omega }{\left(\omega -\nu \right)}^{\mathfrak{a}-1}{\rm Y}\left(\nu \right)d\nu , \left(\omega >t\right),$$

(1)

and

$${\mathcal{I}}_{s^{-}}^{\mathfrak{a}} {\rm Y}\left(\omega \right)=\frac{1}{\Gamma \left(\mathfrak{a}\right)}{{\displaystyle \int }}_{\omega }^s{\left(\nu -\omega \right)}^{\mathfrak{a}-1}{\rm Y}\left(\nu \right)d\nu , (\omega <s),$$

(2)

respectively, where$\Gamma \left(\omega \right)={{\displaystyle \int }}_0^{\infty }{\nu }^{\omega -1}{e}^{-\nu }d\nu$is the Euler gamma function.

Definition 2.

[31] The I∙V-F ${\rm Y}:K\to {\mathcal{X}}_I^{+}$ is named as the left and right convex-I∙V- F on convex set $K$ if the coming inequality,

$${\rm Y}\left(\nu \omega +\left(1-\nu \right)z \right){\le }_p\nu {\rm Y}\left(\omega \right)+\left(1-\nu \zeta \right){\rm Y}\left(z\right),$$

(3)

holds for all$\omega , z\in K$and$\nu \in \left[0, 1\right]$we have. If inequality (3) is reversed, then${\rm Y}$is named as the left and right concave on$K$. ${\rm Y}$is affine, if and only if, it is both left and right convex and left and right concave.

Theorem 2.

[31] Let${\rm Y}:K\to {\mathcal{X}}_I^{+}$be an I∙V-F, such that

$${\rm Y}\left(\omega \right)=\left[{{\rm Y}}_{*}\left(\omega \right), {{\rm Y}}^{*}\left(\omega \right)\right], \forall \omega \in K$$

(4)

for all$\omega \in K$. Then,${\rm Y}$is a left and right convex I∙V-F on$K,$if and only if,${{\rm Y}}_{*}\left(\omega \right)$and${{\rm Y}}^{*}\left(\omega \right)$both are convex functions.

3. Interval Fractional Hermite–Hadamard Inequalities

The major goal, and the main purpose of this section, is to develop a novel version of the 𝓗–𝓗 inequalities in the mode of interval-valued left and right convex functions.

Theorem 3.

Let ${\rm Y}:\left[s, t\right]\to {\mathcal{X}}_I^{+}$ be a left and right convex I∙V-F on $\left[s, t\right]$ and provided by ${\rm Y}\left(\omega \right)=\left[{{\rm Y}}_{*}\left(\omega \right), {{\rm Y}}^{*}\left(\omega \right)\right]$ for all $\omega \in \left[s, t\right]$. If ${\rm Y}\in L\left(\left[s, t\right],{\mathcal{X}}_I^{+}\right)$, then

$${\rm Y}\left(\frac{s+t}{2}\right){\le }_p\frac{\Gamma \left(\mathfrak{a}+1\right)}{2{\left(t-s\right)}^{\mathfrak{a}}}\left[{\mathcal{I}}_{s^{+}}^{\mathfrak{a}} {\rm Y}\left(t\right)+{\mathcal{I}}_{t^{-}}^{\mathfrak{a}} {\rm Y}\left(s\right)\right]{\le }_p\frac{{\rm Y}\left(s\right)+{\rm Y}\left(t\right)}{2}.$$

(5)

If${\rm Y}\left(\omega \right)$is a left and right concave I∙V-F, then

$${\rm Y}\left(\frac{s+t}{2}\right){\ge }_p\frac{\Gamma \left(\mathfrak{a}+1\right)}{2{\left(t-s\right)}^{\mathfrak{a}}}\left[{\mathcal{I}}_{s^{+}}^{\mathfrak{a}} {\rm Y}\left(t\right)+{\mathcal{I}}_{t^{-}}^{\mathfrak{a}} {\rm Y}\left(s\right)\right]{\ge }_p\frac{{\rm Y}\left(s\right)+{\rm Y}\left(t\right)}{2}.$$

(6)

Proof. 

Let ${\rm Y}:\left[s, t\right]\to {\mathcal{X}}_I^{+}$ be a left and right convex I∙V-F. Then, by hypothesis, we have:

$$2{\rm Y}\left(\frac{s+t}{2}\right){\le }_p{\rm Y}\left(\nu s+\left(1-\nu \right)t\right)+{\rm Y}\left(\left(1-\nu \right)s+\nu t\right).$$

Therefore, we have

$$\begin{array}{c}2{{\rm Y}}_{*}\left(\frac{s+t}{2}\right)\le {{\rm Y}}_{*}\left(\nu s+\left(1-\nu \right)t\right)+{{\rm Y}}_{*}\left(\left(1-\nu \right)s+\nu t\right), \\ 2{{\rm Y}}^{*}\left(\frac{s+t}{2}\right)\le {{\rm Y}}^{*}\left(\nu s+\left(1-\nu \right)t\right)+{{\rm Y}}^{*}\left(\left(1-\nu \right)s+\nu t\right).\end{array}$$

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

$$\begin{array}{c}2{{\displaystyle \int }}_0^1{\nu }^{\mathfrak{a}-1}{{\rm Y}}_{*}\left(\frac{s+t}{2}\right)d\nu \\ \le {{\displaystyle \int }}_0^1{\nu }^{\mathfrak{a}-1}{{\rm Y}}_{*}\left(\nu s+\left(1-\nu \right)t\right)d\nu +{{\displaystyle \int }}_0^1{\nu }^{\mathfrak{a}-1}{{\rm Y}}_{*}\left(\left(1-\nu \right)s+\nu t\right)d\nu ,\\ 2{{\displaystyle \int }}_0^1{\nu }^{\mathfrak{a}-1}{{\rm Y}}^{*}\left(\frac{s+t}{2}\right)d\nu \hfill \\ \le {{\displaystyle \int }}_0^1{\nu }^{\mathfrak{a}-1}{{\rm Y}}^{*}\left(\nu s+\left(1-\nu \right)t\right)d\nu +{{\displaystyle \int }}_0^1{\nu }^{\mathfrak{a}-1}{{\rm Y}}^{*}\left(\left(1-\nu \right)s+\nu t\right)d\nu .\end{array}$$

Let $\omega =\nu s+\left(1-\nu \right)t$ and $z=\left(1-\nu \right)s+\nu t.$ Then, we have

$$\begin{array}{c}\frac{2}{\mathfrak{a}}{{\rm Y}}_{*}\left(\frac{s+t}{2}\right) \le \frac{1}{{\left(t-s\right)}^{\mathfrak{a}}} \underset{s}{\overset{t}{{\displaystyle \int }}}{\left(t-z\right)}^{\mathfrak{a}-1}{{\rm Y}}_{*}\left(z\right)dz+\frac{1}{{\left(t-s\right)}^{\mathfrak{a}}}\underset{s}{\overset{t}{{\displaystyle \int }}}{\left(\omega -s\right)}^{\mathfrak{a}-1}{{\rm Y}}_{*}\left(\omega \right)d\omega \\ \frac{2}{\mathfrak{a}}{{\rm Y}}_{*}\left(\frac{s+t}{2}\right)\le \frac{1}{{\left(t-s\right)}^{\mathfrak{a}}} \underset{s}{\overset{t}{{\displaystyle \int }}}{\left(t-z\right)}^{\mathfrak{a}-1}{{\rm Y}}^{*}\left(z\right)dz+\frac{1}{{\left(t-s\right)}^{\mathfrak{a}}}\underset{s}{\overset{t}{{\displaystyle \int }}}{\left(\omega -s\right)}^{\mathfrak{a}-1}{{\rm Y}}^{*}\left(\omega \right)d\omega ,\end{array}$$

$$\begin{array}{c}\le \frac{\Gamma \left(\mathfrak{a}\right)}{{\left(t-s\right)}^{\mathfrak{a}}}\left[{\mathcal{I}}_{s^{+}}^{\mathfrak{a}} {{\rm Y}}_{*}\left(t\right)+{\mathcal{I}}_{t^{-}}^{\mathfrak{a}} {{\rm Y}}_{*}\left(s\right)\right] \\ \le \frac{\Gamma \left(\mathfrak{a}\right)}{{\left(t-s\right)}^{\mathfrak{a}}}\left[{\mathcal{I}}_{s^{+}}^{\mathfrak{a}} {{\rm Y}}^{*}\left(t\right)+{\mathcal{I}}_{t^{-}}^{\mathfrak{a}} {{\rm Y}}^{*}\left(s\right)\right],\end{array}$$

That is,

$$\frac{2}{\mathfrak{a}}\left[{{\rm Y}}_{*}\left(\frac{s+t}{2}\right), {{\rm Y}}^{*}\left(\frac{s+t}{2}\right)\right]{\le }_p\frac{\Gamma \left(\mathfrak{a}\right)}{{\left(t-s\right)}^{\mathfrak{a}}}\left[\left[{\mathcal{I}}_{s^{+}}^{\mathfrak{a}} {{\rm Y}}_{*}\left(t\right)+{\mathcal{I}}_{t^{-}}^{\mathfrak{a}} {{\rm Y}}_{*}\left(s\right)\right], \left[{\mathcal{I}}_{s^{+}}^{\mathfrak{a}} {{\rm Y}}^{*}\left(t\right)+{\mathcal{I}}_{t^{-}}^{\mathfrak{a}} {{\rm Y}}^{*}\left(s\right)\right]\right]$$

Thus:

$$\frac{2}{\mathfrak{a}} {\rm Y}\left(\frac{s+t}{2}\right){\le }_p\frac{\Gamma \left(\mathfrak{a}\right)}{{\left(t-s\right)}^{\mathfrak{a}}}\left[{\mathcal{I}}_{s^{+}}^{\mathfrak{a}} {\rm Y}\left(t\right)+{\mathcal{I}}_{t^{-}}^{\mathfrak{a}} {\rm Y}\left(s\right)\right]$$

(7)

Similar to the above, we have

$$\frac{\Gamma \left(\mathfrak{a}\right)}{{\left(t-s\right)}^{\mathfrak{a}}}\left[{\mathcal{I}}_{s^{+}}^{\mathfrak{a}} {\rm Y}\left(t\right)+{\mathcal{I}}_{t^{-}}^{\mathfrak{a}} {\rm Y}\left(s\right)\right]{\le }_p\frac{{\rm Y}\left(s\right)+{\rm Y}\left(t\right)}{2}$$

(8)

Combining (7) and (8), we have

$${\rm Y}\left(\frac{s+t}{2}\right){\le }_p\frac{\Gamma \left(\mathfrak{a}+1\right)}{2{\left(t-s\right)}^{\mathfrak{a}}}\left[{\mathcal{I}}_{s^{+}}^{\mathfrak{a}} {\rm Y}\left(t\right)+{\mathcal{I}}_{t^{-}}^{\mathfrak{a}} {\rm Y}\left(s\right)\right]{\le }_p\frac{{\rm Y}\left(s\right)+{\rm Y}\left(t\right)}{2}$$

Hence, we achieve the required result. □

Remark 2.

We may observe from Theorem 3 that:

Let one take $\mathsf{\alpha }=1$. Then, from Theorem 1 and (5), we achieve the coming inequality (see [23]):

$${\rm Y}\left(\frac{s+t}{2}\right){\le }_p\frac{1}{t-s}{{\displaystyle \int }}_s^t{\rm Y}\left(\omega \right)d\omega {\le }_p\frac{{\rm Y}\left(s\right)+ {\rm Y}\left(t\right)}{2}$$

If we take ${{\rm Y}}_{*}\left(\omega \right)={{\rm Y}}^{*}\left(\omega \right)$, then from Theorem 3 and (5), we acquire the coming inequality (see [32]):

$${\rm Y}\left(\frac{s+t}{2}\right)\le \frac{\Gamma \left(\mathfrak{a}+1\right)}{2{\left(t-s\right)}^{\mathfrak{a}}}\left[{\mathcal{I}}_{s^{+}}^{\mathfrak{a}} {\rm Y}\left(t\right)+{\mathcal{I}}_{t^{-}}^{\mathfrak{a}} {\rm Y}\left(s\right)\right]\le \frac{{\rm Y}\left(s\right)+{\rm Y}\left(t\right)}{2}$$

Let one take $\mathfrak{a}=1$ and ${{\rm Y}}_{*}\left(\omega \right)={{\rm Y}}^{*}\left(\omega \right)$. Then, from Theorem 1 and (5), we achieve classical 𝓗–𝓗 type inequality.

Example 1.

Let$\mathfrak{a}=\frac{1}{2}, \omega \in \left[2, 3\right]$, and the I∙V-F${\rm Y}:\left[s, t\right]=\left[2, 3\right]\to {\mathcal{X}}_I^{+},$provided by${\rm Y}\left(\omega \right)=\left[1, 2\right]\left(2-{\omega }^{\frac{1}{2}}\right)$. Since the left and right endpoint functions,${{\rm Y}}_{*}\left(\omega \right)=2-{\omega }^{\frac{1}{2}},$ ${{\rm Y}}^{*}\left(\omega \right)=2\left(2-{\omega }^{\frac{1}{2}}\right)$, are left and rightconvex functions, then${\rm Y}\left(\omega \right)$is a left and rightconvex I∙V-F. We clearly see that${\rm Y}\in L\left(\left[s, t\right],{\mathcal{X}}_I^{+}\right)$,and

$$\begin{gathered} {{\rm Y}}_{*}\left(\frac{s+t}{2}\right)={{\rm Y}}_{*}\left(\frac{5}{2}\right)=\frac{4-\sqrt{10}}{2} \\ {{\rm Y}}^{*}\left(\frac{s+t}{2}\right)={{\rm Y}}^{*}\left(\frac{5}{2}\right)=4-\sqrt{10} \\ \frac{{{\rm Y}}_{*}\left(s\right)+{{\rm Y}}_{*}\left(t\right)}{2}=\frac{4-\sqrt{2}-\sqrt{3}}{2} \\ \frac{{{\rm Y}}^{*}\left(s\right)+{{\rm Y}}^{*}\left(t\right)}{2}=4-\sqrt{2}-\sqrt{3} \end{gathered}$$

Note that

$$\begin{gathered} \frac{\Gamma \left(\mathfrak{a}+1\right)}{2{\left(t-s\right)}^{\mathfrak{a}}}\left[{\mathcal{I}}_{s^{+}}^{\mathfrak{a}} {{\rm Y}}_{*}\left(t\right)+{\mathcal{I}}_{t^{-}}^{\mathfrak{a}} {{\rm Y}}_{*}\left(s\right)\right] \\ =\frac{\Gamma \left(\frac{3}{2}\right)}{2}\frac{1}{\sqrt{\pi }}\underset{2}{\overset{3}{{\displaystyle \int }}}{\left(3-\omega \right)}^{\frac{-1}{2}} . \left(2-{\omega }^{\frac{1}{2}}\right)d\omega \\ +\frac{\Gamma \left(\frac{3}{2}\right)}{2}\frac{1}{\sqrt{\pi }}\underset{2}{\overset{3}{{\displaystyle \int }}}{\left(\omega -2\right)}^{\frac{-1}{2}} . \left(2-{\omega }^{\frac{1}{2}}\right)d\omega \\ =\frac{1}{4}\left[\frac{7393}{10,000}+\frac{9501}{10,000}\right] \\ =\frac{8447}{20,000}. \\ \frac{\Gamma \left(\mathfrak{a}+1\right)}{2{\left(t-s\right)}^{\mathfrak{a}}}\left[{\mathcal{I}}_{s^{+}}^{\mathfrak{a}} {{\rm Y}}^{*}\left(t\right)+{\mathcal{I}}_{t^{-}}^{\mathfrak{a}} {{\rm Y}}^{*}\left(s\right)\right] \\ =\frac{\Gamma \left(\frac{3}{2}\right)}{2}\frac{1}{\sqrt{\pi }}\underset{2}{\overset{3}{{\displaystyle \int }}}{\left(3-\omega \right)}^{\frac{-1}{2}} . 2\left(2-{\omega }^{\frac{1}{2}}\right)d\omega \\ +\frac{\Gamma \left(\frac{3}{2}\right)}{2}\frac{1}{\sqrt{\pi }}\underset{2}{\overset{3}{{\displaystyle \int }}}{\left(\omega -2\right)}^{\frac{-1}{2}} . 2\left(2-{\omega }^{\frac{1}{2}}\right)d\omega \\ =\frac{1}{2}\left[\frac{7393}{\mathrm{10,000}}+\frac{9501}{\mathrm{10,000}}\right] \\ =\frac{8447}{\mathrm{10,000}} \end{gathered}$$

Therefore,

$$\left[\frac{4-\sqrt{10}}{2},4-\sqrt{10}\right]{\le }_p\left[\frac{8447}{\mathrm{20,000}},\frac{8447}{\mathrm{10,000}}\right]{\le }_p\left[\frac{4-\sqrt{2}-\sqrt{3}}{2},4-\sqrt{2}-\sqrt{3}\right]$$

and Theorem 3 is verified.

The upcoming two results acquire the fractional inequalities for the product of left and right convex I∙V-Fs.

Theorem 4.

Let${\rm Y},\mathfrak{G}:\left[s, t\right]\to {\mathcal{X}}_I^{+}$be two left and rightconvex I∙V-Fs on$\left[s, t\right]$, provided by${\rm Y}\left(\omega \right)=\left[{{\rm Y}}_{*}\left(\omega \right), {{\rm Y}}^{*}\left(\omega \right)\right]$and$\mathfrak{G}\left(\omega \right)=\left[{\mathfrak{G}}_{*}\left(\omega \right), {\mathfrak{G}}^{*}\left(\omega \right)\right]$for all$\omega \in \left[s, t\right]$. If${\rm Y}\times \mathfrak{G}\in L\left(\left[s, t\right],{\mathcal{X}}_I^{+}\right)$, then

$$\begin{gathered} \frac{\Gamma \left(\mathfrak{a}\right)}{2{\left(t-s\right)}^{\mathfrak{a}}}\left[{\mathcal{I}}_{s^{+}}^{\mathfrak{a}} {\rm Y}\left(t\right)\times \mathfrak{G}\left(t\right)+{\mathcal{I}}_{t^{-}}^{\mathfrak{a}}{\rm Y}\left(s\right)\times \mathfrak{G}\left(s\right)\right] \\ {\le }_p\left(\frac{1}{2}-\frac{\mathfrak{a}}{\left(\mathfrak{a}+1\right)\left(\mathfrak{a}+2\right)}\right)\phi \left(s,t\right)+\left(\frac{\mathfrak{a}}{\left(\mathfrak{a}+1\right)\left(\mathfrak{a}+2\right)}\right)\nabla \left(s,t\right) \end{gathered}$$

where $\phi \left(s,t\right)={\rm Y}\left(s\right)\times \mathfrak{G}\left(s\right)+{\rm Y}\left(t\right)\times \mathfrak{G}\left(t\right),$ $\nabla \left(s,t\right)={\rm Y}\left(s\right)\times \mathfrak{G}\left(t\right)+{\rm Y}\left(t\right)\times \mathfrak{G}\left(s\right),$ and $\phi \left(s,t\right)=\left[{\phi }_{*}\left(s,t\right), {\phi }^{*}\left(s,t\right)\right]$ and $\nabla \left(s,t\right)=\left[{\nabla }_{*}\left(s,t\right), {\nabla }^{*}\left(s,t\right)\right].$

Proof. 

Since ${\rm Y}, \mathfrak{G}$ are both left and right convex I∙V-Fs, then we have

$$\begin{array}{c} \\ {{\rm Y}}_{*}\left(\nu s+\left(1-\nu \right)t\right)\le \nu {{\rm Y}}_{*}\left(s\right)+\left(1-\nu \right){{\rm Y}}_{*}\left(t\right), \\ \\ {{\rm Y}}^{*}\left(\nu s+\left(1-\nu \right)t\right)\le \nu {{\rm Y}}^{*}\left(s\right)+\left(1-\nu \right){{\rm Y}}^{*}\left(t\right). \\ \end{array}$$

and

$$\begin{array}{c} \\ {\mathfrak{G}}_{*}\left(\nu s+\left(1-\nu \right)t\right)\le \nu {\mathfrak{G}}_{*}\left(s\right)+\left(1-\nu \right){\mathfrak{G}}_{*}\left(t\right), \\ \\ {\mathfrak{G}}^{*}\left(\nu s+\left(1-\nu \right)t\right)\le \nu {\mathfrak{G}}^{*}\left(s\right)+\left(1-\nu \right){\mathfrak{G}}^{*}\left(t\right). \\ \end{array}$$

From the definition of left and right convex I∙V-Fs, it follows that $0{\le }_p{\rm Y}\left(\omega \right)$ and $0{\le }_p\mathfrak{G}\left(\omega \right)$, so

$$\begin{gathered} {{\rm Y}}_{*}\left(\nu s+\left(1-\nu \right)t\right)\times {\mathfrak{G}}_{*}\left(\nu s+\left(1-\nu \right)t\right) \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\le \left(\nu {{\rm Y}}_{*}\left(s\right)+\left(1-\nu \right){{\rm Y}}_{*}\left(t\right)\right)\left(\nu {\mathfrak{G}}_{*}\left(s\right)+\left(1-\nu \right){\mathfrak{G}}_{*}\left(t\right)\right) \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}={\nu }^2{{\rm Y}}_{*}\left(s\right)\times {\mathfrak{G}}_{*}\left(s\right)+{\left(1-\nu \right)}^2{{\rm Y}}_{*}\left(t\right)\times {\mathfrak{G}}_{*}\left(t\right) \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\nu \left(1-\nu \right){{\rm Y}}_{*}\left(s\right)\times {\mathfrak{G}}_{*}\left(t\right)+\nu \left(1-\nu \right){{\rm Y}}_{*}\left(t\right)\times {\mathfrak{G}}_{*}\left(s\right) \\ {{\rm Y}}^{*}\left(\nu s+\left(1-\nu \right)t\right)\times {\mathfrak{G}}^{*}\left(\nu s+\left(1-\nu \right)t\right) \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\le \left(\nu {{\rm Y}}^{*}\left(s\right)+\left(1-\nu \right){{\rm Y}}^{*}\left(t\right)\right)\left(\nu {\mathfrak{G}}^{*}\left(s\right)+\left(1-\nu \right){\mathfrak{G}}^{*}\left(t\right)\right) \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}={\nu }^2{{\rm Y}}^{*}\left(s\right)\times {\mathfrak{G}}^{*}\left(s\right)+{\left(1-\nu \right)}^2{{\rm Y}}^{*}\left(t\right)\times {\mathfrak{G}}^{*}\left(t\right) \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\nu \left(1-\nu \right){{\rm Y}}^{*}\left(s\right)\times {\mathfrak{G}}^{*}\left(t\right)+\nu \left(1-\nu \right){{\rm Y}}^{*}\left(t\right)\times {\mathfrak{G}}^{*}\left(s\right), \end{gathered}$$

(9)

Analogously, we have

$$\begin{gathered} {{\rm Y}}_{*}\left(\left(1-\nu \right)s+\nu t\right){\mathfrak{G}}_{*}\left(\left(1-\nu \right)s+\nu t\right) \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\le {\left(1-\nu \right)}^2{{\rm Y}}_{*}\left(s\right)\times {\mathfrak{G}}_{*}\left(s\right)+{\nu }^2{{\rm Y}}_{*}\left(t\right)\times {\mathfrak{G}}_{*}\left(t\right) \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\nu \left(1-\nu \right){{\rm Y}}_{*}\left(s\right)\times {\mathfrak{G}}_{*}\left(t\right)+\nu \left(1-\nu \right){{\rm Y}}_{*}\left(t\right)\times {\mathfrak{G}}_{*}\left(s\right) \\ {{\rm Y}}^{*}\left(\left(1-\nu \right)s+\nu t\right)\times {\mathfrak{G}}^{*}\left(\left(1-\nu \right)s+\nu t\right) \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\le {\left(1-\nu \right)}^2{{\rm Y}}^{*}\left(s\right)\times {\mathfrak{G}}^{*}\left(s\right)+{\nu }^2{{\rm Y}}^{*}\left(t\right)\times {\mathfrak{G}}^{*}\left(t\right) \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\nu \left(1-\nu \right){{\rm Y}}^{*}\left(s\right)\times {\mathfrak{G}}^{*}\left(t\right)+\nu \left(1-\nu \right){{\rm Y}}^{*}\left(t\right)\times {\mathfrak{G}}^{*}\left(s\right). \end{gathered}$$

(10)

Adding (9) and (10), we have

$$\begin{gathered} {{\rm Y}}_{*}\left(\nu s+\left(1-\nu \right)t\right)\times {\mathfrak{G}}_{*}\left(\nu s+\left(1-\nu \right)t\right) \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{{\rm Y}}_{*}\left(\left(1-\nu \right)s+\nu t\right)\times {\mathfrak{G}}_{*}\left(\left(1-\nu \right)s+\nu t\right) \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\le \left[{\nu }^2+{\left(1-\nu \right)}^2\right]\left[{{\rm Y}}_{*}\left(s\right)\times {\mathfrak{G}}_{*}\left(s\right)+{{\rm Y}}_{*}\left(t\right)\times {\mathfrak{G}}_{*}\left(t\right)\right] \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+2\nu \left(1-\nu \right)\left[{{\rm Y}}_{*}\left(t\right)\times {\mathfrak{G}}_{*}\left(s\right)+{{\rm Y}}_{*}\left(s\right)\times {\mathfrak{G}}_{*}\left(t\right)\right] \\ {{\rm Y}}^{*}\left(\nu s+\left(1-\nu \right)t\right)\times {\mathfrak{G}}^{*}\left(\nu s+\left(1-\nu \right)t\right) \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{{\rm Y}}^{*}\left(\left(1-\nu \right)s+\nu t\right)\times {\mathfrak{G}}^{*}\left(\left(1-\nu \right)s+\nu t\right) \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\le \left[{\nu }^2+{\left(1-\nu \right)}^2\right]\left[{{\rm Y}}^{*}\left(s\right)\times {\mathfrak{G}}^{*}\left(s\right)+{{\rm Y}}^{*}\left(t\right)\times {\mathfrak{G}}^{*}\left(t\right)\right] \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+2\nu \left(1-\nu \right)\left[{{\rm Y}}^{*}\left(t\right)\times {\mathfrak{G}}^{*}\left(s\right)+{{\rm Y}}^{*}\left(s\right)\times {\mathfrak{G}}^{*}\left(t\right)\right]. \end{gathered}$$

(11)

Taking the multiplication of (11) by ${\nu }^{\mathfrak{a}-1}$ and integrating the obtained result, with respect to $\nu$ over (0, 1), we have

$$\begin{gathered} {{\displaystyle \int }}_0^1{\nu }^{\mathfrak{a}-1}{{\rm Y}}_{*}\left(\nu s+\left(1-\nu \right)t\right)\times {\mathfrak{G}}_{*}\left(\nu s+\left(1-\nu \right)t\right) \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\nu }^{\mathfrak{a}-1}{{\rm Y}}_{*}\left(\left(1-\nu \right)s+\nu t\right)\times {\mathfrak{G}}_{*}\left(\left(1-\nu \right)s+\nu t\right)d\nu \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\le {\phi }_{*}\left(s,t\right){{\displaystyle \int }}_0^1{\nu }^{\mathfrak{a}-1}\left[{\nu }^2+{\left(1-\nu \right)}^2\right]d\nu +2{\nabla }_{*}\left(s,t\right){{\displaystyle \int }}_0^1{\nu }^{\mathfrak{a}-1}\nu \left(1-\nu \right)d\nu \\ {{\displaystyle \int }}_0^1{\nu }^{\mathfrak{a}-1}{{\rm Y}}^{*}\left(\nu s+\left(1-\nu \right)t\right)\times {\mathfrak{G}}^{*}\left(\nu s+\left(1-\nu \right)t\right) \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\nu }^{\mathfrak{a}-1}{{\rm Y}}^{*}\left(\left(1-\nu \right)s+\nu t\right)\times {\mathfrak{G}}^{*}\left(\left(1-\nu \right)s+\nu t\right)d\nu \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\le {\phi }^{*}\left(s,t\right){{\displaystyle \int }}_0^1{\nu }^{\mathfrak{a}-1}\left[{\nu }^2+{\left(1-\nu \right)}^2\right]d\nu +2{\nabla }^{*}\left(s,t\right){{\displaystyle \int }}_0^1{\nu }^{\mathfrak{a}-1}\nu \left(1-\nu \right)d\nu . \end{gathered}$$

It follows that

$$\begin{gathered} \frac{\Gamma \left(\mathfrak{a}\right)}{{\left(t-s\right)}^{\mathfrak{a}}}\left[{\mathcal{I}}_{s^{+}}^{\mathfrak{a}} {{\rm Y}}_{*}\left(t\right)\times {\mathfrak{G}}_{*}\left(t\right)+{\mathcal{I}}_{t^{-}}^{\mathfrak{a}} {{\rm Y}}_{*}\left(s\right)\times {\mathfrak{G}}_{*}\left(s\right)\right] \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\le \frac{2}{\mathfrak{a}}\left(\frac{1}{2}-\frac{\mathfrak{a}}{\left(\mathfrak{a}+1\right)\left(\mathfrak{a}+2\right)}\right){\phi }_{*}\left(s,t\right)+\frac{2}{\mathfrak{a}}\left(\frac{\mathfrak{a}}{\left(\mathfrak{a}+1\right)\left(\mathfrak{a}+2\right)}\right){\nabla }_{*}\left(s,t\right) \\ \frac{\Gamma \left(\mathfrak{a}\right)}{{\left(t-s\right)}^{\mathfrak{a}}}\left[{\mathcal{I}}_{s^{+}}^{\mathfrak{a}} {{\rm Y}}^{*}\left(t\right)\times {\mathfrak{G}}^{*}\left(t\right)+{\mathcal{I}}_{t^{-}}^{\mathfrak{a}} {{\rm Y}}^{*}\left(s\right)\times {\mathfrak{G}}^{*}\left(s\right)\right] \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\le \frac{2}{\mathfrak{a}}\left(\frac{1}{2}-\frac{\mathfrak{a}}{\left(\mathfrak{a}+1\right)\left(\mathfrak{a}+2\right)}\right){\phi }^{*}\left(s,t\right)+\frac{2}{\mathfrak{a}}\left(\frac{\mathfrak{a}}{\left(\mathfrak{a}+1\right)\left(\mathfrak{a}+2\right)}\right){\nabla }^{*}\left(s,t\right), \end{gathered}$$

$$\begin{array}{c}\frac{\Gamma \left(\mathfrak{a}\right)}{{\left(t-s\right)}^{\mathfrak{a}}}\left[{\mathcal{I}}_{s^{+}}^{\mathfrak{a}} {{\rm Y}}_{*}\left(t\right)\times {\mathfrak{G}}_{*}\left(t\right)+{\mathcal{I}}_{t^{-}}^{\mathfrak{a}} {{\rm Y}}_{*}\left(s\right)\times {\mathfrak{G}}_{*}\left(s\right)\right]\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\le \frac{2}{\mathfrak{a}}\left(\frac{1}{2}-\frac{\mathfrak{a}}{\left(\mathfrak{a}+1\right)\left(\mathfrak{a}+2\right)}\right){\phi }_{*}\left(s,t\right)+\frac{2}{\mathfrak{a}}\left(\frac{\mathfrak{a}}{\left(\mathfrak{a}+1\right)\left(\mathfrak{a}+2\right)}\right){\nabla }_{*}\left(s,t\right)\\ \frac{\Gamma \left(\mathfrak{a}\right)}{{\left(t-s\right)}^{\mathfrak{a}}}\left[{\mathcal{I}}_{s^{+}}^{\mathfrak{a}} {{\rm Y}}^{*}\left(t\right)\times {\mathfrak{G}}^{*}\left(t\right)+{\mathcal{I}}_{t^{-}}^{\mathfrak{a}} {{\rm Y}}^{*}\left(s\right)\times {\mathfrak{G}}^{*}\left(s\right)\right] \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\le \frac{2}{\mathfrak{a}}\left(\frac{1}{2}-\frac{\mathfrak{a}}{\left(\mathfrak{a}+1\right)\left(\mathfrak{a}+2\right)}\right){\phi }^{*}\left(s,t\right)+\frac{2}{\mathfrak{a}}\left(\frac{\mathfrak{a}}{\left(\mathfrak{a}+1\right)\left(\mathfrak{a}+2\right)}\right){\nabla }^{*}\left(s,t\right), \end{array}$$

That is,

$$\begin{gathered} \frac{\Gamma \left(\mathfrak{a}\right)}{{\left(t-s\right)}^{\mathfrak{a}}}\left[{\mathcal{I}}_{s^{+}}^{\mathfrak{a}} {{\rm Y}}_{*}\left(t\right)\times {\mathfrak{G}}_{*}\left(t\right)+{\mathcal{I}}_{t^{-}}^{\mathfrak{a}} {{\rm Y}}_{*}\left(s\right)\times {\mathfrak{G}}_{*}\left(s\right), {\mathcal{I}}_{s^{+}}^{\mathfrak{a}} {{\rm Y}}^{*}\left(t\right)\times {\mathfrak{G}}^{*}\left(t\right)+{\mathcal{I}}_{t^{-}}^{\mathfrak{a}} {{\rm Y}}^{*}\left(s\right)\times {\mathfrak{G}}^{*}\left(s\right)\right] \\ {\le }_p\frac{2}{\mathfrak{a}}\left(\frac{1}{2}-\frac{\mathfrak{a}}{\left(\mathfrak{a}+1\right)\left(\mathfrak{a}+2\right)}\right)\left[{\phi }_{*}\left(s,t\right), {\phi }^{*}\left(s,t\right)\right]+\frac{2}{\mathfrak{a}}\left(\frac{\mathfrak{a}}{\left(\mathfrak{a}+1\right)\left(\mathfrak{a}+2\right)}\right)\left[{\nabla }_{*}\left(s,t\right), {\nabla }^{*}\left(s,t\right)\right] \end{gathered}$$

Thus,

$$\begin{gathered} \frac{\Gamma \left(\mathfrak{a}\right)}{2{\left(t-s\right)}^{\mathfrak{a}}}\left[{\mathcal{I}}_{s^{+}}^{\mathfrak{a}} {\rm Y}\left(t\right)\times \mathfrak{G}\left(t\right)+{\mathcal{I}}_{t^{-}}^{\mathfrak{a}}{\rm Y}\left(s\right)\times \mathfrak{G}\left(s\right)\right] \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\le }_p\left(\frac{1}{2}-\frac{\mathfrak{a}}{\left(\mathfrak{a}+1\right)\left(\mathfrak{a}+2\right)}\right)\phi \left(s,t\right)+\left(\frac{\mathfrak{a}}{\left(\mathfrak{a}+1\right)\left(\mathfrak{a}+2\right)}\right)\nabla \left(s,t\right) \end{gathered}$$

and the theorem has been established. □

Example 2.

Let$\left[s,t\right]=\left[0,2 \right]$,$\mathfrak{a}=\frac{1}{2}, {\rm Y}\left(\omega \right)=\left[\frac{\omega }{2},\frac{3\omega }{2}\right]$,and$\mathfrak{G}\left(\omega \right)=\left[\omega ,3\omega \right].$Since the left and right endpoint functions,${{\rm Y}}_{*}\left(\omega \right)=\frac{\omega }{2},$ ${{\rm Y}}^{*}\left(\omega \right)=\frac{3\omega }{2}$,${\mathfrak{G}}_{*}\left(\omega \right)=\omega$and${\mathfrak{G}}^{*}\left(\omega \right)=3\omega$, are left and rightconvex functions, then${\rm Y}\left(\omega \right)$and$\mathfrak{G}\left(\omega \right)$are both left and rightconvex I∙V-Fs. We clearly see that${\rm Y}\left(\omega \right)\times \mathfrak{G}\left(\omega \right)\in L\left(\left[s, t\right],{\mathcal{X}}_I^{+}\right)$, and

$$\begin{gathered} \frac{\Gamma \left(1+\mathfrak{a}\right)}{2{\left(t-s\right)}^{\mathfrak{a}}}\left[{\mathcal{I}}_{s^{+}}^{\mathfrak{a}} {{\rm Y}}_{*}\left(t\right)\times {\mathfrak{G}}_{*}\left(t\right)+{\mathcal{I}}_{t^{-}}^{\mathfrak{a}} {{\rm Y}}_{*}\left(s\right)\times {\mathfrak{G}}_{*}\left(s\right)\right] \\ =\frac{\Gamma \left(\frac{3}{2}\right)}{2\sqrt{2}}\frac{1}{\sqrt{\pi }}\underset{0}{\overset{2}{{\displaystyle \int }}}{\left(2-\omega \right)}^{\frac{-1}{2}} \left( \frac{1}{2}.{\omega }^2\right)d\omega +\frac{\Gamma \left(\frac{3}{2}\right)}{2\sqrt{2}}\frac{1}{\sqrt{\pi }}\underset{0}{\overset{2}{{\displaystyle \int }}}{\left(\omega \right)}^{\frac{-1}{2}} \left(\frac{1}{2}.{\omega }^2\right)d\omega \approx 0.7333, \\ \frac{\Gamma \left(1+\mathfrak{a}\right)}{2{\left(t-s\right)}^{\mathfrak{a}}}\left[{\mathcal{I}}_{s^{+}}^{\mathfrak{a}} {{\rm Y}}^{*}\left(t\right)\times {\mathfrak{G}}^{*}\left(t\right)+{\mathcal{I}}_{t^{-}}^{\mathfrak{a}} {{\rm Y}}^{*}\left(s\right)\times {\mathfrak{G}}^{*}\left(s\right)\right] \\ =\frac{\Gamma \left(\frac{3}{2}\right)}{2\sqrt{2}}\frac{1}{\sqrt{\pi }}\underset{0}{\overset{2}{{\displaystyle \int }}}{\left(2-\omega \right)}^{\frac{-1}{2}} .\frac{9}{2}{\omega }^{2}d\omega +\frac{\Gamma \left(\frac{3}{2}\right)}{2\sqrt{2}}\frac{1}{\sqrt{\pi }}\underset{0}{\overset{2}{{\displaystyle \int }}}{\left(\omega \right)}^{\frac{-1}{2}} .\frac{9}{2}{\omega }^{2}d\omega \approx 6.5997, \end{gathered}$$

Note that

$$\begin{gathered} \left(\frac{1}{2}-\frac{\mathfrak{a}}{\left(\mathfrak{a}+1\right)\left(\mathfrak{a}+2\right)}\right){\phi }_{*}\left(s,t\right)=\left[{{\rm Y}}_{*}\left(s\right)\times {\mathfrak{G}}_{*}\left(s\right)+{{\rm Y}}_{*}\left(t\right)\times {\mathfrak{G}}_{*}\left(t\right)\right]=\frac{11}{15}, \\ \left(\frac{1}{2}-\frac{\mathfrak{a}}{\left(\mathfrak{a}+1\right)\left(\mathfrak{a}+2\right)}\right){\phi }^{*}\left(s,t\right)=\left[{{\rm Y}}^{*}\left(s\right)\times {\mathfrak{G}}^{*}\left(s\right)+{{\rm Y}}^{*}\left(t\right)\times {\mathfrak{G}}^{*}\left(t\right)\right]=\frac{33}{5}, \\ \left(\frac{\mathfrak{a}}{\left(\mathfrak{a}+1\right)\left(\mathfrak{a}+2\right)}\right){\nabla }_{*}\left(s,t\right)=\left[{{\rm Y}}_{*}\left(s\right)\times {\mathfrak{G}}_{*}\left(t\right)+{{\rm Y}}_{*}\left(t\right)\times {\mathfrak{G}}_{*}\left(s\right)\right]=\frac{2}{15}\left(0\right), \\ \left(\frac{\mathfrak{a}}{\left(\mathfrak{a}+1\right)\left(\mathfrak{a}+2\right)}\right){\nabla }_{*}\left(s,t\right)=\left[{{\rm Y}}^{*}\left(s\right)\times {\mathfrak{G}}^{*}\left(t\right)+{{\rm Y}}^{*}\left(t\right)\times {\mathfrak{G}}^{*}\left(s\right)\right]=\frac{2}{15}\left(0\right). \end{gathered}$$

Therefore, we have

$$\begin{gathered} \left(\frac{1}{2}-\frac{\mathfrak{a}}{\left(\mathfrak{a}+1\right)\left(\mathfrak{a}+2\right)}\right)\phi \left(s,t\right)+\left(\frac{\mathfrak{a}}{\left(\mathfrak{a}+1\right)\left(\mathfrak{a}+2\right)}\right)\nabla \left(s,t\right) \\ =\left[\frac{11}{15},\frac{33}{5}\right]+\frac{2}{15}\left[0,0\right]=\left[\frac{11}{15},\frac{33}{5}\right] \end{gathered}$$

It follows that

$$\left[0.7333, 6.5997\right] {\le }_p\left[\frac{11}{15},\frac{33}{5}\right]$$

and Theorem 4 has been demonstrated.

Theorem 5.

Let ${\rm Y},\mathfrak{G}:\left[s, t\right]\to {\mathcal{X}}_I^{+}$ be two left and right convex I∙V-Fs, provided by ${\rm Y}\left(\omega \right)=\left[{{\rm Y}}_{*}\left(\omega \right), {{\rm Y}}^{*}\left(\omega \right)\right]$ and $\mathfrak{G}\left(\omega \right)=\left[{\mathfrak{G}}_{*}\left(\omega \right), {\mathfrak{G}}^{*}\left(\omega \right)\right]$ for all $\omega \in \left[s, t\right]$. If ${\rm Y}\times \mathfrak{G}\in L\left(\left[s, t\right],{\mathcal{X}}_I^{+}\right)$, then

$$\begin{gathered} \frac{1}{\mathfrak{a}} {\rm Y}\left(\frac{s+t}{2}\right)\times \mathfrak{G}\left(\frac{s+t}{2}\right){\le }_p\frac{\Gamma \left(\mathfrak{a}+1\right)}{4{\left(t-s\right)}^{\mathfrak{a}}}\left[{\mathcal{I}}_{s^{+}}^{\mathfrak{a}} {\rm Y}\left(t\right)\times \mathfrak{G}\left(t\right)+{\mathcal{I}}_{t^{-}}^{\mathfrak{a}} {\rm Y}\left(s\right)\times \mathfrak{G}\left(s\right)\right] \\ +\frac{1}{2\mathfrak{a}}\left(\frac{1}{2}-\frac{\mathfrak{a}}{\left(\mathfrak{a}+1\right)\left(\mathfrak{a}+2\right)}\right)\nabla \left(s,t\right)+\frac{1}{2\mathfrak{a}}\left(\frac{\mathfrak{a}}{\left(\mathfrak{a}+1\right)\left(\mathfrak{a}+2\right)}\right)\phi \left(s,t\right) \end{gathered}$$

where$\phi \left(s,t\right)={\rm Y}\left(s\right)\times \mathfrak{G}\left(s\right)+{\rm Y}\left(t\right)\times \mathfrak{G}\left(t\right),$ $\nabla \left(s,t\right)={\rm Y}\left(s\right)\times \mathfrak{G}\left(t\right)+{\rm Y}\left(t\right)\times \mathfrak{G}\left(s\right),$ $\phi \left(s,t\right)=\left[{\phi }_{*}\left(s,t\right), {\phi }^{*}\left(s,t\right)\right]$, and$\nabla \left(s,t\right)=\left[{\nabla }_{*}\left(s,t\right), {\nabla }^{*}\left(s,t\right)\right].$

Proof. 

Consider that ${\rm Y},\mathfrak{G}:\left[s, t\right]\to {\mathcal{X}}_I^{+}$ are left and right convex I∙V-Fs. Then, by hypothesis, we have

$$\begin{array}{c}{{\rm Y}}_{*}\left(\frac{s+t}{2} \right)\times {\mathfrak{G}}_{*}\left(\frac{s+t}{2}\right) \\ {{\rm Y}}^{*}\left(\frac{s+t}{2}\right)\times {\mathfrak{G}}^{*}\left(\frac{s+t}{2}\right)\end{array}$$

$$\begin{gathered} \le \frac{1}{4}\left[\begin{array}{c}{{\rm Y}}_{*}\left(\nu s+\left(1-\nu \right)t\right)\times {\mathfrak{G}}_{*}\left(\nu s+\left(1-\nu \right)t\right)\\ +{{\rm Y}}_{*}\left(\nu s+\left(1-\nu \right)t\right)\times {\mathfrak{G}}_{*}\left(\left(1-\nu \right)s+\nu t\right)\end{array}\right] \\ +\frac{1}{4}\left[\begin{array}{c}{{\rm Y}}_{*}\left(\left(1-\nu \right)s+\nu t\right)\times {\mathfrak{G}}_{*}\left(\nu s+\left(1-\nu \right)t\right)\\ +{{\rm Y}}_{*}\left(\left(1-\nu \right)s+\nu t\right)\times {\mathfrak{G}}_{*}\left(\left(1-\nu \right)s+\nu t\right)\end{array}\right] \\ \le \frac{1}{4}\left[\begin{array}{c}{{\rm Y}}^{*}\left(\nu s+\left(1-\nu \right)t\right)\times {\mathfrak{G}}^{*}\left(\nu s+\left(1-\nu \right)t\right)\\ +{{\rm Y}}^{*}\left(\nu s+\left(1-\nu \right)t\right)\times {\mathfrak{G}}^{*}\left(\left(1-\nu \right)s+\nu t\right)\end{array}\right] \\ +\frac{1}{4}\left[\begin{array}{c}{{\rm Y}}^{*}\left(\left(1-\nu \right)s+\nu t\right)\times {\mathfrak{G}}^{*}\left(\nu s+\left(1-\nu \right)t\right)\\ +{{\rm Y}}^{*}\left(\left(1-\nu \right)s+\nu t\right)\times {\mathfrak{G}}^{*}\left(\left(1-\nu \right)s+\nu t\right)\end{array}\right], \end{gathered}$$

$$\begin{gathered} \le \frac{1}{4}\left[\begin{array}{c}{{\rm Y}}_{*}\left(\nu s+\left(1-\nu \right)t\right)\times {\mathfrak{G}}_{*}\left(\nu s+\left(1-\nu \right)t\right)\\ +{{\rm Y}}_{*}\left(\left(1-\nu \right)s+\nu t\right)\times {\mathfrak{G}}_{*}\left(\left(1-\nu \right)s+\nu t\right)\end{array}\right] \\ +\frac{1}{4}\left[\begin{array}{c}\left(\nu {{\rm Y}}_{*}\left(s\right)+\left(1-\nu \right){{\rm Y}}_{*}\left(t\right)\right)\\ \times \left(\left(1-\nu \right){\mathfrak{G}}_{*}\left(s\right)+\nu {\mathfrak{G}}_{*}\left(t\right)\right)\\ +\left(\left(1-\nu \right){{\rm Y}}_{*}\left(s\right)+\nu {{\rm Y}}_{*}\left(t\right)\right)\\ \times \left(\nu {\mathfrak{G}}_{*}\left(s\right)+\left(1-\nu \right){\mathfrak{G}}_{*}\left(t\right)\right)\end{array}\right] \\ \le \frac{1}{4}\left[\begin{array}{c}{{\rm Y}}^{*}\left(\nu s+\left(1-\nu \right)t\right)\times {\mathfrak{G}}^{*}\left(\nu s+\left(1-\nu \right)t\right)\\ +{{\rm Y}}^{*}\left(\left(1-\nu \right)s+\nu t\right)\times {\mathfrak{G}}^{*}\left(\left(1-\nu \right)s+\nu t\right)\end{array}\right] \\ +\frac{1}{4}\left[\begin{array}{c}\left(\nu {{\rm Y}}^{*}\left(s\right)+\left(1-\nu \right){{\rm Y}}^{*}\left(t\right)\right)\\ \times \left(\left(1-\nu \right){\mathfrak{G}}^{*}\left(s\right)+\nu {\mathfrak{G}}^{*}\left(t\right)\right)\\ +\left(\left(1-\nu \right){{\rm Y}}^{*}\left(s\right)+\nu {{\rm Y}}^{*}\left(t\right)\right)\\ \times \left(\nu {\mathfrak{G}}^{*}\left(s\right)+\left(1-\nu \right){\mathfrak{G}}^{*}\left(t\right)\right)\end{array}\right], \end{gathered}$$

$$\begin{array}{c}=\frac{1}{4}\left[\begin{array}{c}{{\rm Y}}_{*}\left(\nu s+\left(1-\nu \right)t\right)\times {\mathfrak{G}}_{*}\left(\nu s+\left(1-\nu \right)t\right)\\ +{{\rm Y}}_{*}\left(\left(1-\nu \right)s+\nu t\right)\times {\mathfrak{G}}_{*}\left(\left(1-\nu \right)s+\nu t\right)\end{array}\right]\\ +\frac{1}{4}\left[\begin{array}{c}\left\{{\nu }^2+{\left(1-\nu \right)}^2\right\}{\nabla }_{*}\left(s,t\right)\\ +\left\{\nu \left(1-\nu \right)+\left(1-\nu \right)\nu \right\}{\phi }_{*}\left(s,t\right)\end{array}\right]\\ =\frac{1}{4}\left[\begin{array}{c}{{\rm Y}}^{*}\left(\nu s+\left(1-\nu \right)t\right)\times {\mathfrak{G}}^{*}\left(\nu s+\left(1-\nu \right)t\right)\\ +{{\rm Y}}^{*}\left(\left(1-\nu \right)s+\nu t\right)\times {\mathfrak{G}}^{*}\left(\left(1-\nu \right)s+\nu t\right)\end{array}\right]\\ +\frac{1}{4}\left[\begin{array}{c}\left\{{\nu }^2+{\left(1-\nu \right)}^2\right\}{\nabla }^{*}\left(s,t\right)\\ +\left\{\nu \left(1-\nu \right)+\left(1-\nu \right)\nu \right\}{\phi }^{*}\left(s,t\right)\end{array}\right].\end{array}$$

(12)

Taking the multiplication of (12) with ${\nu }^{\mathfrak{a}-1}$ and integrating over $\left(0, 1\right),$ we get

$$\begin{gathered} \frac{1}{\mathfrak{a}} {{\rm Y}}_{*}\left(\frac{s+t}{2}\right)\times {\mathfrak{G}}_{*}\left(\frac{s+t}{2}\right) \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\le \frac{1}{4{\left(t-s\right)}^{\mathfrak{a}}} \left[\begin{array}{c}{{\displaystyle \int }}_s^t{\left(t-\omega \right)}^{\mathfrak{a}-1}{{\rm Y}}_{*}\left(\omega \right)\times {\mathfrak{G}}_{*}\left(\omega \right)d\omega \\ +{{\displaystyle \int }}_s^t{\left(z-s\right)}^{\mathfrak{a}-1}{{\rm Y}}_{*}\left(z\right)\times {\mathfrak{G}}_{*}\left(z\right)dz \end{array}\right] \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\frac{1}{2\mathfrak{a}}\left(\frac{1}{2}-\frac{\mathfrak{a}}{\left(\mathfrak{a}+1\right)\left(\mathfrak{a}+2\right)}\right){\nabla }_{*}\left(s,t\right)+\frac{1}{2\mathfrak{a}}\left(\frac{\mathfrak{a}}{\left(\mathfrak{a}+1\right)\left(\mathfrak{a}+2\right)}\right){\phi }_{*}\left(s,t\right) \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=\frac{\Gamma \left(\mathfrak{a}+1\right)}{4{\left(t-s\right)}^{\mathfrak{a}}}\left[{\mathcal{I}}_{s^{+}}^{\mathfrak{a}} {{\rm Y}}_{*}\left(t\right)\times {\mathfrak{G}}_{*}\left(t\right)+{\mathcal{I}}_{t^{-}}^{\mathfrak{a}} {{\rm Y}}_{*}\left(s\right)\times {\mathfrak{G}}_{*}\left(s\right)\right] \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\frac{1}{2\mathfrak{a}}\left(\frac{1}{2}-\frac{\mathfrak{a}}{\left(\mathfrak{a}+1\right)\left(\mathfrak{a}+2\right)}\right){\nabla }_{*}\left(s,t\right)+\frac{1}{2\mathfrak{a}}\left(\frac{\mathfrak{a}}{\left(\mathfrak{a}+1\right)\left(\mathfrak{a}+2\right)}\right){\phi }_{*}\left(s,t\right) \\ \frac{1}{\mathfrak{a}} {{\rm Y}}^{*}\left(\frac{s+t}{2}\right)\times {\mathfrak{G}}^{*}\left(\frac{s+t}{2}\right) \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\le \frac{1}{4{\left(t-s\right)}^{\mathfrak{a}}} \left[\begin{array}{c}{{\displaystyle \int }}_s^t{\left(t-\omega \right)}^{\mathfrak{a}-1}{{\rm Y}}^{*}\left(\omega \right)\times {\mathfrak{G}}^{*}\left(\omega \right)d\omega \\ +{{\displaystyle \int }}_s^t{\left(z-s\right)}^{\mathfrak{a}-1}{{\rm Y}}^{*}\left(z\right)\times {\mathfrak{G}}^{*}\left(z\right)dz \end{array}\right] \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\frac{1}{2\mathfrak{a}}\left(\frac{1}{2}-\frac{\mathfrak{a}}{\left(\mathfrak{a}+1\right)\left(\mathfrak{a}+2\right)}\right){\nabla }^{*}\left(s,t\right)+\frac{1}{2\mathfrak{a}}\left(\frac{\mathfrak{a}}{\left(\mathfrak{a}+1\right)\left(\mathfrak{a}+2\right)}\right){\phi }^{*}\left(s,t\right) \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=\frac{\Gamma \left(\mathfrak{a}+1\right)}{4{\left(t-s\right)}^{\mathfrak{a}}}\left[{\mathcal{I}}_{s^{+}}^{\mathfrak{a}} {{\rm Y}}^{*}\left(t\right)\times {\mathfrak{G}}^{*}\left(t\right)+{\mathcal{I}}_{t^{-}}^{\mathfrak{a}} {{\rm Y}}^{*}\left(s\right)\times {\mathfrak{G}}^{*}\left(s\right)\right] \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\frac{1}{2\mathfrak{a}}\left(\frac{1}{2}-\frac{\mathfrak{a}}{\left(\mathfrak{a}+1\right)\left(\mathfrak{a}+2\right)}\right){\nabla }^{*}\left(s,t\right)+\frac{1}{2\mathfrak{a}}\left(\frac{\mathfrak{a}}{\left(\mathfrak{a}+1\right)\left(\mathfrak{a}+2\right)}\right){\phi }^{*}\left(s,t\right), \end{gathered}$$

That is,

$$\begin{gathered} \frac{1}{\mathfrak{a}} {\rm Y}\left(\frac{s+t}{2}\right)\times \mathfrak{G}\left(\frac{s+t}{2}\right){\le }_p\frac{\Gamma \left(\mathfrak{a}+1\right)}{4{\left(t-s\right)}^{\mathfrak{a}}}\left[{\mathcal{I}}_{s^{+}}^{\mathfrak{a}} {\rm Y}\left(t\right)\times \mathfrak{G}\left(t\right)+{\mathcal{I}}_{t^{-}}^{\mathfrak{a}} {\rm Y}\left(s\right)\times \mathfrak{G}\left(s\right)\right] \\ +\frac{1}{2\mathfrak{a}}\left(\frac{1}{2}-\frac{\mathfrak{a}}{\left(\mathfrak{a}+1\right)\left(\mathfrak{a}+2\right)}\right)\nabla \left(s,t\right)+\frac{1}{2\mathfrak{a}}\left(\frac{\mathfrak{a}}{\left(\mathfrak{a}+1\right)\left(\mathfrak{a}+2\right)}\right)\phi \left(s,t\right). \end{gathered}$$

Hence, the required result is achieved. □

The upcoming results discuss the 𝓗–𝓗 Fejér type inequality left and right convex I∙V-F. Firstly, we achieve second 𝓗–𝓗 Fejér type inequality.

Theorem 6.

Let ${\rm Y}:\left[s, t\right]\to {\mathcal{X}}_I^{+}$ be a left and right convex I∙V-F, with $s<t$, provided by ${\rm Y}\left(\omega \right)=\left[{{\rm Y}}_{*}\left(\omega \right), {{\rm Y}}^{*}\left(\omega \right)\right]$ for all $\omega \in \left[s, t\right]$. Let ${\rm Y}\in L\left(\left[s, t\right],{\mathcal{X}}_I^{+}\right)$ and $ℭ:\left[s, t\right]\to ℝ, ℭ\left(\omega \right)\ge 0,$ be symmetric with respect to $\frac{s+t}{2}.$Then,

$$\left[{\mathcal{I}}_{s^{+}}^{\mathfrak{a}} {\rm Y}ℭ\left(t\right)+{\mathcal{I}}_{t^{-}}^{\mathfrak{a}} {\rm Y}ℭ\left(s\right)\right]{\le }_p\frac{{\rm Y}\left(s\right)+{\rm Y}\left(t\right)}{2}\left[{\mathcal{I}}_{s^{+}}^{\mathfrak{a}}ℭ\left(t\right)+{\mathcal{I}}_{t^{-}}^{\mathfrak{a}}ℭ\left(s\right)\right]$$

(13)

If${\rm Y}$is a concave I∙V-F, then inequality (13) is reversed.

Proof. 

Let ${\rm Y}$ be a left and right convex I∙V-F and ${\nu }^{\mathfrak{a}-1}ℭ\left(\nu s+\left(1-\nu \right)t\right)\ge 0$. Then, we have

$$\begin{array}{c}{\nu }^{\mathfrak{a}-1}{{\rm Y}}_{*}\left(\nu s+\left(1-\nu \right)t\right)C\left(\nu s+\left(1-\nu \right)t\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\le {\nu }^{\mathfrak{a}-1}\left(\nu {{\rm Y}}_{*}\left(s\right)+\left(1-\nu \right){{\rm Y}}_{*}\left(t\right)\right)C\left(\nu s+\left(1-\nu \right)t\right)\\ {\nu }^{\mathfrak{a}-1}{{\rm Y}}^{*}\left(\nu s+\left(1-\nu \right)t\right)ℭ\left(\nu s+\left(1-\nu \right)t\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\le {\nu }^{\mathfrak{a}-1}\left(\nu {{\rm Y}}^{*}\left(s\right)+\left(1-\nu \right){{\rm Y}}^{*}\left(t\right)\right)ℭ\left(\nu s+\left(1-\nu \right)t\right).\end{array}$$

(14)

and

$$\begin{array}{c}{\nu }^{\mathfrak{a}-1}{{\rm Y}}_{*}\left(\left(1-\nu \right)s+\nu t\right)C\left(\left(1-\nu \right)s+\nu t\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\le {\nu }^{\mathfrak{a}-1}\left(\left(1-\nu \right){{\rm Y}}_{*}\left(s\right)+\nu {{\rm Y}}_{*}\left(t\right)\right)C\left(\left(1-\nu \right)s+\nu t\right)\\ {\nu }^{\mathfrak{a}-1}{{\rm Y}}^{*}\left(\left(1-\nu \right)s+\nu t\right)ℭ\left(\left(1-\nu \right)s+\nu t\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\le {\nu }^{\mathfrak{a}-1}\left(\left(1-\nu \right){{\rm Y}}^{*}\left(s\right)+\nu {{\rm Y}}^{*}\left(t\right)\right)ℭ\left(\left(1-\nu \right)s+\nu t\right).\end{array}$$

(15)

After adding (14) and (15), and integrating over $\left[0, 1\right],$ we get

$$\begin{array}{c}{{\displaystyle \int }}_0^1{\nu }^{\mathfrak{a}-1}{{\rm Y}}_{*}\left(\nu s+\left(1-\nu \right)t\right)ℭ\left(\nu s+\left(1-\nu \right)t\right)d\nu \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{{\displaystyle \int }}_0^1{\nu }^{\mathfrak{a}-1}{{\rm Y}}_{*}\left(\left(1-\nu \right)s+\nu t\right)ℭ\left(\left(1-\nu \right)s+\nu t\right)d\nu \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\le {{\displaystyle \int }}_0^1\left[\begin{array}{c}{\nu }^{\mathfrak{a}-1}{{\rm Y}}_{*}\left(s\right)\left\{\nu ℭ\left(\nu s+\left(1-\nu \right)t\right)+\left(1-\nu \right)ℭ\left(\left(1-\nu \right)s+\nu t\right)\right\}\\ +{\nu }^{\mathfrak{a}-1}{{\rm Y}}_{*}\left(t\right)\left\{\left(1-\nu \right)ℭ\left(\nu s+\left(1-\nu \right)t\right)+\nu ℭ\left(\left(1-\nu \right)s+\nu t\right)\right\}\end{array}\right]d\nu ,\\ {{\displaystyle \int }}_0^1{\nu }^{\mathfrak{a}-1}{{\rm Y}}^{*}\left(\left(1-\nu \right)s+\nu t\right)ℭ\left(\left(1-\nu \right)s+\nu t\right)d\nu \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{{\displaystyle \int }}_0^1{\nu }^{\mathfrak{a}-1}{{\rm Y}}^{*}\left(\nu s+\left(1-\nu \right)t\right)ℭ\left(\nu s+\left(1-\nu \right)t\right)d\nu \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\le {{\displaystyle \int }}_0^1\left[\begin{array}{c}{\nu }^{\mathfrak{a}-1}{{\rm Y}}^{*}\left(s\right)\left\{\nu ℭ\left(\nu s+\left(1-\nu \right)t\right)+\left(1-\nu \right)ℭ\left(\left(1-\nu \right)s+\nu t\right)\right\}\\ +{\nu }^{\mathfrak{a}-1}{{\rm Y}}^{*}\left(t\right)\left\{\left(1-\nu \right)ℭ\left(\nu s+\left(1-\nu \right)t\right)+\nu ℭ\left(\left(1-\nu \right)s+\nu t\right)\right\}\end{array}\right]d\nu ,\end{array}$$

$$\begin{array}{c}={{\rm Y}}_{*}\left(s\right){{\displaystyle \int }}_0^1{\nu }^{\mathfrak{a}-1}C\left(\nu s+\left(1-\nu \right)t\right)d\nu +{{\rm Y}}_{*}\left(t\right){{\displaystyle \int }}_0^1{\nu }^{\mathfrak{a}-1}C\left(\left(1-\nu \right)s+\nu t\right)d\nu ,\\ ={{\rm Y}}^{*}\left(s\right){{\displaystyle \int }}_0^1{\nu }^{\mathfrak{a}-1}C\left(\nu s+\left(1-\nu \right)t\right)d\nu +{{\rm Y}}^{*}\left(t\right){{\displaystyle \int }}_0^1{\nu }^{\mathfrak{a}-1}C\left(\left(1-\nu \right)s+\nu t\right)d\nu .\end{array}$$

Since $ℭ$ is symmetric, then

$$\begin{array}{c}=\left[{{\rm Y}}_{*}\left(s\right)+{{\rm Y}}_{*}\left(t\right)\right]{{\displaystyle \int }}_0^1\begin{array}{c}{\nu }^{\mathfrak{a}-1}C\left(\left(1-\nu \right)s+\nu t\right)\end{array}d\nu \\ =\left[{{\rm Y}}^{*}\left(s\right)+{{\rm Y}}^{*}\left(t\right)\right]{{\displaystyle \int }}_0^1\begin{array}{c}{\nu }^{\mathfrak{a}-1}C\left(\left(1-\nu \right)s+\nu t\right)\end{array}d\nu .\end{array}$$

$$\begin{array}{c} \\ =\frac{{{\rm Y}}_{*}\left(s\right)+{{\rm Y}}_{*}\left(t\right)}{2}\frac{\Gamma \left(\mathfrak{a}\right)}{{\left(t-s\right)}^{\mathfrak{a}}}\left[{\mathcal{I}}_{s^{+}}^{\mathfrak{a}}ℭ\left(t\right)+{\mathcal{I}}_{t^{-}}^{\mathfrak{a}}ℭ\left(s\right)\right],\\ =\frac{{{\rm Y}}^{*}\left(s\right)+{{\rm Y}}^{*}\left(t\right)}{2}\frac{\Gamma \left(\mathfrak{a}\right)}{{\left(t-s\right)}^{\mathfrak{a}}}\left[{\mathcal{I}}_{s^{+}}^{\mathfrak{a}}ℭ\left(t\right)+{\mathcal{I}}_{t^{-}}^{\mathfrak{a}}ℭ\left(s\right)\right].\end{array}$$

(16)

Since

$$\begin{gathered} {{\displaystyle \int }}_0^1{\nu }^{\mathfrak{a}-1}{{\rm Y}}_{*}\left(\nu s+\left(1-\nu \right)t\right)ℭ\left(\left(1-\nu \right)s+\nu t\right)d\nu \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{{\displaystyle \int }}_0^1{\nu }^{\mathfrak{a}-1}{{\rm Y}}_{*}\left(\left(1-\nu \right)s+\nu t\right)ℭ\left(\left(1-\nu \right)s+\nu t\right)d\nu \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=\frac{1}{{\left(t-s\right)}^{\mathfrak{a}}} {{\displaystyle \int }}_s^t{\left(\omega -s\right)}^{\mathfrak{a}-1}{{\rm Y}}_{*}\left(s+t-\omega \right)ℭ\left(\omega \right)d\omega \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\frac{1}{{\left(t-s\right)}^{\mathfrak{a}}} {{\displaystyle \int }}_s^t{\left(\omega -s\right)}^{\mathfrak{a}-1}{{\rm Y}}_{*}\left(\omega \right)ℭ\left(\omega \right)d\omega \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=\frac{1}{{\left(t-s\right)}^{\mathfrak{a}}} {{\displaystyle \int }}_s^t{\left(\omega -s\right)}^{\mathfrak{a}-1}{{\rm Y}}_{*}\left(\omega \right)ℭ\left(s+t-\omega \right)d\omega \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\frac{1}{{\left(t-s\right)}^{\mathfrak{a}}} {{\displaystyle \int }}_s^t{\left(\omega -s\right)}^{\mathfrak{a}-1}{{\rm Y}}_{*}\left(\omega \right)ℭ\left(\omega \right)d\omega \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=\frac{\Gamma \left(\mathfrak{a}\right)}{{\left(t-s\right)}^{\mathfrak{a}}}\left[{\mathcal{I}}_{s^{+}}^{\mathfrak{a}} {{\rm Y}}_{*}ℭ\left(t\right)+{\mathcal{I}}_{t^{-}}^{\mathfrak{a}} {{\rm Y}}_{*}ℭ\left(s\right)\right], \\ {{\displaystyle \int }}_0^1{\nu }^{\mathfrak{a}-1}{{\rm Y}}^{*}\left(\nu s+\left(1-\nu \right)t\right)ℭ\left(\left(1-\nu \right)s+\nu t\right)d\nu \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{{\displaystyle \int }}_0^1{\nu }^{\mathfrak{a}-1}{{\rm Y}}^{*}\left(\left(1-\nu \right)s+\nu t\right)ℭ\left(\left(1-\nu \right)s+\nu t\right)d\nu \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=\frac{\Gamma \left(\mathfrak{a}\right)}{{\left(t-s\right)}^{\mathfrak{a}}}\left[{\mathcal{I}}_{s^{+}}^{\mathfrak{a}} {{\rm Y}}^{*}ℭ\left(t\right)+{\mathcal{I}}_{t^{-}}^{\mathfrak{a}} {{\rm Y}}^{*}ℭ\left(s\right)\right]. \end{gathered}$$

(17)

then, from (16), we have

$$\begin{gathered} \frac{\Gamma \left(\mathfrak{a}\right)}{{\left(t-s\right)}^{\mathfrak{a}}}\left[{\mathcal{I}}_{s^{+}}^{\mathfrak{a}} {{\rm Y}}^{*}ℭ\left(t\right)+{\mathcal{I}}_{t^{-}}^{\mathfrak{a}} {{\rm Y}}^{*}ℭ\left(s\right)\right] \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\le \frac{{{\rm Y}}^{*}\left(s\right)+{{\rm Y}}^{*}\left(t\right)}{2}\frac{\Gamma \left(\mathfrak{a}\right)}{{\left(t-s\right)}^{\mathfrak{a}}}\left[{\mathcal{I}}_{s^{+}}^{\mathfrak{a}}ℭ\left(t\right)+{\mathcal{I}}_{t^{-}}^{\mathfrak{a}}ℭ\left(s\right)\right] \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\le \frac{{{\rm Y}}^{*}\left(s\right)+{{\rm Y}}^{*}\left(t\right)}{2}\frac{\Gamma \left(\mathfrak{a}\right)}{{\left(t-s\right)}^{\mathfrak{a}}}\left[{\mathcal{I}}_{s^{+}}^{\mathfrak{a}}ℭ\left(t\right)+{\mathcal{I}}_{t^{-}}^{\mathfrak{a}}ℭ\left(s\right)\right], \\ \frac{\Gamma \left(\mathfrak{a}\right)}{{\left(t-s\right)}^{\mathfrak{a}}}\left[{\mathcal{I}}_{s^{+}}^{\mathfrak{a}} {{\rm Y}}^{*}ℭ\left(t\right)+{\mathcal{I}}_{t^{-}}^{\mathfrak{a}} {{\rm Y}}^{*}ℭ\left(s\right)\right] \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\le \frac{{{\rm Y}}^{*}\left(s\right)+{{\rm Y}}^{*}\left(t\right)}{2}\frac{\Gamma \left(\mathfrak{a}\right)}{{\left(t-s\right)}^{\mathfrak{a}}}\left[{\mathcal{I}}_{s^{+}}^{\mathfrak{a}}ℭ\left(t\right)+{\mathcal{I}}_{t^{-}}^{\mathfrak{a}}ℭ\left(s\right)\right] \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\le \frac{{{\rm Y}}^{*}\left(s\right)+{{\rm Y}}^{*}\left(t\right)}{2}\frac{\Gamma \left(\mathfrak{a}\right)}{{\left(t-s\right)}^{\mathfrak{a}}}\left[{\mathcal{I}}_{s^{+}}^{\mathfrak{a}}ℭ\left(t\right)+{\mathcal{I}}_{t^{-}}^{\mathfrak{a}}ℭ\left(s\right)\right], \end{gathered}$$

That is,

$$\frac{\Gamma \left(\mathfrak{a}\right)}{{\left(t-s\right)}^{\mathfrak{a}}}\left[\left[{\mathcal{I}}_{s^{+}}^{\mathfrak{a}} {{\rm Y}}_{*}ℭ\left(t\right)+{\mathcal{I}}_{t^{-}}^{\mathfrak{a}} {{\rm Y}}_{*}ℭ\left(s\right)\right], {\mathcal{I}}_{s^{+}}^{\mathfrak{a}} {{\rm Y}}^{*}ℭ\left(t\right)+{\mathcal{I}}_{t^{-}}^{\mathfrak{a}} {{\rm Y}}^{*}ℭ\left(s\right)\right]$$

$${\le }_p\frac{\Gamma \left(\mathfrak{a}\right)}{{\left(t-s\right)}^{\mathfrak{a}}}\left[\frac{{{\rm Y}}_{*}\left(s\right)+{{\rm Y}}_{*}\left(t\right)}{2}, \frac{{{\rm Y}}^{*}\left(s\right)+{{\rm Y}}^{*}\left(t\right)}{2}\right]\left[{\mathcal{I}}_{s^{+}}^{\mathfrak{a}}ℭ\left(t\right)+{\mathcal{I}}_{t^{-}}^{\mathfrak{a}}ℭ\left(s\right)\right]$$

$${\le }_p\frac{\Gamma \left(\mathfrak{a}\right)}{{\left(t-s\right)}^{\mathfrak{a}}}\left[\frac{{{\rm Y}}_{*}\left(s\right)+{{\rm Y}}_{*}\left(t\right)}{2}, \frac{{{\rm Y}}^{*}\left(s\right)+{{\rm Y}}^{*}\left(t\right)}{2}\right]\left[{\mathcal{I}}_{s^{+}}^{\mathfrak{a}}ℭ\left(t\right)+{\mathcal{I}}_{t^{-}}^{\mathfrak{a}}ℭ\left(s\right)\right],$$

Hence,

$$\left[{\mathcal{I}}_{s^{+}}^{\mathfrak{a}} {\rm Y}ℭ\left(t\right) {\le }_p {\mathcal{I}}_{t^{-}}^{\mathfrak{a}} {\rm Y}ℭ\left(s\right)\right]{\le }_p\frac{{\rm Y}\left(s\right)+{\rm Y}\left(t\right)}{2}\left[{\mathcal{I}}_{s^{+}}^{\mathfrak{a}}ℭ\left(t\right)+{\mathcal{I}}_{t^{-}}^{\mathfrak{a}}ℭ\left(s\right)\right]$$

$${\le }_p\frac{{\rm Y}\left(s\right)+{\rm Y}\left(t\right)}{2}\left[{\mathcal{I}}_{s^{+}}^{\mathfrak{a}}ℭ\left(t\right)+{\mathcal{I}}_{t^{-}}^{\mathfrak{a}}ℭ\left(s\right)\right].$$

Now, we first obtain the 𝓗–𝓗 Fejér type inequality for the left and right convex I∙V-F. □

Theorem 7.

Let ${\rm Y}:\left[s, t\right]\to {\mathcal{X}}_I^{+}$ be a left and right convex I∙V-F, with $s<t$, and defined by ${\rm Y}\left(\omega \right)=\left[{{\rm Y}}_{*}\left(\omega \right), {{\rm Y}}^{*}\left(\omega \right)\right]$ for all $\omega \in \left[s, t\right]$. If ${\rm Y}\in L\left(\left[s, t\right],{\mathcal{X}}_I^{+}\right)$ and $ℭ:\left[s, t\right]\to ℝ, ℭ\left(\omega \right)\ge 0$ are symmetric with respect to $\frac{s+t}{2}$, then

$${\rm Y}\left(\frac{s+t}{2}\right)\left[{\mathcal{I}}_{s^{+}}^{\mathfrak{a}}ℭ\left(t\right)+{\mathcal{I}}_{t^{-}}^{\mathfrak{a}}ℭ\left(s\right)\right]{\le }_p\left[{\mathcal{I}}_{s^{+}}^{\mathfrak{a}} {\rm Y}ℭ\left(t\right)+{\mathcal{I}}_{t^{-}}^{\mathfrak{a}} {\rm Y}ℭ\left(s\right)\right]$$

(18)

If${\rm Y}$is a concave I∙V-F, then inequality (18) is reversed.

Proof. 

Since ${\rm Y}$ is a left and right convex I∙V-F, then we have

$$\begin{array}{c} \\ {{\rm Y}}_{*}\left(\frac{s+t}{2}\right)\le \frac{1}{2}\left({{\rm Y}}_{*}\left(\nu s+\left(1-\nu \right)t\right)+{{\rm Y}}_{*}\left(\left(1-\nu \right)s+\nu t\right)\right)\\ {{\rm Y}}^{*}\left(\frac{s+t}{2}\right)\le \frac{1}{2}\left({{\rm Y}}^{*}\left(\nu s+\left(1-\nu \right)t\right)+{{\rm Y}}^{*}\left(\left(1-\nu \right)s+\nu t\right)\right),\end{array}$$

(19)

Since $ℭ\left(\nu s+\left(1-\nu \right)t\right)=ℭ\left(\left(1-\nu \right)s+\nu t\right)$, then by multiplying (19) by ${\nu }^{\mathfrak{a}-1}ℭ\left(\left(1-\nu \right)s+\nu t\right)$ and integrating it, with respect to $\nu$ over $\left[0, 1\right],$ we obtain

$$\begin{array}{c}{{\rm Y}}_{*}\left(\frac{s+t}{2}\right){{\displaystyle \int }}_0^1{\nu }^{\mathfrak{a}-1}ℭ\left(\left(1-\nu \right)s+\nu t\right)d\nu \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\le \frac{1}{2}\left(\begin{array}{c}{{\displaystyle \int }}_0^1{\nu }^{\mathfrak{a}-1}{{\rm Y}}_{*}\left(\nu s+\left(1-\nu \right)t\right)ℭ\left(\left(1-\nu \right)s+\nu t\right)d\nu \\ +{{\displaystyle \int }}_0^1{\nu }^{\mathfrak{a}-1}{{\rm Y}}_{*}\left(\left(1-\nu \right)s+\nu t\right)ℭ\left(\left(1-\nu \right)s+\nu t\right)d\nu \end{array}\right),\\ {{\rm Y}}^{*}\left(\frac{s+t}{2}\right){{\displaystyle \int }}_0^1ℭ\left(\left(1-\nu \right)s+\nu t\right)d\nu \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\le \frac{1}{2}\left(\begin{array}{c}{{\displaystyle \int }}_0^1{\nu }^{\mathfrak{a}-1}{{\rm Y}}^{*}\left(\nu s+\left(1-\nu \right)t\right)ℭ\left(\left(1-\nu \right)s+\nu t\right)d\nu \\ +{{\displaystyle \int }}_0^1{\nu }^{\mathfrak{a}-1}{{\rm Y}}^{*}\left(\left(1-\nu \right)s+\nu t\right)ℭ\left(\left(1-\nu \right)s+\nu t\right)d\nu \end{array}\right)\end{array}$$

(20)

Let $\omega =\left(1-\nu \right)s+\nu t$. Then, we have

$$\begin{gathered} {{\displaystyle \int }}_0^1{\nu }^{\mathfrak{a}-1}{{\rm Y}}_{*}\left(\nu s+\left(1-\nu \right)t\right)ℭ\left(\left(1-\nu \right)s+\nu t\right)d\nu \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{{\displaystyle \int }}_0^1{\nu }^{\mathfrak{a}-1}{{\rm Y}}_{*}\left(\left(1-\nu \right)s+\nu t\right)ℭ\left(\left(1-\nu \right)s+\nu t\right)d\nu \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=\frac{1}{{\left(t-s\right)}^{\mathfrak{a}}} {{\displaystyle \int }}_s^t{\left(\omega -s\right)}^{\mathfrak{a}-1}{{\rm Y}}_{*}\left(s+t-\omega \right)ℭ\left(\omega \right)d\omega \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\frac{1}{{\left(t-s\right)}^{\mathfrak{a}}} {{\displaystyle \int }}_s^t{\left(\omega -s\right)}^{\mathfrak{a}-1}{{\rm Y}}_{*}\left(\omega \right)ℭ\left(\omega \right)d\omega \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=\frac{1}{{\left(t-s\right)}^{\mathfrak{a}}} {{\displaystyle \int }}_s^t{\left(\omega -s\right)}^{\mathfrak{a}-1}{{\rm Y}}_{*}\left(\omega \right)ℭ\left(s+t-\omega \right)d\omega \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\frac{1}{{\left(t-s\right)}^{\mathfrak{a}}} {{\displaystyle \int }}_s^t{\left(\omega -s\right)}^{\mathfrak{a}-1}{{\rm Y}}_{*}\left(\omega \right)ℭ\left(\omega \right)d\omega \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=\frac{\Gamma \left(\mathfrak{a}\right)}{{\left(t-s\right)}^{\mathfrak{a}}}\left[{\mathcal{I}}_{s^{+}}^{\mathfrak{a}} {{\rm Y}}_{*}ℭ\left(t\right)+{\mathcal{I}}_{t^{-}}^{\mathfrak{a}} {{\rm Y}}_{*}ℭ\left(s\right)\right], \\ {{\displaystyle \int }}_0^1{\nu }^{\mathfrak{a}-1}{{\rm Y}}^{*}\left(\nu s+\left(1-\nu \right)t\right)ℭ\left(\left(1-\nu \right)s+\nu t\right)d\nu \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{{\displaystyle \int }}_0^1{\nu }^{\mathfrak{a}-1}{{\rm Y}}^{*}\left(\left(1-\nu \right)s+\nu t\right)ℭ\left(\left(1-\nu \right)s+\nu t\right)d\nu \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=\frac{\Gamma \left(\mathfrak{a}\right)}{{\left(t-s\right)}^{\mathfrak{a}}}\left[{\mathcal{I}}_{s^{+}}^{\mathfrak{a}} {{\rm Y}}^{*}ℭ\left(t\right)+{\mathcal{I}}_{t^{-}}^{\mathfrak{a}} {{\rm Y}}^{*}ℭ\left(s\right)\right]. \end{gathered}$$

(21)

Then, from (21), we have

$$\begin{array}{c}\frac{\Gamma \left(\mathfrak{a}\right)}{{\left(t-s\right)}^{\mathfrak{a}}}{{\rm Y}}_{*}\left(\frac{s+t}{2}\right)\left[{\mathcal{I}}_{s^{+}}^{\mathfrak{a}}ℭ\left(t\right)+{\mathcal{I}}_{t^{-}}^{\mathfrak{a}}ℭ\left(s\right)\right]\\ \le \frac{\Gamma \left(\mathfrak{a}\right)}{{\left(t-s\right)}^{\mathfrak{a}}} \left[{\mathcal{I}}_{s^{+}}^{\mathfrak{a}} {{\rm Y}}_{*}ℭ\left(t\right)+{\mathcal{I}}_{t^{-}}^{\mathfrak{a}} {{\rm Y}}_{*}ℭ\left(s\right)\right]\\ \frac{\Gamma \left(\mathfrak{a}\right)}{{\left(t-s\right)}^{\mathfrak{a}}}{{\rm Y}}^{*}\left(\frac{s+t}{2}\right)\left[{\mathcal{I}}_{s^{+}}^{\mathfrak{a}}ℭ\left(t\right)+{\mathcal{I}}_{t^{-}}^{\mathfrak{a}}ℭ\left(s\right)\right]\\ \le \frac{\Gamma \left(\mathfrak{a}\right)}{{\left(t-s\right)}^{\mathfrak{a}}} \left[{\mathcal{I}}_{s^{+}}^{\mathfrak{a}} {{\rm Y}}^{*}ℭ\left(t\right)+{\mathcal{I}}_{t^{-}}^{\mathfrak{a}} {{\rm Y}}^{*}ℭ\left(s\right)\right]\end{array}$$

from which, we have

$$\begin{array}{c}\frac{\Gamma \left(\mathfrak{a}\right)}{{\left(t-s\right)}^{\mathfrak{a}}}\left[{{\rm Y}}_{*}\left(\frac{s+t}{2}\right), {{\rm Y}}^{*}\left(\frac{s+t}{2}\right)\right]\left[{\mathcal{I}}_{s^{+}}^{\mathfrak{a}}ℭ\left(t\right)+{\mathcal{I}}_{t^{-}}^{\mathfrak{a}}ℭ\left(s\right)\right]\\ {\begin{array}{c}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\begin{array}{c}\le \end{array}\end{array}}_p\frac{\Gamma \left(\mathfrak{a}\right)}{{\left(t-s\right)}^{\mathfrak{a}}}\left[ {\mathcal{I}}_{s^{+}}^{\mathfrak{a}} {{\rm Y}}_{*}ℭ\left(t\right)+{\mathcal{I}}_{t^{-}}^{\mathfrak{a}} {{\rm Y}}_{*}ℭ\left(s\right), {\mathcal{I}}_{s^{+}}^{\mathfrak{a}} {{\rm Y}}^{*}ℭ\left(t\right)+{\mathcal{I}}_{t^{-}}^{\mathfrak{a}} {{\rm Y}}^{*}ℭ\left(s\right)\right],\end{array}$$

That is,

$$\frac{\Gamma \left(\mathfrak{a}\right)}{{\left(t-s\right)}^{\mathfrak{a}}}{\rm Y}\left(\frac{s+t}{2}\right)\left[{\mathcal{I}}_{s^{+}}^{\mathfrak{a}}ℭ\left(t\right)+{\mathcal{I}}_{t^{-}}^{\mathfrak{a}}ℭ\left(s\right)\right]$$

$${\le }_p\frac{\Gamma \left(\mathfrak{a}\right)}{{\left(t-s\right)}^{\mathfrak{a}}} \left[{\mathcal{I}}_{s^{+}}^{\mathfrak{a}} {\rm Y}ℭ\left(t\right)+{\mathcal{I}}_{t^{-}}^{\mathfrak{a}} {\rm Y}ℭ\left(s\right)\right].$$

This completes the proof. □

Example 3.

We consider the I∙V-F${\rm Y}:\left[0, 2\right]\to {\mathcal{X}}_I^{+}$, defined by${\rm Y}\left(\omega \right)=\left[\left(2-\sqrt{\omega }\right),2\left(2-\sqrt{\omega }\right)\right]$. Since endpoint functions${{\rm Y}}_{*}\left(\omega \right),$ ${{\rm Y}}^{*}\left(\omega \right)$are convex functions, then${\rm Y}\left(\omega \right)$is a left and rightconvex I∙V-F. If

$$ℭ\left(\omega \right)=\left\{\begin{array}{c}\sqrt{\omega }, \omega \in \left[0,1\right],\\ \sqrt{2-\omega }, \omega \in \left(1, 2\right], \end{array}\right.$$

then$ℭ\left(2-\omega \right)=ℭ\left(\omega \right)\ge 0$for all$\omega \in \left[0,2\right]$. Since${{\rm Y}}_{*}\left(\omega \right)=2-\sqrt{\omega }$and${{\rm Y}}^{*}\left(\omega \right)=2\left(2-\sqrt{\omega }\right)$, if$\mathfrak{a}=\frac{1}{2}$, then we compute the following:

$$\left[{\mathcal{I}}_{s^{+}}^{\mathfrak{a}} {\rm Y}ℭ\left(t\right)\widetilde{+}{\mathcal{I}}_{t^{-}}^{\mathfrak{a}} {\rm Y}ℭ\left(s\right)\right]{\le }_p\frac{{\rm Y}\left(s\right)+{\rm Y}\left(t\right)}{2}\left[{\mathcal{I}}_{s^{+}}^{\mathfrak{a}}ℭ\left(t\right)+{\mathcal{I}}_{t^{-}}^{\mathfrak{a}}ℭ\left(s\right)\right]$$

$$\begin{array}{c}\frac{{{\rm Y}}_{*}\left(s\right)+{{\rm Y}}_{*}\left(t\right)}{2}\left[{\mathcal{I}}_{s^{+}}^{\mathfrak{a}}ℭ\left(t\right)+{\mathcal{I}}_{t^{-}}^{\mathfrak{a}}ℭ\left(s\right)\right]=\frac{\pi }{\sqrt{2}}\left(\frac{4-\sqrt{2}}{2}\right)\\ \frac{{{\rm Y}}^{*}\left(s\right)\widetilde{+}{{\rm Y}}^{*}\left(t\right)}{2}\left[{\mathcal{I}}_{s^{+}}^{\mathfrak{a}}ℭ\left(t\right)+{\mathcal{I}}_{t^{-}}^{\mathfrak{a}}ℭ\left(s\right)\right]=\frac{\pi }{\sqrt{2}}\left(4-\sqrt{2}\right),\end{array}$$

(22)

$$\begin{array}{c}\frac{{{\rm Y}}_{*}\left(s\right)+{{\rm Y}}_{*}\left(t\right)}{2}\left[{\mathcal{I}}_{s^{+}}^{\mathfrak{a}}ℭ\left(t\right)+{\mathcal{I}}_{t^{-}}^{\mathfrak{a}}ℭ\left(s\right)\right]=\frac{\pi }{\sqrt{2}}\left(\frac{4-\sqrt{2}}{2}\right)\\ \frac{{{\rm Y}}^{*}\left(s\right)\widetilde{+}{{\rm Y}}^{*}\left(t\right)}{2}\left[{\mathcal{I}}_{s^{+}}^{\mathfrak{a}}ℭ\left(t\right)+{\mathcal{I}}_{t^{-}}^{\mathfrak{a}}ℭ\left(s\right)\right]=\frac{\pi }{\sqrt{2}}\left(4-\sqrt{2}\right),\end{array}$$

(23)

$$\begin{array}{c}\left[{\mathcal{I}}_{s^{+}}^{\mathfrak{a}} {{\rm Y}}_{*}ℭ\left(t\right)+{\mathcal{I}}_{t^{-}}^{\mathfrak{a}} {{\rm Y}}_{*}ℭ\left(s\right)\right]=\frac{1}{\sqrt{\pi }}\left(2\pi +\frac{4-8\sqrt{2}}{3}\right),\\ \left[{\mathcal{I}}_{s^{+}}^{\mathfrak{a}} {{\rm Y}}^{*}ℭ\left(t\right)+{\mathcal{I}}_{t^{-}}^{\mathfrak{a}} {{\rm Y}}^{*}ℭ\left(s\right)\right]=\frac{2}{\sqrt{\pi }}\left(2\pi +\frac{4-8\sqrt{2}}{3}\right). \end{array}$$

(24)

From (22)–(24), (13) we have

$$\frac{1}{\sqrt{\pi }}\left[\left(2\pi +\frac{4-8\sqrt{2}}{3}\right), 2\left(2\pi +\frac{4-8\sqrt{2}}{3}\right)\right]{\begin{array}{c} \begin{array}{c}\le \end{array}\end{array}}_p\frac{\pi }{\sqrt{2}}\left[\frac{4-\sqrt{2}}{2}, 4-\sqrt{2}\right]=\frac{\pi }{\sqrt{2}}\left[\frac{4-\sqrt{2}}{2}, 4-\sqrt{2}\right].$$

Hence, Theorem 6 is verified.

For Theorem 7, we have

$$\begin{array}{c}{{\rm Y}}_{*}\left(\frac{s+t}{2}\right)\left[{\mathcal{I}}_{s^{+}}^{\mathfrak{a}}ℭ\left(t\right)+{\mathcal{I}}_{t^{-}}^{\mathfrak{a}}ℭ\left(s\right)\right]=\sqrt{\pi }, \\ {{\rm Y}}^{*}\left(\frac{s+t}{2}\right)\left[{\mathcal{I}}_{s^{+}}^{\mathfrak{a}}ℭ\left(t\right)+{\mathcal{I}}_{t^{-}}^{\mathfrak{a}}ℭ\left(s\right)\right]=2\sqrt{\pi }. \end{array}$$

(25)

From (24) and (25), we have

$$\sqrt{\pi }\left[1, 2\right]{\begin{array}{c} \begin{array}{c}\le \end{array}\end{array}}_p\frac{1}{\sqrt{\pi }}\left[2\pi +\frac{4-8\sqrt{2}}{3}, 2\left(2\pi +\frac{4-8\sqrt{2}}{3}\right)\right]$$

Hence, (18) has been verified.

Remark 3.

If one takes$ℭ\left(\omega \right)=1$, then, from (13) and (18), we acquire (5).

Let us take$\mathfrak{a}=1$. Then, we achieve the coming inequality (see [22]).

$${\rm Y}\left(\frac{s+t}{2}\right){\le }_p\frac{1}{{{\displaystyle \int }}_s^tℭ\left(\omega \right)d\omega }{{\displaystyle \int }}_s^t{\rm Y}\left(\omega \right)ℭ\left(\omega \right)d\omega {\le }_p\frac{{\rm Y}\left(s\right)+{\rm Y}\left(t\right)}{2}$$

If we take${{\rm Y}}_{*}\left(\omega \right)={{\rm Y}}^{*}\left(\omega \right)$, then from (13) and (18), we acquire the coming inequality (see [33]).

$${\rm Y}\left(\frac{s+t}{2}\right)\left[{\mathcal{I}}_{s^{+}}^{\mathfrak{a}}ℭ\left(t\right)+{\mathcal{I}}_{t^{-}}^{\mathfrak{a}}ℭ\left(s\right)\right]{\le }_p\left[{\mathcal{I}}_{s^{+}}^{\mathfrak{a}} {\rm Y}ℭ\left(t\right)+{\mathcal{I}}_{t^{-}}^{\mathfrak{a}} {\rm Y}ℭ\left(s\right)\right]$$

$${\le }_p\frac{{\rm Y}\left(s\right)+{\rm Y}\left(t\right)}{2}\left[{\mathcal{I}}_{s^{+}}^{\mathfrak{a}}ℭ\left(t\right)+{\mathcal{I}}_{t^{-}}^{\mathfrak{a}}ℭ\left(s\right)\right]$$

If one takes${{\rm Y}}_{*}\left(\omega \right)={{\rm Y}}^{*}\left(\omega \right)$with$\mathfrak{a}=1$, then from (13) and (18), we achieve the classical 𝓗–𝓗 Fejér inequality (see [26]).

4. Conclusions

In applied sciences, convex functions and fractional calculus are essential. The new interval-valued left and right convex functions are presented in this article. Some novel Riemann–Liouville fractional integral 𝓗–𝓗 and Fejér-type inequalities are provided, utilizing the idea of interval-valued left and right convex functions and some supplementary interval analysis findings. Our results are a generalization of a number of previously published findings. In the future, we will use generalized interval and fuzzy Riemann–Liouville fractional operators to investigate this concept for generalized left and right convex I∙V-Fs and F-I∙V-Fs by using interval Katugampola fractional integrals and fuzzy Katugampola fractional integrals. For applications, see [53,54,55,56].