Abstract
The framework of fuzzy-interval-valued functions (FIVFs) is a generalization of interval-valued functions (IVF) and single-valued functions. To discuss convexity with these kinds of functions, in this article, we introduce and investigate the harmonically \(\mathsf{h}\)-convexity for FIVFs through fuzzy-order relation (FOR). Using this class of harmonically \(\mathsf{h}\)-convex FIVFs (\(\mathcal{H}-\mathsf{h}\)-convex FIVFs), we prove some Hermite–Hadamard (H⋅H) and Hermite–Hadamard–Fejér (H⋅H Fejér) type inequalities via fuzzy interval Riemann–Liouville fractional integral (FI Riemann–Liouville fractional integral). The concepts and techniques of this paper are refinements and generalizations of many results which are proved in the literature.
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1 Introduction
Fractional calculus dates back to the seventeenth century, when Leibniz and Marquis de l'Hospital began a conversation about semi-derivatives. Many well-known mathematicians were inspired by this subject to investigate modern views of the area. The theory of fractional calculus grew greatly in the late nineteenth century. Math, physics, viscoelasticity, rheology, chemistry, and statistical physics, as well as electrical and mechanical engineering, are now covered.
The application of integral inequalities in mathematical analysis has seen an exponential increase in publications. Riemann–Liouville, Caputo, Katugampola, and Caputo–Fabrizio are just a few of the integral inequalities that have been developed in recent years using a variety of fractional-order operator definitions. Researchers have obtained several versions of well-known inequalities of Hermite–Hadamard, Hardy, Opial, Ostrowski, and Grüss (see [1,2,3,4,5,6,7,8,9,10]) using these integrals.
On the other hand, Costa [11] just uncovered Jensen’s type inequality in FIVF. Costa and Roman–Flores [12, 13] looked at the characteristics of several types of inequalities in the context of FIVF and IVF. Roman–Flores et al. [14] established Gronwall inequality for IVFs. Furthermore, Chalco-Cano et al. [15, 16] employed the generalized Hukuhara derivative to demonstrate Ostrowski-type inequalities for IVFs, as well as numerical integration applications in IVF. Nikodem et al. [17] and Matkowski and Nikodem [18] proposed new versions of Jensen’s inequality for strongly convex and convex functions.
Zhao et al. [19, 20] were employed IVFs to generate Chebyshev, Jensen’s, and HH type inequalities. Zhang et al. [21] recently employed a pseudo order relation to extend Jensen’s inequalities for set-valued and fuzzy-set-valued functions and develop a novel form of Jensen’s inequalities. Budek [22] subsequently established an interval-valued fractional Riemann–Liouville HH inequality for convex IVF using an inclusion relation. For further detail, see [23,24,25,26,27] and the references therein.
Recently, Khan et al. [28] used FOR to construct a new class of convex FIVFs which is known as (\({\mathsf{h}}_{1},{\mathsf{h}}_{2}\))-convex FIVFs, as well as some new versions of the H⋅H type inequality for (\({\mathsf{h}}_{1},{\mathsf{h}}_{2}\))-convex FIVFs that incorporates the FI Riemann integral. Khan et al. went a step further by providing new convex and extended convex FIVF classes, as well as new fractional H⋅H type and H⋅H type inequalities for convex FIVF [29], \(\mathsf{h}\)-convex FIVF [30], (\({\mathsf{h}}_{1},{\mathsf{h}}_{2}\))-preinvex FIVF [31], log-s-convex FIVFs in the second sense [32], harmonically convex FIVFs [33], coordinated convex FIVFs [34] and the references therein. We suggest readers to [35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58] and the references therein for more study of literature on the applications and properties of FI, as well as inequalities and extended convex fuzzy mappings.
Motivated and inspired by ongoing research work, we have introduced the new generation of harmonic functions is known as ℋ − h-convex functions using FOR in Sect. 2. In Sect. 3, we have used FI fractional operators to derive new versions of Hermite–Hadamard inequalities with the help of this class. Furthermore, we have examined the study’s special circumstances as applications. In the end, we have given conclusion and future plan.
2 Preliminaries
We will start by reviewing the fundamental notations and definitions.
The collection of all closed and bounded intervals of \({\mathbb{R}}\) is denoted and defined as
The set of all positive interval is denoted by \({{\mathcal{K}}_{C}}^{+}\) and defined as
We will now look at some of the properties of intervals using arithmetic operations. Let \(\left[{\zeta }_{*}, {\zeta }^{*}\right], \left[{\lambda }_{*}, {\lambda }^{*}\right]\in {\mathcal{K}}_{C}\) and \(\rho \in {\mathbb{R}}\), then we have
where
and
For \(\left[{\zeta }_{*}, {\zeta }^{*}\right], \left[{\lambda }_{*}, {\lambda }^{*}\right]\in {\mathcal{K}}_{C},\) the inclusion \("\subseteq "\) is defined by
Remark 2.1.
The relation \({"\le }_{I}"\) defined on \({\mathcal{K}}_{C}\) by
for all \(\left[{\zeta }_{*}, {\zeta }^{*}\right], \left[{\lambda }_{*}, {\lambda }^{*}\right]\in {\mathcal{K}}_{C},\) it is an order relation, see [35].
Let \({\mathbb{R}}\) be the set of real numbers. A mapping \(\zeta :{\mathbb{R}}\to [\mathrm{0,1}]\) called the membership function distinguishes a fuzzy subset set \(\mathcal{A}\) of \({\mathbb{R}}\). This representation is found to be acceptable in this study. \({\mathbb{F}}_{0}\) also stands for the collection of all fuzzy subsets of \({\mathbb{R}}\).
Proposition 2.2.
[18] Let \(\stackrel{\sim }{\zeta },\stackrel{\sim }{\lambda }\in {\mathbb{F}}_{0}\). Then FOR \("\preccurlyeq "\) given on \({\mathbb{F}}_{0}\) by
it is partial order relation.
We will now look at some of the properties of FIs using arithmetic operations. Let \(\stackrel{\sim }{\zeta },\stackrel{\sim }{\lambda }\in {\mathbb{F}}_{0}\) and \(\rho \in {\mathbb{R}}\), then we have
Definition 2.3.
[36] A FIV map \(\mathfrak{G}:K\subset {\mathbb{R}}\to {\mathbb{F}}_{0}\) is called FIVF. For each \(\varphi \in (0, 1],\) whose \(\varphi \)-cuts derive the following IVFs \({\mathfrak{G}}_{\varphi }:K\subset {\mathbb{R}}\to {\mathcal{K}}_{C}\) are given by \({\mathfrak{G}}_{\varphi }\left(\varpi \right)=\left[{\mathfrak{G}}_{*}\left(\varpi ,\varphi \right), {\mathfrak{G}}^{*}\left(\varpi ,\varphi \right)\right]\) for all \(\varpi \in K.\) Here, for each \(\varphi \in (0, 1],\) the end point real functions \({\mathfrak{G}}_{*}\left(.,\varphi \right), {\mathfrak{G}}^{*}\left(.,\varphi \right):K\to {\mathbb{R}}\) are called lower and upper functions of \(\mathfrak{G}\).
The following FI Riemann–Liouville fractional integral operators were introduced by Allahviranloo et al. [10]:
Definition 2.4.
Let \(\alpha >0\) and \(L\left(\left[\mu , \upsilon \right],{\mathbb{F}}_{0}\right)\) be the collection of all Lebesgue measurable FIVFs on \([\mu ,\upsilon ]\). Then, the FI left and right Riemann–Liouville fractional integral of \(\stackrel{\sim }{\mathfrak{G}}\in \) \(L\left(\left[\mu , \upsilon \right],{\mathbb{F}}_{0}\right)\) with order \(\alpha >0\) are defined by.
and
respectively, where \(\Gamma \left(\varpi \right)={\int }_{0}^{\infty }{\varsigma }^{\varpi -1}{\mu }^{-\varsigma }d\varsigma \) is the Euler gamma function. The FI left and right Riemann–Liouville fractional integral \(\varpi \) based on left and right end point functions can be defined, that is.
where
and
Similarly, the left and right end point functions can be used to define the right Riemann–Liouville fractional integral \(\mathfrak{G}\) of \(\varpi \).
Definition 2.5.
[7] A set \(K=\left[\mu , \upsilon \right]\subset {\mathbb{R}}^{+}=\left(0,\infty \right)\) is said to be harmonically convex set, if, for all \(\varpi , \mathcal{Z}\in K, \varsigma \in \left[0, 1\right]\), we have
Definition 2.6.
[7] The \(\mathfrak{G}:\left[\mu , \upsilon \right]\to {\mathbb{R}}^{+}\) is called harmonically convex function on \(\left[\mu , \upsilon \right]\) if
for all \(\varpi , \mathcal{Z}\in \left[\mu , \upsilon \right], \varsigma \in \left[0, 1\right],\) where \(\mathfrak{G}\left(\varpi \right)\ge 0\) for all \(\varpi \in \left[\mu , \upsilon \right].\) If (14) is reversed then, \(\mathfrak{G}\) is called harmonically concave FIVF on \(\left[\mu , \upsilon \right]\).
Definition 2.7.
[43] The positive real-valued function \(\mathfrak{G}:\left[\mu , \upsilon \right]\to {\mathbb{R}}^{+}\) is called ℋ − -convex function on \(\left[\mu , \upsilon \right]\) if
for all \(\varpi , \mathcal{Z}\in \left[\mu , \upsilon \right], \varsigma \in \left[0, 1\right],\) where \(\mathfrak{G}\left(\varpi \right)\ge 0\) for all \(\varpi \in \left[\mu , \upsilon \right]\) and and \(\mathsf{h}:[0, 1]\subseteq [\mu , \upsilon ]\to {\mathbb{R}}^{+}\) such that \(\mathsf{h}\not\equiv 0\). If (15) is reversed then, \(\mathfrak{G}\) is called ℋ − -concave function on \(\left[\mu , \upsilon \right]\). The set of all ℋ − -convex (ℋ − -concave) functions is denoted by
Definition 2.8.
[28] The FIVF \(\stackrel{\sim }{\mathfrak{G}}:\left[\mu , \upsilon \right]\to {\mathbb{F}}_{0}\) is called \(\mathsf{h}\)-convex FIVF on \(\left[\mu , \upsilon \right]\) if
for all \(\varpi , \mathcal{Z}\in \left[\mu , \upsilon \right], \varsigma \in \left[0, 1\right],\) where \(\mathfrak{G}\left(\varpi \right)\ge 0\) for all \(\varpi \in \left[\mu , \upsilon \right]\) and and \(\mathsf{h}:[0, 1]\subseteq [\mu , \upsilon ]\to {\mathbb{R}}^{+}\) such that \(\mathsf{h}\not\equiv 0\). If (16) is reversed then, \(\stackrel{\sim }{\mathfrak{G}}\) is called \(\mathsf{h}\)-concave FIVF on \(\left[\mu , \upsilon \right]\). The set of all \(\mathsf{h}\)-convex (\(\mathsf{h}\)-concave) FIVF is denoted by
Definition 2.9.
[34] The FIVF \(\stackrel{\sim }{\mathfrak{G}}:\left[\mu , \upsilon \right]\to {\mathbb{F}}_{0}\) is called harmonically convex FIVF on \(\left[\mu , \upsilon \right]\) if
for all \(\varpi , \mathcal{Z}\in \left[\mu , \upsilon \right], \varsigma \in \left[0, 1\right],\) where \(\stackrel{\sim }{\mathfrak{G}}\left(\varpi \right)\succcurlyeq \tilde{0 }\), for all \(\varpi \in \left[\mu , \upsilon \right].\) If (17) is reversed then, \(\stackrel{\sim }{\mathfrak{G}}\) is called harmonically concave FIVF on \(\left[\mu , \upsilon \right]\).
Definition 2.10.
[47] The FIVF \(\stackrel{\sim }{\mathfrak{G}}:\left[\mu , \upsilon \right]\to {\mathbb{F}}_{0}\) is called harmonically -convex (ℋ − -convex) FIVF on \(\left[\mu , \upsilon \right]\) if
for all \(\varpi , \mathcal{Z}\in \left[\mu , \upsilon \right], \varsigma \in \left[0, 1\right],\) where \(\stackrel{\sim }{\mathfrak{G}}\left(\varpi \right)\succcurlyeq \tilde{0 }\), for all \(\varpi \in \left[\mu , \upsilon \right]\) and \(\mathsf{h}:[0, 1]\subseteq [\mu , \upsilon ]\to {\mathbb{R}}^{+}\) such that \(\mathsf{h}\not\equiv 0\). If (18) is reversed then, \(\stackrel{\sim }{\mathfrak{G}}\) is called ℋ − -concave FIVF on \(\left[\mu , \upsilon \right]\). The set of all ℋ − -convex (ℋ − -concave) FIVF is denoted by
Theorem 2.11.
[47] Let \(\left[\mu , \upsilon \right]\) be harmonically convex set, and let \(\stackrel{\sim }{\mathfrak{G}}:\left[\mu , \upsilon \right]\to {\mathbb{F}}_{C}\left({\mathbb{R}}\right)\) be a FIVF, whose \(\varphi \)-cuts derive the following IVFs \({\mathfrak{G}}_{\varphi }:\left[\mu , \upsilon \right]\subset {\mathbb{R}}\to {\mathcal{K}}_{C}^{+}\subset {\mathcal{K}}_{C}\) are given by
for all \(\varpi \in \left[\mu , \upsilon \right]\), \(\varphi \in \left[0, 1\right]\). Then, \(\stackrel{\sim }{\mathfrak{G}}\in HFSX\left(\left[\mu , \upsilon \right], {\mathbb{F}}_{0}, \mathsf{h}\right),\) if and only if, for all \(\in \left[0, 1\right],\) \({\mathfrak{G}}_{*}\left(\varpi , \varphi \right)\), \({\mathfrak{G}}^{*}\left(\varpi , \varphi \right)\in HSX\left(\left[\mu , \upsilon \right], {\mathbb{R}}^{+},\mathsf{h}\right).\)
Proof.
Assume that for each \(\in \left[0, 1\right],\) \({\mathfrak{G}}_{*}\left(\varpi , \varphi \right)\), \({\mathfrak{G}}^{*}\left(\varpi , \varphi \right)\in HSX\left(\left[\mu , \upsilon \right], {\mathbb{R}}^{+},\mathsf{h}\right).\) Then from (15), we have
for all \(\varpi ,\mathcal{Z}\in \left[\mu , \upsilon \right], \varsigma \in \left[0, 1\right], \)and
for all \(\varpi ,\mathcal{Z}\in \left[\mu , \upsilon \right], \varsigma \in \left[0, 1\right].\)
Then, by (19), (5) and (6), we obtain
\({\mathfrak{G}}_{\varphi }\left(\frac{\varpi \mathcal{Z}}{\varsigma \varpi +\left(1-\varsigma \right)\mathcal{Z}}\right)=\left[{\mathfrak{G}}_{*}\left(\varsigma \varpi +\left(1-\varsigma \right)\mathcal{Z}, \varphi \right), {\mathfrak{G}}^{*}\left(\varsigma \varpi +\left(1-\varsigma \right)\mathcal{Z}, \varphi \right)\right]\),
\({\le }_{I}\mathsf{h}\left(1-\varsigma \right)\left[{\mathfrak{G}}_{*}\left(\varpi , \varphi \right), {\mathfrak{G}}^{*}\left(\varpi , \varphi \right)\right]\)
that is
for all \(\varpi ,\mathcal{Z}\in \left[\mu , \upsilon \right], \varsigma \in \left[0, 1\right].\)
Hence,\(\stackrel{\sim }{\mathfrak{G}}\in HFSX\left(\left[\mu , \upsilon \right], {\mathbb{F}}_{0}, \mathsf{h}\right).\)
Conversely, let \(\stackrel{\sim }{\mathfrak{G}}\in HFSX\left(\left[\mu , \upsilon \right], {\mathbb{F}}_{0}, \mathsf{h}\right).\) Then for all \(\varpi ,\mathcal{Z}\in \left[\mu , \upsilon \right]\), \(\varsigma \in \left[0, 1\right],\) we have.
Therefore, from (19), for each \(\varphi \in \left[0, 1\right]\), left side of above inequality, we have
Again, from (19), we obtain
\(\mathsf{h}\left(1-\varsigma \right){\mathfrak{G}}_{\varphi }\left(\varpi \right)+\mathsf{h}\left(\varsigma \right){\mathfrak{G}}_{\varphi }\left(\mathcal{Z}\right)=\mathsf{h}\left(1-\varsigma \right)\left[{\mathfrak{G}}_{*}\left(\varpi , \varphi \right), {\mathfrak{G}}^{*}\left(\varpi , \varphi \right)\right]\)
for all \(\varpi ,\mathcal{Z}\in \left[\mu , \upsilon \right]\), \(\varsigma \in \left[0, 1\right].\) Then by ℋ − -convexity of \(\stackrel{\sim }{\mathfrak{G}}\), we have for all \(\varpi ,\mathcal{Z}\in \left[\mu , \upsilon \right]\), \(\varsigma \in \left[0, 1\right]\) such that
and
for each \(\varphi \in \left[0, 1\right].\) Hence, for each \(\varphi \in \left[0, 1\right]\), \({\mathfrak{G}}_{*}\left(\varpi , \varphi \right)\), \({\mathfrak{G}}^{*}\left(\varpi , \varphi \right)\in HSX\left(\left[\mu , \upsilon \right], {\mathbb{R}}^{+},\mathsf{h}\right)\).
Remark 2.12.
On fixing \({\mathfrak{G}}_{*}\left(\varpi ,\varphi \right)={\mathfrak{G}}^{*}\left(\varpi ,\varphi \right)\) and \(\varphi =1\), then from Definition 2.10, we obtain Definition 2.7.
On fixing \(\mathsf{h}\left(\varsigma \right)=\varsigma \), then from Definition 2.10, we obtain Definition 2.9.
Example 2.13.
We consider the FIVFs \(\stackrel{\sim }{\mathfrak{G}}:[0, 2]\to {\mathbb{F}}_{C}\left({\mathbb{R}}\right)\) defined by.
Then, for each \(\varphi \in \left[0, 1\right],\) we have \({\mathfrak{G}}_{\varphi }\left(\varpi \right)=\left[\varphi \sqrt{\varpi },(2-\varphi )\sqrt{\varpi }\right]\). Since \({\mathfrak{G}}_{*}\left(\varpi , \varphi \right)\), \({\mathfrak{G}}^{*}\left(\varpi , \varphi \right)\in HSX\left(\left[\mu , \upsilon \right], {\mathbb{R}}^{+},\mathsf{h}\right)\) with \(\mathsf{h}\left(\varsigma \right)=\varsigma \), for each \(\varphi \in [0, 1]\). Hence \(\stackrel{\sim }{\mathfrak{G}}\in HFSX\left(\left[\mu , \upsilon \right], {\mathbb{F}}_{0}, \mathsf{h}\right)\).
We shall develop a relationship between \(\mathsf{h}\)-convex FIVF and \(\mathcal{H}-\mathsf{h}\)-convex FIVF in the next finding.
Theorem 2.14.
Let \(\stackrel{\sim }{\mathfrak{G}}:\left[\mu , \upsilon \right]\to {\mathbb{F}}_{C}\left({\mathbb{R}}\right)\) be a FIVF, where for all \(\varphi \in \left[0, 1\right]\), whose \(\varphi \)-cuts derive the following IVFs \({\mathfrak{G}}_{\varphi }:\left[\mu , \upsilon \right]\subset {\mathbb{R}}\to {\mathcal{K}}_{C}^{+}\subset {\mathcal{K}}_{C}\) are given by \({\mathfrak{G}}_{\varphi }\left(\varpi \right)=\left[{\mathfrak{G}}_{*}\left(\varpi ,\varphi \right), {\mathfrak{G}}^{*}\left(\varpi ,\varphi \right)\right],\) for all \(\varpi \in \left[\mu , \upsilon \right]\). Then \(\stackrel{\sim }{\mathfrak{G}}\in HFSX\left(\left[\mu , \upsilon \right], {\mathbb{F}}_{0}, \mathsf{h}\right),\) if and only if,\(\stackrel{\sim }{\mathfrak{G}}\left(\frac{1}{\varpi }\right)\in FSX\left(\left[\mu , \upsilon \right], {\mathbb{F}}_{0}, \mathsf{h}\right),.\)
Proof.
Since \(\stackrel{\sim }{\mathfrak{G}}\in HFSX\left(\left[\mu , \upsilon \right], {\mathbb{F}}_{0}, \mathsf{h}\right),\) then, for \(\varpi , \mathcal{Z}\in \left[\mu , \upsilon \right], \varsigma \in \left[0, 1\right]\), we have.
Therefore, for each \(\varphi \in [0, 1]\), we have
Consider \(\stackrel{\sim }{\varphi }\left(\varpi \right)=\stackrel{\sim }{\mathfrak{G}}\left(\frac{1}{\varpi }\right)\). Taking \(m=\frac{1}{\varpi }\) and \(n=\frac{1}{\mathcal{Z}}\) to replace \(\varpi \) and \(\mathcal{Z}\), respectively. Then for each \(\varphi \in [0, 1]\), applying (20)
It follows that
which implies that
that is
This concludes that \(\stackrel{\sim }{\varphi }\left(\varpi \right)\in FSX\left(\left[\mu , \upsilon \right], {\mathbb{F}}_{0}, \mathsf{h}\right)\).
Conversely, let \(\stackrel{\sim }{\varphi }\in FSX\left(\left[\mu , \upsilon \right], {\mathbb{F}}_{0}, \mathsf{h}\right).\) Then, for all \(\varpi ,\mathcal{Z}\in \left[\mu , \upsilon \right]\), \(\varsigma \in \left[0, 1\right],\) we have.
Using same steps as above, for each \(\varphi \in [0, 1]\), we have
It follows that.
that is
the proof the theorem has been completed.
Remark 2.15.
If \(\mathsf{h}\left(\varsigma \right)=\varsigma \), and \({\mathfrak{G}}_{*}\left(\varpi ,\varphi \right)={\mathfrak{G}}^{*}\left(\varpi ,\varphi \right)\) with \(\varphi =1\), then from Theorem 2.14, we obtain Lemma 2.1 of [9].
3 Hermite–Hadamard Inequalities for Harmonically \(\mathsf{h}\)-Convex Fuzzy-Interval-Valued Functions
We shall prove two forms of inequalities in this section. The first is H⋅H and its variant forms, while the second is H⋅H Fejér inequalities for ℋ − -convex FIVFs with FIVFs as integrands. In the following, \(L\left(\left[\mu , \upsilon \right],{\mathbb{F}}_{0}\right)\) denotes the family of Lebesgue measureable FIVFs.
Theorem 3.1.
Let \(\stackrel{\sim }{\mathfrak{G}}\in HFSX\left(\left[\mu , \upsilon \right], {\mathbb{F}}_{0}, \mathsf{h}\right),\) whose \(\varphi \)-cuts derive the following IVFs \({\mathfrak{G}}_{\varphi }:\left[\mu , \upsilon \right]\subset {\mathbb{R}}\to {\mathcal{K}}_{C}^{+}\) are given by \({\mathfrak{G}}_{\varphi }\left(\varpi \right)=\left[{\mathfrak{G}}_{*}\left(\varpi ,\varphi \right), {\mathfrak{G}}^{*}\left(\varpi ,\varphi \right)\right]\) for all \(\varpi \in \left[\mu , \upsilon \right]\), \(\varphi \in \left[0, 1\right]\). If \(\stackrel{\sim }{\mathfrak{G}}\in L\left(\left[\mu , \upsilon \right],{\mathbb{F}}_{0}\right)\), then.
If \(\stackrel{\sim }{\mathfrak{G}}(\varpi )\) is concave FIVF then
where \(\psi \left(\varpi \right)=\frac{1}{\varpi }\).
Proof.
Let \(\stackrel{\sim }{\mathfrak{G}}\in HFSX\left(\left[\mu , \upsilon \right], {\mathbb{F}}_{0}, \mathsf{h}\right),\). Then, by hypothesis, we have.
Therefore, for each \(\varphi \in [0, 1]\), we have
Consider \(\stackrel{\sim }{\varphi }\left(\varpi \right)=\stackrel{\sim }{\mathfrak{G}}\left(\frac{1}{\varpi }\right).\) By Theorem 2.14, we have \(\stackrel{\sim }{\varphi }\left(\varpi \right)\in FSX\left(\left[\mu , \upsilon \right], {\mathbb{F}}_{0}, \mathsf{h}\right),\) then for each \(\varphi \in [0, 1]\), above inequality, we have
Multiplying both sides by \({\varsigma }^{\alpha -1}\) and integrating the obtained result with respect to \(\varsigma \) over \((\mathrm{0,1})\), we have
Let \(\varpi =\frac{\left(1-\varsigma \right)\mu +\varsigma \upsilon }{\mu \upsilon }\) and \(\mathcal{Z}=\frac{\varsigma \mu +\left(1-\varsigma \right)\upsilon }{\mu \upsilon }.\) Then, we have
\(\begin{array}{c}\frac{1}{\alpha \mathsf{h}\left(\frac{1}{2}\right)}{\varphi }_{*}\left(\frac{\mu +\upsilon }{2\mu \upsilon }, \varphi \right) \le {\left(\frac{\mu \upsilon }{\upsilon -\mu }\right)}^{\alpha } \underset{\frac{1}{\upsilon }}{\overset{\frac{1}{\mu }}{\int }}{\left(\frac{1}{\mu }-\mathcal{Z}\right)}^{\alpha -1}{\varphi }_{*}\left(\mathcal{Z},\varphi \right)dz\\ +{\left(\frac{\mu \upsilon }{\upsilon -\mu }\right)}^{\alpha }\underset{\frac{1}{\upsilon }}{\overset{\frac{1}{\mu }}{\int }}{(\varpi -\frac{1}{\upsilon })}^{\alpha -1}{\varphi }_{*}(\varpi ,\varphi )d\varpi \\ \frac{1}{\alpha \mathsf{h}\left(\frac{1}{2}\right)}{\varphi }_{*}\left(\frac{\mu +\upsilon }{2\mu \upsilon }, \varphi \right)\le {\left(\frac{\mu \upsilon }{\upsilon -\mu }\right)}^{\alpha } \underset{\frac{1}{\upsilon }}{\overset{\frac{1}{\mu }}{\int }}{\left(\frac{1}{\mu }-\mathcal{Z}\right)}^{\alpha -1}{\varphi }^{*}\left(\mathcal{Z},\varphi \right)dz\\ +{\left(\frac{\mu \upsilon }{\upsilon -\mu }\right)}^{\alpha }\underset{\frac{1}{\upsilon }}{\overset{\frac{1}{\mu }}{\int }}{(\varpi -\frac{1}{\upsilon })}^{\alpha -1}{\varphi }^{*}\left(\varpi ,\varphi \right)d\varpi ,\end{array}\)
It follows that
That is
In a similar way as above, we have
Combining (23) and (24), we have
that is
Hence, the required result.
Remark 3.2.
Followings results can be obtained through inequality (21):
On fixing \(\mathsf{h}\left(\varsigma \right)=\varsigma \), the following H⋅H inequality is obtained, see [34];
On fixing \(\mathsf{h}\left(\varsigma \right)=\varsigma \) and \(\alpha =1\), the following H⋅H inequality is obtained, see [34];
On fixing \(\mathsf{h}\left(\varsigma \right)=\varsigma \) and \({\mathfrak{G}}_{*}\left(\varpi ,\varphi \right)={\mathfrak{G}}^{*}\left(\varpi ,\varphi \right)\) with \(\varphi =1\) the following H⋅H inequality is obtained, see [9]:
On fixing \(\mathsf{h}\left(\varsigma \right)=\varsigma \) and \({\mathfrak{G}}_{*}\left(\varpi ,\varphi \right)={\mathfrak{G}}^{*}\left(\varpi ,\varphi \right)\) with \(\varphi =1\) and \(\alpha =1\), the following H⋅H inequality is obtained, see [7].
For the product of ℋ − -convex FIVFs, we now have some H⋅H inequalities. These inequalities are modifications of previously published inequalities [34, 38, 43].
Theorem 3.3.
Let \(\stackrel{\sim }{\mathfrak{G}}\in HFSX\left(\left[\mu , \upsilon \right], {\mathbb{F}}_{0}, {\mathsf{h}}_{1}\right)\) and \(\stackrel{\sim }{\mathcal{P}}\in HFSX\left(\left[\mu , \upsilon \right], {\mathbb{F}}_{0}, {\mathsf{h}}_{2}\right)\), whose \(\varphi \)-cuts constitute the following IVFs \({\mathfrak{G}}_{\varphi }, {\mathcal{P}}_{\varphi }:\left[\mu , \upsilon \right]\subset {\mathbb{R}}\to {\mathcal{K}}_{C}^{+}\) are defined by \({\mathfrak{G}}_{\varphi }\left(\varpi \right)=\left[{\mathfrak{G}}_{*}\left(\varpi ,\varphi \right), {\mathfrak{G}}^{*}\left(\varpi ,\varphi \right)\right]\) and \({\mathcal{P}}_{\varphi }\left(\varpi \right)=\left[{\mathcal{P}}_{*}\left(\varpi ,\varphi \right), {\mathcal{P}}^{*}\left(\varpi ,\varphi \right)\right]\) for all \(\varpi \in \left[\mu , \upsilon \right]\), \(\varphi \in \left[0, 1\right]\), respectively. If \(\stackrel{\sim }{\mathfrak{G}}\stackrel{\sim }{\times }\stackrel{\sim }{\mathcal{P}}\in L\left(\left[\mu , \upsilon \right],{\mathbb{F}}_{0}\right)\), then.
\(\Gamma \left(\alpha \right){\left(\frac{\mu \upsilon }{\upsilon -\mu }\right)}^{\alpha }\left[\begin{array}{c}{\mathcal{I}}_{{\left(\frac{1}{\upsilon }\right)}^{+}}^{\alpha } \stackrel{\sim }{\mathfrak{G}}o\psi \left(\frac{1}{\mu }\right)\stackrel{\sim }{\times }\stackrel{\sim }{\mathcal{P}}o\psi \left(\frac{1}{\mu }\right)\\ +{\mathcal{I}}_{{\left(\frac{1}{\mu }\right)}^{-}}^{\alpha }\stackrel{\sim }{\mathfrak{G}}o\psi \left(\frac{1}{\upsilon }\right)\stackrel{\sim }{\times }\stackrel{\sim }{\mathcal{P}}o\psi \left(\frac{1}{\upsilon }\right)\end{array}\right]\)
\(\preccurlyeq \tilde{M }\left(\mu ,\upsilon \right){\int }_{0}^{1}\left[{\varsigma }^{\alpha -1}+{\left(1-\varsigma \right)}^{\alpha -1}\right]{\mathsf{h}}_{1}\left(\varsigma \right){\mathsf{h}}_{2}\left(\varsigma \right)d\varsigma \)
where \(\tilde{M }\left(\mu ,\upsilon \right)=\stackrel{\sim }{\mathfrak{G}}\left(\mu \right)\stackrel{\sim }{\times }\stackrel{\sim }{\mathcal{P}}\left(\mu \right) \stackrel{\sim }{+} \stackrel{\sim }{\mathfrak{G}}\left(\upsilon \right)\stackrel{\sim }{\times }\stackrel{\sim }{\mathcal{P}}\left(\upsilon \right),\) \(\tilde{N }\left(\mu ,\upsilon \right)=\stackrel{\sim }{\mathfrak{G}}\left(\mu \right)\stackrel{\sim }{\times }\stackrel{\sim }{\mathcal{P}}\left(\upsilon \right) \stackrel{\sim }{+} \stackrel{\sim }{\mathfrak{G}}\left(\upsilon \right)\stackrel{\sim }{\times }\stackrel{\sim }{\mathcal{P}}\left(\mu \right),\) and \({M}_{\varphi }\left(\mu ,\upsilon \right)=\left[{M}_{*}\left(\left(\mu ,\upsilon \right), \varphi \right), {M}^{*}\left(\left(\mu ,\upsilon \right), \varphi \right)\right]\) and \({N}_{\varphi }\left(\mu ,\upsilon \right)=\left[{N}_{*}\left(\left(\mu ,\upsilon \right), \varphi \right), {N}^{*}\left(\left(\mu ,\upsilon \right), \varphi \right)\right].\)
Proof.
Since \(\stackrel{\sim }{\mathfrak{G}}, \stackrel{\sim }{\mathcal{P}}\) are \(\mathcal{H}-{\mathsf{h}}_{1}\) and \(\mathcal{H}-{\mathsf{h}}_{2}\)-convex FIVFs then, for each \(\varphi \in \left[0, 1\right],\) we have
and
From the definition of ℋ − -convex FIVFs, it follows that \(\tilde{0 }\preccurlyeq \stackrel{\sim }{\mathfrak{G}}\left(\varpi \right)\) and \(\tilde{0 }\preccurlyeq \stackrel{\sim }{\mathcal{P}}\left(\varpi \right)\), so
Analogously, we have
Adding (28) and (29), we have
Taking the result of multiplying (30) by \({\varsigma }^{\alpha -1}\) and integrating it with respect to \(\varsigma \) over (0, 1), we get
It follows that
That is
\(\Gamma \left(\alpha \right){\left(\frac{\mu \upsilon }{\upsilon -\mu }\right)}^{\alpha }\left[{\mathcal{I}}_{{\left(\frac{1}{\upsilon }\right)}^{+}}^{\alpha } {\mathfrak{G}}_{*}\left(\frac{1}{\mu }, \varphi \right)\times {\mathcal{P}}_{*}\left(\frac{1}{\mu }, \varphi \right)+{\mathcal{I}}_{{\left(\frac{1}{\mu }\right)}^{-}}^{\alpha } {\mathfrak{G}}_{*}\left(\frac{1}{\upsilon },\varphi \right)\times {\mathcal{P}}_{*}\left(\frac{1}{\upsilon },\varphi \right), {\mathcal{I}}_{{\left(\frac{1}{\upsilon }\right)}^{+}}^{\alpha } {\mathfrak{G}}^{*}\left(\frac{1}{\mu }, \varphi \right)\times {\mathcal{P}}^{*}\left(\frac{1}{\mu }, \varphi \right)+{\mathcal{I}}_{{\left(\frac{1}{\mu }\right)}^{-}}^{\alpha } {\mathfrak{G}}^{*}\left(\frac{1}{\upsilon },\varphi \right)\times {\mathcal{P}}^{*}\left(\frac{1}{\upsilon }, \varphi \right)\right]{\le }_{I}\left[{M}_{*}\left(\left(\mu ,\upsilon \right), \varphi \right), {M}^{*}\left(\left(\mu ,\upsilon \right), \varphi \right)\right]{\int }_{0}^{1}\left[\begin{array}{c}{\varsigma }^{\alpha -1}+{\left(1-\varsigma \right)}^{\alpha -1}\end{array}\right]{\mathsf{h}}_{1}\left(\varsigma \right){\mathsf{h}}_{2}\left(\varsigma \right)d\varsigma \)
Thus,
As a result, the theorem has been proven.
Theorem 3.4.
Let \(\stackrel{\sim }{\mathfrak{G}}\in HFSX\left(\left[\mu , \upsilon \right], {\mathbb{F}}_{0}, {\mathsf{h}}_{1}\right)\) and \(\stackrel{\sim }{\mathcal{P}}\in HFSX\left(\left[\mu , \upsilon \right], {\mathbb{F}}_{0}, {\mathsf{h}}_{2}\right)\) with \({\mathsf{h}}_{1}\left(\frac{1}{2}\right){\mathsf{h}}_{2}\left(\frac{1}{2}\right)\ne 0\), whose \(\varphi \)-cuts derive the following IVFs \({\mathfrak{G}}_{\varphi }, {\mathcal{P}}_{\varphi }:\left[\mu , \upsilon \right]\subset {\mathbb{R}}\to {\mathcal{K}}_{C}^{+}\) are given by \({\mathfrak{G}}_{\varphi }\left(\varpi \right)=\left[{\mathfrak{G}}_{*}\left(\varpi ,\varphi \right), {\mathfrak{G}}^{*}\left(\varpi ,\varphi \right)\right]\) and \({\mathcal{P}}_{\varphi }\left(\varpi \right)=\left[{\mathcal{P}}_{*}\left(\varpi ,\varphi \right), {\mathcal{P}}^{*}\left(\varpi ,\varphi \right)\right]\) for all \(\varpi \in \left[\mu , \upsilon \right]\), \(\varphi \in \left[0, 1\right]\). If \(\stackrel{\sim }{\mathfrak{G}}\stackrel{\sim }{\times }\stackrel{\sim }{\mathcal{P}}\in L\left(\left[\mu , \upsilon \right],{\mathbb{F}}_{0}\right)\), then.
\(\frac{1}{\alpha {\mathsf{h}}_{1}\left(\frac{1}{2}\right){\mathsf{h}}_{2}\left(\frac{1}{2}\right)} \stackrel{\sim }{\mathfrak{G}}\left(\frac{2\mu \upsilon }{\mu +\upsilon }\right)\stackrel{\sim }{\times }\stackrel{\sim }{\mathcal{P}}\left(\frac{2\mu \upsilon }{\mu +\upsilon }\right)\preccurlyeq \Gamma \left(\alpha \right){\left(\frac{\mu \upsilon }{\upsilon -\mu }\right)}^{\alpha }\left[\begin{array}{c}{\mathcal{I}}_{{\left(\frac{1}{\upsilon }\right)}^{+}}^{\alpha } \stackrel{\sim }{\mathfrak{G}}o\psi \left(\frac{1}{\mu }\right)\stackrel{\sim }{\times }\stackrel{\sim }{\mathcal{P}}o\psi \left(\frac{1}{\mu }\right)\\ \stackrel{\sim }{+}{\mathcal{I}}_{{\left(\frac{1}{\mu }\right)}^{-}}^{\alpha } \stackrel{\sim }{\mathfrak{G}}o\psi \left(\frac{1}{\upsilon }\right)\stackrel{\sim }{\times }\stackrel{\sim }{\mathcal{P}}o\psi \left(\frac{1}{\upsilon }\right)\end{array}\right]\) \(+\tilde{M }\left(\mu ,\upsilon \right){\int }_{0}^{1}\left[\begin{array}{c}{\varsigma }^{\alpha -1}\\ +{\left(1-\varsigma \right)}^{\alpha -1}\end{array}\right]{\mathsf{h}}_{1}\left(\varsigma \right){\mathsf{h}}_{2}\left(1-\varsigma \right)d\varsigma \)
where \(\tilde{M }\left(\mu ,\upsilon \right)=\stackrel{\sim }{\mathfrak{G}}\left(\mu \right)\stackrel{\sim }{\times }\stackrel{\sim }{\mathcal{P}}\left(\mu \right) \stackrel{\sim }{+} \stackrel{\sim }{\mathfrak{G}}\left(\upsilon \right)\stackrel{\sim }{\times }\stackrel{\sim }{\mathcal{P}}\left(\upsilon \right),\) \(\tilde{N }\left(\mu ,\upsilon \right)=\stackrel{\sim }{\mathfrak{G}}\left(\mu \right)\stackrel{\sim }{\times }\stackrel{\sim }{\mathcal{P}}\left(\upsilon \right) \stackrel{\sim }{+} \stackrel{\sim }{\mathfrak{G}}\left(\upsilon \right)\stackrel{\sim }{\times }\stackrel{\sim }{\mathcal{P}}\left(\mu \right),\) and \({M}_{\varphi }\left(\mu ,\upsilon \right)=\left[{M}_{*}\left(\left(\mu ,\upsilon \right), \varphi \right), {M}^{*}\left(\left(\mu ,\upsilon \right), \varphi \right)\right]\) and \({N}_{\varphi }\left(\mu ,\upsilon \right)=\left[{N}_{*}\left(\left(\mu ,\upsilon \right), \varphi \right), {N}^{*}\left(\left(\mu ,\upsilon \right), \varphi \right)\right].\)
Proof.
Consider \(\stackrel{\sim }{\mathfrak{G}},\stackrel{\sim }{\mathcal{P}} :\left[\mu , \upsilon \right]\to {\mathbb{F}}_{0}\) are \(\mathcal{H}-{\mathsf{h}}_{1}\) and \(\mathcal{H}-{\mathsf{h}}_{2}\)-convex FIVFs. Then, by hypothesis, for each \(\varphi \in \left[0, 1\right],\) we have
\(\begin{array}{c}{\mathfrak{G}}_{*}\left(\frac{2\mu \upsilon }{\mu +\upsilon },\varphi \right)\times {\mathcal{P}}_{*}\left(\frac{2\mu \upsilon }{\mu +\upsilon },\varphi \right) \\ {\mathfrak{G}}^{*}\left(\frac{2\mu \upsilon }{\mu +\upsilon },\varphi \right)\times {\mathcal{P}}^{*}\left(\frac{2\mu \upsilon }{\mu +\upsilon },\varphi \right)\end{array}\)
\(\begin{array}{c}\le {\mathsf{h}}_{1}\left(\frac{1}{2}\right){\mathsf{h}}_{2}\left(\frac{1}{2}\right)\left[\begin{array}{c}{\mathfrak{G}}_{*}\left(\frac{\mu \upsilon }{\varsigma \mu +\left(1-\varsigma \right)\upsilon }, \varphi \right)\times {\mathcal{P}}_{*}\left(\frac{\mu \upsilon }{\varsigma \mu +\left(1-\varsigma \right)\upsilon }, \varphi \right)\\ +{\mathfrak{G}}_{*}\left(\frac{\mu \upsilon }{\varsigma \mu +\left(1-\varsigma \right)\upsilon }, \varphi \right)\times {\mathcal{P}}_{*}\left(\frac{\mu \upsilon }{\left(1-\varsigma \right)\mu +\varsigma \upsilon }, \varphi \right)\end{array}\right] \\ + {\mathsf{h}}_{1}\left(\frac{1}{2}\right){\mathsf{h}}_{2}\left(\frac{1}{2}\right)\left[\begin{array}{c}{\mathfrak{G}}_{*}\left(\frac{\mu \upsilon }{\left(1-\varsigma \right)\mu +\varsigma \upsilon }, \varphi \right){\times \mathcal{P}}_{*}\left(\frac{\mu \upsilon }{\varsigma \mu +\left(1-\varsigma \right)\upsilon }, \varphi \right)\\ +{\mathfrak{G}}_{*}\left(\frac{\mu \upsilon }{\left(1-\varsigma \right)\mu +\varsigma \upsilon }, \varphi \right){\times \mathcal{P}}_{*}\left(\frac{\mu \upsilon }{\left(1-\varsigma \right)\mu +\varsigma \upsilon }, \varphi \right)\end{array}\right] \\ \le {\mathsf{h}}_{1}\left(\frac{1}{2}\right){\mathsf{h}}_{2}\left(\frac{1}{2}\right)\left[\begin{array}{c}{\mathfrak{G}}^{*}\left(\frac{\mu \upsilon }{\varsigma \mu +\left(1-\varsigma \right)\upsilon }, \varphi \right){\times \mathcal{P}}^{*}\left(\frac{\mu \upsilon }{\varsigma \mu +\left(1-\varsigma \right)\upsilon }, \varphi \right)\\ +{\mathfrak{G}}^{*}\left(\frac{\mu \upsilon }{\varsigma \mu +\left(1-\varsigma \right)\upsilon }, \varphi \right){\times \mathcal{P}}^{*}\left(\frac{\mu \upsilon }{\left(1-\varsigma \right)\mu +\varsigma \upsilon }, \varphi \right)\end{array}\right] \\ +{\mathsf{h}}_{1}\left(\frac{1}{2}\right){\mathsf{h}}_{2}\left(\frac{1}{2}\right)\left[\begin{array}{c}{\mathfrak{G}}^{*}\left(\frac{\mu \upsilon }{\left(1-\varsigma \right)\mu +\varsigma \upsilon }, \varphi \right){\times \mathcal{P}}^{*}\left(\frac{\mu \upsilon }{\varsigma \mu +\left(1-\varsigma \right)\upsilon }, \varphi \right)\\ +{\mathfrak{G}}^{*}\left(\frac{\mu \upsilon }{\left(1-\varsigma \right)\mu +\varsigma \upsilon }, \varphi \right){\times \mathcal{P}}^{*}\left(\frac{\mu \upsilon }{\left(1-\varsigma \right)\mu +\varsigma \upsilon }, \varphi \right)\end{array}\right],\end{array}\)
Inequality (31) is multiplied by \({\varsigma }^{\alpha -1}\) and integrated over \((0, 1),\)
Taking \(\varpi =\frac{\mu \upsilon }{\varsigma \mu +\left(1-\varsigma \right)\upsilon }\) and \(\mathcal{Z}=\frac{\mu \upsilon }{\left(1-\varsigma \right)\mu +\varsigma \upsilon }\), then we get
that is
\(\frac{1}{\alpha {\mathsf{h}}_{1}\left(\frac{1}{2}\right){\mathsf{h}}_{2}\left(\frac{1}{2}\right)} \stackrel{\sim }{\mathfrak{G}}\left(\frac{2\mu \upsilon }{\mu +\upsilon }\right)\stackrel{\sim }{\times }\stackrel{\sim }{\mathcal{P}}\left(\frac{2\mu \upsilon }{\mu +\upsilon }\right)\)
\(\preccurlyeq \Gamma \left(\alpha \right){\left(\frac{\mu \upsilon }{\upsilon -\mu }\right)}^{\alpha }\left[\begin{array}{c}{\mathcal{I}}_{{\left(\frac{1}{\upsilon }\right)}^{+}}^{\alpha } \stackrel{\sim }{\mathfrak{G}}o\psi \left(\frac{1}{\mu }\right)\stackrel{\sim }{\times }\stackrel{\sim }{\mathcal{P}}o\psi \left(\frac{1}{\mu }\right)\\ \stackrel{\sim }{+}{\mathcal{I}}_{{\left(\frac{1}{\mu }\right)}^{-}}^{\alpha } \stackrel{\sim }{\mathfrak{G}}o\psi \left(\frac{1}{\upsilon }\right)\stackrel{\sim }{\times }\stackrel{\sim }{\mathcal{P}}o\psi \left(\frac{1}{\upsilon }\right)\end{array}\right]+\tilde{M }\left(\mu ,\upsilon \right){\int }_{0}^{1}\left[\begin{array}{c}{\varsigma }^{\alpha -1}\\ +{\left(1-\varsigma \right)}^{\alpha -1}\end{array}\right]{\mathsf{h}}_{1}\left(\varsigma \right){\mathsf{h}}_{2}\left(1-\varsigma \right)d\varsigma \)
As a result, the desired outcome has been achieved.
For ℋ − -convex FIVFs, we now have H⋅H Fejér inequalities. For ℋ − -convex FIVF, we first get the second H⋅H Fejér inequality.
Theorem 3.5.
Let \(\stackrel{\sim }{\mathfrak{G}}\in HFSX\left(\left[\mu , \upsilon \right], {\mathbb{F}}_{0}, \mathsf{h}\right)\), whose \(\varphi \)-cuts derive the following IVFs \({\mathfrak{G}}_{\varphi }:\left[\mu , \upsilon \right]\subset {\mathbb{R}}\to {\mathcal{K}}_{C}^{+}\) are given by \({\mathfrak{G}}_{\varphi }\left(\varpi \right)=\left[{\mathfrak{G}}_{*}\left(\varpi ,\varphi \right), {\mathfrak{G}}^{*}\left(\varpi ,\varphi \right)\right]\) for all \(\varpi \in \left[\mu , \upsilon \right]\), \(\varphi \in \left[0, 1\right]\). If \(\stackrel{\sim }{\mathfrak{G}}\in L\left(\left[\mu , \upsilon \right],{\mathbb{F}}_{0}\right)\) and \(\Omega :\left[\mu , \upsilon \right]\to {\mathbb{R}}, \Omega \left(\frac{1}{\frac{1}{\mu }+\frac{1}{\upsilon }-\frac{1}{\varpi }}\right)=\Omega (\varpi )\ge 0,\) then
If \(\stackrel{\sim }{\mathfrak{G}}\in HFSV\left(\left[\mu , \upsilon \right], {\mathbb{F}}_{0}, \mathsf{h}\right)\), then inequality (32) is reversed such that
Proof.
Let \(\stackrel{\sim }{\mathfrak{G}}\in HFSX\left(\left[\mu , \upsilon \right], {\mathbb{F}}_{0}, \mathsf{h}\right)\) and \({\varsigma }^{\alpha -1}\Omega \left(\frac{\mu \upsilon }{\varsigma \mu +\left(1-\varsigma \right)\upsilon }\right)\ge 0\). Then, for each \(\varphi \in \left[0, 1\right],\) we have
and
After adding (33) and (34), and integrating over \(\left[0, 1\right],\) we get
From which, we have
that is
As a result, the desired result has been achieved.
Following result obtains the first FI fractional \(H\cdot H\) Fejér inequality.
Theorem 3.6.
Let \(\stackrel{\sim }{\mathfrak{G}}\in HFSX\left(\left[\mu , \upsilon \right], {\mathbb{F}}_{0}, \mathsf{h}\right)\), whose \(\varphi \)-cuts derive the following IVFs \({\mathfrak{G}}_{\varphi }:\left[\mu , \upsilon \right]\subset {\mathbb{R}}\to {\mathcal{K}}_{C}^{+}\) are given by \({\mathfrak{G}}_{\varphi }\left(\varpi \right)=\left[{\mathfrak{G}}_{*}\left(\varpi ,\varphi \right), {\mathfrak{G}}^{*}\left(\varpi ,\varphi \right)\right]\) for all \(\varpi \in \left[\mu , \upsilon \right]\), \(\varphi \in \left[0, 1\right]\). If \(\stackrel{\sim }{\mathfrak{G}}\in L\left(\left[\mu , \upsilon \right],{\mathbb{F}}_{0}\right)\) and \(\Omega :\left[\mu , \upsilon \right]\to {\mathbb{R}}, \Omega \left(\frac{1}{\frac{1}{\mu }+\frac{1}{\upsilon }-\frac{1}{\varpi }}\right)=\Omega (\varpi )\ge 0,\) then.
If \(\stackrel{\sim }{\mathfrak{G}}\in HFSV\left(\left[\mu , \upsilon \right], {\mathbb{F}}_{0}, \mathsf{h}\right)\), then inequality (36) is reversed such that
Proof.
Since \(\stackrel{\sim }{\mathfrak{G}}\in HFSX\left(\left[\mu , \upsilon \right], {\mathbb{F}}_{0}, \mathsf{h}\right)\), then for \(\varphi \in \left[0, 1\right],\) we have.
Multiplying both sides by (37) by \({\varsigma }^{\alpha -1}\Omega \left(\frac{\mu \upsilon }{\left(1-\varsigma \right)\mu +\varsigma \upsilon }\right)\) and then integrating the resultant with respect to \(\varsigma \) over \(\left[0, 1\right],\) we obtain
Let \(\varpi =\frac{\mu \upsilon }{\varsigma \mu +\left(1-\varsigma \right)\upsilon }\). Then, we have
From (39), we have
that is
The theorem has been proved.
Remark 3.7.
From Theorem 3.5 and Theorem 3.6, following result can be obtained:
On fixing \(\mathsf{h}\left(\varsigma \right)=\varsigma \) and \(\alpha =1\) then following H⋅H inequality is obtained, see [33]:
\(\stackrel{\sim }{\mathfrak{G}}\left(\frac{2\mu \upsilon }{\mu +\upsilon }\right){\int }_{\mu }^{\upsilon }\frac{\Omega \left(\varpi \right)}{{\varpi }^{2}}d\varpi \preccurlyeq {\int }_{\mu }^{\upsilon }\frac{\stackrel{\sim }{\mathfrak{G}}\left(\varpi \right)}{{\varpi }^{2}}\Omega \left(\varpi \right)d\varpi \)
On fixing \(\mathsf{h}\left(\varsigma \right)=\varsigma \) and \(\Omega \left(\varpi \right)=1\), the inequality (21) is obtained.
On fixing \(\mathsf{h}\left(\varsigma \right)=\varsigma \), \(\Omega \left(\varpi \right)=1\) and \(\alpha =1\), the following H⋅H inequality is obtained:
4 Conclusion and Future Study
This study introduced the ℋ − -convex FIVFs, a new family of harmonically convex functions. We discovered a link between Riemann–Liouville fractional integral inequalities with FIs and ℋ − -convex FIVFs. Furthermore, as applications of ℋ − -convex FIVFs and Riemann–Liouville fractional integral inequalities, we derived certain previously defined and novel specific instances. In the future, we will use generalized interval and FI Riemann–Liouville fractional operators to investigate this concept for generalized ℋ − -convex interval-valued and FIVFs.
Availability of data and materials
Not applicable.
Abbreviations
- Fuzzy-IVFs:
-
Fuzzy-interval-valued functions
- IVFs:
-
Interval-valued functions
- \(\mathcal{H}-\mathsf{h}\)-convex FIVFs:
-
Harmonically \(\mathsf{h}\)-convex FIVFs
- \(H\cdot H\)-inequality:
-
Hermite–Hadamard inequality
- \(H\cdot H\) Fejér inequality:
-
Hermite–Hadamard–Fejér inequality
- \(\left(FR\right)\)-integrable:
-
Fuzzy Riemann integrable
- FI Riemann–Liouville Fractional:
-
Fuzzy interval Riemann–Liouville Fractional
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Acknowledgements
This work was funded by Taif University Researchers Supporting Project number (TURSP-2020/345), Taif University, Taif, Saudi Arabia. The research of Santos-García was funded by the Spanish MINECO project TRACES TIN2015–67522– C3–3–R.
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Khan, M.B., Zaini, H.G., Santos-García, G. et al. Riemann–Liouville Fractional Integral Inequalities for Generalized Harmonically Convex Fuzzy-Interval-Valued Functions. Int J Comput Intell Syst 15, 28 (2022). https://doi.org/10.1007/s44196-022-00081-w
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DOI: https://doi.org/10.1007/s44196-022-00081-w