Abstract
In this study, we use the fuzzy order relation to show some novel variants of Hermite–Hadamard inequalities for pre-invex fuzzy-interval-valued mappings (F-I∙V-Ms), which we term fuzzy-interval Hermite–Hadamard inequalities and fuzzy-interval Hermite–Hadamard–Fejér inequalities. This fuzzy order relation is defined as the level of the fuzzy-interval space by the Kulisch–Miranker order relation. There are also some new exceptional instances mentioned. The theory proposed in this research is shown with practical examples that demonstrate its usefulness. This paper's approaches and methodologies might serve as a springboard for future study in this field.
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1 Introduction
Convexity is a well-known notion in optimization ideas, and it plays an important role in operations research, economics, decision making, and management sciences. Many convex mapping extensions and refinements have recently been discovered. See [1,2,3,4,5,6,7,8,9], and the references therein for further information. In classical approach, a real-valued mapping \(\mathfrak{U}:K\to {\mathbb{R}}\) is called convex if
for all \(\varkappa , \mathsf{z}\in K, \mathsf{r}\in \left[0, 1\right].\)
The idea of convexity in the context of integral problems is a fascinating field of study. Integral inequalities are a good tool for establishing convexity and nonconvexity's qualitative and quantitative features. Because of the numerous uses of these disparities in various industries, there has been a steady increase in interest in this field of study. As a result, numerous inequalities have been suggested as applications of convex mappings and extended convex mappings, as shown in [7, 10,11,12,13], and the references therein. The Hermite–Hadamard inequality (abbreviated as HH-inequality) is a well-known integral inequality in the literature (see [14, 15]):
where \(\mathfrak{U}:K\to {\mathbb{R}}\) is a convex mapping on the interval \(K=\left[\varsigma , \rho \right]\) with \(\varsigma <\rho .\) In 2007, Noor [16] derived the following HH-inequality for pre-invex mapping:
where \(\mathfrak{U}:K\to {\mathbb{R}}\) is a pre-invex mapping on the invex set \(K=\left[\varsigma , \varsigma +\varpi \left(\rho , \varsigma \right)\right]\) with \(\varsigma <\varsigma +\varpi \left(\rho , \varsigma \right).\)
In addition, Moore [17] and Kulish and W. Miranker [18] introduced and examined the notion of interval analysis. It is a discipline in which a real-number interval is used to represent an uncertain variable. In 2018, Zhao et al. [19] developed h-convex interval-valued mappings (I∙V-Ms) and established that the HH-inequality for convex I∙V-Ms is a special case of the HH-inequality for convex I∙V-Ms.
Theorem I.1.
Let \(\mathfrak{U}:\left[\varsigma , \rho \right]\subset {\mathbb{R}}\to {{\mathcal{K}}_{C}}^{+}\) be a convex I∙V-M given by \(\mathfrak{U}\left(\varkappa \right)=\left[{\mathfrak{U}}_{*}\left(\varkappa \right), {\mathfrak{U}}^{*}\left(\varkappa \right)\right]\) for all \(\varkappa \in \left[\varsigma , \rho \right]\), where \({\mathfrak{U}}_{*}\left(\varkappa \right)\) is a convex mapping and \({\mathfrak{U}}^{*}\left(\varkappa \right)\) is a concave mapping. If \(\mathfrak{U}\) is Riemann integrable, then:
We suggest readers to [10, 20,21,22,23,24,25,26,27,28,29] and the references therein for more examination of literature on the applications and properties of extended convex mappings and HH-integral inequalities.
As a novel non-probabilistic approach, interval analysis is a special instance of fuzzy-interval-valued analysis. There is no doubt that fuzzy-interval analysis is extremely important in both pure and practical research. One of the initial aims of the fuzzy-interval analysis process was to analyze the error estimations of finite state machines' numerical solutions. However, the fuzzy-interval analysis technique, which has been used in mathematical models in engineering for over 50 years as one of the ways to solve interval uncertain structural systems, is a critical cornerstone. It is worth noting that applications in automatic error analysis, operation research, computer science, management sciences, artificial intelligence, control engineering, and decision sciences, see [30]. Furthermore, [31,32,33,34,35,36,37,38,39,40,41,42,43] has a number of applications in optimization theory relating to fuzzy interval-valued mappings. We refer interested readers to [43, 44] and the bibliographies cited in them for recent developments in the field of interval-valued mappings.
Moreover, Jensen's integral inequality for F-I∙V-M was derived by Oseuna-Gomez et al. [13] and Costa et al. [45]. Costa and Floures used the same method to show Minkowski and Beckenbach's inequalities, with F-I∙V-Ms as integrands.
Moreover, we make generalizations of integral inequality (1.2) by constructing fuzzy-interval integral inequality for convex F-I∙V-M, where the integrands are convex F-I∙V-M, using established relation between elements of fuzzy-interval space and interval space, in other words, fuzzy order relation on fuzzy-interval space is distinguished by level-wise through Kulisch–Miranker order relation defined on interval space, as motivated by [13, 45, 46], and [19]. See [29, 44, 47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62] for a more comprehensive discussion of the literature on fuzzy inter inequalities.
This article is organized as follows: in the second section, we review and discuss the basic concepts of interval and fuzzy intervals, as well as a class of modified convex F-I∙V-Ms known as pre-invex F-I∙V-Ms. In the third section, we obtain fuzzy interval HH-inequalities and verify these inequalities with the help of examples by employing this class. Furthermore, pre-invex F-I∙V-Ms introduce certain HH-Fejér inequalities. The final portion of this study concludes with findings and future plans.
2 Preliminaries
We will begin by introducing interval analysis theory, which will be used throughout this article.
Let \({\mathcal{K}}_{C}\) be the collection of all closed and bounded intervals of \({\mathbb{R}}\) that is \({\mathcal{K}}_{C}=\left\{\left[{\mathfrak{C}}_{*}, {\mathfrak{C}}^{*}\right]:{\mathfrak{C}}_{*}, {\mathfrak{C}}^{*}\in {\mathbb{R}}\mathrm{and} {\mathfrak{C}}_{*}\le {\mathfrak{C}}^{*}\right\}.\) If \({\mathfrak{C}}_{*}\ge 0\), then \(\left[{\mathfrak{C}}_{*}, {\mathfrak{C}}^{*}\right]\) is called positive interval. The set of all positive interval is denoted by \({{\mathcal{K}}_{C}}^{+}\) and defined as \({{\mathcal{K}}_{C}}^{+}=\left\{\left[{\mathfrak{C}}_{*}, {\mathfrak{C}}^{*}\right]:\left[{\mathfrak{C}}_{*}, {\mathfrak{C}}^{*}\right]\in {\mathcal{K}}_{C} \mathrm{and}{\mathfrak{C}}_{*}\ge 0\right\}.\)
We now discuss some properties of intervals under the arithmetic operations addition, multiplication and scalar multiplication. If \(\left[{\mathfrak{G}}_{*}, {\mathfrak{G}}^{*}\right], \left[{\mathfrak{C}}_{*}, {\mathfrak{C}}^{*}\right]\in {\mathcal{K}}_{C}\) and \(\rho \in {\mathbb{R}}\), then arithmetic operations are defined by
For \(\left[{\mathfrak{G}}_{*}, {\mathfrak{G}}^{*}\right], \left[{\mathfrak{C}}_{*}, {\mathfrak{C}}^{*}\right]\in {\mathcal{K}}_{C},\) the inclusion \("\subseteq "\) is defined by:
\(\left[{\mathfrak{G}}_{*}, {\mathfrak{G}}^{*}\right]\subseteq \left[{\mathfrak{C}}_{*}, {\mathfrak{C}}^{*}\right]\), if and only if \({\mathfrak{C}}_{*}\le {\mathfrak{G}}_{*}\), \({\mathfrak{G}}^{*}\le {\mathfrak{C}}^{*}.\)
Remark 1
The relation \({"\le }_{I}"\) defined on \({\mathcal{K}}_{C}\) by.
\(\left[{\mathfrak{G}}_{*}, {\mathfrak{G}}^{*}\right]{\le }_{I}\left[{\mathfrak{C}}_{*}, {\mathfrak{C}}^{*}\right]\) if and only if \({\mathfrak{G}}_{*}{\le \mathfrak{C}}_{*}, {\mathfrak{G}}^{*}{\le \mathfrak{C}}^{*},\)
for all \(\left[{\mathfrak{G}}_{*}, {\mathfrak{G}}^{*}\right], \left[{\mathfrak{C}}_{*}, {\mathfrak{C}}^{*}\right]\in {\mathcal{K}}_{C},\) it is an order relation, see [18]. For given \(\left[{\mathfrak{G}}_{*}, {\mathfrak{G}}^{*}\right], \left[{\mathfrak{C}}_{*}, {\mathfrak{C}}^{*}\right]\in {\mathcal{K}}_{C},\) we say that \(\left[{\mathfrak{G}}_{*}, {\mathfrak{G}}^{*}\right]{\le }_{I}\left[{\mathfrak{C}}_{*}, {\mathfrak{C}}^{*}\right]\) if and only if \({\mathfrak{G}}_{*}{\le \mathfrak{C}}_{*}, {\mathfrak{G}}^{*}{\le \mathfrak{C}}^{*}\) or \({\mathfrak{G}}_{*}{\le \mathfrak{C}}_{*}, {\mathfrak{G}}^{*}{<\mathfrak{C}}^{*}\).
The concept of Riemann integral for I∙V-M first introduced by Moore [17] is defined as follows:
Theorem 2
[17] If \(\mathfrak{G}:[\varsigma ,\rho ]\subset {\mathbb{R}}\to {\mathcal{K}}_{C}\) is an I∙V-M on such that \(\left[{\mathfrak{U}}_{*}, {\mathfrak{U}}^{*}\right].\) Then \(\mathfrak{U}\) is Riemann integrable over \(:[\varsigma ,\rho ]\) if and only if, \({\mathfrak{U}}_{*}\) and \({\mathfrak{U}}^{*}\) both are Riemann integrable over \(\left[\varsigma ,\rho \right]\) such that:
The collection of all Riemann integrable real-valued mappings and Riemann integrable I∙V-Ms is denoted by \({\mathcal{R}}_{[\varsigma , \rho ]}\) and \({\mathcal{I}\mathcal{R}}_{[\varsigma , \rho ]},\) respectively.
Let \({\mathbb{E}}\) represent the collection of all real fuzzy intervals. let \(\mathfrak{C}\in \boldsymbol{ }{\mathbb{E}}\) be real fuzzy interval, if and only if, \(\Upsilon\)-levels \({\left[\mathfrak{C}\right]}^{\Upsilon}\) is a nonempty compact convex set of \({\mathbb{R}}\). This is represented by
from these definitions, we have
where
Proposition 1
[46] Let \(\mathfrak{C},\mathfrak{G}\in {\mathbb{E}}\). Then relation \("\preccurlyeq "\) given on \({\mathbb{E}}\) by:
\(\mathfrak{C}\preccurlyeq \mathfrak{G}\) if and only if, \({{\left[\mathfrak{C}\right]}^{\Upsilon}\le }_{I}{\left[\mathfrak{G}\right]}^{\Upsilon}\) for all \(\Upsilon\in [0, 1],\)
it is partial order relation.
We now discuss some properties of real fuzzy intervals under addition, scalar multiplication, multiplication and division. If \(\mathfrak{C},\mathfrak{G}\in {\mathbb{E}}\) and \(\rho \in {\mathbb{R}}\), then arithmetic operations are defined by
Definition 1
[46] A fuzzy-interval-valued map \(\mathfrak{U}:K\subset {\mathbb{R}}\to {\mathbb{E}}\) is called F-I∙V-M. For each \(\Upsilon\in \left[0, 1\right],\) whose \(\Upsilon\)-levels define the family of I∙V-Ms \({\mathfrak{U}}_{\Upsilon}:K\subset {\mathbb{R}}\to {\mathcal{K}}_{C}\) are given by \({\mathfrak{U}}_{\Upsilon}\left(\varkappa \right)=\left[{\mathfrak{U}}_{*}\left(\varkappa ,\Upsilon\right), {\mathfrak{U}}^{*}\left(\varkappa ,\Upsilon\right)\right]\) for all \(\varkappa \in K.\) Here, for each \(\Upsilon\in \left[0, 1\right],\) the left and right mappings \({\mathfrak{U}}_{*}\left(.,\Upsilon\right), {\mathfrak{U}}^{*}\left(.,\Upsilon\right):K\to {\mathbb{R}}\) are called lower and upper mappings of \(\mathfrak{U}\).
Remark 3
Let \(\mathfrak{U}:K\subset {\mathbb{R}}\to {\mathbb{E}}\) be a F-I∙V-M. Then, \(\mathfrak{U}\left(\varkappa \right)\) is said to be continuous at \(\varkappa \in K,\) if for each \(\Upsilon\in \left[0, 1\right],\) left and right mappings \({\mathfrak{U}}_{*}\left(\varkappa ,\Upsilon\right)\) and \({\mathfrak{U}}^{*}\left(\varkappa ,\Upsilon\right)\) are continuous at \(\varkappa \in K\), respectively.
From the above literature review, following results can be concluded, see [14, 17, 25, 46]:
Definition 2
Let \(\mathfrak{U}:[\varsigma , \rho ]\subset {\mathbb{R}}\to {\mathbb{E}}\) is called F-I∙V-M. The fuzzy Riemann integral of \(\mathfrak{U}\) over \(\left[\varsigma , \rho \right],\) denoted by \(\left(FR\right){\int }_{\varsigma }^{\rho }\mathfrak{U}\left(\varkappa \right)d\varkappa \), it is defined level-wise by:
for all \(\Upsilon\in \left[0, 1\right],\) where \({\mathcal{R}}_{[\varsigma , \rho ]}\) is the collection of left and right mappings of I∙V-Ms. \(\mathfrak{U}\) is \(\left(FR\right)\)-integrable over \([\varsigma , \rho ]\) if \(\left(FR\right){\int }_{\varsigma }^{\rho }\mathfrak{U}\left(\varkappa \right)d\varkappa \in {\mathbb{E}}.\) Note that, if both left and right mappings are Lebesgue-integrable, then \(\mathfrak{U}\) is fuzzy Aumann-integrable, see [14, 17, 25].
Theorem 5
Let \(\mathfrak{U}:[\varsigma , \rho ]\subset {\mathbb{R}}\to {\mathbb{E}}\) be a F-I∙V-M, whose \(\Upsilon\)-levels define the family of I∙V-Ms \({\mathfrak{U}}_{\Upsilon}:[\varsigma , \rho ]\subset {\mathbb{R}}\to {\mathcal{K}}_{C}\) are given by \({\mathfrak{U}}_{\Upsilon}\left(\varkappa \right)=\left[{\mathfrak{U}}_{*}\left(\varkappa ,\Upsilon\right), {\mathfrak{U}}^{*}\left(\varkappa ,\Upsilon\right)\right]\) for all \(\varkappa \in [\varsigma , \rho ]\) and for all \(\Upsilon\in \left[0, 1\right].\) Then \(\mathfrak{U}\) is \(\left(FR\right)\)-integrable over \([\varsigma , \rho ]\) if and only if, \({\mathfrak{U}}_{*}\left(\varkappa ,\Upsilon\right)\) and \({\mathfrak{U}}^{*}\left(\varkappa ,\Upsilon\right)\) both are \(R\)-integrable over \([\varsigma , \rho ]\). Moreover, if \(\mathfrak{U}\) is \(\left(FR\right)\)-integrable over \(\left[\varsigma , \rho \right],\) then:
for all \(\Upsilon\in \left[0, 1\right].\)
The family of all \(\left(FR\right)\)-integrable F-I∙V-Ms and \(R\)-integrable mappings over \([\varsigma , \rho ]\) are denoted by \({\mathcal{F}\mathcal{R}}_{\left(\left[\varsigma , \rho \right], \Upsilon\right)}\) and \({\mathcal{R}}_{\left(\left[\varsigma , \rho \right], \Upsilon\right)},\) for all \(\Upsilon\in \left[0, 1\right].\)
Definition 3
[8] Let \(K\) be an invex set. Then F-I∙V-M \(\mathfrak{U}:K\to {\mathbb{E}}\) is said to be:
-
convex on \(K\) if
for all \(\varkappa , \mathsf{z}\in K, \mathsf{r}\in \left[0, 1\right],\) where \(\mathfrak{U}\left(\varkappa \right)\succcurlyeq \widetilde{0},\) \(\varpi :K\times K\to {\mathbb{R}}.\)
-
concave on \(K\) if inequality (11) is reversed.
-
affine on \(K\) if
for all \(\varkappa , \mathsf{z}\in K, \mathsf{r}\in \left[0, 1\right],\) where \(\mathfrak{U}\left(\varkappa \right)\succcurlyeq \widetilde{0}, \varpi :K\times K\to {\mathbb{R}}.\)
Definition 4
[38] Let \(K\) be an invex set. Then F-I∙V-M \(\mathfrak{U}:K\to {\mathbb{E}}\) is said to be:
-
pre-invex on \(K\) with respect to \(\varpi \) if
for all \(\varkappa , \mathsf{z}\in K, \mathsf{r}\in \left[0, 1\right],\) where \(\mathfrak{U}\left(\varkappa \right)\succcurlyeq \widetilde{0},\) \(\varpi :K\times K\to {\mathbb{R}}.\)
-
preconcave on \(K\) with respect to \(\varpi \) if inequality (13) is reversed.
-
affine on \(K\) with respect to \(\varpi \) if
for all \(\varkappa , \mathsf{z}\in K, \mathsf{r}\in \left[0, 1\right],\) where \(\mathfrak{U}\left(\varkappa \right)\succcurlyeq \widetilde{0}, \varpi :K\times K\to {\mathbb{R}}.\)
Remark 4
The pre-invex F-I∙V-Ms have some very nice properties similar to convex F-I∙V-M,
If \(\mathfrak{U}\) is pre-invex F-I∙V-M, then \(\Upsilon\mathfrak{U}\) is also pre-invex for \(\Upsilon\ge 0\).
If \(\mathfrak{U}\) and \(\mathcal{T}\) both are pre-invex F-I∙V-Ms, then \(\mathrm{max}\left(\mathfrak{U}(\varkappa ),\mathcal{T}(\varkappa )\right)\) is also pre-invex F-I∙V-M.
In the case of \(\varpi \left(\mathsf{z},\varkappa \right)=\mathsf{z}-\varkappa ,\) we obtain the definition of convex F-I∙V-M, see [37].
Theorem 6
Suppose that \(K\) be an invex set and \(\mathfrak{U}:K\to {\mathbb{E}}\) is a F-I∙V-Ms along with family of I∙V-Ms \({\mathfrak{U}}_{\Upsilon}:K\subset {\mathbb{R}}\to {{\mathcal{K}}_{C}}^{+}\subset {\mathcal{K}}_{C}\) as well as:
Example 1
We consider the F-I∙V-M \(\mathfrak{U}:\left[0, 1\right]\to {\mathbb{E}}\) defined by,
Then, for each \(\Upsilon\in \left[0, 1\right],\) we have \({\mathfrak{U}}_{\Upsilon}\left(\varkappa \right)=\left[2\Upsilon{\varkappa }^{2},(4-2\Upsilon){\varkappa }^{2}\right]\). Since left and right mappings \({\mathfrak{U}}_{*}\left(\varkappa ,\Upsilon\right),\) \({\mathfrak{U}}^{*}\left(\varkappa ,\Upsilon\right)\) are pre-invex mappings along with \(\varpi \left(\mathsf{z},\varkappa \right)=\mathsf{z}-\varkappa ,\) for each \(\Upsilon\in [0, 1]\). Hence, \(\mathfrak{U}\left(\varkappa \right)\) is pre-invex F-I∙V-M.
3 Main Results
Since \(\varpi :K\times K\to {\mathbb{R}}\) is a bi-mapping, then we requiring following condition to prove the upcoming results:
Condition C. Let \(K\) be an invex set with respect to \(\varpi .\) For any \(\varsigma , \rho \in K\) and \(\mathsf{r}\in \left[0, 1\right]\),
From Condition C, it can be easily seen that when \(\mathsf{r}\) = 0, then \(\varpi \left(\rho ,\varsigma \right)\) = 0 if and only if,\(\rho =\varsigma \), for all \(\varsigma , \rho \in K\). For more useful details and the applications of Condition C, see [36, 38,39,40,41,42].
Theorem 7
(The fuzzy-interval HH-type inequality for pre-invex F-I∙V-M). Suppose that \(\mathfrak{U}:\left[\varsigma , \varsigma +\varpi (\rho , \varsigma )\right]\to {\mathbb{E}}\) is a pre-invex F-I∙V-M along with family of I∙V-Ms \({\mathfrak{U}}_{\Upsilon}:\left[\varsigma , \varsigma +\varpi (\rho , \varsigma )\right]\subset {\mathbb{R}}\to {{\mathcal{K}}_{C}}^{+}\) as well as \({\mathfrak{U}}_{\Upsilon}\left(\varkappa \right)=\left[{\mathfrak{U}}_{*}\left(\varkappa ,\Upsilon\right), {\mathfrak{U}}^{*}\left(\varkappa ,\Upsilon\right)\right]\) for all \(\varkappa \in \left[\varsigma , \varsigma +\varpi (\rho , \varsigma )\right]\) and for all \(\Upsilon\in \left[0, 1\right]\). If \(\varpi \) satisfies the Condition C and \(\mathfrak{U}\in {\mathcal{F}\mathcal{R}}_{\left(\left[\varsigma , \varsigma +\varpi (\rho , \varsigma )\right], \Upsilon\right)},\) then:
Proof
Let \(\mathfrak{U}:\left[\varsigma , \varsigma +\varpi (\rho , \varsigma )\right]\to {\mathbb{E}}\) be a pre-invex F-I∙V-M. Then, by hypothesis, we have
Therefore, for every \(\Upsilon\in [0, 1]\), we have
Then
It follows that
That is
Thus,
In a similar way as above, we have
Combining (17) and (18), we have
This completes the proof.
Remark 5.
If \(\xi \left(\rho ,\varsigma \right)=\rho - \varsigma \), then Theorem 7 reduces to the result for convex F-I∙V-M:
If \({\mathfrak{U}}_{*}\left(\varsigma , \Upsilon\right)={\mathfrak{U}}^{*}\left(\rho , \Upsilon\right)\) with \(\Upsilon=1\) then Theorem 7 reduces to the result for pre-invex mapping, see [16]:
If \({\mathfrak{U}}_{*}\left(\varsigma , \Upsilon\right)={\mathfrak{U}}^{*}\left(\rho , \Upsilon\right)\) with \(\xi \left(\rho ,\varsigma \right)=\rho - \varsigma \) and \(\Upsilon=1\) then Theorem 7 reduces to the result for convex mapping, see [12, 15]:
Example 2
We consider the F-I∙V-M \(\mathfrak{U}:\left[\varsigma , \varsigma +\varpi (\rho , \varsigma )\right]=[0, \varpi (2, 0)]\to {\mathbb{E}}\) defined by,
Then, for each \(\Upsilon\in \left[0, 1\right],\) we have \({\mathfrak{U}}_{\Upsilon}\left(\varkappa \right)=\left[2\Upsilon{\varkappa }^{2},(4-2\Upsilon){\varkappa }^{2}\right]\). Since left and right mappings \({\mathfrak{U}}_{*}\left(\varkappa ,\Upsilon\right)=2\Upsilon{\varkappa }^{2},\) \({\mathfrak{U}}^{*}\left(\varkappa , \Upsilon\right)=(4-2\Upsilon){\varkappa }^{2}\) are pre-invex mappings along with \(\varpi \left(\rho , \varsigma \right)=\rho -\varsigma \) for each \(\Upsilon\in [0, 1]\). Hence \(\mathfrak{U}\left(\varkappa \right)\) is pre-invex F-I∙V-M with respect to \(\varpi \left(\rho , \varsigma \right)=\rho -\varsigma \). We now compute the following:
for all \(\Upsilon\in \left[0, 1\right].\) That means
Similarly, it can be easily shown that
for all \(\Upsilon\in \left[0, 1\right],\) such that
From which, it follows that
that is.
\(\left[2\Upsilon, \left(4-2\Upsilon\right)\right]{\begin{array}{c}\begin{array}{c}\le \end{array}\end{array}}_{I}\left[\frac{8\Upsilon}{3}, \frac{4\left(4-2\Upsilon\right)}{3}\right]{\begin{array}{c}\begin{array}{c}\le \end{array}\end{array}}_{I}\left[4\Upsilon, 2\left(4-2\Upsilon\right)\right],\) for all \(\Upsilon\in \left[0, 1\right],\)
hence,
Further, we also offer the fuzzy integral relations of the product of two pre-invex F-I∙V-Ms.
Theorem 8
Suppose that \(\mathfrak{U},\mathfrak{N} :\left[\varsigma , \varsigma +\varpi (\rho , \varsigma )\right]\to {\mathbb{E}}\) are two pre-invex F-I∙V-Ms along with family of I∙V-Ms \({\mathfrak{U}}_{\Upsilon}, {\mathfrak{N}}_{\Upsilon}:\left[\varsigma , \varsigma +\varpi (\rho , \varsigma )\right]\subset {\mathbb{R}}\to {{\mathcal{K}}_{C}}^{+}\) as well as \({\mathfrak{U}}_{\Upsilon}\left(\varkappa \right)=\left[{\mathfrak{U}}_{*}\left(\varkappa ,\Upsilon\right), {\mathfrak{U}}^{*}\left(\varkappa ,\Upsilon\right)\right]\) and \({\mathfrak{N}}_{\Upsilon}\left(\varkappa \right)=\left[{\mathfrak{N}}_{*}\left(\varkappa ,\Upsilon\right), {\mathfrak{N}}^{*}\left(\varkappa ,\Upsilon\right)\right]\) for all \(\varkappa \in \left[\varsigma , \varsigma +\varpi (\rho , \varsigma )\right]\) and for all\(\Upsilon\in \left[0, 1\right]\). If \(\varpi \) satisfies the Condition C and\(\mathfrak{U}\left(\varkappa \right)\widetilde{\times }\mathfrak{N}\left(\varkappa \right)\in {\mathcal{F}\mathcal{R}}_{\left(\left[\varsigma , \varsigma +\varpi (\rho , \varsigma )\right], \Upsilon\right)}\), then.
where \(\mathfrak{J}\left(\varsigma ,\rho \right)=\mathfrak{U}\left(\varsigma \right)\widetilde{\times }\mathfrak{N}\left(\varsigma \right) \widetilde{+} \mathfrak{U}\left(\rho \right)\widetilde{\times }\mathfrak{N}\left(\rho \right),\) \(\mathfrak{M}\left(\varsigma ,\rho \right)=\mathfrak{U}\left(\varsigma \right)\widetilde{\times }\mathfrak{N}\left(\rho \right) \widetilde{+} \mathfrak{U}\left(\rho \right)\widetilde{\times }\mathfrak{N}\left(\varsigma \right),\) and \({\mathfrak{J}}_{\Upsilon}\left(\varsigma ,\rho \right)=\left[{\mathfrak{J}}_{*}\left(\left(\varsigma ,\rho \right), \Upsilon\right), {\mathfrak{J}}^{*}\left(\left(\varsigma ,\rho \right), \Upsilon\right)\right]\) and \({\mathfrak{M}}_{\Upsilon}\left(\varsigma ,\rho \right)=\left[{\mathfrak{M}}_{*}\left(\left(\varsigma ,\rho \right), \Upsilon\right), {\mathfrak{M}}^{*}\left(\left(\varsigma ,\rho \right), \Upsilon\right)\right].\)
Example 3
We consider the F-I∙V-Ms \(\mathfrak{U}, \mathfrak{N}:\left[\varsigma , \varsigma +\varpi (\rho , \varsigma )\right]=[0, \varpi (1, 0)]\to {\mathbb{E}}\) defined by,
then, for each \(\Upsilon\in \left[0, 1\right],\) we have \({\mathfrak{U}}_{\Upsilon}\left(\varkappa \right)=\left[2\Upsilon{\varkappa }^{2},(4-2\Upsilon){\varkappa }^{2}\right]\) and \({\mathfrak{N}}_{\Upsilon}\left(\varkappa \right)=\left[\Upsilon\varkappa ,(2-\Upsilon)\varkappa \right].\) Since left and right mappings \({\mathfrak{U}}_{*}\left(\varkappa ,\Upsilon\right)=2\Upsilon{\varkappa }^{2},\) \({\mathfrak{U}}^{*}\left(\varkappa , \Upsilon\right)=(4-2\Upsilon){\varkappa }^{2}\) and \({\mathfrak{N}}_{*}\left(\varkappa ,\Upsilon\right)=\Upsilon\varkappa \), \({\mathfrak{N}}^{*}\left(\varkappa , \Upsilon\right)=(2-\Upsilon)\varkappa \) pre-invex mappings along with \(\varpi \left(\rho , \varsigma \right)=\rho -\varsigma \), for each \(\Upsilon\in [0, 1]\). Hence \(\mathfrak{U},\) \(\mathfrak{N}\) both are pre-invex F-I∙V-Ms. We now compute the following:
for each \(\Upsilon\in \left[0, 1\right],\) that means
Hence, Theorem 8 is verified.
Theorem 9
Suppose that \(\mathfrak{U},\mathfrak{N} :\left[\varsigma , \varsigma +\varpi (\rho , \varsigma )\right]\to {\mathbb{E}}\) are two pre-invex F-I∙V-Ms along with family of I∙V-Ms \({\mathfrak{U}}_{\Upsilon}, {\mathfrak{N}}_{\Upsilon}:\left[\varsigma , \varsigma +\varpi (\rho , \varsigma )\right]\subset {\mathbb{R}}\to {{\mathcal{K}}_{C}}^{+}\) as well as \({\mathfrak{U}}_{\Upsilon}\left(\varkappa \right)=\left[{\mathfrak{U}}_{*}\left(\varkappa ,\Upsilon\right), {\mathfrak{U}}^{*}\left(\varkappa ,\Upsilon\right)\right]\) and \({\mathfrak{N}}_{\Upsilon}\left(\varkappa \right)=\left[{\mathfrak{N}}_{*}\left(\varkappa ,\Upsilon\right), {\mathfrak{N}}^{*}\left(\varkappa ,\Upsilon\right)\right]\) for all \(\varkappa \in \left[\varsigma , \varsigma +\varpi (\rho , \varsigma )\right]\) and for all\(\Upsilon\in \left[0, 1\right]\). If \(\varpi \) satisfies the Condition C and\(\mathfrak{U}\left(\varkappa \right)\widetilde{\times }\mathfrak{N}\left(\varkappa \right)\in {\mathcal{F}\mathcal{R}}_{\left(\left[\varsigma , \varsigma +\varpi (\rho , \varsigma )\right], \Upsilon\right)}\), then:
where \(\mathfrak{J}\left(\varsigma ,\rho \right)=\mathfrak{U}\left(\varsigma \right)\widetilde{\times }\mathfrak{N}\left(\varsigma \right) \widetilde{+} \mathfrak{U}\left(\rho \right)\widetilde{\times }\mathfrak{N}\left(\rho \right),\) \(\mathfrak{M}\left(\varsigma ,\rho \right)=\mathfrak{U}\left(\varsigma \right)\widetilde{\times }\mathfrak{N}\left(\rho \right) \widetilde{+} \mathfrak{U}\left(\rho \right)\widetilde{\times }\mathfrak{N}\left(\varsigma \right),\) and \({\mathfrak{J}}_{\Upsilon}\left(\varsigma ,\rho \right)=\left[{\mathfrak{J}}_{*}\left(\left(\varsigma ,\rho \right), \Upsilon\right), {\mathfrak{J}}^{*}\left(\left(\varsigma ,\rho \right), \Upsilon\right)\right]\) and \({\mathfrak{M}}_{\Upsilon}\left(\varsigma ,\rho \right)=\left[{\mathfrak{M}}_{*}\left(\left(\varsigma ,\rho \right), \Upsilon\right), {\mathfrak{M}}^{*}\left(\left(\varsigma ,\rho \right), \Upsilon\right)\right].\)
Proof
Using condition C, we can write.
By hypothesis, for each \(\Upsilon\in \left[0, 1\right],\) we have
Integrating over \(\left[0, 1\right],\) we have
from which, we have
that is
This completes the proof.
Example 4
We consider the F-I∙V-Ms \(\mathfrak{U}, \mathfrak{N}:\left[\varsigma , \varsigma +\varpi \left(\rho , \varsigma \right)\right]=[0, \varpi \left(1, 0\right)]\to {\mathbb{E}}\) defined by, for each \(\Upsilon\in \left[0, 1\right],\) we have \({\mathfrak{U}}_{\Upsilon}\left(\varkappa \right)=\left[2\Upsilon{\varkappa }^{2},(4-2\Upsilon){\varkappa }^{2}\right]\) and \({\mathfrak{N}}_{\Upsilon}\left(\varkappa \right)=\left[\Upsilon\varkappa ,(2-\Upsilon)\varkappa \right],\) as in Example 3, then \(\mathfrak{U}\left(\varkappa \right), \mathfrak{N}(\varkappa )\) both are pre-invex F-I∙V-Ms with respect to \(\varpi \left(\rho , \varsigma \right)=\rho -\varsigma \). We have \({\mathfrak{U}}_{*}\left(\varkappa ,\Upsilon\right)=2\Upsilon{\varkappa }^{2},\) \({\mathfrak{U}}^{*}\left(\varkappa , \Upsilon\right)=(4-2\Upsilon){\varkappa }^{2}\) and \({\mathfrak{N}}_{*}\left(\varkappa ,\Upsilon\right)=\Upsilon\varkappa \), \({\mathfrak{N}}^{*}\left(\varkappa , \Upsilon\right)=(2-\Upsilon)\varkappa \). We now compute the following:
for each \(\Upsilon\in \left[0, 1\right],\) that means
Hence, Theorem 9 is verified.
The next results, which are linked with the well-known Fejér–Hermite–Hadamard type inequalities, will be obtained using symmetric mappings of one variable forms.
Theorem 10
Suppose that \(\mathfrak{U}:\left[\varsigma , \varsigma +\varpi (\rho , \varsigma )\right]\to {\mathbb{E}}\) is a pre-invex F-I∙V-M along with family of I∙V-Ms \({\mathfrak{U}}_{\Upsilon}:\left[\varsigma , \varsigma +\varpi (\rho , \varsigma )\right]\subset {\mathbb{R}}\to {{\mathcal{K}}_{C}}^{+}\) as well as \({\mathfrak{U}}_{\Upsilon}\left(\varkappa \right)=\left[{\mathfrak{U}}_{*}\left(\varkappa ,\Upsilon\right), {\mathfrak{U}}^{*}\left(\varkappa ,\Upsilon\right)\right]\) for all \(\varkappa \in \left[\varsigma , \varsigma +\varpi (\rho , \varsigma )\right]\) and for all \(\Upsilon\in \left[0, 1\right]\). If \(\varpi \) satisfies the Condition C and \(\mathfrak{U}\in {\mathcal{F}\mathcal{R}}_{\left(\left[\varsigma , \varsigma +\varpi (\rho , \varsigma )\right], \Upsilon\right)}\), and \(\mathfrak{V}:\left[\varsigma , \varsigma +\varpi (\rho , \varsigma )\right]\to {\mathbb{R}}, \mathfrak{V}(\varkappa )\ge 0,\) symmetric with respect to \(\varsigma +\frac{1}{2}\varpi \left(\rho , \varsigma \right),\) then:
Proof
Let \(\mathfrak{U}\) be a pre-invex F-I∙V-M. Then, for each \(\Upsilon\in \left[0, 1\right],\) we have.
And
After adding (23) and (24), and integrating over \(\left[0, 1\right],\) we get
Since \(\mathfrak{V}\) is symmetric, then
Since
From (26), we have
that is
hence
Next, we construct first HH-Fejér inequality for pre-invex F-I∙V-M, which generalizes first HH-Fejér inequalities for pre-invex mapping, see [16, 43].
Theorem 11.
Suppose that \(\mathfrak{U}:\left[\varsigma , \varsigma +\varpi (\rho , \varsigma )\right]\to {\mathbb{E}}\) are two pre-invex F-I∙V-Ms along with \(\varsigma < \varsigma +\varpi (\rho , \varsigma )\) and family of I∙V-Ms \({\mathfrak{U}}_{\Upsilon}:\left[\varsigma , \varsigma +\varpi (\rho , \varsigma )\right]\subset {\mathbb{R}}\to {{\mathcal{K}}_{C}}^{+}\) as well as \({\mathfrak{U}}_{\Upsilon}\left(\varkappa \right)=\left[{\mathfrak{U}}_{*}\left(\varkappa ,\Upsilon\right), {\mathfrak{U}}^{*}\left(\varkappa ,\Upsilon\right)\right]\) for all \(\varkappa \in \left[\varsigma , \varsigma +\varpi (\rho , \varsigma )\right]\) and for all \(\Upsilon\in \left[0, 1\right]\). If \(\mathfrak{U}\in {\mathcal{F}\mathcal{R}}_{\left(\left[\varsigma , \varsigma +\varpi (\rho , \varsigma )\right], \Upsilon\right)}\) and \(\mathfrak{V}:\left[\varsigma , \varsigma +\varpi (\rho , \varsigma )\right]\to {\mathbb{R}}, \mathfrak{V}(\varkappa )\ge 0,\) symmetric with respect to \(\varsigma +\frac{1}{2}\varpi \left(\rho , \varsigma \right),\) and \({\int }_{\varsigma }^{\varsigma +\varpi (\rho , \varsigma )}\mathfrak{V}(\varkappa )d\varkappa >0\), and Condition C for \(\varpi \), then.
Proof. Using Condition C, we can write
Since \(\mathfrak{U}\) is a pre-invex, then for \(\Upsilon\in \left[0, 1\right],\) we have
By multiplying (27) by \(\mathfrak{V}\left(\varsigma +\left(1-\mathsf{r}\right)\varpi \left(\rho ,\varsigma \right)\right)=\mathfrak{V}\left(\varsigma +\mathsf{r}\varpi \left(\rho ,\varsigma \right)\right)\) and integrate it by \(\mathsf{r}\) over \(\left[0, 1\right],\) we obtain
Since
From (29), we have
From which, we have
that is
This completes the proof.
Remark 6
If \(\left(\rho , \varsigma \right)=\rho -\varsigma \), then inequalities in Theorem 10 and 11 reduce for convex F-I∙V-Ms which are also new one.
If \({\mathfrak{U}}_{*}\left(\varsigma ,\Upsilon\right)={\mathfrak{U}}^{*}\left(\varsigma , \Upsilon\right)\) with \(\Upsilon=1\), then Theorem 10 and 11 reduces to classical first and second HH-Fejér inequality for pre-invex mapping, see[16].
If \({\mathfrak{U}}_{*}\left(\varsigma ,\Upsilon\right)={\mathfrak{U}}^{*}\left(\varsigma , \Upsilon\right)\) with \(\Upsilon=1\) and \(\varpi \left(\rho , \varsigma \right)=\rho -\varsigma \) then Theorem 10 and 11reduce to classical first and second HH-Fejér inequality for convex mapping, see [43].
Example 5
We consider the F-I∙V-M \(\mathfrak{U}:\left[1, 1+ \varpi (4, 1)\right]\to {\mathbb{E}}\) defined by,
Then, for each \(\Upsilon\in \left[0, 1\right],\) we have\({\mathfrak{U}}_{\Upsilon}\left(\varkappa \right)=\left[(1+\Upsilon){e}^{\varkappa },2(2-\Upsilon){e}^{\varkappa }\right]\). Since left and right mappings \({\mathfrak{U}}_{*}\left(\varkappa ,\Upsilon\right),\) \({\mathfrak{U}}^{*}\left(\varkappa ,\Upsilon\right)\) are pre-invex mappings along with \(\varpi \left(\mathsf{z},\varkappa \right)=\mathsf{z}-\varkappa \) for each \(\in [0, 1]\), respectively, then \(\mathfrak{U}\left(\varkappa \right)\) is pre-invex F-I∙V-M. If
Then, we have
And
From (30) and (31), we have
for each \(\Upsilon\in \left[0, 1\right].\)
Hence, Theorem 10 is verified.
For Theorem 11, we have
From (32) and (33), we have
Hence, Theorem 11 is verified.
4 Conclusion
In this work, some new HH-inequalities are established by means of fuzzy order relation on fuzzy-interval space for pre-invex F-I∙V-Ms. Useful examples that verify the applicability of theory developed in this study are presented. In future, we intend to use various types of pre-invex F-I∙V-Ms to construct fuzzy-interval inequalities of F-I∙V-Ms. We hope that this concept will be helpful for other authors to play their roles in different fields of knowledge creation.
Availability of data and material
Not applicable.
Abbreviations
- \(HH\)-inequality:
-
Hermite–Hadamard inequality
- \(HH\)-Fejér inequality:
-
Hermite–Hadamard–Fejér inequality
- I∙V-Ms :
-
Interval-valued mappings
- F-I∙V-Ms :
-
Fuzzy-interval-valued mappings
- \(\left(FR\right)\)-integrable:
-
Fuzzy Riemann integrable
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Acknowledgements
The research of Santos-García was funded by the Spanish MINECO project TRACES TIN2015–67522–C3–3–R. This work was funded by Taif University Researchers Supporting Project number (TURSP-2020/345), Taif University, Taif, Saudi Arabia.
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Khan, M.B., Noor, M.A., Zaini, H.G. et al. The New Versions of Hermite–Hadamard Inequalities for Pre-invex Fuzzy-Interval-Valued Mappings via Fuzzy Riemann Integrals. Int J Comput Intell Syst 15, 66 (2022). https://doi.org/10.1007/s44196-022-00127-z
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DOI: https://doi.org/10.1007/s44196-022-00127-z
Keywords
- Pre-invex fuzzy-interval-valued mappings
- Fuzzy Riemann integral
- Fuzzy-interval Hermite-Hadamard type inequality
- Fuzzy-interval Hermite-Hadamard-Fejér inequality