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Article

Some Certain Fuzzy Aumann Integral Inequalities for Generalized Convexity via Fuzzy Number Valued Mappings

by
Muhammad Bilal Khan
1,
Hakeem A. Othman
2,
Michael Gr. Voskoglou
3,*,
Lazim Abdullah
4,* and
Alia M. Alzubaidi
2
1
Department of Mathematics, COMSATS University Islamabad, Islamabad 44000, Pakistan
2
Department of Mathematics, AL-Qunfudhah University College, Umm Al-Qura University, Makkah 24382, Saudi Arabia
3
Mathematical Sciences, Graduate TEI of Western Greece, 26334 Patras, Greece
4
Management Science Research Group, Faculty of Ocean Engineering Technology and Informatics, Universiti Malaysia Terengganu, Kuala Nerus 21030, Terengganu, Malaysia
*
Authors to whom correspondence should be addressed.
Mathematics 2023, 11(3), 550; https://doi.org/10.3390/math11030550
Submission received: 9 December 2022 / Revised: 30 December 2022 / Accepted: 13 January 2023 / Published: 19 January 2023
(This article belongs to the Special Issue Fuzzy Sets, Fuzzy Logic and Their Applications 2021)

Abstract

:
The topic of convex and nonconvex mapping has many applications in engineering and applied mathematics. The Aumann and fuzzy Aumann integrals are the most significant interval and fuzzy operators that allow the classical theory of integrals to be generalized. This paper considers the well-known fuzzy Hermite–Hadamard (HH) type and associated inequalities. With the help of fuzzy Aumann integrals and the newly introduced fuzzy number valued up and down convexity ( U D -convexity), we increase this mileage even further. Additionally, with the help of definitions of lower U D -concave (lower U D -concave) and upper U D -convex (concave) fuzzy number valued mappings ( F N V M s), we have gathered a sizable collection of both well-known and new extraordinary cases that act as applications of the main conclusions. We also offer a few examples of fuzzy number valued U D -convexity to further demonstrate the validity of the fuzzy inclusion relations presented in this study.

1. Introduction

Many fields make use of the convexity of functions such as game theory, variational science, mathematical programming theory, economics, and optimal control theory. Convex analysis, a brand-new mathematics branch, started taking shape in the 1960s. Many writers have employed related concepts of convexity during the past 20 years and generalized other inequalities, including h-convex functions (see References [1,2,3,4,5,6,7,8,9,10]), log convex functions (see References [11,12,13,14,15,16,17,18,19], and coordinated convex functions (see References [20,21]). Convexity is a fundamental term in optimization theory applied in operations research, economics, control theory, decision-making, and management. Several writers have expanded and generalized integral inequalities using various convex functions; see Refs. [22,23]. For more information, see [24,25,26,27,28,29,30,31,32,33] and references therein.
Calculating mistakes in a numerical analysis has always been difficult. The interval analysis has received a lot of attention as a novel method for resolving uncertainty issues because of its capacity to reduce calculation errors and make calculations meaningless. Set-valued analysis, a set-centric approach to mathematics and topology, includes interval analysis. It deals with interval variables rather than point variables, and the computation results are expressed as intervals; therefore, it removes mistakes that lead to incorrect conclusions. Moore [34] first adapted an interval analysis to automatic error analysis to deal with data uncertainty in 1966. The work garnered a lot of attention from academics and led to an improvement in calculation performance. They are helpful in many applications because of their capacity to be expressed as uncertain variables, including computer graphics [35], automatic error analysis [36], decision analysis [37], etc. There are numerous great applications and results for readers interested in interval analysis in other branches of mathematics; see References [38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53].
On the other hand, a generalized convexity mapping has the potential to solve a wide range of issues in both a nonlinear and pure analysis. Recently, well-known inequalities such as Jensen, Simpson, Opial, Ostrowski, Bullen, and the famous Hermite–Hadamard that are extended in the setting of interval-valued functions ( I V M ) have been constructed using a variety of related classes of convexity. Chalco-Cano [54] established interval-based inequalities for the Ostrowski type using a derivative of the Hukuhara type. Opial-type inequalities for I V M s were developed by Costa in [55]. The Minkowski inequalities for I V M s were one of the inequalities suggested by Beckenbach and Roman-Flores [56]. According to the literature assessment, the majority of authors used an inclusion connection, similarly to in 2018, to evaluate inequality. These inequalities were created by Zhao et al. [57] for the harmonic h-convex I V M s and the h-convex I V M s. The authors who came after used both harmonical ( h 1 , h 2 )-convex functions and ( h 1 , h 2 )-convex functions to create these inequalities; for more information, see Refs. [58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75].
Using the radius and interval midpoint, Bhunia and his co-author defined the center-radius order in 2014; see Ref. [76]. The following findings for the cr-h-convex, harmonically cr-h-convex, and cr-h-GL functions were developed in 2022 by Wei Liu and his co-authors; see References [77,78,79,80,81,82,83,84,85,86,87,88]. Our examination of the literature showed that inclusion and fuzzy inclusion relations are the main sources of the majority of these discrepancies. The fundamental benefit of the up and down fuzzy relation for up and down functions is that the inequality term generated by employing these conceptions is more exact, and the argument’s validity can be supported by intriguing examples of illustrated theorems. For further study related to interval-valued functions and fuzzy mappings, see [89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111].
This study provides an introduced class of convexity based on the fuzzy inclusion order and is known as U D -convex F N V M s, and is inspired by Refs. [56,57]. We create new H.H. inequalities with the aid of these innovative ideas, and eventually, the Jensen inequality is developed. The study includes a variety of examples to help bolster the results reached.
The article is formatted as follows, in order: Section 2 gives some background information. Section 3 each provide an overview of the primary conclusions. A succinct conclusion is explored in Section 4.

2. Preliminaries

We recall a few definitions, which can be found in the literature and that will be relevant in the follow-up.
Let us consider that X o is the space of all closed and bounded intervals of , and that S X o is given by
S = S * ,   S * = w |   S * w S * ,   S * ,   S * ,
If S * = S * , then S is degenerate. In the follow-up, all intervals are considered non-degenerate. If S * 0 , then S is positive. We denote by X o + = S * ,   S * : S * ,   S * X o   and   S * 0 the set of all positive intervals.
Let ɷ and ɷ S be given by
ɷ S = ɷ S * ,   ɷ S *   if   ɷ > 0 , 0               if f   ɷ = 0 , ɷ S * , ɷ S *   if   ɷ < 0 .
We consider the Minkowski sum, S + O , product, S × O , and difference, O S , for S , O X o , as
O * ,   O * + S * ,   S * = O * + S * ,     O * + S * ,
O * ,   O * × S * ,   S * = min O * S * ,   O * S * ,   O * S * ,   O * S * ,   max O * S * ,   O * S * ,   O * S * ,   O * S *
O * ,   O * S * ,   S * = O * S * ,     O * S * .
Remark 1.
 
(i) For given  O * ,   O * ,   S * ,   S * I ,  the relation  I , defined on  I  by
S * ,   S * I O * ,   O *   if   and   only   if   S * O * ,     O * S * ,
for all  O * ,   O * ,   S * ,   S * I ,  is a partial interval inclusion relation. Moreover,  S * ,   S * I O * ,   O *  coincides with  S * ,   S * O * ,   O *  on  I .  The relation  I  is of UD order [105].
(ii) For given  O * ,   O * ,   S * ,   S * I ,  the relation  I , defined on  I  by  O * ,   O * I S * ,   S *  if and only if  O * S * ,   O * S *  or  O * S * ,   O * < S * , is a partial interval order relation. Plus, we have  O * ,   O * I S * ,   S *  that coincides with  O * ,   O * S * ,   S *  on  I .  The relation  I  is of the left and right (LR) type [104,105].
Given the intervals  O * ,   O * ,   S * ,   S * X o ,  their Hausdorff–Pompeiu distance is
d H O * ,   O * ,   S * ,   S * = m a x O * S * ,   O * S * .  
We have  X o , d H  that is a complete metric space [94,102,103].
Definition 1
([93,94]). A fuzzy subset  L  of   is a mapping  S ˜ : 0 , 1 , a denoted membership mapping of  L . We adopt the symbol   to represent the set of all fuzzy subsets of  .
Let us consider S ˜ . If the following properties hold, then S ˜ is a fuzzy number:
(1)
S ˜ is normal if there exists w and S ˜ w = 1 ;
(2)
S ˜ is upper semi-continuous on if for a w there exists ε > 0 and δ > 0 yielding S ˜ w S ˜ y < ε for all y with w y < δ ;
(3)
S ˜ is a fuzzy convex, meaning that S ˜ 1 ɷ w + ɷ y min S ˜ w ,   S ˜ y , for all w , y , and ɷ 0 ,   1 ;
(4)
S ˜ is compactly supported, which means that cl w   S ˜ w 0 is compact.
The symbol o will be adopted to designate the set of all fuzzy numbers of .
Definition 2.
([93,94]). For  S ˜ o , the  ʚ -level, or  ʚ -cut, sets of  S ˜  are  S ˜ ʚ = w   S ˜ w ʚ  for all  ʚ 0 ,   1 , and  S ˜ 0 = w   S ˜ w 0 .
Proposition 1.
([96]). Let  S ˜ , O ˜ o . The relation  F , defined on  o  by
S ˜ F O ˜   when   and   only   when   S ˜ ʚ I O ˜ ʚ ,   for   every   ʚ 0 ,   1 ,
is a LR order relation.
Proposition 2.
([79]). Let  S ˜ , O ˜ o . The relation  F , defined on  o  by
S ˜ F O ˜   when   and   only   when   S ˜ ʚ I O ˜ ʚ ,   for   every   ʚ 0 ,   1 ,
is an UD order relation.
If  S ˜ , O ˜ o  and  ʚ , then, for every  ʚ 0 ,   1 ,
S ˜ O ˜ ʚ = S ˜ ʚ + O ˜ ʚ ,  
S ˜ O ˜ ʚ = S ˜ ʚ ×   O ˜ ʚ ,  
ɷ S ˜ ʚ = ʚ . S ˜ ʚ  
result from Equations (4)–(6), respectively.
Theorem 1
([94]). For  S ˜ ,   O ˜ o , the supremum metric
d S ˜ ,   O ˜ = sup 0 ʚ 1 d H S ˜ ʚ ,   O ˜ ʚ
is a complete metric space, where  H  stands for the Hausdorff metric on a space of intervals.
Theorem 2
([94,95]). If  H : b , z X o  is an  I V M  satisfying  H w = H * w ,   H * w , then  H  is Aumann integrable (IA-integrable) over  b , z  when and only when  H * w  and  H * w  are integrable over  b , z , meaning
I A b z H w d w = b   z H * w d w ,   b z H * w d w
Definition 3
([104]). Let  H ˜ : I o  be a  F N V M . The family of  I V M s, for every  ʚ 0 ,   1 , is  H ʚ : I X o  satisfying  H ʚ w = H * w , ʚ ,   H * w , ʚ  for every  w I .  For every  ʚ 0 ,   1 ,  the lower and upper mappings of  H ʚ  are the endpoint real-valued mappings  H * · , ʚ ,   H * · , ʚ : I .
Definition 4
([104]). Let  H ˜ : I o  be a  F N V M . Then,  H ˜ w  is continuous at  w I ,  if for every  ʚ 0 ,   1 ,   H ʚ w  is continuous when and only when  H * w , ʚ  and  H * w , ʚ  are continuous at  w I .
Definition 5
([95]). Let  H ˜ : b ,   z o  be a  F N V M . The fuzzy Aumann integral ( F A -integral) of  H ˜  over  b ,   z  is
F A b z H ˜ w d w   ʚ = I A b z H ʚ w d w = b z H w , ʚ d w : H w , ʚ S H ʚ ,
where  S H ʚ = H . , ʚ : H . , ʚ   i s   i n t e g r a b l e ,   a n d   H w , ʚ H ʚ w ,   for every  ʚ 0 ,   1 .  Moreover,  H ˜  is  F A -integrable over  b ,   z  if  F A b z H ˜ w d w o .
Theorem 3
[96]. Let  H ˜ : b ,   z o  be a  F N V M , whose  ʚ -levels define the family of  I V M s  H ʚ : b ,   z X o  satisfying  H ʚ w = H * w , ʚ ,   H * w , ʚ  for every  w b ,   z  and  ʚ 0 ,   1 .   H ˜  is  F A -integrable over  b ,   z  when and only when  H * w , ʚ  and  H * w , ʚ  are integrable over  b ,   z . Moreover, if  H ˜  is  F A -integrable over  b ,   z , then we have
F A b z H ˜ w d w   ʚ = b z H * w , ʚ d w ,   b z H * w , ʚ d w = I A b z H ʚ w d w
for every  ʚ 0 ,   1 .
Breckner discussed the coming emerging idea of interval-valued convexity in [97].
An I∙V∙M  H : I = b ,   z X o  is called convex I∙V∙M if
H ɷ w + 1 ɷ s ɷ H w + 1 ɷ H s ,  
for all  w ,   y b ,   z ,   ɷ 0 ,   1 , where  X o  is the collection of real-valued intervals. If (17) is reversed, then  H  is called concave.
Definition 6
([89]). The  F N V M   H ˜ : b ,   z o  is called convex  F N V M  on   b ,   z  if
H ˜ ɷ w + 1 ɷ s   F ɷ H ˜ w 1 ɷ H ˜ s ,
for all   w ,   s b ,   z ,   ɷ 0 ,   1 ,  where  H ˜ w F 0 ˜  for all  w b ,   z .  If (18) is reversed, then  H ˜  is called concave  F N V M  on  b ,   z . H ˜  is affine if and only if it is both convex and concave  F N V M .
Definition 7
([105]). The  F N V M   H ˜ : b ,   z o  is called  U D -convex  F N V M  on   b ,   z  if
H ˜ ɷ w + 1 ɷ s F ɷ H ˜ w 1 ɷ H ˜ s ,
for all   w ,   s b ,   z ,   ɷ 0 ,   1 ,  where  H ˜ w F 0 ˜  for all  w b ,   z .  If (19) is reversed then,  H ˜  is called  U D -concave  F N V M  on  b ,   z . H ˜  is  U D -affine  F N V M  if and only if it is both  U D -convex and  U D -concave  F N V M .
Theorem 4
([105]). Let  H ˜ : b , z o  be a  F N V M , whose  ʚ -cuts define the family of inteval-valued mappings  H ʚ : b , z X o + X o  are given by
H ʚ w = H * w , ʚ ,   H * w , ʚ ,  
for all  w b , z  and for all  ʚ 0 ,   1 . Then,  H ˜  is  U D -convex  F N V M  on  b , z ,  if and only if, for all  ʚ 0 ,   1 ,   H * w ,   ʚ  is a convex mapping and  H * w ,   ʚ  is a concave mapping.
Remark 2.
If  H * w , ʚ H * w , ʚ  and  ʚ   =   1 , then we obtain the inequality (17).
If H * w , ʚ = H * w , ʚ and ʚ   =   1 , then we obtain the classical definition of convex mappings.
Now we have obtained some new definitions from the literature which will be helpful to investigate some classical and new results as special cases of main results.
Definition 8.
([79]). Let  H ˜ : b , z o  be a  F N V M , whose  ʚ -cuts define the family of  I V M s  H ʚ : b , z X o + X o  are given by
H ʚ w = H * w , ʚ ,   H * w , ʚ ,  
for all  w b , z  and for all  ʚ 0 ,   1 . Then,  H ˜  is lower  U D -convex (concave)  F N V M  on  b , z ,  if and only if, for all  ʚ 0 ,   1 ,   H * w ,   ʚ  is a convex (concave) mapping and  H * w ,   ʚ  is an affine mapping.
Definition 9.
([79]). Let  H ˜ : b , z o  be a  F N V M , whose  ʚ -cuts define the family of  I V M s  H ʚ : b , z X o + X o  are given by
H ʚ w = H * w , ʚ ,   H * w , ʚ ,  
for all  w b , z  and for all  ʚ 0 ,   1 . Then,  H  is upper  U D -convex (concave)  F N V M  on  b , z ,  if and only if, for all  ʚ 0 ,   1 ,   H * w ,   ʚ  is an affine mapping and  H * w ,   ʚ  is a convex (concave) mapping.
Remark 3.
Both concepts “ U D -convex  F N V M ” and classical “convex  F N V M , see [41]” behave alike when  H ˜  is lower  U D -convex  F N V M .

3. Fuzzy Number Hermite–Hadamard Inequalities

In this section, we propose Hermite–Hadamard and Hermite–Hadamard–Fejér inequalities for U D -convex F N V M s, and verify with the help of nontrivial examples.
Theorem 5.
Let  H ˜ : b ,   z o  be a  U D -convex  F N V M  on  b ,   z ,  whose  ʚ -cuts define the family of  I V M s  H ʚ : b ,   z X o +  are given by  H ʚ w = H * w , ʚ ,   H * w , ʚ  for all  w b ,   z  and for all  ʚ 0 ,   1 . If  H ˜ F A b ,   z ,   ʚ , then
  H ˜ b + z 2 F 1 z b F A b z H ˜ w d w F H ˜ b     H ˜ z 2 .
If  H ˜ w  concave  F N V M , then (23) is reversed.
Proof. 
Let H ˜ : b ,   z o be a U D -convex F N V M . Then, by hypothesis, we have
2 H ˜ b + z 2 F H ˜ ɷ b + 1 ɷ z H ˜ 1 ɷ b + ɷ z .
Therefore, for every ʚ 0 ,   1 , we have
2 H * b + z 2 ,   ʚ H * ɷ b + 1 ɷ z ,   ʚ + H * 1 ɷ b + ɷ z ,   ʚ ,     2 H * b + z 2 ,   ʚ H * ɷ b + 1 ɷ z ,   ʚ + H * 1 ɷ b + ɷ z , ʚ .
Then
2 0 1 H * b + z 2 ,   ʚ d ɷ 0 1 H * ɷ b + 1 ɷ z , ʚ d ɷ + 0 1 H * 1 ɷ b + ɷ z ,   ʚ d ɷ ,     2 0 1 H * b + z 2 , ʚ d ɷ 0 1 H * ɷ b + 1 ɷ z ,   ʚ d ɷ + 0 1 H * 1 ɷ b + ɷ z , ʚ d ɷ .
It follows that
H * b + z 2 ,   ʚ 1 z b   b z H * w ,   ʚ d w , H * b + z 2 ,   ʚ 1 z b   b z H * w ,   ʚ d w .
That is
H * b + z 2 ,   ʚ ,   H * b + z 2 ,   ʚ I 1 z b b z H * w ,   ʚ d w ,   b z H * w ,   ʚ d w .
Thus,
  H ˜ b + z 2 F 1 z b   F A b z H ˜ w d w .
In a similar way as above, we have
1 z b F A b z H ˜ w d w F H ˜ b     H ˜ z 2 .
Combining (24) and (25), we have
H ˜ b + z 2 F 1 z b F A b z H ˜ w d w F H ˜ b     H ˜ z 2 .
Hence, the required result. □
Remark 4.
The following are some exceptional cases which can be obtained from inequality (23):
If one lays H is lower U D -convex F N V M on b , z , then one acquires the following coming inequality, see [90]:
H b + z 2 F 1 z b   F A b z H w d w F H b H z 2
If one takes H is lower U D -convex F N V M on b , z and ʚ   =   , then one achieves the following coming inequality, see [98]:
H b + z 2 I 1 z b   I A b z H w d w I H b + H z 2
Let ʚ   =   1 . Then, from Theorem 5, we acquire the following inequality, see [99]:
H b + z 2 1 z b   I A b z H w d w H b + H z 2 .
Let ʚ   =   and H * w ,   ʚ = H * w ,   ʚ . Then, from Theorem 5, we achieve the classical Hermite–Hadamard inequality.
Example 1.
Let  w 2 , 3 , and the F N V M   H ˜ : b ,   z = 2 ,   3 o , defined by
H ˜ w θ = θ 2 + w 1 2 1 w 1 2       θ 2 w 1 2 ,   3 ,   2 + w 1 2 θ w 1 2 1       θ 3 ,   2 + w 1 2 , 0               otherwise ,
Then, for each  ʚ 0 ,   1 ,  we have  H ʚ w = 1 ʚ 2 w 1 2 + 3 ʚ , 1 ʚ 2 + w 1 2 + 3 ʚ . Since left and right end point mappings  H * w , ʚ = 1 ʚ 2 w 1 2 + 3 ʚ ,  and  H * w ,   ʚ = 1 ʚ 2 + w 1 2 + 3 ʚ , are convex and concave mappings, respectively, for each  ʚ 0 ,   1 , then  H ˜ w  is  U D -convex  F N V M . We clearly see that  H ˜ L b ,   z , o  and
H * b + z 2 ,   ʚ = H * 5 2 ,   ʚ = 1 ʚ 4 10 2 + 3 ʚ ,  
H * b + z 2 ,   ʚ = H * 5 2 ,   ʚ = 1 ʚ 4 + 10 2 + 3 ʚ .  
Note that
1 z b   b z H * w ,   ʚ d w = 2 3 1 ʚ 2 w 1 2 + 3 ʚ d w 0.4215 1 ʚ + 3 ʚ ,  
1 z b   b z H * w ,   ʚ d w = 2 3 1 + ʚ 2 + w 1 2 + 3 ʚ d w 3.58 1 ʚ + 3 ʚ ,
and
H * b , ʚ + H * z ,   ʚ 2 = 1 ʚ 4 2 3 2 + 3 ʚ ,
H * b , ʚ + H * z ,   ʚ 2 = 1 ʚ 4 + 2 + 3 2 + 3 ʚ .
Therefore,
1 ʚ 4 10 2 + 3 ʚ , 1 ʚ 4 + 10 2 + 3 ʚ I 843 2000 1 ʚ + 3 ʚ , 179 50 1 ʚ + 3 ʚ I 1 ʚ 4 2 3 2 + 3 ʚ , 1 ʚ 4 + 2 + 3 2 + 3 ʚ ,
Hence,
H ˜ b + z 2 F 1 z b F A b z H ˜ w d w F H ˜ b H ˜ z 2 ,
and Theorem 5 is verified.
Theorem 6.
Let  H ˜ : b ,   z o  be a  U D -convex  F N V M  on  b ,   z ,  whose  ʚ -cuts define the family of  I V M s  H ʚ : b ,   z X o +  are given by  H ʚ w = H * w , ʚ ,   H * w , ʚ  for all  w b ,   z  and for all  ʚ 0 ,   1 . If  H ˜   F A b ,   z ,   ʚ , then
  H ˜ b + z 2 F   T 2 F 1 z b F A b z H ˜ w d w F T 1 F H ˜ b     H ˜ z 2 ,
where
T 1 = H ˜ b     H ˜ z 2     H ˜ b + z 2 2 ,   T 2 = H ˜ 3 b + z 4     H ˜ b + 3 z 4 2
and  T 1 = T 1 * ,   T 1 * ,  T 2 = T 2 * ,   T 2 * .
Proof. 
Take b ,   b + z 2 , we have
2 H ˜ ɷ b + 1 ɷ b + z 2 2 + 1 ɷ b + ɷ b + z 2 2 F H ˜ ɷ b + 1 ɷ b + z 2   H ˜ 1 ɷ b + ɷ b + z 2 .
Therefore, for every ʚ 0 ,   1 , we have
2 H * ɷ b + 1 ɷ b + z 2 2 + 1 ɷ b + ɷ b + z 2 2 ,   ʚ H * ɷ b + 1 ɷ b + z 2 ,   ʚ + H * 1 ɷ b + ɷ b + z 2 ,   ʚ , 2 H * ɷ b + 1 ɷ b + z 2 2 + 1 ɷ b + ɷ b + z 2 2 ,   ʚ H * ɷ b + 1 ɷ b + z 2 ,   ʚ + H * 1 ɷ b + ɷ b + z 2 , ʚ .
In consequence, we obtain
H * 3 b + z 4 ,   ʚ 2 1 z b   b b + z 2 H * w ,   ʚ d w , H * 3 b + z 4 ,   ʚ 2 1 z b   b b + z 2 H * w ,   ʚ d w .
That is
H * 3 b + z 4 ,   ʚ ,     H * 3 b + z 4 ,   ʚ 2 1 z b b b + z 2 H * w ,   ʚ d w ,   b b + z 2 H * w ,   ʚ d w .
It follows that
H ˜ 3 b + z 4 2 F 1 z b F A b b + z 2 H ˜ w d w .
In a similar way as above, we have
H ˜ b + 3 z 4 2 F 1 z b F A b + z 2 z H ˜ w d w .
Combining (31) and (32), we have
H ˜ 3 b + z 4 H ˜ b + 3 z 4 2 F 1 z b F A b z H ˜ w d w .
By using Theorem 5, we have
  H ˜ b + z 2 = H ˜ 1 2 . 3 b + z 4 + 1 2 . b + 3 z 4 .
Therefore, for every ʚ 0 ,   1 , we have
H * b + z 2 ,   ʚ = H * 1 2 . 3 b + z 4 + 1 2 . b + 3 z 4 ,   ʚ H * b + z 2 ,   ʚ = H * 1 2 . 3 b + z 4 + 1 2 . b + 3 z 4 ,   ʚ ,
1 2 H * 3 b + z 4 ,   ʚ + 1 2 H * b + 3 z 4 ,   ʚ 1 2 H * 3 b + z 4 ,   ʚ + 1 2 H * b + 3 z 4 ,   ʚ ,
1 z b   b z H * w ,   ʚ d w     1 z b   b z H * w ,   ʚ d w ,
= T 2 * = T 2 * ,
1 2 H * b ,   ʚ + H * z ,   ʚ 2 + H * b + z 2 ,   ʚ 1 2 H * b ,   ʚ + H * z ,   ʚ 2 + H * b + z 2 ,   ʚ ,
= T 1 * = T 1 * ,
1 2 H * b ,   ʚ + H * z ,   ʚ 2 + H * b ,   ʚ + H * z ,   ʚ 2 1 2 H * b ,   ʚ + H * z ,   ʚ 2 + H * b ,   ʚ + H * z ,   ʚ 2 ,
= H * b ,   ʚ + H * z ,   ʚ 2 = H * b ,   ʚ + H * z ,   ʚ 2 ,
that is
H ˜ b + z 2 F   T 2 F 1 z b F A b z H ˜ w d w F T 1 F H ˜ b     H ˜ z 2 ,
hence, the result follows. □
Example 2.
We consider the  F N V M   H ˜ : b ,   z = 2 ,   3 o  defined by,  H ʚ w = 1 ʚ 2 w 1 2 + 3 ʚ , 1 + ʚ 2 + w 1 2 + 3 ʚ ,  as in Example 1, then  H ˜ w  is  U D -convex  F N V M  and satisfying (10). We have  H * w ,   ʚ = 1 ʚ 2 w 1 2 + 3 ʚ  and  H * w ,   ʚ = 1 + ʚ 2 + w 1 2 + 3 ʚ . We now compute the following
H * b ,   ʚ + H * z ,   ʚ 2 = 4 + 2 ʚ 1 ʚ 2 + 3 2 H * b , ʚ   + ˜   H * z , ʚ 2 = 4 + 10 ʚ + 1 + ʚ 2 + 3 2 ,
T 1 * = H * b ,   ʚ + H * z ,   ʚ 2 + H * b + z 2 ,   ʚ 2 = 8 + 4 ʚ 1 ʚ 2 + 3 + 2 × 5 4 T 1 * = H * b ,   ʚ + H * z ,   ʚ 2 + H * b + z 2 ,   ʚ 2 = 8 + 20 ʚ + 1 + ʚ 2 + 3 + 2 × 5 4 ,
T 2 * = H * 3 b + z 4 ,   ʚ + H * b + 3 z 4 ,   ʚ 2 = 5 + 7 ʚ 11 1 ʚ 4   T 2 * = H * 3 b + z 4 ,   ʚ + H * b + 3 z 4 ,   ʚ 2 = 11 + 23 ʚ + 11 1 + ʚ 4 ,
Then we obtain that
1 ʚ 4 10 2 + 3 ʚ 5 + 7 ʚ 11 1 ʚ 4 843 2000 1 ʚ + 3 ʚ 8 + 4 ʚ 1 ʚ 2 + 3 + 2 × 5 4 1 ʚ 4 2 3 2 + 3 ʚ 1 + ʚ 4 + 10 2 + 3 ʚ 11 + 23 ʚ + 11 1 + ʚ 4 179 50 1 + ʚ + 3 ʚ 8 + 20 ʚ + 1 + ʚ 2 + 3 + 2 × 5 4 1 + ʚ 4 + 2 + 3 2 + 3 ʚ .
Hence, Theorem 6 is verified.
We now obtain some HH-inequalities for the product of U D -convex F N V M s. These inequalities are refinements of some known inequalities, see [57].
Theorem 7.
Let  H ˜ , Τ ˜   : b ,   z o  be two  U D -convex  F N V M s on  b ,   z , whose  ʚ -cuts  H ʚ ,   Τ ʚ : b ,   z X o +  are defined by  H ʚ w = H * w , ʚ ,   H * w , ʚ  and  Τ ʚ w = Τ * w , ʚ ,   Τ * w , ʚ  for all  w b ,   z  and for all  ʚ 0 ,   1 . If  H ˜ Τ ˜ F A b ,   z ,   ʚ , then
1 z b F A b z H ˜ w Τ ˜ w d w F M ˜ b , z 3 N ˜ b , z 6 .
where  M ˜ b , z = H ˜ b Τ ˜ b H ˜ z Τ ˜ z ,   N ˜ b , z = H ˜ b Τ ˜ z H ˜ z Τ ˜ b ,  and  M ʚ b , z = M * b , z ,   ʚ ,   M * b , z ,   ʚ  and  N ʚ b , z = N * b , z ,   ʚ ,   N * b , z ,   ʚ .
Proof. 
Since H ˜ , Τ ˜ F A b ,   z , then we have
H * ς b + 1 ς z ,   ʚ ς H * b ,   ʚ + 1 ς H * z ,   ʚ ,   H * ς b + 1 ς z ,   ʚ ς H * b ,   ʚ + 1 ς H * z ,   ʚ .
And
Τ * ς b + 1 ς z ,   ʚ ς Τ * b ,   ʚ + 1 ς Τ * z ,   ʚ , Τ * ς b + 1 ς z ,   ʚ ς Τ * b ,   ʚ + 1 ς Τ * z ,   ʚ .
From the definition of U D -convex F N V M s, it follows that 0 ˜ F H ˜ w and 0 ˜ F Τ ˜ w , so
H * ς b + 1 ς z ,   ʚ × Τ * ς b + 1 ς z ,   ʚ ς H * b ,   ʚ + 1 ς H * z ,   ʚ × ς Τ * b ,   ʚ + 1 ς Τ * z ,   ʚ = H * b ,   ʚ × Τ * b ,   ʚ ς 2 + H * z ,   ʚ × Τ * z ,   ʚ ς 2 + H * b ,   ʚ × Τ * z ,   ʚ ς 1 ς + H * z ,   ʚ × Τ * b ,   ʚ ς 1 ς H * ς b + 1 ς z ,   ʚ × Τ * ς b + 1 ς z ,   ʚ ς H * b ,   ʚ + 1 ς H * z ,   ʚ × ς Τ * b ,   ʚ + 1 ς Τ * z ,   ʚ = H * b ,   ʚ × Τ * b ,   ʚ ς 2 + H * z ,   ʚ × Τ * z ,   ʚ ς 2 + H * b ,   ʚ Τ * × z ,   ʚ ς 1 ς + H * z ,   ʚ × Τ * b ,   ʚ ς 1 ς ,  
Integrating both sides of the above inequality over [0, 1], we get
0 1 H * ς b + 1 ς z ,   ʚ × Τ * ς b + 1 ς z ,   ʚ d ς = 1 z b b z H * w ,   ʚ × Τ * w ,   ʚ d w H * b ,   ʚ × Τ * b ,   ʚ + H * z ,   ʚ × Τ * z ,   ʚ 0 1 ς 2 d ς + H * b ,   ʚ × Τ * z ,   ʚ + H * z ,   ʚ × Τ * b ,   ʚ 0 1 ς 1 ς d ς , 0 1 H * ς b + 1 ς z ,   ʚ × Τ * ς b + 1 ς z ,   ʚ d ς = 1 z b b z H * w ,   ʚ × Τ * w ,   ʚ d w H * b ,   ʚ × Τ * b ,   ʚ + H * z ,   ʚ × Τ * z ,   ʚ 0 1 ς 2 d ς + H * b ,   ʚ × Τ * z ,   ʚ + H * z ,   ʚ × Τ * b ,   ʚ 0 1 ς 1 ς d ς .
It follows that,
1 z b b z H * w ,   ʚ × Τ * w ,   ʚ d w B * b , z ,   ʚ 0 1 ς 2 d ς + * b , z ,   ʚ 0 1 ς 1 ς d ς , 1 z b b z H * w ,   ʚ × Τ * w ,   ʚ d w B * b , z ,   ʚ 0 1 ς 2 d ς + * b , z ,   ʚ 0 1 ς 1 ς d ς ,
that is
1 z b b z H * w ,   ʚ × Τ * w ,   ʚ d w ,   b z H * w ,   ʚ × Τ * w ,   ʚ d w
I B * b , z ,   ʚ 3 ,   B * b , z ,   ʚ 3 + * b , z ,   ʚ 6 ,   * b , z ,   ʚ 6 .
Thus,
1 z b F A b z H ˜ w Τ ˜ w d w F M ˜ b , z 3 N ˜ b , z 6 .
And the theorem has been established. □
Example 3.
Let  b , z = 0 , 2 ,   and the  F N V M s  H , Τ : b ,   z = 0 , 2 o ,  defined by
H w θ = θ w                     θ 0 ,   w , 2 w θ w             θ w ,   2 w , 0                   otherwise ,
Τ w θ = θ w 2 w                   θ w ,   2 ,   8 e w θ 8 e w 2       θ 2 ,   8 e w , 0                       otherwise .
Then, for each  ʚ 0 ,   1 ,  we have  H ʚ w = ʚ w , 2 ʚ w  and  Τ ʚ w = 1 ʚ w + 2 ʚ , 1 ʚ 8 e w + 2 ʚ .  Since left and right end point mappings  H * w , ʚ = ʚ w ,  and  H * w ,   ʚ = 2 ʚ w , are convex and concave mappings, respectively, and  Τ * w , ʚ = 1 ʚ w + 2 ʚ  and  Τ * w ,   ʚ = 1 ʚ 8 e w + 2 ʚ  are convex and concave mappings, respectively, for each  ʚ 0 ,   1 , then  H ˜ w  and  Τ ˜ w  both are  U D -convex  F N V M s. We clearly see that  H ˜ Τ ˜ L b ,   z , o  and
1 z b   b z H * w , ʚ × Τ * w , ʚ d w = 1 2 0 2   ʚ 1 ʚ w 2 + 2 ʚ 2 w d w = 2 3 ʚ 2 + ʚ ,
1 z b b z H * w , ʚ × Τ * w , ʚ d w = 1 2 0 2 1 ʚ 2 ʚ w 8 e w + 2 ʚ 2 ʚ w d w
2 ʚ 2 1903 250 903 250 ʚ .
Note that
Δ * b , z = H * b × Τ * b + H * z × Τ * z = 4 ʚ ,
Δ * b , z = H * b × Τ * b + H * z × Τ * z = 2 2 ʚ 1 ʚ 8 e 2 + 2 ʚ ,
* b , z = H * b × Τ * z + H * z × Τ * b = 4 ʚ 2 ,
* b , z = H * b × Τ * z + H * z × Τ * b = 2 2 ʚ 7 5 ʚ .
Therefore, we have
1 3 Δ ʚ b , z + 1 6 ʚ b , z
= 1 3 4 ʚ , 2 2 ʚ 1 ʚ 8 e 2 + 2 ʚ + 1 3 2 ʚ 2 , 2 ʚ 7 5 ʚ
= 1 3 2 ʚ 2 + ʚ , 2 ʚ 2 1 ʚ 8 e 2 ʚ + 7 .
It follows that
[ 2 3 ʚ 1 + 2 ʚ , 2 ʚ 2 1903 250 903 250 ʚ ] I 1 3 2 ʚ 2 + ʚ , 2 ʚ 2 1 ʚ 8 e 2 ʚ + 7 ,
and Theorem 7 has been demonstrated.
Theorem 8.
Let  H ˜ , Τ ˜   : b ,   z o  be two  U D -convex  F N V M s, whose  ʚ -cuts define the family of  I V M s  H ʚ ,   Τ ʚ : b ,   z X o +  are given by  H ʚ w = H * w , ʚ ,   H * w , ʚ  and  Τ ʚ w = Τ * w , ʚ ,   Τ * w , ʚ  for all  w b ,   z  and for all  ʚ 0 ,   1 , respectively. If  H ˜ Τ ˜ F A b ,   z ,   ʚ , then
2   H ˜ b + z 2 Τ ˜ b + z 2 F 1 z b F A b z H ˜ w Τ ˜ w d w M ˜ b , z 6 N ˜ b , z 3 .
where  M ˜ b , z = H ˜ b Τ ˜ b H ˜ z Τ ˜ z ,   N ˜ b , z = H ˜ b Τ ˜ z H ˜ z Τ ˜ b ,  and  M ʚ b , z = M * b , z ,   ʚ ,   M * b , z ,   ʚ  and  N ʚ b , z = N * b , z ,   ʚ ,   N * b , z ,   ʚ .
Proof. 
By hypothesis, for each ʚ 0 ,   1 , we have
H * b + z 2 , ʚ × Τ * b + z 2 , ʚ H * b + z 2 , ʚ × Τ * b + z 2 , ʚ
1 4 H * ɷ b + 1 ɷ z ,   ʚ × Τ * ɷ b + 1 ɷ z ,   ʚ + H * ɷ b + 1 ɷ z ,   ʚ × Τ * 1 ɷ b + ɷ z ,   ʚ + 1 4 H * 1 ɷ b + ɷ z ,   ʚ × Τ * ɷ b + 1 ɷ z ,   ʚ + H * 1 ɷ b + ɷ z ,   ʚ × Τ * 1 ɷ b + ɷ z ,   ʚ , 1 4 H * ɷ b + 1 ɷ z ,   ʚ × Τ * ɷ b + 1 ɷ z ,   ʚ + H * ɷ b + 1 ɷ z ,   ʚ × Τ * 1 ɷ b + ɷ z ,   ʚ + 1 4 H * 1 ɷ b + ɷ z ,   ʚ × Τ * ɷ b + 1 ɷ z ,   ʚ + H * 1 ɷ b + ɷ z ,   ʚ × Τ * 1 ɷ b + ɷ z ,   ʚ ,
1 4 H * ɷ b + 1 ɷ z ,   ʚ × Τ * ɷ b + 1 ɷ z ,   ʚ + H * 1 ɷ b + ɷ z ,   ʚ × Τ * 1 ɷ b + ɷ z ,   ʚ + 1 4 ɷ H * b ,   ʚ + 1 ɷ H * z ,   ʚ × 1 ɷ Τ * b ,   ʚ + ɷ Τ * z ,   ʚ + 1 ɷ H * b ,   ʚ + ɷ H * z ,   ʚ × ɷ Τ * b ,   ʚ + 1 ɷ Τ * z ,   ʚ , 1 4 H * ɷ b + 1 ɷ z ,   ʚ × Τ * ɷ b + 1 ɷ z ,   ʚ + H * 1 ɷ b + ɷ z ,   ʚ × Τ * 1 ɷ b + ɷ z ,   ʚ + 1 4 ɷ H * b ,   ʚ + 1 ɷ H * z ,   ʚ × 1 ɷ Τ * b ,   ʚ + ʚ Τ * z ,   ʚ + 1 ɷ H * b ,   ʚ + ʚ H * z ,   ʚ × ɷ Τ * b ,   ʚ + 1 ɷ Τ * z ,   ʚ ,
= 1 4 H * ɷ b + 1 ɷ z ,   ʚ × Τ * ɷ b + 1 ɷ z ,   ʚ + H * 1 ɷ b + ɷ z ,   ʚ × Τ * 1 ɷ b + ɷ z ,   ʚ + 1 2 ɷ 2 + 1 ɷ 2 N * b , z ,   ʚ + ɷ 1 ɷ + 1 ɷ ɷ M * b , z ,   ʚ , = 1 4 H * ɷ b + 1 ɷ z ,   ʚ × Τ * ɷ b + 1 ɷ z ,   ʚ + H * 1 ɷ b + ɷ z ,   ʚ × Τ * 1 ɷ b + ɷ z ,   ʚ + 1 2 ɷ 2 + 1 ɷ 2 N * b , z ,   ʚ + ɷ 1 ɷ + 1 ɷ ɷ M * b , z ,   ʚ .
Taking integration over 0 ,   1 , we have
2   H * b + z 2 , ʚ × Τ * b + z 2 , ʚ 1 z b   b z H * w , ʚ × Τ * w , ʚ d w + M * b , z ,   ʚ 6 + N * b , z ,   ʚ 3 ,     2   H * b + z 2 , ʚ × Τ * b + z 2 , ʚ 1 z b   b z H * w , ʚ × Τ * w , ʚ d w + M * b , z ,   ʚ 6 + N * b , z ,   ʚ 3 ,
that is
2   H ˜ b + z 2 Τ ˜ b + z 2 F 1 z b F A b z H ˜ w Τ ˜ w d w   M ˜ b , z 6 N ˜ b , z 3 .
Hence, the required result. □
Example 4.
We consider the  F N V M s  H ˜ ,   Τ ˜ : b ,   z = 0 ,   2 o . Then, for each  ʚ 0 ,   1 ,  we have  H ʚ w = ʚ w , 2 ʚ w  and  Τ ʚ w = 1 ʚ w + 2 ʚ , 1 ʚ 8 e w + 2 ʚ ,  as in Example 3, then  H ˜  and  Τ ˜  both are  U D -convex mappings. We have  H * w , ʚ = ʚ w ,   H * w ,   ʚ = 2 ʚ w  and  Τ * w , ʚ = 1 ʚ w + 2 ʚ ,  Τ * w ,   ʚ = 1 ʚ 8 e w + 2 ʚ , then
2 H * b + z 2 , ʚ × Τ * b + z 2 , ʚ = 2 ʚ 1 + ʚ , 2 H * b + z 2 , ʚ × Τ * b + z 2 , ʚ = 2 16 20 ʚ + 6 ʚ 2 + 2 3 ʚ + ʚ 2 e ,
1 z b   b z H * w , ʚ × Τ * w , ʚ d w = 1 2 0 2   ʚ 1 ʚ w 2 + 2 ʚ 2 w d w = 4 3 ʚ 3 ʚ ,
1 z b b z H * w , ʚ × Τ * w , ʚ d w = 1 2 0 2 1 ʚ 2 ʚ w 8 e w + 2 ʚ 2 ʚ w d w
2 ʚ 2 1903 250 903 250 ʚ .
Δ * b , z = H * b × Τ * b + H * z × Τ * z = 4 ʚ ,
Δ * b , z = H * b × Τ * b + H * z × Τ * z = 2 2 ʚ 1 ʚ 8 e 2 + 2 ʚ ,
* b , z = H * b × Τ * z + H * z × Τ * b = 4 ʚ 2 ,
* b , z = H * b × Τ * z + H * z × Τ * b = 2 2 ʚ 7 5 ʚ .
Therefore, we have
1 6 Δ ʚ b , z ,   ʚ + 1 3 ʚ b , z ,   ʚ
= 1 3 2 ʚ , 2 ʚ 1 ʚ 8 e 2 + 2 ʚ + 2 3 2 ʚ 2 , 2 ʚ 7 5 ʚ
= 1 3 2 ʚ 1 + 2 ʚ , 2 ʚ 1 ʚ 8 e 2 8 ʚ + 14 .
It follows that
2 ʚ 1 + ʚ , 16 20 ʚ + 6 ʚ 2 + 2 3 ʚ + ʚ 2 e I 2 3 ʚ 2 + ʚ , 2 ʚ 2 1903 250 903 250 ʚ
+ 1 3 2 ʚ 1 + 2 ʚ , 2 ʚ 1 ʚ 8 e 2 8 ʚ + 14 ,
and Theorem 8 has been demonstrated.
We now give HH-Fejér inequalities for U D -convex F N V M s. Firstly, we obtain the second HH-Fejér inequality for U D -convex F N V M .
Theorem 9.
Let  H ˜ : b ,   z o  be a  U D -convex  F N V M  with  b < z , whose  ʚ -cuts define the family of  I V M s  H ʚ : b ,   z X o +  are given by  H ʚ w = H * w , ʚ ,   H * w , ʚ  for all  w b ,   z  and for all  ʚ 0 ,   1 . If  H ˜   F A b ,   z ,   ʚ  and  B : b ,   z ,   B w 0 ,  symmetric with respect to  b + z 2 ,  then
1 z b F A b z H w B w d w F H b H z 0 1 ɷ B 1 ɷ b + ɷ z d ɷ .
Proof. 
Let H ˜ be a U D -convex F N V M . Then, for each ʚ 0 ,   1 , we have
H * ɷ b + 1 ɷ z ,   ʚ B ɷ b + 1 ɷ z ɷ H * b ,   ʚ + 1 ɷ H * z ,   ʚ B ɷ b + 1 ɷ z , H * ɷ b + 1 ɷ z ,   ʚ B ɷ b + 1 ɷ z ɷ H * b ,   ʚ + 1 ɷ H * z ,   ʚ B ɷ b + 1 ɷ z .
And
H * 1 ɷ b + ɷ z ,   ʚ B 1 ɷ b + ɷ z 1 ɷ H * b ,   ʚ + ɷ H * z ,   ʚ B 1 ɷ b + ɷ z , H * 1 ɷ b + ɷ z ,   ʚ B 1 ɷ b + ɷ z 1 ɷ H * b ,   ʚ + ɷ H * z ,   ʚ B 1 ɷ b + ɷ z .
After adding (36) and (37), and integrating over 0 ,   1 , we get
0 1 H * ɷ b + 1 ɷ z ,   ʚ B ɷ b + 1 ɷ z d ɷ + 0 1 H * 1 ɷ b + ɷ z ,   ʚ B 1 ɷ b + ɷ z d ɷ 0 1 H * b ,   ʚ ɷ B ɷ b + 1 ɷ z + 1 ɷ B 1 ɷ b + ɷ z + H * z ,   ʚ 1 ɷ B ɷ b + 1 ɷ z + ɷ B 1 ɷ b + ɷ z d ɷ , 0 1 H * 1 ɷ b + ɷ z ,   ʚ B 1 ɷ b + ɷ z d ɷ + 0 1 H * ɷ b + 1 ɷ z ,   ʚ B ɷ b + 1 ɷ z d ɷ 0 1 H * b ,   ʚ ɷ B ɷ b + 1 ɷ z + 1 ɷ B 1 ɷ b + ɷ z + H * z ,   ʚ 1 ɷ B ɷ b + 1 ɷ z + ɷ B 1 ɷ b + ɷ z d ɷ ,
= 2 H * b ,   ʚ 0 1 ɷ B ɷ b + 1 ɷ z d ɷ + 2 H * z ,   ʚ 0 1 ɷ B 1 ɷ b + ɷ z d ɷ , = 2 H * b ,   ʚ 0 1 ɷ B ɷ b + 1 ɷ z d ɷ + 2 H * z ,   ʚ 0 1 ɷ B 1 ɷ b + ɷ z d ɷ .
Since B is symmetric, then
0 1 H * ɷ b + 1 ɷ z ,   ʚ B ɷ b + 1 ɷ z d ɷ + 0 1 H * ɷ b + 1 ɷ z ,   ʚ B ɷ b + 1 ɷ z d ɷ 2 H * b ,   ʚ + H * z ,   ʚ 0 1 ɷ B 1 ɷ b + ɷ z d ɷ , 0 1 H * 1 ɷ b + ɷ z ,   ʚ B 1 ɷ b + ɷ z d ɷ + 0 1 H * ɷ b + 1 ɷ z ,   ʚ B ɷ b + 1 ɷ z d ɷ 2 H * b ,   ʚ + H * z ,   ʚ 0 1 ɷ B 1 ɷ b + ɷ z d ɷ .
Since
0 1 H * ɷ b + 1 ɷ z ,   ʚ B ɷ b + 1 ɷ z d ɷ = 0 1 H * 1 ɷ b + ɷ z ,   ʚ B 1 ɷ b + ɷ z d ɷ = 1 z b   b z H * w , ʚ B w d w 0 1 H * 1 ɷ b + ɷ z ,   ʚ B 1 ɷ b + ɷ z d ɷ = 0 1 H * ɷ b + 1 ɷ z ,   ʚ B ɷ b + 1 ɷ z d ɷ = 1 z b   b z H * w ,   ʚ B w d w
Then from (38), we have
1 z b   b z H * w , ʚ B w d w H * b ,   ʚ + H * z ,   ʚ 0 1 ɷ B 1 ɷ b + ɷ z d ɷ , 1 z b   b z H * w , ʚ B w d w   H * b ,   ʚ + H * z ,   ʚ 0 1 ɷ B 1 ɷ b + ɷ z d ɷ ,
that is
1 z b   b z H * w , ʚ B w d w ,   1 z b   b z H * w , ʚ B w d w
I [ H * b ,   ʚ + H * z ,   ʚ ,   H * b ,   ʚ + H * z ,   ʚ ] 0 1 ɷ B 1 ɷ b + ɷ z d ɷ ,
hence
1 z b F A b z H ˜ w B w d w F H ˜ b H ˜ z 0 1 ɷ B 1 ɷ b + ɷ z d ɷ .
Next, we construct first HH-Fejér inequality for U D -convex F N V M , which generalizes first HH-Fejér inequalities for classical convex mapping. □
Theorem 10.
Let  H ˜ : b ,   z o  be a  U D -convex  F N V M  with  b < z , whose  ʚ -cuts define the family of  I V M s  H ʚ : b ,   z X o +  are given by  H ʚ w = H * w , ʚ ,   H * w , ʚ  for all  w b ,   z  and for all  ʚ 0 ,   1 . If  H ˜ F A b ,   z ,   ʚ  and  B : b ,   z ,   B w 0 ,  symmetric with respect to  b + z 2 ,  and  b z B w d w > 0 , then
  H ˜ b + z 2 F 1 b z B w d w F A b z H ˜ w B w d w .
Proof. 
Since H ˜ is a U D -convex, then for ʚ 0 ,   1 , we have
H * b + z 2 ,   ʚ 1 2 H * ɷ b + 1 ɷ z ,   ʚ + H * 1 ɷ b + ɷ z ,   ʚ , H * b + z 2 ,   ʚ 1 2 H * ɷ b + 1 ɷ z ,   ʚ + H * 1 ɷ b + ɷ z ,   ʚ ,
Since B ɷ b + 1 ɷ z = B 1 ɷ b + ɷ z , then by multiplying (41) by B 1 ɷ b + ɷ z and integrating it with respect to ɷ over 0 ,   1 , we obtain
H * b + z 2 ,   ʚ 0 1 B 1 ɷ b + ɷ z d ɷ 1 2 0 1 H * ɷ b + 1 ɷ z ,   ʚ B ɷ b + 1 ɷ z d ɷ + 0 1 H * 1 ɷ b + ɷ z ,   ʚ B 1 ɷ b + ɷ z d ɷ , H * b + z 2 ,   ʚ 0 1 B 1 ɷ b + ɷ z d ɷ 1 2 0 1 H * ɷ b + 1 ɷ z ,   ʚ B ɷ b + 1 ɷ z d ɷ + 0 1 H * 1 ɷ b + ɷ z ,   ʚ B 1 ɷ b + ɷ z d ɷ .
Since
0 1 H * ɷ b + 1 ɷ z ,   ʚ B ɷ b + 1 ɷ; z d ɷ = 0 1 H * 1 ɷ b + ɷ z ,   ʚ B 1 ɷ b + ɷ z d ɷ = 1 z b   b z H * w , ʚ B w d w 0 1 H * 1 ɷ b + ɷ z ,   ʚ B 1 ɷ b + ɷ z d ɷ = 0 1 H * ɷ b + 1 ɷ z ,   ʚ B ɷ b + 1 ɷ z d ɷ = 1 z b   b z H * w , ʚ B w d w .
Then from (43), we have
H * b + z 2 ,   ʚ 1 b z B w d w   b z H * w , ʚ B w d w ,     H * b + z 2 ,   ʚ 1 b z B w d w   b z H * w , ʚ B w d w ,
from which, we have
H * b + z 2 ,   ʚ ,     H * b + z 2 ,   ʚ I 1 b z B w d w b z H * w , ʚ B w d w ,     b z H * w , ʚ B w d w ,
that is
H ˜ b + z 2 F 1 b z B w d w   F A b z H ˜ w B w d w .
This completes the proof. □
Remark 5.
From Theorem 9 and Theorem 10, we clearly see that:
If  W w = 1 , then we acquire the inequality (23).
If  H  is lower  U D -convex  F N V M  on  b , z ,  then we acquire the following coming inequality, see [90]:
H b + z 2 F 1 b z W w d w F A b z H w W w d w F H b H z 2 .
If  H  is lower  U D -convex  F N V M  on  b , z  with  ʚ   =   , then from (35) and (40) we acquire the following coming inequality, see [99]:
H b + z 2 I 1 z b   I A b z H w d w I H b + H z 2 .
If  H  is lower  U D -convex  F N V M  on  b , z  with  ʚ   =   , then from (35) and (40) we acquire the following coming inequality, see [99]:
H b + z 2 I 1 b z W w d w   I A b z H w W w d w I H b + H z 2 .
Let  ʚ   =   . Then from (35) and (40), we acquire the following inequality, see [56]:
H b + z 2 1 b z W w d w   I A b z H w W w d w H b + H z 2 .
Let  ʚ = 1  and  H * w ,   ʚ = H * w ,   ʚ . Then, from(35) and (40), we obtain the following classical Fejér inequality:
H b + z 2 1 b z W w d w b z H w W w d w H b + H z 2 .
Example 5.
We consider the  F N V M   H : 0 ,   2 I  defined by
H w θ = θ 2 + w 1 2 3 2 2 w 1 2             θ 2 w 1 2 ,   3 2 , 2 + w 1 2 θ 2 + w 1 2 3 2           θ 3 2 ,   2 + w 1 2 , 0                     otherwise ,
Then, for each  ʚ 0 ,   1 ,  we have  H ʚ w = 1 ʚ 2 w 1 2 + 3 2 ʚ , 1 + ʚ 2 + w 1 2 + 3 2 ʚ . Since end point mappings  H * w , ʚ ,  and  H * w , ʚ  are convex and concave mappings, respectively, for each  ʚ 0 ,   1 , then  H w  is  U D -convex  F N V M . If
B w = w ,             σ 0 , 1 , 2 w ,       σ 1 ,   2 ,  
then  B 2 w = B w 0 , for all  w 0 ,   2 .
Since H * w , ʚ = 1 ʚ 2 w 1 2 + 3 2 ʚ and  H * w , ʚ = 1 + ʚ 2 + w 1 2 + 3 2 ʚ . Now
we compute the following:
1 z b b z H * w , ʚ B w d w = 1 2 0 2 H * w , ʚ B w d w                 = 1 2 0 1 H * w , ʚ B w d w + 1 2 1 2 H * w , ʚ B w d w , 1 z b b z H * w , ʚ B w d w = 1 2 0 2 H * w , ʚ B w d w                 = 1 2 0 1 H * w , ʚ B w d w + 1 2 1 2 H * w , ʚ B w d w ,
= 1 2 0 1 1 ʚ 2 w 1 2 + 3 2 ʚ w d w + 1 2 1 2 1 ʚ 2 w 1 2 + 3 2 ʚ 2 w d w = 1 4 13 3 π 2 + ʚ π 8 1 12 , = 1 2 0 1 1 + ʚ 2 + w 1 2 + 3 2 ʚ w d w + 1 2 1 2 1 + ʚ 2 + w 1 2 + 3 2 ʚ 2 w d w = 1 4 19 3 + π 2 + ʚ π 8 + 31 12 .
And
H * b ,   ʚ + H * z ,   ʚ 0 1 ɷ B 1 ɷ b + ɷ z d ɷ = 4 1 ʚ 2 1 ʚ + 3 ʚ 0 1 2 ɷ 2 ɷ d ɷ + 1 2 1 ɷ 2 1 ɷ d ɷ = 1 3 4 1 ʚ 2 1 ʚ + 3 ʚ , H * b ,   ʚ + H * z ,   ʚ 0 1 ɷ B 1 ɷ b + ɷ z d ɷ = 4 1 + ʚ + 2 1 + ʚ + 3 ʚ 0 1 2 ɷ 2 ɷ d ɷ + 1 2 1 ɷ 2 1 ɷ d ɷ = 1 3 4 1 + ʚ + 2 1 + ʚ + 3 ʚ .
From (49) and (50), we have
1 4 13 3 π 2 + ʚ π 4 7 6 ,   1 4 19 3 + π 2 + ʚ π 4 + 25 6
I 1 3 4 1 ʚ 2 1 ʚ + 3 ʚ ,   1 3 4 1 + ʚ + 2 1 + ʚ + 3 ʚ ,   for   all   ʚ 0 ,   1 .
Hence, Theorem 9 is verified.
For Theorem 10, we have
H * b + z 2 ,   ʚ = H * 1 ,   ʚ = 2 + ʚ 2 , H * b + z 2 ,   ʚ = H * 1 ,   ʚ = 3 2 + 3 ʚ 2 ,
b z B w d w = 0   1 w d w + 1 2 2 w d w = 4 3 ,
1 b z B w d w   b z H * w , ʚ B w d w = 3 8 13 3 π 2 + 3 ʚ 2 π 8 1 12 ,   1 b z B w d w   b z H * w , ʚ B w d w = 3 8 19 3 + π 2 + 3 ʚ 2 π 8 + 31 12 .
From (51) and (52), we have
2 + ʚ 2 ,   3 2 + 3 ʚ 2 I 3 8 13 3 π 2 + 3 ʚ 2 π 8 1 12 ,   3 8 19 3 + π 2 + 3 ʚ 2 π 8 + 31 12 .
Hence, Theorem 10 has been verified.

4. Conclusions

This paper provides the introduced class U D -convex concept for F N V M s. The H.H. and Jensen-type inequalities were developed utilizing this idea and a fuzzy-inclusion relation. This study expands on several recent findings made by Zhao et al. [56,57] and the writers who came after them, Refs. [61,62]. Furthermore, some nontrivial cases are provided to verify our primary conclusions’ accuracy. In the future, it will be fascinating to look into how analogous inequalities are established for other convexity types and by employing various integral operators. Our study of interval integral operator-type integral inequalities will broaden their practical applications because integral operators are widely used in engineering technology, such as various forms of mathematical modeling, and because different integral operators are suitable for different forms of practical problems. Convex optimization theory may take a new turn as a result of this idea. Other researchers working on a range of scientific subjects may probably find the idea useful.

Author Contributions

Conceptualization, M.B.K.; methodology, M.B.K.; validation, A.M.A.; formal analysis, H.A.O. and A.M.A.; investigation, M.B.K.; resources, H.A.O. and A.M.A.; data curation, M.G.V.; writing—original draft preparation, M.B.K.; writing—review and editing, M.B.K. and A.M.A.; visualization, M.G.V.; supervision, M.B.K. and L.A.; project administration, M.B.K. and L.A.; funding acquisition, A.M.A. and M.G.V. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Not applicable.

Acknowledgments

The authors would like to thank the Rector, COMSATS University Islamabad, Islamabad, Pakistan. The authors would like to thank the Deanship of Scientific Research at Umm Al-Qura University for supporting this work by Grant Code: 22UQU4330052DSR11. This research was also supported by Office of Research Management, Universiti Malaysia Terengganu, Malaysia.

Conflicts of Interest

The authors declare no conflict of interest.

References

  1. Sarikaya, M.Z.; Saglam, A.; Yildirim, H. On some Hadamard-type inequalities for h-convex functions. J. Math. Inequal. 2008, 2, 335–341. [Google Scholar] [CrossRef] [Green Version]
  2. Bombardelli, M.; Varoşanec, S. Properties of h-convex functions related to the Hermite-Hadamard-Fejer inequalities. Comput. Math. Appl. 2009, 58, 1869–1877. [Google Scholar] [CrossRef] [Green Version]
  3. Noor, M.A.; Noor, K.I.; Awan, M.U. A new Hermite-Hadamard type inequality for h-convex functions. Creat. Math. Inform. 2015, 2, 191–197. [Google Scholar]
  4. Khan, M.B.; Santos-García, G.; Treanțǎ, S.; Noor, M.A.; Soliman, M.S. Perturbed Mixed Variational-Like Inequalities and Auxiliary Principle Pertaining to a Fuzzy Environment. Symmetry 2022, 14, 2503. [Google Scholar] [CrossRef]
  5. Khan, M.B.; Santos-García, G.; Noor, M.A.; Soliman, M.S. New Class of Preinvex Fuzzy Mappings and Related Inequalities. Mathematics 2022, 10, 3753. [Google Scholar] [CrossRef]
  6. Khan, M.B.; Macías-Díaz, J.E.; Treanțǎ, S.; Soliman, M.S. Some Fejér-Type Inequalities for Generalized Interval-Valued Convex Functions. Mathematics 2022, 10, 3851. [Google Scholar] [CrossRef]
  7. Liu, Z.-H.; Sofonea, M.T. Differential quasivariational inequalities in contact mechanics, Math. Mech. Solids. 2019, 24, 845–861. [Google Scholar] [CrossRef]
  8. Zeng, S.-D.; Migórski, S.; Liu, Z.-H.; Yao, J.-C. Convergence of a generalized penalty method for variational-hemivariational inequalities. Commun. Nonlinear Sci. Numer. Simul. 2021, 92, 105476. [Google Scholar] [CrossRef]
  9. Li, X.-W.; Li, Y.-X.; Liu, Z.-H.; Li, J. Sensitivity analysis for optimal control problems described by nonlinear fractional evolution inclusions. Fract. Calc. Appl. Anal. 2018, 21, 1439–1470. [Google Scholar] [CrossRef]
  10. Liu, Z.-H.; Papageorgiou, N.S. Positive solutions for resonant (p,q)-equations with convection. Adv. Nonlinear Anal. 2021, 10, 217–232. [Google Scholar] [CrossRef]
  11. Dragomir, S.S.; Mond, B. Integral inequalities of Hadamard type for log-convex functions. Demonstr. Math. 1998, 31, 355–364. [Google Scholar] [CrossRef]
  12. Dragomir, S.S. Refinements of the Hermite-Hadamard integral inequality for log-convex functions. RGMIA Res. Rep. Collect. 2000, 3, 219–225. [Google Scholar]
  13. Niculescu, C.P. The Hermite–Hadamard inequality for log-convex functions. Nonlinear Anal. 2000, 3, 219–225. [Google Scholar] [CrossRef]
  14. Khan, M.B.; Zaini, H.G.; Santos-García, G.; Noor, M.A.; Soliman, M.S. New Class Up and Down λ-Convex Fuzzy-Number Valued Mappings and Related Fuzzy Fractional Inequalities. Fractal Fract. 2022, 6, 679. [Google Scholar] [CrossRef]
  15. Khan, M.B.; Zaini, H.G.; Macías-Díaz, J.E.; Soliman, M.S. Up and Down -Pre-Invex Fuzzy-Number Valued Mappings and Some Certain Fuzzy Integral Inequalities. Axioms 2023, 12, 1. [Google Scholar] [CrossRef]
  16. Khan, M.B.; Noor, M.A.; Macías-Díaz, J.E.; Soliman, M.S.; Zaini, H.G. Some integral inequalities for generalized left and right log convex interval-valued functions based upon the pseudo-order relation. Demonstr. Math. 2022, 55, 387–403. [Google Scholar] [CrossRef]
  17. Khan, M.B.; Noor, M.A.; Zaini, H.G.; Santos-García, G.; Soliman, M.S. The New Versions of Hermite–Hadamard Inequalities for Pre-invex Fuzzy-Interval-Valued Mappings via Fuzzy Riemann Integrals. Int. J. Comput. Intell. Syst. 2022, 15, 66. [Google Scholar] [CrossRef]
  18. Khan, M.B.; Santos-García, G.; Noor, M.A.; Soliman, M.S. New Hermite–Hadamard Inequalities for Convex Fuzzy-Number-Valued Mappings via Fuzzy Riemann Integrals. Mathematics 2022, 10, 3251. [Google Scholar] [CrossRef]
  19. Khan, M.B.; Treanțǎ, S.; Soliman, M.S. Generalized Preinvex Interval-Valued Functions and Related Hermite–Hadamard Type Inequalities. Symmetry 2022, 14, 1901. [Google Scholar] [CrossRef]
  20. Dragomir, S.S. On the Hadamard’s inequlality for convex functions on the co-ordinates in a rectangle from the plane. Taiwan J. Math. 2001, 5, 775–788. [Google Scholar] [CrossRef]
  21. Zhao, D.; Zhao, G.; Ye, G.; Liu, W.; Dragomir, S.S. On Hermite–Hadamard-Type Inequalities for Coordinated h-Convex Interval-Valued Functions. Mathematics 2001, 9, 2352. [Google Scholar] [CrossRef]
  22. Faisal, S.; Khan, M.A.; Iqbal, S. Generalized Hermite-Hadamard-Mercer type inequalities via majorization. Filomat 2022, 36, 469–483. [Google Scholar] [CrossRef]
  23. Faisal, S.; Adil Khan, M.; Khan, T.U.; Saeed, T.; Alshehri, A.M.; Nwaeze, E.R. New “Conticrete” Hermite–Hadamard–Jensen–Mercer Fractional Inequalities. Symmetry 2022, 14, 294. [Google Scholar] [CrossRef]
  24. Dragomir, S.S. Inequalities of Hermite–Hadamard type for functions of selfadjoint operators and matrices. J. Math. Inequalities 2017, 11, 241–259. [Google Scholar] [CrossRef] [Green Version]
  25. Stojiljković, V.; Ramaswamy, R.; Ashour Abdelnaby, O.A.; Radenović, S. Riemann-Liouville Fractional Inclusions for Convex Functions Using Interval Valued Setting. Mathematics 2022, 10, 3491. [Google Scholar] [CrossRef]
  26. Stojiljković, V.; Ramaswamy, R.; Alshammari, F.; Ashour, O.A.; Alghazwani, M.L.H.; Radenović, S. Hermite–Hadamard Type Inequalities Involving (k-p) Fractional Operator for Various Types of Convex Functions. Fractal Fract. 2022, 6, 376. [Google Scholar] [CrossRef]
  27. Wang, H. Certain integral inequalities related to (φ,ϱα)–Lipschitzian mappings and generalized h–convexity on fractal sets. J. Nonlinear Funct. Anal. 2021, 2021, 12. [Google Scholar]
  28. Tam, N.; Wen, C.; Yao, J.; Yen, N. Structural convexity and ravines of quadratic functions. J. Appl. Numer. Optim. 2021, 3, 425–434. [Google Scholar]
  29. Khan, M.B.; Alsalami, O.M.; Treanțǎ, S.; Saeed, T.; Nonlaopon, K. New class of convex interval-valued functions and Riemann Liouville fractional integral inequalities. AIMS Math. 2022, 7, 15497–15519. [Google Scholar] [CrossRef]
  30. Saeed, T.; Khan, M.B.; Treanțǎ, S.; Alsulami, H.H.; Alhodaly, M.S. Interval Fejér-Type Inequalities for Left and Right-λ-Preinvex Functions in Interval-Valued Settings. Axioms 2022, 11, 368. [Google Scholar] [CrossRef]
  31. Khan, M.B.; Cătaş, A.; Alsalami, O.M. Some New Estimates on Coordinates of Generalized Convex Interval-Valued Functions. Fractal Fract. 2022, 6, 415. [Google Scholar] [CrossRef]
  32. Santos-García, G.; Khan, M.B.; Alrweili, H.; Alahmadi, A.A.; Ghoneim, S.S. Hermite–Hadamard and Pachpatte type inequalities for coordinated preinvex fuzzy-interval-valued functions pertaining to a fuzzy-interval double integral operator. Mathematics 2022, 10, 2756. [Google Scholar] [CrossRef]
  33. Macías-Díaz, J.E.; Khan, M.B.; Alrweili, H.; Soliman, M.S. Some Fuzzy Inequalities for Harmonically s-Convex Fuzzy Number Valued Functions in the Second Sense Integral. Symmetry 2022, 14, 1639. [Google Scholar] [CrossRef]
  34. Moore, R.E. Interval Analysis; Prentice Hall: Hoboken, NJ, USA, 1966. [Google Scholar]
  35. Snyder, J.M. Interval analysis for computer graphics. In Proceedings of the 19th Annual Conference on Computer Graphics and Interactive Techniques, Chicago, IL, USA, 27–31 July 1992; pp. 121–130. [Google Scholar]
  36. Rahman, M.S.; Shaikh, A.A.; Bhunia, A.K. Necessary and sufficient optimality conditions for non-linear unconstrained and constrained optimization problem with interval valued objective function. Comput. Ind. Eng. 2020, 147, 106634. [Google Scholar] [CrossRef]
  37. Qian, Y.; Liang, J.; Dang, C. Interval ordered information systems. Comput. Math. Appl. 2008, 56, 1994–2009. [Google Scholar] [CrossRef] [Green Version]
  38. De Weerdt, E.; Chu, Q.P.; Mulder, J.A. Neural network output optimization using interval analysis. IEEE Trans. Neural Netw. 2009, 20, 638–653. [Google Scholar] [CrossRef] [Green Version]
  39. Gao, W.; Song, C.; Tin-Loi, F. Probabilistic interval analysis for strucqrres with uncertainty. Struct. Saf. 2010, 32, 191–199. [Google Scholar] [CrossRef]
  40. Wang, X.; Wang, L.; Qiu, Z. A feasible implementation procedure for interval analysis method from measurement data. Appl. Math. Model. 2014, 38, 2377–2397. [Google Scholar] [CrossRef]
  41. Mizukoshi, M.T.; Lodwick, W.A. The interval eigenvalue problem using constraint interval analysis with an application to linear differential equations. Fuzzy Sets Syst. 2021, 419, 141–157. [Google Scholar] [CrossRef]
  42. Jiang, C.; Han, X.; Guan, F.J.; Li, Y.H. An uncertain structural optimization method based on nonlinear interval number programming and interval analysis method. Eng. Struct. 2007, 29, 3168–3177. [Google Scholar] [CrossRef]
  43. Wang, C.; Li, J.; Guo, P. The normalized interval regression model with outlier detection and its real-world application to house pricing problems. Fuzzy Sets Syst. 2015, 274, 109–123. [Google Scholar] [CrossRef]
  44. Zhao, T.H.; Castillo, O.; Jahanshahi, H.; Yusuf, A.; Alassafi, M.O.; Alsaadi, F.E.; Chu, Y.M. A fuzzy-based strategy to suppress the novel coronavirus (2019-NCOV) massive outbreak. Appl. Comput. Math. 2021, 20, 160–176. [Google Scholar]
  45. Zhao, T.H.; Wang, M.K.; Chu, Y.M. On the bounds of the perimeter of an ellipse. Acta Math. Sci. 2022, 42B, 491–501. [Google Scholar] [CrossRef]
  46. Zhao, T.H.; Wang, M.K.; Hai, G.J.; Chu, Y.M. Landen inequalities for Gaussian hypergeometric function. RACSAM Rev. R. Acad. A 2022, 116, 1–23. [Google Scholar] [CrossRef]
  47. Wang, M.K.; Hong, M.Y.; Xu, Y.F.; Shen, Z.H.; Chu, Y.M. Inequalities for generalized trigonometric and hyperbolic functions with one parameter. J. Math. Inequal. 2020, 14, 1–21. [Google Scholar] [CrossRef]
  48. Zhao, T.H.; Qian, W.M.; Chu, Y.M. Sharp power mean bounds for the tangent and hyperbolic sine means. J. Math. Inequal. 2021, 15, 1459–1472. [Google Scholar] [CrossRef]
  49. Chu, Y.M.; Wang, G.D.; Zhang, X.H. The Schur multiplicative and harmonic convexities of the complete symmetric function. Math. Nachr. 2011, 284, 53–663. [Google Scholar] [CrossRef]
  50. Chu, Y.M.; Xia, W.F.; Zhang, X.H. The Schur concavity, Schur multiplicative and harmonic convexities of the second dual form of the Hamy symmetric function with applications. J. Multivariate Anal. 2012, 105, 412–421. [Google Scholar] [CrossRef] [Green Version]
  51. Hajiseyedazizi, S.N.; Samei, M.E.; Alzabut, J.; Chu, Y.M. On multi-step methods for singular fractional q-integro-differential equations. Open Math. 2021, 19, 1378–1405. [Google Scholar] [CrossRef]
  52. Jin, F.; Qian, Z.S.; Chu, Y.M.; Rahman, M. On nonlinear evolution model for drinking behavior under Caputo-Fabrizio derivative. J. Appl. Anal. Comput. 2022, 12, 790–806. [Google Scholar] [CrossRef]
  53. Wang, F.Z.; Khan, M.N.; Ahmad, I.; Ahmad, H.; Abu-Zinadah, H.; Chu, Y.M. Numerical solution of traveling waves in chemical kinetics: Time-fractional fisher’s equations. Fractals 2022, 30, 2240051. [Google Scholar] [CrossRef]
  54. Chalco-Cano, Y.; Flores-Franulic, A.; Román-Flores, H. Ostrowski type inequalities for interval-valued functions using generalized Hukuhara derivative. Comput. Appl. Math. 2012, 31, 457–472. [Google Scholar]
  55. Costa, T.M.; Román-Flores, H.; Chalco-Cano, Y. Opial-type inequalities for interval-valued functions. Fuzzy Sets Syst. 2019, 358, 48–63. [Google Scholar] [CrossRef]
  56. Román-Flores, H.; Chalco-Cano, Y.; Lodwick, W. Some integral inequalities for interval-valued functions. Comput. Appl. Math. 2016, 37, 22–29. [Google Scholar] [CrossRef]
  57. Zhao, D.; An, T.; Ye, G.; Liu, W. New Jensen and Hermite-Hadamard type inequalities for h-convex interval-valued functions. J. Inequalities Appl. 2018, 2018, 302. [Google Scholar] [CrossRef] [Green Version]
  58. Ibrahim, M.; Nabi, S.; Baz, A.; Alhakami, H.; Raza, M.S.; Hussain, A.; Salah, K.; Djemame, K. An In-Depth Empirical Investigation of State-of-the-Art Scheduling Approaches for Cloud Computing. IEEE Access 2020, 8, 128282–128294. [Google Scholar] [CrossRef]
  59. Talpur, N.; Abdulkadir, S.J.; Alhussian, H.; Hasan, M.H.; Aziz, N.; Bamhdi, A. A comprehensive review of deep neuro-fuzzy system architectures and their optimization methods. Neural Comput. Applic. 2022, 34, 1837–1875. [Google Scholar] [CrossRef]
  60. Alsaedi, A.; Ahmad, B.; Assolami, A.; Ntouyas, S.K. On a nonlinear coupled system of differential equations involving Hilfer fractional derivative and Riemann-Liouville mixed operators with nonlocal integro-multi-point boundary conditions. AIMS Mathematics 2022, 7, 12718–12741. [Google Scholar] [CrossRef]
  61. Khan, M.B.; Noor, M.A.; Al-Bayatti, H.M.; Noor, K.I. Some New Inequalities for LR-Log-h-Convex Interval-Valued Functions by Means of Pseudo Order Relation. Appl. Math. 2021, 15, 459–470. [Google Scholar]
  62. Khan, M.B.; Noor, M.A.; Abdeljawad, T.; Mousa, A.A.A.; Abdalla, B.; Alghamdi, S.M. LR-Preinvex Interval-Valued Functions and Riemann-Liouville Fractional Integral Inequalities. Fractal Fract. 2021, 5, 243. [Google Scholar] [CrossRef]
  63. Bhunia, A.K.; Samanta, S.S. A study of interval metric and its application in multi-objective optimization with interval objectives. Comput. Ind. Eng. 2014, 74, 169–178. [Google Scholar] [CrossRef]
  64. Zhao, T.H.; Bhayo, B.A.; Chu, Y.M. Inequalities for generalized Grötzsch ring function. Comput. Meth. Funct. Theory 2022, 22, 559–574. [Google Scholar] [CrossRef]
  65. Zhao, T.H.; He, Z.Y.; Chu, Y.M. Sharp bounds for the weighted Hölder mean of the zero-balanced generalized complete elliptic integrals. Comput. Meth. Funct. Theory 2021, 21, 413–426. [Google Scholar] [CrossRef]
  66. Zhao, T.H.; Wang, M.K.; Chu, Y.M. Concavity and bounds involving generalized elliptic integral of the first kind. J. Math. Inequal. 2021, 15, 701–724. [Google Scholar] [CrossRef]
  67. Zhao, T.H.; Wang, M.K.; Chu, Y.M. Monotonicity and convexity involving generalized elliptic integral of the first kind. RACSAM Rev. R. Acad. A 2021, 115, 1–13. [Google Scholar] [CrossRef]
  68. Chu, H.H.; Zhao, T.H.; Chu, Y.M. Sharp bounds for the Toader mean of order 3 in terms of arithmetic, quadratic and contra harmonic means. Math. Slovaca 2020, 70, 1097–1112. [Google Scholar] [CrossRef]
  69. Zhao, T.H.; He, Z.Y.; Chu, Y.M. On some refinements for inequalities involving zero-balanced hyper geometric function. AIMS Math. 2020, 5, 6479–6495. [Google Scholar] [CrossRef]
  70. Zhao, T.H.; Wang, M.K.; Chu, Y.M. A sharp double inequality involving generalized complete elliptic integral of the first kind. AIMS Math. 2020, 5, 4512–4528. [Google Scholar] [CrossRef]
  71. Zhao, T.H.; Shi, L.; Chu, Y.M. Convexity and concavity of the modified Bessel functions of the first kind with respect to Hölder means. RACSAM Rev. R. Acad. A 2020, 114, 1–14. [Google Scholar] [CrossRef]
  72. Zhao, T.H.; Zhou, B.C.; Wang, M.K.; Chu, Y.M. On approximating the quasi-arithmetic mean. J. Inequal. Appl. 2019, 2019, 42. [Google Scholar] [CrossRef] [Green Version]
  73. Zhao, T.H.; Wang, M.K.; Zhang, W.; Chu, Y.M. Quadratic transformation inequalities for Gaussian hyper geometric function. J. Inequal. Appl. 2018, 2018, 251. [Google Scholar] [CrossRef] [PubMed] [Green Version]
  74. Chu, Y.M.; Zhao, T.H. Concavity of the error function with respect to Hölder means. Math. Inequal. Appl. 2016, 19, 589–595. [Google Scholar] [CrossRef]
  75. Qian, W.M.; Chu, H.H.; Wang, M.K.; Chu, Y.M. Sharp inequalities for the Toader mean of order –1 in terms of other bivariate means. J. Math. Inequal. 2022, 16, 127–141. [Google Scholar] [CrossRef]
  76. Othman, H.A. On fuzzy θ-generalized-semi-closed sets. J. Adv. Stud. Topol. 2016, 7, 84–92. [Google Scholar] [CrossRef] [Green Version]
  77. Elghribi, M.; Othman, H.A.; Al-Nashri, A.H.A. Homogeneous functions: New characterization and applications. Trans. A. Razmadze Math. Inst. 2017, 171, 1–10. [Google Scholar] [CrossRef]
  78. Khan, M.B.; Noor, M.A.; Abdullah, L.; Chu, Y.M. Some new classes of preinvex fuzzy-interval-valued functions and inequalities. Int. J. Comput. Intell. Syst. 2021, 14, 1403–1418. [Google Scholar] [CrossRef]
  79. Khan, M.B.; Santos-García, G.; Noor, M.A.; Soliman, M.S. Some new concepts related to fuzzy fractional calculus for up and down convex fuzzy-number valued functions and inequalities. Chaos Solitons Fractals 2022, 164, 112692. [Google Scholar] [CrossRef]
  80. Zhao, T.H.; Chu, H.H.; Chu, Y.M. Optimal Lehmer mean bounds for the nth power-type Toader mean of n = −1, 1, 3. J. Math. Inequal. 2022, 16, 157–168. [Google Scholar] [CrossRef]
  81. Zhao, T.H.; Wang, M.K.; Dai, Y.Q.; Chu, Y.M. On the generalized power-type Toader mean. J. Math. Inequal. 2022, 16, 247–264. [Google Scholar] [CrossRef]
  82. Huang, T.R.; Chen, L.; Chu, Y.M. Asymptotically sharp bounds for the complete p-elliptic integral of the first kind. Hokkaido Math. J. 2022, 51, 189–210. [Google Scholar] [CrossRef]
  83. Zhao, T.H.; Qian, W.M.; Chu, Y.M. On approximating the arc lemniscate functions. Indian J. Pure Appl. Math. 2022, 53, 316–329. [Google Scholar] [CrossRef]
  84. Liu, Z.H.; Motreanu, D.; Zeng, S.D. Generalized penalty and regularization method for differential variational- hemivariational inequalities. SIAM J. Optim. 2021, 31, 1158–1183. [Google Scholar] [CrossRef]
  85. Liu, Y.J.; Liu, Z.H.; Wen, C.F.; Yao, J.C.; Zeng, S.D. Existence of solutions for a class of noncoercive variational-hemivariational inequalities arising in contact problems. Appl. Math. Optim. 2021, 84, 2037–2059. [Google Scholar] [CrossRef]
  86. Zeng, S.D.; Migorski, S.; Liu, Z.H. Well-posedness, optimal control, and sensitivity analysis for a class of differential variational-hemivariational inequalities. SIAM J. Optim. 2021, 31, 2829–2862. [Google Scholar] [CrossRef]
  87. Liu, Y.J.; Liu, Z.H.; Motreanu, D. Existence and approximated results of solutions for a class of nonlocal elliptic variational-hemivariational inequalities. Math. Method Appl. Sci. 2020, 43, 9543–9556. [Google Scholar] [CrossRef]
  88. Liu, Y.J.; Liu, Z.H.; Wen, C.F. Existence of solutions for space-fractional parabolic hemivariational inequalities. Discrete Contin. Dyn. Syst. Ser. B 2019, 24, 1297–1307. [Google Scholar] [CrossRef] [Green Version]
  89. Nanda, N.; Kar, K. Convex fuzzy mappings. Fuzzy Sets Syst. 1992, 48, 129–132. [Google Scholar] [CrossRef]
  90. Khan, M.B.; Noor, M.A.; Noor, K.I.; Chu, Y.M. New Hermite–Hadamard–type inequalities for (h1, h2)-convex fuzzy-interval-valued functions. Adv. Differ. Equ. 2021, 2021, 149. [Google Scholar] [CrossRef]
  91. Kulish, U.; Miranker, W. Computer Arithmetic in Theory and Practice; Academic Press: New York, NY, USA, 2014. [Google Scholar]
  92. Khan, M.B.; Noor, M.A.; Shah, N.A.; Abualnaja, K.M.; Botmart, T. Some New Versions of Hermite–Hadamard Integral Inequalities in Fuzzy Fractional Calculus for Generalized Pre-Invex Functions via Fuzzy-Interval-Valued Settings. Fractal Fract. 2022, 6, 83. [Google Scholar] [CrossRef]
  93. Bede, B. Mathematics of Fuzzy Sets and Fuzzy Logic. Studies in Fuzziness and Soft Computing; Springer: Berlin, Germany, 2013; p. 295. [Google Scholar]
  94. Diamond, P.; Kloeden, P.E. Metric Spaces of Fuzzy Sets: Theory and Applications; World Scientific: Singapore, 1994. [Google Scholar]
  95. Kaleva, O. Fuzzy differential equations. Fuzzy Sets Syst. 1987, 24, 301–317. [Google Scholar] [CrossRef]
  96. Costa, T.M.; Roman-Flores, H. Some integral inequalities for fuzzy-interval-valued functions. Inf. Sci. 2017, 420, 110–125. [Google Scholar] [CrossRef]
  97. Breckner, W.W. Continuity of generalized convex and generalized concave set–valued functions. Rev. Anal. Numér. Théor. Approx. 1993, 22, 39–51. [Google Scholar]
  98. Sadowska, E. Hadamard inequality and a refinement of Jensen inequality for set-valued functions. Result Math. 1997, 32, 332–337. [Google Scholar] [CrossRef]
  99. Khan, M.B.; Treanțǎ, S.; Soliman, M.S.; Nonlaopon, K.; Zaini, H.G. Some Hadamard–Fejér Type Inequalities for LR-Convex Interval-Valued Functions. Fractal Fract. 2022, 6, 6. [Google Scholar] [CrossRef]
  100. Khan, M.B.; Santos-García, G.; Treanțǎ, S.; Soliman, M.S. New Class Up and Down Pre-Invex Fuzzy Number Valued Mappings and Related Inequalities via Fuzzy Riemann Integrals. Symmetry 2022, 14, 2322. [Google Scholar] [CrossRef]
  101. Khan, M.B.; Macías-Díaz, J.E.; Soliman, M.S.; Noor, M.A. Some New Integral Inequalities for Generalized Preinvex Functions in Interval-Valued Settings. Axioms 2022, 11, 622. [Google Scholar] [CrossRef]
  102. Aubin, J.P.; Cellina, A. Differential Inclusions: Set-Valued Maps and Viability Theory, Grundlehren der Mathematischen Wissenschaften; Springer: New York, NY, USA, 1984. [Google Scholar]
  103. Aubin, J.P.; Frankowska, H. Set-Valued Analysis; Birkhäuser: Boston, MA, USA, 1990. [Google Scholar]
  104. Costa, T.M. Jensen’s inequality type integral for fuzzy-interval-valued functions. Fuzzy Sets Syst. 2017, 327, 31–47. [Google Scholar] [CrossRef]
  105. Zhang, D.; Guo, C.; Chen, D.; Wang, G. Jensen’s inequalities for set-valued and fuzzy set-valued functions. Fuzzy Sets Syst. 2020, 2020, 1–27. [Google Scholar] [CrossRef]
  106. Liu, Z.H.; Loi, N.V.; Obukhovskii, V. Existence and global bifurcation of periodic solutions to a class of differential variational inequalities. Int. J. Bifurcat. Chaos Appl. Sci. Eng. 2013, 23, 1350125. [Google Scholar] [CrossRef]
  107. Ashpazzadeh, E.; Chu, Y.-M.; Hashemi, M.S.; Moharrami, M.; Inc, M. Hermite multiwavelets representation for the sparse solution of nonlinear Abel’s integral equation. Appl. Math. Comput. 2022, 427, 127171. [Google Scholar] [CrossRef]
  108. Chu, Y.-M.; Ullah, S.; Ali, M.; Tuzzahrah, G.F.; Munir, T. Numerical investigation of Volterra integral equations of second kind using optimal homotopy asymptotic methd. Appl. Math. Comput. 2022, 430, 127304. [Google Scholar]
  109. Chu, Y.-M.; Inc, M.; Hashemi, M.S.; Eshaghi, S. Analytical treatment of regularized Prabhakar fractional differential equations by invariant subspaces. Comput. Appl. Math. 2022, 41, 271. [Google Scholar] [CrossRef]
  110. Zeng, S.-D.; Migórski, S.; Liu, Z.-H. Nonstationary incompressible Navier-Stokes system governed by a quasilinear reaction-diffusion equation. Sci. Sin. Math. 2022, 52, 331–354. [Google Scholar]
  111. Deveci, M.; Gokasar, I.; Castillo, O.; Daim, T. Evaluation of Metaverse integration of freight fluidity measurement alternatives using fuzzy Dombi EDAS model. Comput. Ind. Eng. 2022, 174, 108773. [Google Scholar] [CrossRef]
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MDPI and ACS Style

Khan, M.B.; Othman, H.A.; Voskoglou, M.G.; Abdullah, L.; Alzubaidi, A.M. Some Certain Fuzzy Aumann Integral Inequalities for Generalized Convexity via Fuzzy Number Valued Mappings. Mathematics 2023, 11, 550. https://doi.org/10.3390/math11030550

AMA Style

Khan MB, Othman HA, Voskoglou MG, Abdullah L, Alzubaidi AM. Some Certain Fuzzy Aumann Integral Inequalities for Generalized Convexity via Fuzzy Number Valued Mappings. Mathematics. 2023; 11(3):550. https://doi.org/10.3390/math11030550

Chicago/Turabian Style

Khan, Muhammad Bilal, Hakeem A. Othman, Michael Gr. Voskoglou, Lazim Abdullah, and Alia M. Alzubaidi. 2023. "Some Certain Fuzzy Aumann Integral Inequalities for Generalized Convexity via Fuzzy Number Valued Mappings" Mathematics 11, no. 3: 550. https://doi.org/10.3390/math11030550

APA Style

Khan, M. B., Othman, H. A., Voskoglou, M. G., Abdullah, L., & Alzubaidi, A. M. (2023). Some Certain Fuzzy Aumann Integral Inequalities for Generalized Convexity via Fuzzy Number Valued Mappings. Mathematics, 11(3), 550. https://doi.org/10.3390/math11030550

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