scholarly journals New Hermite–Hadamard and Jensen inequalities for log-$$s$$-convex fuzzy-interval-valued functions in the second sense

2021 ◽  
Author(s):  
Peide Liu ◽  
Muhammad Bilal Khan ◽  
Muhammad Aslam Noor ◽  
Khalida Inayat Noor
Keyword(s):  
Order Relation ◽  
New Class ◽  
Starting Point ◽  
Special Cases ◽  
Interval Valued Function ◽  
Interval Space ◽  
Fuzzy Order ◽  
Interval Valued ◽  
Fuzzy Interval

AbstractIn this paper, our aim is to consider the new class of log-convex fuzzy-interval-valued function known as log-s-convex fuzzy-interval-valued functions (log-s-convex fuzzy-IVFs). By this concept, we have introduced Hermite–Hadamard inequalities (HH-inequalities) by means of fuzzy order relation. This fuzzy order relation is defined level-wise through Kulisch–Miranker order relation defined on interval space. Moreover, some new Hermite–Hadamard–Fejér inequalities (HH–Fejér-inequalities) and Jensen’s inequalities via log-s-convex fuzzy-IVFs are also established and verified with the support of useful examples. Some special cases are also discussed which can be viewed as applications of fuzzy-interval HH-inequalities. The concepts and approaches of this paper may be the starting point for further research in this area.

scholarly journals Fuzzy integral inequalities on coordinates of convex fuzzy interval-valued functions

2021 ◽  
Vol 18 (5) ◽  
pp. 6552-6580
Author(s):  
Muhammad Bilal Khan ◽  
◽  
Pshtiwan Othman Mohammed ◽  
Muhammad Aslam Noor ◽  
Khadijah M. Abualnaja ◽  
...  
Keyword(s):  
Order Relation ◽  
Strong Relationship ◽  
Double Integral ◽  
New Class ◽  
Fundamental Properties ◽  
Starting Point ◽  
Special Cases ◽  
Fuzzy Order ◽  
Interval Valued ◽  
Fuzzy Interval

<abstract> <p>In this study, we introduce and study new fuzzy-interval integral is known as fuzzy-interval double integral, where the integrand is fuzzy-interval-valued functions (FIVFs). Also, some fundamental properties are also investigated. Moreover, we present a new class of convex fuzzy-interval-valued functions is known as coordinated convex fuzzy-interval-valued functions (coordinated convex FIVFs) through fuzzy order relation (FOR). The FOR $\left(\preccurlyeq \right)$ and fuzzy inclusion relation (⊇) are two different concepts. With the help of fuzzy-interval double integral and FOR, we have proved that coordinated convex fuzzy-IVF establish a strong relationship between Hermite-Hadamard (<italic>HH</italic>-) and Hermite-Hadamard-Fejér (<italic>HH</italic>-Fejér) inequalities. With the support of this relation, we also derive some related <italic>HH</italic>-inequalities for the product of coordinated convex FIVFs. Some special cases are also discussed. Useful examples that verify the applicability of the theory developed in this study are presented. The concepts and techniques of this paper may be a starting point for further research in this area.</p> </abstract>

scholarly journals Some Hadamard–Fejér Type Inequalities for LR-Convex Interval-Valued Functions

2021 ◽  
Vol 6 (1) ◽  
pp. 6
Author(s):  
Muhammad Bilal Khan ◽  
Savin Treanțǎ ◽  
Mohamed S. Soliman ◽  
Kamsing Nonlaopon ◽  
Hatim Ghazi Zaini
Keyword(s):  
Order Relation ◽  
Inclusion Relation ◽  
Hadamard Inequality ◽  
New Class ◽  
Starting Point ◽  
Interval Valued Function ◽  
Fejér Inequality ◽  
Interval Space ◽  
Interval Valued

The purpose of this study is to introduce the new class of Hermite–Hadamard inequality for LR-convex interval-valued functions known as LR-interval Hermite–Hadamard inequality, by means of pseudo-order relation ( ≤p ). This order relation is defined on interval space. We have proved that if the interval-valued function is LR-convex then the inclusion relation “ ⊆ ” coincident to pseudo-order relation “ ≤p ” under some suitable conditions. Moreover, the interval Hermite–Hadamard–Fejér inequality is also derived for LR-convex interval-valued functions. These inequalities also generalize some new and known results. Useful examples that verify the applicability of the theory developed in this study are presented. The concepts and techniques of this paper may be a starting point for further research in this area.

scholarly journals Some Integral Inequalities for Generalized Convex Fuzzy-Interval-Valued Functions via Fuzzy Riemann Integrals

2021 ◽  
Vol 14 (1) ◽  
Author(s):  
Muhammad Bilal Khan ◽  
Muhammad Aslam Noor ◽  
Pshtiwan Othman Mohammed ◽  
Juan L. G. Guirao ◽  
Khalida Inayat Noor
Keyword(s):  
Type Inequality ◽  
Order Relation ◽  
Strong Relationship ◽  
Inclusion Relation ◽  
Special Cases ◽  
Interval Space ◽  
Fuzzy Order ◽  
Riemann Integrals ◽  
Interval Valued ◽  
Fuzzy Interval

AbstractIn this study, we introduce the new concept of $$h$$ h -convex fuzzy-interval-valued functions. Under the new concept, we present new versions of Hermite–Hadamard inequalities (H–H inequalities) are called fuzzy-interval Hermite–Hadamard type inequalities for $$h$$ h -convex fuzzy-interval-valued functions ($$h$$ h -convex FIVF) by means of fuzzy order relation. This fuzzy order relation is defined level wise through Kulisch–Miranker order relation defined on fuzzy-interval space. Fuzzy order relation and inclusion relation are two different concepts. With the help of fuzzy order relation, we also present some H–H type inequalities for the product of $$h$$ h -convex FIVFs. Moreover, we have also established strong relationship between Hermite–Hadamard–Fej´er (H–H–Fej´er) type inequality and $$h$$ h -convex FIVF. There are also some special cases presented that can be considered applications. There are useful examples provided to demonstrate the applicability of the concepts proposed in this study. This paper's thoughts and methodologies could serve as a springboard for more research in this field.

scholarly journals New Hermite–Hadamard-type inequalities for "Equation missing" No EquationSource Format="TEX", only image -convex fuzzy-interval-valued functions

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Muhammad Bilal Khan ◽  
Muhammad Aslam Noor ◽  
Khalida Inayat Noor ◽  
Yu-Ming Chu
Keyword(s):  
Order Relation ◽  
Interval Space ◽  
Fuzzy Order ◽  
Interval Valued ◽  
Fuzzy Interval

AbstractIn this paper, we introduce the non-convex interval-valued functions for fuzzy-interval-valued functions, which are called "Equation missing"-convex fuzzy-interval-valued functions, by means of fuzzy order relation. This fuzzy order relation is defined level-wise through Kulisch–Miranker order relation given on the interval space. By using the "Equation missing"-convexity concept, we present fuzzy-interval Hermite–Hadamard inequalities for fuzzy-interval-valued functions. Several exceptional cases are debated, which can be viewed as useful applications. Interesting examples that verify the applicability of the theory developed in this study are presented. The results of this paper can be considered as extensions of previously established results.

scholarly journals Some Inequalities for LR-$$\left({h}_{1}, {h}_{2}\right)$$-Convex Interval-Valued Functions by Means of Pseudo Order Relation

2021 ◽  
Vol 14 (1) ◽  
Author(s):  
Muhammad Bilal Khan ◽  
Muhammad Aslam Noor ◽  
Khalida Inayat Noor ◽  
Kottakkaran Sooppy Nisar ◽  
Khadiga Ahmed Ismail ◽  
...  
Keyword(s):  
Applied Mathematics ◽  
Critical Role ◽  
Order Relation ◽  
Strong Relationship ◽  
New Class ◽  
Starting Point ◽  
Interval Valued Function ◽  
Fejér Inequality ◽  
Definition Of ◽  
Interval Valued

AbstractIn both theoretical and applied mathematics fields, integral inequalities play a critical role. Due to the behavior of the definition of convexity, both concepts convexity and integral inequality depend on each other. Therefore, the relationship between convexity and symmetry is strong. Whichever one we work on, we introduced the new class of generalized convex function is known as LR-$$\left({h}_{1}, {h}_{2}\right)$$ h 1 , h 2 -convex interval-valued function (LR-$$\left({h}_{1}, {h}_{2}\right)$$ h 1 , h 2 -IVF) by means of pseudo order relation. Then, we established its strong relationship between Hermite–Hadamard inequality (HH-inequality)) and their variant forms. Besides, we derive the Hermite–Hadamard–Fejér inequality (HH–Fejér inequality)) for LR-$$\left({h}_{1}, {h}_{2}\right)$$ h 1 , h 2 -convex interval-valued functions. Several exceptional cases are also obtained which can be viewed as its applications of this new concept of convexity. Useful examples are given that verify the validity of the theory established in this research. This paper’s concepts and techniques may be the starting point for further research in this field.

scholarly journals New fuzzy-interval inequalities in fuzzy-interval fractional calculus by means of fuzzy order relation

2021 ◽  
Vol 6 (10) ◽  
pp. 10964-10988
Author(s):  
Muhammad Bilal Khan ◽  
◽  
Pshtiwan Othman Mohammed ◽  
Muhammad Aslam Noor ◽  
Abdullah M. Alsharif ◽  
...  
Keyword(s):  
Interval Analysis ◽  
Order Relation ◽  
Strong Relationship ◽  
Data Uncertainty ◽  
Hadamard Inequality ◽  
Special Cases ◽  
Fejér Inequality ◽  
Fuzzy Order ◽  
Interval Valued ◽  
Fuzzy Interval

<abstract> <p>It is well-known that interval analysis provides tools to deal with data uncertainty. In general, interval analysis is typically used to deal with the models whose data are composed of inaccuracies that may occur from certain kinds of measurements. In interval analysis and fuzzy-interval analysis, the inclusion relation (⊆) and fuzzy order relation $\left(\preccurlyeq \right)$ both are two different concepts, respectively. In this article, with the help of fuzzy order relation, we introduce fractional Hermite-Hadamard inequality (<italic>HH</italic>-inequality) for <italic>h</italic>-convex fuzzy-interval-valued functions (<italic>h</italic>-convex-IVFs). Moreover, we also establish a strong relationship between <italic>h</italic>-convex fuzzy-IVFs and Hermite-Hadamard Fejér inequality (<italic>HH</italic>-Fejér inequality) via fuzzy Riemann Liouville fractional integral operator. It is also shown that our results include a wide class of new and known inequalities for <italic>h</italic>-convex fuzz-IVFs and their variant forms as special cases. Nontrivial examples are presented to illustrate the validity of the concept suggested in this review. This paper's techniques and approaches may serve as a springboard for further research in this field.</p> </abstract>

scholarly journals Some fuzzy-interval integral inequalities for harmonically convex fuzzy-interval-valued functions

2021 ◽  
Vol 7 (1) ◽  
pp. 349-370
Author(s):  
Muhammad Bilal Khan ◽  
◽  
Muhammad Aslam Noor ◽  
Thabet Abdeljawad ◽  
Bahaaeldin Abdalla ◽  
...  
Keyword(s):  
Order Relation ◽  
Integral Inequalities ◽  
Fuzzy Integral ◽  
Inclusion Relation ◽  
Fuzzy Space ◽  
Interval Space ◽  
Fuzzy Order ◽  
Riemann Integrals ◽  
Interval Valued ◽  
Fuzzy Interval

<abstract> <p>It is well-known fact that fuzzy interval-valued functions (F-I-V-Fs) are generalizations of interval-valued functions (I-V-Fs), and inclusion relation and fuzzy order relation on interval space and fuzzy space are two different concepts. Therefore, by using fuzzy order relation (FOR), we derive inequalities of Hermite-Hadamard (<italic>H</italic>·<italic>H</italic>) and Hermite-Hadamard Fejér (<italic>H</italic>·<italic>H</italic> Fejér) like for harmonically convex fuzzy interval-valued functions by applying fuzzy Riemann integrals. Moreover, we establish the relation between fuzzy integral inequalities and fuzzy products of harmonically convex fuzzy interval-valued functions. The outcomes of this study are generalizations of many known results which can be viewed as an application of a defined new version of inequalities.</p> </abstract>

scholarly journals New Hermite–Hadamard Inequalities in Fuzzy-Interval Fractional Calculus and Related Inequalities

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 673
Author(s):  
Muhammad Bilal Khan ◽  
Pshtiwan Othman Mohammed ◽  
Muhammad Aslam Noor ◽  
Y. S. Hamed
Keyword(s):  
Order Relation ◽  
Fractional Integrals ◽  
Order Relations ◽  
Inequality Theory ◽  
Real World Applications ◽  
Mathematical Problems ◽  
Interval Space ◽  
Fuzzy Order ◽  
Interval Valued ◽  
Fuzzy Interval

It is a familiar fact that inequalities have become a very popular method using fractional integrals, and that this method has been the driving force behind many studies in recent years. Many forms of inequality have been studied, resulting in the introduction of new trend in inequality theory. The aim of this paper is to use a fuzzy order relation to introduce various types of inequalities. On the fuzzy interval space, this fuzzy order relation is defined level by level. With the help of this relation, firstly, we derive some discrete Jensen and Schur inequalities for convex fuzzy interval-valued functions (convex fuzzy-IVF), and then, we present Hermite–Hadamard inequalities (-inequalities) for convex fuzzy-IVF via fuzzy interval Riemann–Liouville fractional integrals. These outcomes are a generalization of a number of previously known results, and many new outcomes can be deduced as a result of appropriate parameter and real valued function selections. We hope that our fuzzy order relations results can be used to evaluate a number of mathematical problems related to real-world applications.

scholarly journals Integral Inequalities for Generalized Harmonically Convex Functions in Fuzzy-Interval-Valued Settings

Symmetry ◽  
2021 ◽  
Vol 13 (12) ◽  
pp. 2352
Author(s):  
Muhammad Bilal Khan ◽  
Pshtiwan Othman Mohammed ◽  
José António Tenreiro Machado ◽  
Juan L. G. Guirao
Keyword(s):  
Critical Role ◽  
Fuzzy Optimization ◽  
Order Relation ◽  
Riemann Integral ◽  
New Class ◽  
Tight Connection ◽  
The Subject ◽  
Fuzzy Order ◽  
Interval Valued ◽  
Fuzzy Interval

It is a well-known fact that convex and non-convex fuzzy mappings play a critical role in the study of fuzzy optimization. Due to the behavior of its definition, the idea of convexity also plays a significant role in the subject of inequalities. The concepts of convexity and symmetry have a tight connection. We may use whatever we learn from both the concepts, owing to the significant correlation that has developed between both in recent years. In this paper, we introduce a new class of harmonically convex fuzzy-interval-valued functions which is known as harmonically h-convex fuzzy-interval-valued functions (abbreviated as harmonically h-convex F-I-V-Fs) by means of fuzzy order relation. This fuzzy order relation is defined level-wise through Kulisch–Miranker order relation defined on interval space. Some properties of this class are investigated. BY using fuzzy order relation and h-convex F-I-V-Fs, Hermite–Hadamard type inequalities for harmonically are developed via fuzzy Riemann integral. We have also obtained some new inequalities for the product of harmonically h-convex F-I-V-Fs. Moreover, we establish Hermite–Hadamard–Fej’er inequality for harmonically h-convex F-I-V-Fs via fuzzy Riemann integral. These outcomes are a generalization of a number of previously known results, as well as many new outcomes can be deduced as a result of appropriate parameter “θ” and real valued function “∇” selections. For the validation of the main results, we have added some nontrivial examples. We hope that the concepts and techniques of this study may open new directions for research.

scholarly journals New Hermite–Hadamard and Jensen Inequalities for Log-h-Convex Fuzzy-Interval-Valued Functions

2021 ◽  
Vol 14 (1) ◽  
Author(s):  
Muhammad Bilal Khan ◽  
Lazim Abdullah ◽  
Muhammad Aslam Noor ◽  
Khalida Inayat Noor
Keyword(s):  
Order Relation ◽  
New Class ◽  
Fuzzy Order ◽  
Interval Valued ◽  
Fuzzy Interval

AbstractIn the preset study, we introduce the new class of convex fuzzy-interval-valued functions which is called log-h-convex fuzzy-interval-valued functions (log-h-convex FIVFs) by means of fuzzy order relation. We have also investigated some properties of log-h-convex FIVFs. Using this class, we present Jensen and Hermite–Hadamard inequalities (HH-inequalities). Moreover, some useful examples are presented to verify HH-inequalities for log-h-convex FIVFs. Several new and known special results are also discussed which can be viewed as an application of this concept.

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