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 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 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 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 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 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 new Jensen, Schur and Hermite-Hadamard inequalities for log convex fuzzy interval-valued functions

2021 ◽  
Vol 7 (3) ◽  
pp. 4338-4358
Author(s):  
Muhammad Bilal Khan ◽  
◽  
Hari Mohan Srivastava ◽  
Pshtiwan Othman Mohammed ◽  
Kamsing Nonlaopon ◽  
...  
Keyword(s):  
Convex Functions ◽  
Optimization Problems ◽  
Fuzzy Optimization ◽  
Order Relation ◽  
Inclusion Relation ◽  
Nonconvex Functions ◽  
New Directions ◽  
Fuzzy Order ◽  
Interval Valued ◽  
Fuzzy Interval

<abstract> <p>The inclusion relation and the order relation are two distinct ideas in interval analysis. Convexity and nonconvexity create a significant link with different sorts of inequalities under the inclusion relation. For many classes of convex and nonconvex functions, many works have been devoted to constructing and refining classical inequalities. However, it is generally known that log-convex functions play a significant role in convex theory since they allow us to deduce more precise inequalities than convex functions. Because the idea of log convexity is so important, we used fuzzy order relation $\left(\preceq \right)$ to establish various discrete Jensen and Schur, and Hermite-Hadamard (H-H) integral inequality for log convex fuzzy interval-valued functions (L-convex F-I-V-Fs). Some nontrivial instances are also offered to bolster our findings. Furthermore, we show that our conclusions include as special instances some of the well-known inequalities for L-convex F-I-V-Fs and their variant forms. Furthermore, we show that our conclusions include as special instances some of the well-known inequalities for L-convex F-I-V-Fs and their variant forms. These results and different approaches may open new directions for fuzzy optimization problems, modeling, and interval-valued functions.</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 Fuzzy-interval inequalities for generalized convex fuzzy-interval-valued functions via fuzzy Riemann integrals

2021 ◽  
Vol 7 (1) ◽  
pp. 1507-1535
Author(s):  
Muhammad Bilal Khan ◽  
◽  
Hari Mohan Srivastava ◽  
Pshtiwan Othman Mohammed ◽  
Dumitru Baleanu ◽  
...  
Keyword(s):  
Optimization Problems ◽  
Fuzzy Optimization ◽  
Type Inequality ◽  
Discrete Form ◽  
Basic Properties ◽  
New Class ◽  
Riemann Integrals ◽  
Interval Valued ◽  
Fuzzy Interval

<abstract> <p>The objective of the authors is to introduce the new class of convex fuzzy-interval-valued functions (convex-FIVFs), which is known as $ p $-convex fuzzy-interval-valued functions ($ p $-convex-FIVFs). Some of the basic properties of the proposed fuzzy-interval-valued functions are also studied. With the help of $ p $-convex FIVFs, we have presented some Hermite-Hadamard type inequalities ($ H-H $ type inequalities), where the integrands are FIVFs. Moreover, we have also proved the Hermite-Hadamard-Fejér type inequality ($ H-H $ Fejér type inequality) for $ p $-convex-FIVFs. To prove the validity of main results, we have provided some useful examples. We have also established some discrete form of Jense's type inequality and Schur's type inequality for $ p $-convex-FIVFs. The outcomes of this paper are generalizations and refinements of different results which are proved in literature. These results and different approaches may open new direction for fuzzy optimization problems, modeling, and interval-valued functions.</p> </abstract>

scholarly journals Fuzzy-interval inequalities for generalized preinvex fuzzy interval valued functions

2022 ◽  
Vol 19 (1) ◽  
pp. 812-835
Author(s):  
Muhammad Bilal Khan ◽  
◽  
Hari Mohan Srivastava ◽  
Pshtiwan Othman Mohammed ◽  
Juan L. G. Guirao ◽  
...  
Keyword(s):  
Order Relation ◽  
Fuzzy Order ◽  
Interval Valued ◽  
Fuzzy Interval

<abstract> <p>In this paper, firstly we define the concept of <italic>h</italic>-preinvex fuzzy-interval-valued functions (<italic>h</italic>-preinvex FIVF). Secondly, some new Hermite-Hadamard type inequalities (<italic>H</italic>-<italic>H</italic> type inequalities) for <italic>h</italic>-preinvex FIVFs via fuzzy integrals are established by means of fuzzy order relation. Finally, we obtain Hermite-Hadamard Fejér type inequalities (<italic>H</italic>-<italic>H</italic> Fejér type inequalities) for <italic>h</italic>-preinvex FIVFs by using above relationship. To strengthen our result, we provide some examples to illustrate the validation of our results, and several new and previously known results are obtained.</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.

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