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 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 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 Some New Fractional Estimates of Inequalities for LR-p-Convex Interval-Valued Functions by Means of Pseudo Order Relation

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 175
Author(s):  
Muhammad Bilal Khan ◽  
Pshtiwan Othman Mohammed ◽  
Muhammad Aslam Noor ◽  
Dumitru Baleanu ◽  
Juan Luis García Guirao
Keyword(s):  
Interval Analysis ◽  
Order Relation ◽  
Strong Relationship ◽  
Data Uncertainty ◽  
Nonconvex Functions ◽  
New Class ◽  
Starting Point ◽  
Special Cases ◽  
Katugampola Fractional Integral ◽  
Interval Valued

It is a familiar fact 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, both the inclusion relation ⊆ and pseudo order relation ≤p are two different concepts. In this article, by using pseudo order relation, we introduce the new class of nonconvex functions known as LR-p-convex interval-valued functions (LR-p-convex-IVFs). With the help of this relation, we establish a strong relationship between LR-p-convex-IVFs and Hermite-Hadamard type inequalities (HH-type inequalities) via Katugampola fractional integral operator. Moreover, we have shown that our results include a wide class of new and known inequalities for LR-p-convex-IVFs and their variant forms as special cases. Useful examples that demonstrate the applicability of the theory proposed in this study are given. The concepts and techniques of this paper may be a starting point for further research in this area.

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 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 Fractional Hermite-Hadamard Integral Inequalities for a New Class of Convex Functions

Symmetry ◽  
2020 ◽  
Vol 12 (9) ◽  
pp. 1485
Author(s):  
Pshtiwan Othman Mohammed ◽  
Thabet Abdeljawad ◽  
Shengda Zeng ◽  
Artion Kashuri
Keyword(s):  
Convex Functions ◽  
Fractional Integral ◽  
Special Functions ◽  
Applied Mathematics ◽  
Strong Relationship ◽  
Integral Inequalities ◽  
Mathematical Methods ◽  
Fractional Integral Operators ◽  
New Class ◽  
Definition Of

Fractional integral inequality plays a significant role in pure and applied mathematics fields. It aims to develop and extend various mathematical methods. Therefore, nowadays we need to seek accurate fractional integral inequalities in obtaining the existence and uniqueness of the fractional methods. Besides, the convexity theory plays a concrete role in the field of fractional integral inequalities due to the behavior of its definition and properties. There is also a strong relationship between convexity and symmetric theories. So, whichever one we work on, we can then apply it to the other one due to the strong correlation produced between them, specifically in the last few decades. First, we recall the definition of φ-Riemann–Liouville fractional integral operators and the recently defined class of convex functions, namely the σ˘-convex functions. Based on these, we will obtain few integral inequalities of Hermite–Hadamard’s type for a σ˘-convex function with respect to an increasing function involving the φ-Riemann–Liouville fractional integral operator. We can conclude that all derived inequalities in our study generalize numerous well-known inequalities involving both classical and Riemann–Liouville fractional integral inequalities. Finally, application to certain special functions are pointed out.

scholarly journals A New Version of the Hermite–Hadamard Inequality for Riemann–Liouville Fractional Integrals

Symmetry ◽  
2020 ◽  
Vol 12 (4) ◽  
pp. 610 ◽  
Author(s):  
Pshtiwan Othman Mohammed ◽  
Iver Brevik
Keyword(s):  
Bessel Functions ◽  
Applied Mathematics ◽  
Critical Role ◽  
Strong Relationship ◽  
Fractional Integrals ◽  
Mathematical Methods ◽  
Digamma Function ◽  
Hadamard Inequality ◽  
The Past ◽  
Riemann Integrals

Integral inequalities play a critical role in both theoretical and applied mathematics fields. It is clear that inequalities aim to develop different mathematical methods. Thus, the present days need to seek accurate inequalities for proving the existence and uniqueness of the mathematical methods. The concept of convexity plays a strong role in the field of inequalities due to the behavior of its definition. There is a strong relationship between convexity and symmetry. Whichever one we work on, we can apply it to the other one due the strong correlation produced between them, especially in the past few years. In this article, we firstly point out the known Hermite–Hadamard (HH) type inequalities which are related to our main findings. In view of these, we obtain a new inequality of Hermite–Hadamard type for Riemann–Liouville fractional integrals. In addition, we establish a few inequalities of Hermite–Hadamard type for the Riemann integrals and Riemann–Liouville fractional integrals. Finally, three examples are presented to demonstrate the application of our obtained inequalities on modified Bessel functions and q-digamma function.

Newton’s divided gH-difference interpolation formula with unequal spacing for interval-valued function using Hukuhara difference

2021 ◽  
pp. 2250093
Author(s):  
Bapin Mondal ◽  
Md Sadikur Rahman
Keyword(s):  
Significant Role ◽  
Unknown Function ◽  
Interpolation Formula ◽  
Interval Uncertainty ◽  
Hukuhara Difference ◽  
Numerical Examples ◽  
Generalized Hukuhara Difference ◽  
Interval Valued Function ◽  
Definition Of ◽  
Interval Valued

Interval interpolation formulae play a significant role to find the value of an unknown function at some points under interval uncertainty. The objective of this paper is to establish Newton’s divided interpolation formula for interval-valued functions using generalized Hukuhara difference of intervals. For this purpose, arithmetic of intervals, Hukuhara difference and its some properties and concept of interval-valued function have been discussed briefly. Using Hukuhara difference of intervals, the definition of Newton’s divided gH-difference for interval-valued function has been introduced. Then Newton’s divided gH-differences interpolation formula has been derived. Finally, with the help of some numerical examples, the proposed interpolation formula has been illustrated.

scholarly journals On the Generalized Hermite–Hadamard Inequalities via the Tempered Fractional Integrals

Symmetry ◽  
2020 ◽  
Vol 12 (4) ◽  
pp. 595 ◽  
Author(s):  
Pshtiwan Othman Mohammed ◽  
Mehmet Zeki Sarikaya ◽  
Dumitru Baleanu
Keyword(s):  
Bessel Functions ◽  
Applied Mathematics ◽  
Critical Role ◽  
Strong Relationship ◽  
Fractional Integrals ◽  
Modified Bessel Functions ◽  
Mathematical Methods ◽  
Digamma Function ◽  
Riemann Integrals ◽  
Published Result

Integral inequality plays a critical role in both theoretical and applied mathematics fields. It is clear that inequalities aim to develop different mathematical methods (numerically or analytically) and to dedicate the convergence and stability of the methods. Unfortunately, mathematical methods are useless if the method is not convergent or stable. Thus, there is a present day need for accurate inequalities in proving the existence and uniqueness of the mathematical methods. Convexity play a concrete role in the field of inequalities due to the behaviour of its definition. There is a strong relationship between convexity and symmetry. Which ever one we work on, we can apply to the other one due to the strong correlation produced between them especially in recent few years. In this article, we first introduced the notion of λ -incomplete gamma function. Using the new notation, we established a few inequalities of the Hermite–Hadamard (HH) type involved the tempered fractional integrals for the convex functions which cover the previously published result such as Riemann integrals, Riemann–Liouville fractional integrals. Finally, three example are presented to demonstrate the application of our obtained inequalities on modified Bessel functions and q-digamma function.

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