scholarly journals Liapounoff type inequality for pseudo-integral of interval-valued function

2022 ◽  
Vol 7 (4) ◽  
pp. 5444-5462
Author(s):  
Tatjana Grbić ◽  
◽  
Slavica Medić ◽  
Nataša Duraković ◽  
Sandra Buhmiler ◽  
...  
Keyword(s):  
Type Inequality ◽  
Interval Valued Function ◽  
Interval Valued

<abstract><p>In this paper, two new Liapounoff type inequalities in terms of pseudo-analysis dealing with set-valued functions are given. The first one is given for a pseudo-integral of set-valued function where pseudo-operations are given by a generator $ g:[0, \infty]\to [0, \infty] $ and the second one is given for the semiring $ ([0, \infty], \sup, \odot) $ with generated pseudo-multiplication. The interval Liapounoff inequality is applied for estimation of interval-valued central $ g $-moment of order $ n $ for interval-valued functions in a $ g $-semiring.</p></abstract>

scholarly journals New Hermite-Hadamard inequalities in fuzzy-interval fractional calculus via exponentially convex fuzzy interval-valued function

2021 ◽  
Vol 6 (11) ◽  
pp. 12260-12278
Author(s):  
Yanping Yang ◽  
◽  
Muhammad Shoaib Saleem ◽  
Waqas Nazeer ◽  
Ahsan Fareed Shah ◽  
...  
Keyword(s):  
Fractional Calculus ◽  
Exponential Type ◽  
Fractional Integral ◽  
Present Note ◽  
Type Inequality ◽  
Interval Valued Function ◽  
Fejér Inequality ◽  
Interval Valued ◽  
Fuzzy Interval

<abstract><p>In the present note, we develop Hermite-Hadamard type inequality and He's inequality for exponential type convex fuzzy interval-valued functions via fuzzy Riemann-Liouville fractional integral and fuzzy He's fractional integral. Moreover, we establish Hermite-Fejér inequality via fuzzy Riemann-Liouville fractional integral.</p></abstract>

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 On Properties of the Choquet Integral of Interval-Valued Functions

2011 ◽  
Vol 2011 ◽  
pp. 1-10 ◽  
Author(s):  
Lee-Chae Jang
Keyword(s):  
Convergence Theorem ◽  
Choquet Integral ◽  
Fuzzy Measure ◽  
Interval Valued Function ◽  
Interval Valued

Based on the concept of an interval-valued function which is motivated by the goal to represent an uncertain function, we define the Choquet integral with respect to a fuzzy measure of interval-valued functions. We also discuss convergence in the(C)mean and convergence in a fuzzy measure of sequences of measurable interval-valued functions. In particular, we investigate the convergence theorem for the Choquet integral of measurable interval-valued functions.

scholarly journals Fractional inclusions of the Hermite–Hadamard type for m-polynomial convex interval-valued functions

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Eze R. Nwaeze ◽  
Muhammad Adil Khan ◽  
Yu-Ming Chu
Keyword(s):  
Fractional Integral ◽  
Convex Set ◽  
Beta Function ◽  
Integral Operators ◽  
Fractional Integral Operators ◽  
Interval Valued Function ◽  
Class Of Functions ◽  
Interval Valued

Abstract The notion of m-polynomial convex interval-valued function $\Psi =[\psi ^{-}, \psi ^{+}]$ Ψ = [ ψ − , ψ + ] is hereby proposed. We point out a relationship that exists between Ψ and its component real-valued functions $\psi ^{-}$ ψ − and $\psi ^{+}$ ψ + . For this class of functions, we establish loads of new set inclusions of the Hermite–Hadamard type involving the ρ-Riemann–Liouville fractional integral operators. In particular, we prove, among other things, that if a set-valued function Ψ defined on a convex set S is m-polynomial convex, $\rho,\epsilon >0$ ρ , ϵ > 0 and $\zeta,\eta \in {\mathbf{S}}$ ζ , η ∈ S , then $$\begin{aligned} \frac{m}{m+2^{-m}-1}\Psi \biggl(\frac{\zeta +\eta }{2} \biggr)& \supseteq \frac{\Gamma _{\rho }(\epsilon +\rho )}{(\eta -\zeta )^{\frac{\epsilon }{\rho }}} \bigl[{_{\rho }{\mathcal{J}}}_{\zeta ^{+}}^{\epsilon } \Psi (\eta )+_{ \rho }{\mathcal{J}}_{\eta ^{-}}^{\epsilon }\Psi (\zeta ) \bigr] \\ & \supseteq \frac{\Psi (\zeta )+\Psi (\eta )}{m}\sum_{p=1}^{m}S_{p}( \epsilon;\rho ), \end{aligned}$$ m m + 2 − m − 1 Ψ ( ζ + η 2 ) ⊇ Γ ρ ( ϵ + ρ ) ( η − ζ ) ϵ ρ [ ρ J ζ + ϵ Ψ ( η ) + ρ J η − ϵ Ψ ( ζ ) ] ⊇ Ψ ( ζ ) + Ψ ( η ) m ∑ p = 1 m S p ( ϵ ; ρ ) , where Ψ is Lebesgue integrable on $[\zeta,\eta ]$ [ ζ , η ] , $S_{p}(\epsilon;\rho )=2-\frac{\epsilon }{\epsilon +\rho p}- \frac{\epsilon }{\rho }\mathcal{B} (\frac{\epsilon }{\rho }, p+1 )$ S p ( ϵ ; ρ ) = 2 − ϵ ϵ + ρ p − ϵ ρ B ( ϵ ρ , p + 1 ) and $\mathcal{B}$ B is the beta function. We extend, generalize, and complement existing results in the literature. By taking $m\geq 2$ m ≥ 2 , we derive loads of new and interesting inclusions. We hope that the idea and results obtained herein will be a catalyst towards further investigation.

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>

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