On the Chebyshev type inequality for seminormed fuzzy integral

Ouyang, Yao; Mesiar, Radko

doi:10.1016/j.aml.2009.06.024

Cited by 59 publications

(28 citation statements)

References 21 publications

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“…Let ϕ 1 (x) = ϕ 2 (x) = ϕ 3 (x) = x s for all 0 < s < ∞, then we get the reverse Minkowski inequality for semiconormed non-additive integrals (if s = 1, then we have the reverse Chebyshev inequality for semiconormed non-additive integrals [18]). …”

Section: Hölder and Minkowski Type Inequalities For Semiconormed Non-mentioning

confidence: 99%

“…We can use the same examples in [18] to show that the condition of ϕ −1 1 (T (ϕ 1 (a b) , c)) ≥ ϕ −1 2 (T (ϕ 2 (a) , c)) b ∨ a ϕ −1 3 (T (ϕ 3 (b) , c)) in Theorem 4.1 cannot be abandoned, and so we omit them here.…”

Section: Reverse Inequalities For Seminormed Non-additive Integralsmentioning

confidence: 99%

“…Recently, the authors generalized several classical inequalities, including Minkowski's and Chebyshev's inequalities, to the frame of Sugeno integral [1,4,20]. Therefore, it would be an interesting topic to generalize an inequality from the frame of Sugeno integral to some integrals which contains Sugeno integral as special cases, such as it was done in [6,18]. Recently, the study of integral inequalities has been extended to a more general class of pseudo-integral operators [2,23].…”

Section: Introductionmentioning

confidence: 99%

“…Corollary 3.8 [18]. Let (X, F, μ) be a non-additive measure space and f, g :X → [0, 1] two comonotone measurable functions.…”

mentioning

confidence: 99%

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On Non-additive Probabilistic Inequalities of Hölder-type

et al. 2011

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Non-additive measure is a generalization of additive probability measure. Integral inequalities play important roles in classical probability and measure theory. Some well-known inequalities such as the Minkowski inequality and the Hölder inequality play important roles not only in the theoretical area but also in application. Non-additive integrals are useful tools in several theoretical and applied statistics which have been built on non-additive measure. For instance, in decision theory and applied statistics, the use of the non-additive integrals can be envisaged from two points of view: decision under uncertainty and multi-criteria decision-making. In fact, the non-additive integrals provide useful tools in many problems in engineering and social choice where the aggregation of data is required. In this paper, Hölder and Minkowski type inequalities for semi(co)normed non-additive integrals are discussed. The main results of this paper generalize some previous results obtained by the authors. Mathematics Subject Classification (2000). Primary 99Z99; Secondary 00A00.

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Section: Hölder and Minkowski Type Inequalities For Semiconormed Non-mentioning

confidence: 99%

Section: Reverse Inequalities For Seminormed Non-additive Integralsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…Corollary 3.8 [18]. Let (X, F, μ) be a non-additive measure space and f, g :X → [0, 1] two comonotone measurable functions.…”

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confidence: 99%

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On Non-additive Probabilistic Inequalities of Hölder-type

et al. 2011

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“…In recent years, some famous integral inequalities have been generalized to fuzzy integrals (cf. [13][14][15][16]). The study of inequalities for the Sugeno integral, which was initiated by Roman-Flores et al [17], is the most popular.…”

Section: Introductionmentioning

confidence: 99%

Sandor Type Fuzzy Inequality Based on the (s,m)-Convex Function in the Second Sense

2017

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Abstract:Integral inequalities play critical roles in measure theory and probability theory. Given recent profound discoveries in the field of fuzzy set theory, fuzzy inequality has become a hot research topic in recent years. For classical Sandor type inequality, this paper intends to extend the Sugeno integral. Based on the (s,m)-convex function in the second sense, a new Sandor type inequality is proposed for the Sugeno integral. Examples are given to verify the conclusion of this paper.

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