General Minkowski type inequalities for Sugeno integrals

doi:10.1016/j.fss.2009.10.007

Fuzzy Sets and Systems

Volume 161, Issue 5, 1 March 2010, Pages 708-715

https://doi.org/10.1016/j.fss.2009.10.007 Get rights and content

Abstract

Minkowski type inequalities for the Sugeno integral on abstract spaces are studied in a rather general form, thus closing the series of papers on the topic dealing with special cases restricted to the (pseudo-)additive operation.

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Citation Excerpt :
To build up various kinds of integral inequalities is a contemporary issue. Recently, based on different non-additive integrals, such as Sugeno integral [1,3,4,18,38], generalized Sugeno integral [23], pseudo-integral [2,34], Choquet integral [27,42,55] and so on, a lot of valuable work had been carried out. Set-valued functions [6] (multifunctions [9], correspondences [24], random sets [29]) as a notion of generalization of functions (single-valued), besides being an important mathematical field, have increasingly become an essential tool for solving problems in practice, especially for mathematical economy (problems of individual demand, mean demand, competitive equilibrium, coalition production economies) [24].
Being an important part of classical analysis, Jensen's inequality has drawn much attention recently. Due to its generality, the inequality based on non-additive integrals appears in many forms, such as Sugeno integrals, Choquet integrals and pseudo-integrals. As a well-known generalization of classical one, the set-valued analysis is frequently applied to the research of mathematical economy, control theory and so on. Thus, it is of great necessity to generalize the set-valued case. Motivated by the pioneering work of Costa's Jensen's fuzzy-interval-valued inequality and Štrboja et al.'s Jensen's set-valued inequality based on Aumann integrals and pseudo-integrals respectively, this paper focuses particularly on proving certain kinds of Jensen's set-valued inequalities and fuzzy set-valued inequalities. These inequalities consist of two families: the related convex (or concave) function is a set-valued or fuzzy set-valued function and the integrand is a real-valued function; the related convex (or concave) function is a real-valued function and the integrand is a set-valued or fuzzy set-valued function. Particularly, Jensen's interval-valued and fuzzy-interval-valued inequalities, including Costa's, are obtained as corollaries.
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We give a necessary and sufficient condition for validity of the Hölder-Minkowski type inequality for the generalized Sugeno integral, any monotone measure and the class of ⋆-associated functions. We also show a connection between integral inequalities and the problem of finding binary operations satisfying the distributivity inequalities. We characterize all semicopulas satisfying Hölder and Minkowski inequalities for the seminormed fuzzy integral. One result has been obtained for the recently introduced q-integral. We describe the relationships among the existing sufficient conditions and show that our result generalizes most known Hölder-Minkowski type inequalities for comonotone functions in the available literature. We also provide two other methods to get the above-mentioned inequality for the Sugeno integral and the class of m-subadditive functions as well as for the Sugeno integral of $[0, 1]$ -valued functions using a duality approach. We point out some flaws and correct the corresponding false statements appearing in the literature. At the end, we pose several problems and questions for further research.
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2019, International Journal of Approximate Reasoning
Let $f, g$ be two comonotonic functions on the unit interval. The classical Chebyshev inequality states that the product of the integrals of f and g is a lower bound of the integral of the product of f and g. This study proves a Chebyshev inequality on an abstract space X for q-integral, which was recently introduced by D. Dubois et al. as a generalization of the Sugeno integral as well as the seminormed integral. It also proves a related inequality for q-cointegrals. Our results generalize previous ones obtained in the framework of Sugeno integral as well as seminormed fuzzy integrals.
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This study presents new fuzzy versions of Minkowski and Beckenbach’s integral inequalities without making use of the Sugeno integral. These new inequalities generalize the interval versions of the Minkowski and Beckenbach inequalities recently published and are obtained by introducing the $p - F A -$ integrability concept for fuzzy-interval-valued functions by means of the Kaleva integral and a fuzzy order relation. This fuzzy order relation is defined level-wise through the Kulisch–Miranker order relation given on the interval space. Numerical examples that illustrate the applicability of the theory developed in this study are presented.

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General Minkowski type inequalities for Sugeno integrals

Abstract

European Journal of Operational Research

Applied Mathematics and Computation

Fuzzy Sets and Systems

Mechanical Systems and Signal Processing

Pattern Recognition

International Journal of Approximate Reasoning

Fuzzy Sets and Systems

Information Sciences

Computers and Mathematics with Applications

International Journal of Approximate Reasoning

Applied Mathematics Letters

Applied Mathematics and Computation

Journal of Mathematical Analysis and Applications