On Chebyshev type inequalities for generalized Sugeno integrals

doi:10.1016/j.fss.2013.10.015

Fuzzy Sets and Systems

Volume 244, 1 June 2014, Pages 51-62

https://doi.org/10.1016/j.fss.2013.10.015 Get rights and content

Abstract

We give the necessary and sufficient conditions guaranteeing the validity of Chebyshev type inequalities for generalized Sugeno integrals in the case of functions belonging to a much wider class than the comonotone functions. For several choices of operators, we characterize the classes of functions for which the Chebyshev type inequality for the classical Sugeno integral is satisfied.

Introduction

The pioneering concept of the fuzzy integral was introduced by Sugeno [24] as a tool for modeling non-deterministic problems. Theoretical investigations of the integral and its generalizations have been pursued by many researchers. Among others, Ralescu and Adams [18] provided several characterizations of the Sugeno integral and proved a monotone convergence theorem for this integral, Román-Flores et al. [19], [20] discussed level-continuity of fuzzy integrals and H-continuity of fuzzy measures, while Wang and Klir [26] presented an excellent general overview on fuzzy measurement and fuzzy integration theory. On the other hand, fuzzy integrals have also been successfully applied to various fields (see, e.g., [6], [8], [15], [25]).

The study of inequalities for Sugeno integral was initiated by Román-Flores et al. [21]. Since then, the fuzzy integral counterparts of several classical inequalities, including Chebyshev's, Jensen's, Minkowski's and Hölder's inequalities, are given by Flores-Franulić and Román-Flores [7], Agahi et al. [2], [3], [4], Mesiar and Ouyang [12], Román-Flores et al. [22], and other researchers.

The purpose of this paper is to study not only the sufficient conditions for the validity of Chebyshev type and reverse Chebyshev type inequalities for the generalized Sugeno integral (cf. [1], [3], [5], [7], [12] and [16]), but also the necessary ones. The results are obtained for quite universal integrals and a rich class of functions, including the comonotone functions as a special case. For some specific choices of operators, the corresponding characterizations of classes of functions for which Chebyshev's inequality for the classical Sugeno integral is satisfied are also presented.

The paper is organized as follows. In Sections 2 and 3 we set up notation and terminology and present our main results, some related results as well as several illustrative examples. Concluding remarks are given in Section 4.

Section snippets

Main results

Let $(X, F)$ be a measurable space and $μ : F \to [0, \infty]$ be a monotone measure, i.e., $μ (\emptyset) = 0$ , $μ (X) > 0$ and $μ (A) \leq μ (B)$ whenever $A \subset B$ . In the literature monotone measures are also called fuzzy measures (see, e.g., [26]). Let $Y \subset R$ be an arbitrary nonempty interval; usually $Y = [0, 1]$ , $Y = [0, \infty]$ , $Y = [0, \infty)$ or $Y = R$ . We denote the range of μ by $μ (F)$ . For a measurable function $h : X \to Y$ , we will define the Sugeno integral of h on a set $A \in F$ with respect to μ and an operator $▿ : Y \times μ (F) \to Y$ as $\int_{A} h ▿ μ = \sup_{a \in Y} [a ▿ μ (A \cap {h \geq a})],$ where ${h \geq a}$

Related results

Throughout this section we shall use the notation and work under the assumptions introduced before the statement of Theorem 2.1. The following theorem gives the sufficient conditions to the validity of the reverse Chebyshev-type inequality, i.e., the inequality opposite to (4).

Theorem 3.1

Assume Y is monotonically closed, $b \mapsto a \circ_{1} b$ is nondecreasing for $a \in Y$ , ⋆ is an increasing operator and $Y ⋆ Y = Y$ , where $Y ⋆ Y = {a ⋆ b : a, b \in Y}$ . Assume also that

(B1)
for a fixed $A \in F$ and for any $a, b \in Y$ and $▵ : μ (F) \times μ (F) \to μ (F)$ $μ ({f_{|_{A}} \geq a} \cup {g_{|_{A}} \geq b}) \leq μ ({f$

Conclusions

We have presented the necessary and sufficient conditions to the validity of Chebyshev type and reverse Chebyshev type inequalities for generalized Sugeno integrals in the case of functions belonging to an essentially wider class than the comonotone functions. For specific choices of operators, we have characterized classes of functions for which the Chebyshev type inequality for the classical Sugeno integral is satisfied. The proofs consist in solving some functional inequalities.

Acknowledgements

The authors would like to thank the referees for their valuable comments which led to improvements in the paper. The work on this paper was partially supported by the grant for young researchers from Lodz University of Technology I-2 501 plan 2 number 12/2013.

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On Chebyshev type inequalities for generalized Sugeno integrals

Abstract

Introduction

Section snippets

Main results

Related results

Conclusions

Acknowledgements

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Inf. Sci.

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Eur. J. Oper. Res.

Appl. Math. Comput.

Inf. Sci.

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Fuzzy Sets Syst.

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Appl. Math. Lett.

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