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International journal of fuzzy logic and intelligent systems : IJFIS v.9 no.2, 2009년, pp.71 - 75  
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A note on Jensen type inequality for Choquet integrals

Jang, Lee-Chae    (Department of Mathematics and Computer Science, Konkuk University  );
  • 초록

    The purpose of this paper is to prove a Jensen type inequality for Choquet integrals with respect to a non-additive measure which was introduced by Choquet [1] and Sugeno [20]; $$\Phi((C)\;{\int}\;fd{\mu})\;{\leq}\;(C)\;\int\;\Phi(f)d{\mu},$$ where f is Choquet integrable, ${\Phi}\;:\;[0,\;\infty)\;\rightarrow\;[0,\;\infty)$ is convex, $\Phi(\alpha)\;\leq\;\alpha$ for all $\alpha\;{\in}\;[0,\;{\infty})$ and ${\mu}_f(\alpha)\;{\leq}\;{\mu}_{\Phi(f)}(\alpha)$ for all ${\alpha}\;{\in}\;[0,\;{\infty})$ . Furthermore, we give some examples assuring both satisfaction and dissatisfaction of Jensen type inequality for the Choquet integral.


  • 주제어

    Choquet integral .   non-additive measure .   Jensen type inequality.  

  • 참고문헌 (25)

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    15. T. Murofushi and M. Sugeno, "A theory of fuzzy measures: representations, the Choquet integral, and null sets," J. Math. Anal. Appl., vol. 159, pp. 532-549, 1991 
    16. L.C. Jang, T. Kim, and J.D. Jeon, "On comonotonically additive interval-valued functionals and interval valued Choquet integrals (II)," J. of Fuzzy Logic and Intelligent Systems, vol. 14, no. 1, pp. 33-38, 2004     
    17. L.C. Jang and W.Kim, "Some properties of Choquet distance measures for interval-valued fuzzy numbers," J. of Fuzzy Logic and Intelligent Systems. vol. 15, no. 7, pp. 789-793, 2005     
    18. S. Ovchinikov and A. Dukhovny, "Tolerant or intorant character of interacting criteria in aggregation by the Choquet integral," European J. of Operational Research, vol. 155, pp. 771-791, 2004 
    19. L.C. Jang, "A note on distance measure and similarity measure defined by Choquet integral on interval valued fuzzy sets," J. of Fuzzy L Intelligent Systems, vol. 17, no. 4, pp. 455-459, 2007     
    20. L.C. Jang, T. Kim, and D.W. Park, "A note on convexity and comonotonically additivity of set-valued Choquet integrals," Far East J. Appl. Math., vol. 11, pp. 137-148, 2003 
    21. L.C. Jang, "Structural characterizations of monotone interval valued set functions defined by the intervalvalued Choquet integral," J. of Fuzzy Logic and Intelligent Systems, vol. 18, no. 3, pp. 311-315, 2008     
    22. H. Rom$\acute{a}$n-Flores, A. Flores-Franulic, Y. Chalco-Cano, "A Jensen type inequality for fuzzy integrals," Information Sciences, vol. 177, pp. 3192-3201, 2007 
    23. L.C. Jang, T. Kim, J.D. Jeon, and W.J. Kim, "On Choquet integrals of measurable fuzzy number-valued functions," Bull. Korean Math. Soc., vol. 41, no. 1, pp. 95-107, 2004     
    24. L.C. Jang, "Interval-valued Choquet integrals and their applications," J. Appl. Math. and Computing, vol. 16, no. 1- no. 2, pp. 429-443     
    25. H.L. Royden, Real Analysis, New York 1988 

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