Hölder Type Inequality and Jensen Type Inequality for Choquet Integral

  • Xuan Zhao
  • Qiang Zhang
Part of the Advances in Intelligent and Soft Computing book series (AINSC, volume 123)

Abstract

The integral inequalities play important roles in classic measure theory. With the development of fuzzy measure theory, experts want to seek for the integral inequalities of fuzzy integral. We concern on the inequalities of Choquet integral. In this paper, Hölder type inequality and Jensen type inequality for Choquet integral are presented. As the fuzzy measure are not additive, thus what is the other conditions for integral inequalities are discussed. Besides, examples are given to show that the conditions can’t be omitted.

Keywords

Type Inequality Integral Inequality Fuzzy Measure Classic Measure Hardy Type Inequality 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Agahi, H., Mesiar, R., Ouyang, Y.: General Minkowski type inequalities for Sugeno integrals. Fuzzy Sets and Systems 161(5), 708–715 (2010)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Baǐnov, D., Simeonov, P.: Integral inequalities and applications. Kluwer Academic Publishers, Dordrecht (1992)Google Scholar
  3. 3.
    Choquet, G.: Theory of capacities. Ann. Inst. Fourier 5(131-295), 54 (1953)MathSciNetGoogle Scholar
  4. 4.
    de Campos, L.M., Jorge, M.: Characterization and comparison of sugeno and choquet integrals. Fuzzy Sets and Systems 52(1), 61–67 (1992)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Flores-Franulic, A., Román-Flores, H.: A Chebyshev type inequality for fuzzy integrals. Applied Mathematics and Computation 190(2), 1178–1184 (2007)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Littlewood, J.E., Hardy, G.H., Polya, G.: Inequalities. Cambridge University Press (1952)Google Scholar
  7. 7.
    Grabisch, M.: The application of fuzzy integrals in multicriteria decision making. European Journal of Operational Research 89(3), 445–456 (1996)MATHCrossRefGoogle Scholar
  8. 8.
    Grabisch, M., Sugeno, M., Murofushi, T.: Fuzzy measures and integrals: theory and applications. Springer-Verlag New York, Inc., Secaucus (2000)MATHGoogle Scholar
  9. 9.
    Wang, Z.Y., Klir, G.J.: Fuzzy measure theory. Plenum Press, New York (1992)MATHCrossRefGoogle Scholar
  10. 10.
    Mesiar, R., Li, J., Pap, E.: The choquet integral as lebesgue integral and related inequalities. Kybernetika 46(6), 1098–1107 (2010)MathSciNetMATHGoogle Scholar
  11. 11.
    Mesiar, R., Ouyang, Y.: General Chebyshev type inequalities for Sugeno integrals. Fuzzy Sets and Systems 160(1), 58–64 (2009)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Ouyang, Y., Mesiar, R.: On the Chebyshev type inequality for seminormed fuzzy integral. Applied Mathematics Letters 22(12), 1810–1815 (2009)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Pap, E.: Null-additive Set Functions. Kluwer, Dordrecht (1995)MATHGoogle Scholar
  14. 14.
    Román-Flores, H., Flores-Franulic, A., Chalco-Cano, Y.: A Jensen type inequality for fuzzy integrals. Information Sciences 177(15), 3192–3201 (2007)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Román-Flores, H., Flores-Franulic, A., Chalco-Cano, Y.: The fuzzy integral for monotone functions. Applied Mathematics and Computation 185(1), 492–498 (2007)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Romano, C.: Applying copula function to risk management. University of Rome, La Sapienza, Working Paper (2002)Google Scholar
  17. 17.
    Sugeno, M.: Theory of fuzzy integrals and its application. PhD thesis (1974)Google Scholar
  18. 18.
    Wang, R.-S.: Some inequalities and convergence theorems for choquet integrals. Journal of Applied Mathematics and Computing, 1–17 (2009), doi:10.1007/s12190-009-0358-yGoogle Scholar
  19. 19.
    Xie, X.L., Beni, G.: A validity measure for fuzzy clustering. IEEE Transactions on Pattern Analysis and Machine Intelligence 13(8), 841–847 (2002)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Xuan Zhao
    • 1
  • Qiang Zhang
    • 1
  1. 1.School of Management and EconomicsBeijing Institute of TechnologyBeijingChina

Personalised recommendations