Kybernetika 52 no. 3, 329-347, 2016

On the Minkowski-Hölder type inequalities for generalized Sugeno integrals with an application

Michał Boczek and Marek KaluszkaDOI: 10.14736/kyb-2016-3-0329


In this paper, we use a new method to obtain the necessary and sufficient condition guaranteeing the validity of the Minkowski-Hölder type inequality for the generalized upper Sugeno integral in the case of functions belonging to a wider class than the comonotone functions. As a by-product, we show that the Minkowski type inequality for seminormed fuzzy integral presented by Daraby and Ghadimi \cite{daraby4} is not true. Next, we study the Minkowski-Hölder inequality for the lower Sugeno integral and the class of $\mu$-subadditive functions introduced in \cite{boczek1}. The results are applied to derive new metrics on the space of measurable functions in the setting of nonadditive measure theory. We also give a partial answer to the open problem $2.22$ posed in \cite{hutnik2}.


semicopula, monotone measure, seminormed fuzzy integral, Minkowski's inequality, Hölder's inequality, convergence in mean


26E50, 28E10


  1. H. Agahi, R. Mesiar and Y. Ouyang: General Minkowski type inequalities for Sugeno integrals. Fuzzy Sets and Systems 161 (2010), 708-715.   DOI:10.1016/j.fss.2009.10.007
  2. H. Agahi and R. Mesiar: On Cauchy-Schwarz's inequality for Choquet-like integrals without the comonotonicity condition. Soft Computing 19 (2015), 1627-1634.   DOI:10.1007/s00500-014-1578-0
  3. B. Bassan and F. Spizzichino: Relations among univariate aging, bivariate aging and dependence for exchangeable lifetimes. J. Multivariate Analysis 93 (2005), 313-339.   DOI:10.1016/j.jmva.2004.04.002
  4. J. Borzová-Molnárová, L. Hal\u{c}inová and O. Hutník: The smallest semicopula-based universal integrals I: Properties and characterizations. Fuzzy Sets and Systems 271 (2015), 1-17.   CrossRef
  5. J. Borzová-Molnárová, L. Hal\u{c}inová and O. Hutník: The smallest semicopula-based universal integrals II: Convergence theorems. Fuzzy Sets and Systems 271 (2015), 18-30.   CrossRef
  6. J. Borzová-Molnárová, L. Hal\u{c}inová and O. Hutník: The smallest semicopula-based universal integrals III: Topology determined by the integral. Fuzzy Sets and Systems (2016).   CrossRef
  7. N. L. Carothers: Real Analysis. University Press, Cambridge 2000.   DOI:10.1017/cbo9780511814228
  8. M. E. Cattaneo: On maxitive integration. Department of Statistics University of Munich 2013,   CrossRef
  9. J. Cerdá: Lorentz capacity spaces. Contemporary Math. 445 (2007), 45-59.   DOI:10.1090/conm/445/08592
  10. A. Chateauneuf, M. Grabisch and A. Rico: Modeling attitudes toward uncertainty through the use of the Sugeno integral. J. Math. Econom. 44 (2008), 1084-1099.   DOI:10.1016/j.jmateco.2007.09.003
  11. B. Daraby and F. Ghadimi: General Minkowski type and related inequalities for seminormed fuzzy integrals. Sahand Commun. Math. Analysis 1 (2014), 9-20.   CrossRef
  12. N. Dunford and J. T. Schwartz: Linear Operators, Part I General Theory. A Wiley Interscience Publishers, New York 1988.   CrossRef
  13. F. Durante and C. Sempi: Semicopul\ae. Kybernetika 41 (2005), 315-328.   CrossRef
  14. K. Fan: Entfernung zweier zufälligen Grössen und die Konvergenz nach Wahrscheinlichkeit. Math. Zeitschrift 49 (1944), 681-683.   DOI:10.1007/bf01174225
  15. H. Föllmer and A. Schied: Stochastic Finance. An Introduction In Discrete Time. De Gruyter, Berlin 2011.   DOI:10.1515/9783110218053
  16. M. Fréchet: Sur divers modes de convergence d'une suite de fonctions d'une variable. Bull. Calcutta Math. Soc. 11 (1919-20), 187-206.   CrossRef
  17. S. Greco, R. Mesiar, F. Rindone and L. \u{S}ipeky: Superadditive and subadditive transformations of integrals and aggregation functions. Fuzzy Sets and Systems 291 (2016), 40-53.   CrossRef
  18. H. Imaoka: On a subjective evaluation model by a generalized fuzzy integral. Int. J. Uncertainty, Fuzziness and Knowledge-Based Systems 5 (1997), 517-529.   DOI:10.1142/s0218488597000403
  19. O. Kallenberg: Foundations of Modern Probability. Second edition. Springer, Berlin 2002.   DOI:10.1007/978-1-4757-4015-8
  20. M. Kaluszka, A. Okolewski and M. Boczek: On Chebyshev type inequalities for generalized Sugeno integrals. Fuzzy Sets and Systems 244 (2014), 51-62.   DOI:10.1016/j.fss.2013.10.015
  21. A. Kandel and W. J. Byatt: Fuzzy sets, fuzzy algebra, and fuzzy statistics. Proc. IEEE 66 (1978), 1619-1639.   DOI:10.1109/proc.1978.11171
  22. E. P. Klement, R. Mesiar and E. Pap: Triangular Norms. Kluwer Academic Publishers, Dordrecht 2000.   DOI:10.1007/978-94-015-9540-7
  23. E. P. Klement, R. Mesiar and E. Pap: A universal integral as common frame for Choquet and Sugeno integral. IEEE Trans. Fuzzy Systems 18 (2010), 178-187.   DOI:10.1109/tfuzz.2009.2039367
  24. G. Li: A metric on space of measurable functions and the related convergence. Int. J. Uncertainty, Fuzziness and Knowledge-Based Systems 20 (2012), 211-222.   DOI:10.1142/s0218488512500109
  25. B. Liu: Uncertainty Theory. Fourth edition. Springer 2015.   DOI:10.1007/978-3-662-44354-5
  26. T. Murofushi: A note on upper and lower Sugeno integrals. Fuzzy Sets and Systems 138 (2003), 551-558.   DOI:10.1016/s0165-0114(02)00375-5
  27. Y. Ouyang and R. Mesiar: On the Chebyshev type inequality for seminormed fuzzy integral. Applied Math. Lett. 22 (2009), 1810-1815.   DOI:10.1016/j.aml.2009.06.024
  28. Y. Ouyang, R. Mesiar and H. Agahi: An inequality related to Minkowski type for Sugeno integrals. Inform. Sci. 180 (2010), 2793-2801.   DOI:10.1016/j.ins.2010.03.018
  29. E. Pap and ed.: Handbook of Measure Theory. Elsevier Science, Amsterdam 2002.   CrossRef
  30. H. Román-Flores, A. Flores-Franulič and Y. Chalco-Cano: The fuzzy integral for monotone functions. Applied Math. Comput. 185 (2007), 492-498.   DOI:10.1016/j.amc.2006.07.066
  31. L. Rüschendorf: Mathematical Risk Analysis: Dependence, Risk Bounds, Optimal Allocations and Portfolios. Springer Science and Business Media, Berlin 2013.   DOI:10.1007/978-3-642-33590-7
  32. N. Shilkret: Maxitive measure and integration. Indagationes Math. 33 (1971), 109-116.   DOI:10.1016/s1385-7258(71)80017-3
  33. F. Suárez García and P. Gil Álvarez: Two families of fuzzy integrals. Fuzzy Sets and Systems 18 (1986), 67-81.   DOI:10.1016/0165-0114(86)90028-x
  34. M. Sugeno: Theory of Fuzzy Integrals and its Applications. Ph.D. Dissertation, Tokyo Institute of Technology 1974.   CrossRef
  35. Z. Wang and G. Klir: Generalized Measure Theory. Springer, New York 2009.   DOI:10.1007/978-0-387-76852-6
  36. Ch. Wu, X. Rena and C. Wu: A note on the space of fuzzy measurable functions for a monotone measure. Fuzzy Sets and Systems 182 (2011), 2-12.   DOI:10.1016/j.fss.2010.10.006
  37. L. Wu, J. Sun, X. Ye and L. Zhu: Hölder type inequality for Sugeno integral. Fuzzy Sets and Systems 161 (2010), 2337-2347.   DOI:10.1016/j.fss.2010.04.017