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On the Minkowski-Hölder type inequalities for generalized Sugeno integrals with an application
Kybernetika, Tome 52 (2016) no. 3, pp. 329-347 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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 [11] is not true. Next, we study the Minkowski-Hölder inequality for the lower Sugeno integral and the class of $\mu$-subadditive functions introduced in [20]. 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 [5].
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 [11] is not true. Next, we study the Minkowski-Hölder inequality for the lower Sugeno integral and the class of $\mu$-subadditive functions introduced in [20]. 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 [5].
DOI : 10.14736/kyb-2016-3-0329
Classification : 26E50, 28E10
Keywords: seminormed fuzzy integral; semicopula; monotone measure; Minkowski's inequality; Hölder's inequality; convergence in mean 
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Boczek, Michał; Kaluszka, Marek. On the Minkowski-Hölder type inequalities for generalized Sugeno integrals with an application. Kybernetika, Tome 52 (2016) no. 3, pp. 329-347. doi: 10.14736/kyb-2016-3-0329

[1] Agahi, H., Mesiar, R., Ouyang, Y.: General Minkowski type inequalities for Sugeno integrals. Fuzzy Sets and Systems 161 (2010), 708-715. | DOI | MR | Zbl

[2] Agahi, H., Mesiar, R.: On Cauchy-Schwarz's inequality for Choquet-like integrals without the comonotonicity condition. Soft Computing 19 (2015), 1627-1634. | DOI

[3] Bassan, B., Spizzichino, F.: Relations among univariate aging, bivariate aging and dependence for exchangeable lifetimes. J. Multivariate Analysis 93 (2005), 313-339. | DOI | MR | Zbl

[4] Borzová-Molnárová, J., Halčinová, L., Hutník, O.: The smallest semicopula-based universal integrals I: Properties and characterizations. Fuzzy Sets and Systems 271 (2015), 1-17. | MR

[5] Borzová-Molnárová, J., Halčinová, L., Hutník, O.: The smallest semicopula-based universal integrals II: Convergence theorems. Fuzzy Sets and Systems 271 (2015), 18-30. | MR

[6] Borzová-Molnárová, J., Halčinová, L., Hutník, O.: The smallest semicopula-based universal integrals III: Topology determined by the integral. Fuzzy Sets and Systems (2016).

[7] Carothers, N. L.: Real Analysis. University Press, Cambridge 2000. | DOI | MR | Zbl

[8] Cattaneo, M. E.: On maxitive integration. Department of Statistics University of Munich 2013, http://www.stat.uni-muenchen.de

[9] Cerdá, J.: Lorentz capacity spaces. Contemporary Math. 445 (2007), 45-59. | DOI | MR | Zbl

[10] Chateauneuf, A., Grabisch, M., Rico, A.: Modeling attitudes toward uncertainty through the use of the Sugeno integral. J. Math. Econom. 44 (2008), 1084-1099. | DOI | MR | Zbl

[11] Daraby, B., Ghadimi, F.: General Minkowski type and related inequalities for seminormed fuzzy integrals. Sahand Commun. Math. Analysis 1 (2014), 9-20. | Zbl

[12] Dunford, N., Schwartz, J. T.: Linear Operators, Part I General Theory. A Wiley Interscience Publishers, New York 1988. | MR | Zbl

[13] Durante, F., Sempi, C.: Semicopulae. Kybernetika 41 (2005), 315-328. | MR | Zbl

[14] Fan, K.: Entfernung zweier zufälligen Grössen und die Konvergenz nach Wahrscheinlichkeit. Math. Zeitschrift 49 (1944), 681-683. | DOI | MR

[15] Föllmer, H., Schied, A.: Stochastic Finance. An Introduction In Discrete Time. De Gruyter, Berlin 2011. | DOI | MR | Zbl

[16] Fréchet, M.: Sur divers modes de convergence d'une suite de fonctions d'une variable. Bull. Calcutta Math. Soc. 11 (1919-20), 187-206.

[17] Greco, S., Mesiar, R., Rindone, F., Šipeky, L.: Superadditive and subadditive transformations of integrals and aggregation functions. Fuzzy Sets and Systems 291 (2016), 40-53. | MR

[18] Imaoka, H.: On a subjective evaluation model by a generalized fuzzy integral. Int. J. Uncertainty, Fuzziness and Knowledge-Based Systems 5 (1997), 517-529. | DOI | MR | Zbl

[19] Kallenberg, O.: Foundations of Modern Probability. Second edition. Springer, Berlin 2002. | DOI | MR

[20] Kaluszka, M., Okolewski, A., Boczek, M.: On Chebyshev type inequalities for generalized Sugeno integrals. Fuzzy Sets and Systems 244 (2014), 51-62. | DOI | MR | Zbl

[21] Kandel, A., Byatt, W. J.: Fuzzy sets, fuzzy algebra, and fuzzy statistics. Proc. IEEE 66 (1978), 1619-1639. | DOI | MR

[22] Klement, E. P., Mesiar, R., Pap, E.: Triangular Norms. Kluwer Academic Publishers, Dordrecht 2000. | DOI | MR | Zbl

[23] Klement, E. P., Mesiar, R., Pap, E.: A universal integral as common frame for Choquet and Sugeno integral. IEEE Trans. Fuzzy Systems 18 (2010), 178-187. | DOI

[24] Li, G.: A metric on space of measurable functions and the related convergence. Int. J. Uncertainty, Fuzziness and Knowledge-Based Systems 20 (2012), 211-222. | DOI | MR | Zbl

[25] Liu, B.: Uncertainty Theory. Fourth edition. Springer 2015. | DOI | MR

[26] Murofushi, T.: A note on upper and lower Sugeno integrals. Fuzzy Sets and Systems 138 (2003), 551-558. | DOI | MR | Zbl

[27] Ouyang, Y., Mesiar, R.: On the Chebyshev type inequality for seminormed fuzzy integral. Applied Math. Lett. 22 (2009), 1810-1815. | DOI | MR | Zbl

[28] Ouyang, Y., Mesiar, R., Agahi, H.: An inequality related to Minkowski type for Sugeno integrals. Inform. Sci. 180 (2010), 2793-2801. | DOI | MR | Zbl

[29] Pap, E., ed.: Handbook of Measure Theory. Elsevier Science, Amsterdam 2002.

[30] Román-Flores, H., Flores-Franulič, A., Chalco-Cano, Y.: The fuzzy integral for monotone functions. Applied Math. Comput. 185 (2007), 492-498. | DOI | MR | Zbl

[31] Rüschendorf, L.: Mathematical Risk Analysis: Dependence, Risk Bounds, Optimal Allocations and Portfolios. Springer Science and Business Media, Berlin 2013. | DOI | MR | Zbl

[32] Shilkret, N.: Maxitive measure and integration. Indagationes Math. 33 (1971), 109-116. | DOI | MR | Zbl

[33] García, F. Suárez, Álvarez, P. Gil: Two families of fuzzy integrals. Fuzzy Sets and Systems 18 (1986), 67-81. | DOI | MR

[34] Sugeno, M.: Theory of Fuzzy Integrals and its Applications. Ph.D. Dissertation, Tokyo Institute of Technology 1974.

[35] Wang, Z., Klir, G.: Generalized Measure Theory. Springer, New York 2009. | DOI | MR | Zbl

[36] Wu, Ch., Rena, X., Wu, C.: A note on the space of fuzzy measurable functions for a monotone measure. Fuzzy Sets and Systems 182 (2011), 2-12. | DOI | MR

[37] Wu, L., Sun, J., Ye, X., Zhu, L.: Hölder type inequality for Sugeno integral. Fuzzy Sets and Systems 161 (2010), 2337-2347. | DOI | MR | Zbl

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