| Title:
|
On the Minkowski-Hölder type inequalities for generalized Sugeno integrals with an application (English) |
| Author:
|
Boczek, Michał |
| Author:
|
Kaluszka, Marek |
| Language:
|
English |
| Journal:
|
Kybernetika |
| ISSN:
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0023-5954 (print) |
| ISSN:
|
1805-949X (online) |
| Volume:
|
52 |
| Issue:
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3 |
| Year:
|
2016 |
| Pages:
|
329-347 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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]. (English) |
| Keyword:
|
seminormed fuzzy integral |
| Keyword:
|
semicopula |
| Keyword:
|
monotone measure |
| Keyword:
|
Minkowski's inequality |
| Keyword:
|
Hölder's inequality |
| Keyword:
|
convergence in mean |
| MSC:
|
26E50 |
| MSC:
|
28E10 |
| idZBL:
|
Zbl 06644298 |
| idMR:
|
MR3532510 |
| DOI:
|
10.14736/kyb-2016-3-0329 |
| . |
| Date available:
|
2016-07-17T12:11:48Z |
| Last updated:
|
2018-01-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/145778 |
| . |
| Reference:
|
[1] Agahi, H., Mesiar, R., Ouyang, Y.: General Minkowski type inequalities for Sugeno integrals..Fuzzy Sets and Systems 161 (2010), 708-715. Zbl 1183.28027, MR 2578627, 10.1016/j.fss.2009.10.007 |
| Reference:
|
[2] Agahi, H., Mesiar, R.: On Cauchy-Schwarz's inequality for Choquet-like integrals without the comonotonicity condition..Soft Computing 19 (2015), 1627-1634. 10.1007/s00500-014-1578-0 |
| Reference:
|
[3] Bassan, B., Spizzichino, F.: Relations among univariate aging, bivariate aging and dependence for exchangeable lifetimes..J. Multivariate Analysis 93 (2005), 313-339. Zbl 1070.60015, MR 2162641, 10.1016/j.jmva.2004.04.002 |
| Reference:
|
[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 3336136 |
| Reference:
|
[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 3336137 |
| Reference:
|
[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). |
| Reference:
|
[7] Carothers, N. L.: Real Analysis..University Press, Cambridge 2000. Zbl 0997.26003, MR 1772332, 10.1017/cbo9780511814228 |
| Reference:
|
[8] Cattaneo, M. E.: On maxitive integration..Department of Statistics University of Munich 2013, http://www.stat.uni-muenchen.de. |
| Reference:
|
[9] Cerdá, J.: Lorentz capacity spaces..Contemporary Math. 445 (2007), 45-59. Zbl 1141.46313, MR 2381885, 10.1090/conm/445/08592 |
| Reference:
|
[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. Zbl 1152.28331, MR 2456469, 10.1016/j.jmateco.2007.09.003 |
| Reference:
|
[11] Daraby, B., Ghadimi, F.: General Minkowski type and related inequalities for seminormed fuzzy integrals..Sahand Commun. Math. Analysis 1 (2014), 9-20. Zbl 1317.26023 |
| Reference:
|
[12] Dunford, N., Schwartz, J. T.: Linear Operators, Part I General Theory..A Wiley Interscience Publishers, New York 1988. Zbl 0635.47001, MR 1009162 |
| Reference:
|
[13] Durante, F., Sempi, C.: Semicopulae..Kybernetika 41 (2005), 315-328. Zbl 1249.26021, MR 2181421 |
| Reference:
|
[14] Fan, K.: Entfernung zweier zufälligen Grössen und die Konvergenz nach Wahrscheinlichkeit..Math. Zeitschrift 49 (1944), 681-683. MR 0011903, 10.1007/bf01174225 |
| Reference:
|
[15] Föllmer, H., Schied, A.: Stochastic Finance. An Introduction In Discrete Time..De Gruyter, Berlin 2011. Zbl 1126.91028, MR 2779313, 10.1515/9783110218053 |
| Reference:
|
[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. |
| Reference:
|
[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 3463652 |
| Reference:
|
[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. Zbl 1232.68132, MR 1480749, 10.1142/s0218488597000403 |
| Reference:
|
[19] Kallenberg, O.: Foundations of Modern Probability. Second edition..Springer, Berlin 2002. MR 1876169, 10.1007/978-1-4757-4015-8 |
| Reference:
|
[20] Kaluszka, M., Okolewski, A., Boczek, M.: On Chebyshev type inequalities for generalized Sugeno integrals..Fuzzy Sets and Systems 244 (2014), 51-62. Zbl 1315.28013, MR 3192630, 10.1016/j.fss.2013.10.015 |
| Reference:
|
[21] Kandel, A., Byatt, W. J.: Fuzzy sets, fuzzy algebra, and fuzzy statistics..Proc. IEEE 66 (1978), 1619-1639. MR 0707701, 10.1109/proc.1978.11171 |
| Reference:
|
[22] Klement, E. P., Mesiar, R., Pap, E.: Triangular Norms..Kluwer Academic Publishers, Dordrecht 2000. Zbl 1087.20041, MR 1790096, 10.1007/978-94-015-9540-7 |
| Reference:
|
[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. 10.1109/tfuzz.2009.2039367 |
| Reference:
|
[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. Zbl 1242.28028, MR 2911763, 10.1142/s0218488512500109 |
| Reference:
|
[25] Liu, B.: Uncertainty Theory. Fourth edition..Springer 2015. MR 3307516, 10.1007/978-3-662-44354-5 |
| Reference:
|
[26] Murofushi, T.: A note on upper and lower Sugeno integrals..Fuzzy Sets and Systems 138 (2003), 551-558. Zbl 1094.28012, MR 1998678, 10.1016/s0165-0114(02)00375-5 |
| Reference:
|
[27] Ouyang, Y., Mesiar, R.: On the Chebyshev type inequality for seminormed fuzzy integral..Applied Math. Lett. 22 (2009), 1810-1815. Zbl 1185.28026, MR 2558545, 10.1016/j.aml.2009.06.024 |
| Reference:
|
[28] Ouyang, Y., Mesiar, R., Agahi, H.: An inequality related to Minkowski type for Sugeno integrals..Inform. Sci. 180 (2010), 2793-2801. Zbl 1193.28016, MR 2644587, 10.1016/j.ins.2010.03.018 |
| Reference:
|
[29] Pap, E., ed.: Handbook of Measure Theory..Elsevier Science, Amsterdam 2002. |
| Reference:
|
[30] Román-Flores, H., Flores-Franulič, A., Chalco-Cano, Y.: The fuzzy integral for monotone functions..Applied Math. Comput. 185 (2007), 492-498. Zbl 1116.26024, MR 2297820, 10.1016/j.amc.2006.07.066 |
| Reference:
|
[31] Rüschendorf, L.: Mathematical Risk Analysis: Dependence, Risk Bounds, Optimal Allocations and Portfolios..Springer Science and Business Media, Berlin 2013. Zbl 1266.91001, MR 3051756, 10.1007/978-3-642-33590-7 |
| Reference:
|
[32] Shilkret, N.: Maxitive measure and integration..Indagationes Math. 33 (1971), 109-116. Zbl 0218.28005, MR 0288225, 10.1016/s1385-7258(71)80017-3 |
| Reference:
|
[33] García, F. Suárez, Álvarez, P. Gil: Two families of fuzzy integrals..Fuzzy Sets and Systems 18 (1986), 67-81. MR 0825620, 10.1016/0165-0114(86)90028-x |
| Reference:
|
[34] Sugeno, M.: Theory of Fuzzy Integrals and its Applications..Ph.D. Dissertation, Tokyo Institute of Technology 1974. |
| Reference:
|
[35] Wang, Z., Klir, G.: Generalized Measure Theory..Springer, New York 2009. Zbl 1184.28002, MR 2453907, 10.1007/978-0-387-76852-6 |
| Reference:
|
[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. MR 2825769, 10.1016/j.fss.2010.10.006 |
| Reference:
|
[37] Wu, L., Sun, J., Ye, X., Zhu, L.: Hölder type inequality for Sugeno integral..Fuzzy Sets and Systems 161 (2010), 2337-2347. Zbl 1194.28019, MR 2658037, 10.1016/j.fss.2010.04.017 |
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