A Chebyshev type inequality for fuzzy integrals. (English) Zbl 1129.26021
The classical Chebyshev’s integral inequality \(\int_0^1 fg \,d\mu \geq(\int_0^1 f \,d\mu )(\int_0^1 g \,d\mu)\) for \(f, g: [0,1] \to [0, \infty )\) of the same monotonicity type is shown for the case, when the Lebesgue integral is replaced by the Sugeno integral. As a corollary an analogous result for finite number of fuctions is included.
Reviewer: Vladimír Janiš (Banská Bystrica)
References:
| [1] | M. Sugeno, Theory of fuzzy integrals and its applications, Ph.D. thesis. Tokyo Institute of Technology, 1974.; M. Sugeno, Theory of fuzzy integrals and its applications, Ph.D. thesis. Tokyo Institute of Technology, 1974. |
| [2] | Ralescu, D.; Adams, G., The fuzzy integral, J. Math. Anal. Appl., 75, 562-570 (1980) · Zbl 0438.28007 |
| [3] | Román-Flores, H.; Flores-Franulič, A.; Bassanezi, R.; Rojas-Medar, M., On the level-continuity of fuzzy integrals, Fuzzy Sets Syst., 80, 339-344 (1996) · Zbl 0872.28013 |
| [4] | Román-Flores, H.; Chalco-Cano, Y., H-continuity of fuzzy measures and set defuzzification, Fuzzy Sets Syst., 157, 230-242 (2006) · Zbl 1084.28010 |
| [5] | Román-Flores, H.; Chalco-Cano, Y., Sugeno integral and geometric inequalities, Int. J. Uncertain. Fuzz. Knowledge-Based Syst., 15, 1, 1-11 (2007) · Zbl 1118.28012 |
| [6] | Wang, Z.; Klir, G., Fuzzy measure theory (1992), Plenum: Plenum New York · Zbl 0812.28010 |
| [7] | H. Román-Flores, A. Flores-Franulič, Y. Chalco-Cano, The fuzzy integral for monotone functions, Appl. Math. Comput., in press, doi:10.1016/j.amc.2006.07.066.; H. Román-Flores, A. Flores-Franulič, Y. Chalco-Cano, The fuzzy integral for monotone functions, Appl. Math. Comput., in press, doi:10.1016/j.amc.2006.07.066. · Zbl 1116.26024 |
| [8] | H. Román-Flores, A. Flores-Franulič, Y. Chalco-Cano, A Jensen type inequality for fuzzy integrals, Inform. Sciences, in press, doi:10.1016/j.ins.2007.02.006.; H. Román-Flores, A. Flores-Franulič, Y. Chalco-Cano, A Jensen type inequality for fuzzy integrals, Inform. Sciences, in press, doi:10.1016/j.ins.2007.02.006. · Zbl 1127.28013 |
| [9] | Tong, Y. L., Relationship between stochastic inequalities and some classical mathematical inequalities, J. Inequal. Appl., 1, 85-98 (1997) · Zbl 0882.60015 |
| [10] | Wagener, A., Chebyshev’s algebraic inequality and comparative statics under uncertainty, Math. Social Sci., 52, 217-221 (2006) · Zbl 1142.60320 |
| [11] | Gauchman, H., Some integral inequalities involving Taylor’s remainder II, J. Inequal. Pure Appl. Math., 4, 1 (2003), Art.1 · Zbl 1032.26015 |
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