Some General Inequalities for Choquet Integral

Xiuli Yang, Xiaoqiu Song^*, Leilei Huang

College of Science, China University of Mining and Technology, Xuzhou, China.

DOI: 10.4236/am.2015.614201                

Abstract

With the development of fuzzy measure theory, the integral inequalities based on Sugeno integral are extensively investigated. We concern on the inequalities of Choquuet integral. The main purpose of this paper is to prove the H?lder inequality for any arbitrary fuzzy measure-based Choquet integral whenever any two of these integrated functions f, g and h are comonotone, and there are three weights. Then we prove Minkowski inequality and Lyapunov inequality for Choquet integral. Moreover, when any two of these integrated functions f₁, f₂, …, f_n are comonotone, we also obtain the Hölder inequality, Minkowski inequality and Lyapunov inequality hold for Choquet integral.

Keywords

Choquet Integral, Fuzzy Measure, Comonotone, Hölder Inequality, Minkowski Inequality, Lyapunov Inequality

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Choquet, G. (1954) Theory of Capacities. Annales de l’institut Fourier (Grenoble), 5, 131-292. http://dx.doi.org/10.5802/aif.53
[2]	Ralescu, D. and Adams, G. (1980) The Fuzzy Integral. Journal of Mathematical Analysis and Applications, 75, 562-570. http://dx.doi.org/10.1016/0022-247X(80)90101-8
[3]	Pap, E. (1995) Null-Additive Set Functions. Kluwer Academic, Dordrecht.
[4]	Wang, Z.Y. and Klir, G. (2008) Generalized Measure Theory. Springer Verlag, New York.
[5]	Murofushi, T., Sugeno, M. and Machida, M. (1994) Non-Monotonic Fuzzy Measures and the Choquet Integral. Fuzzy Sets and Systems, 64, 73-86. http://dx.doi.org/10.1016/0165-0114(94)90008-6
[6]	Sugeno, M. (1974) Theory of Fuzzy Integrals and Its Applications. Ph.D. Thesis, Tokyo Institute of Technology, Tokyo.
[7]	Narukawa, Y. and Torra, V. (2007) Fuzzy Measures and Integrals in Evaluation of Strategies. Information Sciences, 177, 4686-4695. http://dx.doi.org/10.1016/j.ins.2007.05.010
[8]	Schmeidler, D. (1986) Integral Representation without Additivity. Proceedings of the American Mathematical Society, 97, 255-261. http://dx.doi.org/10.1090/S0002-9939-1986-0835875-8
[9]	Hardy, G.H., Littlewood, J.E. and Polya, G. (1952) Inequalities. 2nd Edition, Cambridge University Press, Cambridge.
[10]	Li, Z.G., Song, X.Q. and Yang, X.L. (2014) On Nonuniform Polynomial Trichotomy of Linear Discreat-Time Systems in Banach Spaces. Journal of Applied Mathematics, 2014, Article ID: 807265.
[11]	Wu, L.M., Sun, J.B., Ye, X.Q. and Zhu, L.P. (2010) Hölder Type Inequality for Sugeno Integral. Fuzzy Sets and Systems, 161, 2337-2347. http://dx.doi.org/10.1016/j.fss.2010.04.017
[12]	Roman-Flores, H. and Chalco-Cano, Y. (2007) Sugeno Integral and Geometric Inequalities. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 15, 1-11.
[13]	Roman-Flores, H., Flores-Franulic, A. and Chalco-Cano, Y. (2007) A Jensen Type Inequality for Fuzzy Integrals. Information Sciences, 177, 3192-3201.
[14]	Song, Y.Z., Song, X.Q., et al. (2015) Berwald Type Inequality for Extremal Universal Integrals Based on (α,m)-Concave Function. Journal of Mathematical Inequalities, 1, 1-15. http://dx.doi.org/10.7153/jmi-09-01
[15]	Song, X.Q. and Pan, Z. (1998) Fuzzy Algebra in Triangular Norm System. Fuzzy Sets and Systems, 93, 331-335. http://dx.doi.org/10.1016/S0165-0114(96)00195-9
[16]	Li, D.Q., Song, X.Q. and Yue, T. (2014) Hermite-Hadamard Type Inequality for Sugeno Integrals. Applied Mathematics and Computation, 237, 632-638. http://dx.doi.org/10.1016/j.amc.2014.03.144
[17]	Li, D.Q., Song, X.Q., et al. (2014) Generalization of Liyapunov Type Inequality for Pseudo-Integrals. Applied Mathematics and Computation, 241, 64-69. http://dx.doi.org/10.1016/j.amc.2014.05.006
[18]	Yang, X.L., Song, X.Q. and Lu, W. (2015) Sandor’s Type Inequality for Fuzzy Integrals. Journal of Nanjing University (Natural Sciences), Accepted.
[19]	Zhu, L.P. and Ouyang, Y. (2001) Hölder Inequality for Choquet Integral and Its Application. Fuzzy Systems and Mathematics, 25, 146-151.