×

zbMATH — the first resource for mathematics

Inequalities of Hölder and Minkowski type for pseudo-integrals with respect to interval-valued \(\oplus\)-measures. (English) Zbl 1368.26024
Summary: In the present paper, the Hölder and Minkowski type of inequality for the pseudo-integral of a real-valued function with respect to the interval-valued pseudo-additive measure is proven. Two cases of semirings are considered. In the first case, pseudo-operations (pseudo-addition and pseudo-multiplication) are set by a strictly monotone continuous function. In the second case, the pseudo-addition is the idempotent operation sup, and pseudo-multiplication is specified by a strictly monotone continuous function, as in the first case. Trivial and nontrivial examples of interval-valued pseudo-additive measures are provided, as well as Hölder and Minkowski type of inequalities with respect to those measures.

MSC:
26D15 Inequalities for sums, series and integrals
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Agahi, H.; Mesiar, R.; Ouyang, Y., Chebyshev type inequalities for pseudo-integrals, Nonlinear Anal., 72, 2737-2743, (2010) · Zbl 1195.28014
[2] Agahi, H.; Mesiar, R.; Ouyang, Y., General Minkowski type inequalities for sugeno integrals, Fuzzy Sets Syst., 161, 708-715, (2010) · Zbl 1183.28027
[3] Agahi, H.; Mesiar, R.; Ouyang, Y.; Pap, E.; Štrboja, M., Hölder and Minkowski type inequalities for pseudo-integral, Appl. Math. Comput., 217, 8630-8639, (2011) · Zbl 1217.26039
[4] Agahi, H.; Mesiar, R.; Ouyang, Y.; Pap, E.; Štrboja, M., Berwald type inequality for sugeno integral, Appl. Math. Comput., 217, 4100-4108, (2010) · Zbl 1205.28006
[5] Caballero, J.; Sadarangani, K., A Cauchy-Schwarz type inequality for fuzzy integrals, Nonlinear Anal., 73, 3329-3335, (2010) · Zbl 1202.26027
[6] Chrisman, L., Independence with lower and upper probabilities, (Proceedings of the Twelfth International Conference on Uncertainty in Artificial Intelligence, (1996)), 169-177
[7] Grbić, T.; Medić, S.; Štajner-Papuga, I.; Došenović, T., Inequalities of Jensen and Chebyshev type for interval-valued measures based on pseudo-integrals, (Pap, E., Intelligent Systems: Models and Applications, (2013), Springer-Verlag), 23-41 · Zbl 1294.28014
[8] Grbić, T.; Štajner-Papuga, I.; Nedović, Lj., Pseudo-integral of set-valued functions, (Proceedings of EUSFLAT 2007, Vol. I, (2007)), 221-225
[9] Grbić, T.; Štajner-Papuga, I.; Štrboja, M., An approach to pseudo-integration of set-valued functions, Inf. Sci., 181, 2278-2292, (2011) · Zbl 1217.28030
[10] Gou, C.; Zhang, D., On set-valued fuzzy measures, Inf. Sci., 160, 13-25, (2004) · Zbl 1048.28011
[11] Hong, D. H., A Liapunov type inequality for sugeno integrals, Nonlinear Anal., 74, 7296-7303, (2011) · Zbl 1228.26039
[12] Jang, L. C., A note on convergence properties of interval-valued capacity functionals and Choquet integrals, Inf. Sci., 183, 151-158, (2012) · Zbl 1241.28014
[13] Litvinov, G. L.; Maslov, V. P., Correspondence principle for idempotent calculus and some computer applications, (1995), Institut des Hautes Etudes Scientifiques Bures-sur-Yvette, IHES/M/95/33 · Zbl 0916.49019
[14] Litvinov, G. L.; Maslov, V. P., Idempotent mathematics: correspondence principle and applications, Russ. Math. Surv., 51, 6, 1210-1211, (1996) · Zbl 0916.49019
[15] Medić, S.; Grbić, T.; Perović, A.; Duraković, N., Interval-valued Chebyshev, Hölder and Minkowski inequalities based on g-integrals, (Proceeding of SISY 2014, Subotica, (2014))
[16] Mesiar, R.; Pap, E., Idempotent integral as limit of g-integrals, Fuzzy Sets Syst., 102, 385-392, (1999) · Zbl 0953.28010
[17] Miranda, E.; Couso, I.; Gil, P., Approximations of upper and lower probabilities by measurable selections, Inf. Sci., 180, 1407-1417, (2010) · Zbl 1202.60011
[18] Pap, E., g-calculus, Univ. Novom Sadu, Zb. Rad. Prir.-Mat. Fak., Ser. Mat., 23, 145-156, (1993) · Zbl 0823.28011
[19] Pap, E., Null-additive set functions, (1995), Kluwer Academic Publishers Dordrecht-Boston-London · Zbl 0856.28001
[20] Pap, E., Application of the generated pseudo-analysis on nonlinear partial differential equations, (Litvinov, Eds G. L.; Maslov, V. P., Proceedings of the Conference on Idempotent Mathematics and Mathematical Physics, Contemporary Mathematics, vol. 377, (2005), American Mathematical Society), 239-259 · Zbl 1084.35004
[21] Pap, E., Generalized real analysis and its applications, Int. J. Approx. Reason., 47, 368-386, (2008) · Zbl 1189.26055
[22] Pap, E.; Štrboja, M., Generalization of the Jensen inequality for pseudo-integral, Inf. Sci., 180, 543-548, (2010) · Zbl 1183.26039
[23] Román-Flores, H.; Flores-Franulič, A.; Chalco-Cano, Y., A Jensen type inequality for fuzzy integrals, Inf. Sci., 177, 3192-3201, (2007) · Zbl 1127.28013
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.