Document Type : Research Paper
Authors
Department of Mathematics, University of Maragheh, P. O. Box 55181- 83111, Maragheh, Iran.
Abstract
Minkowski type inequalities for the seminormed fuzzy integrals on abstract spaces are studied in a rather general form. Also related inequalities to Minkowski type inequality for the seminormed fuzzy integrals on abstract spaces are studied. Several examples are given to illustrate the validity of theorems. Some results on Chebyshev and Minkowski type inequalities are obtained.
Keywords
[1] H. Agahi, R. Mesiar, Y. Ouyang, New general Chebyshev type inequalities for Sugeno integrals, International Journal of Approximate Reasoning, 51 (2009) 135-140.
[2] H. Agahi, R. Mesiar, Y. Ouyang, General Minkowski type inequalities for Sugeno integrals, Fuzzy Sets and Systems, 161 (2010) 708-715.
[3] H. Agahi, M. A. Yaghoobi, A Minkowski type inequality for fuzzy integrals, Journal of Uncertain Systems, Vol. 4, No. 3, (2010) 187-194.
[4] B. Daraby, Investigation of a Stolarsky type inequality for integrals in Pseudo-analysis, Fractional Calculus and Applied Analysis, Vol. 13, No. 4 (2010) 1-7.
[5] B. Daraby, Stplarsky type inequality for seminormed fuzzy integral., 30 November - 3 December 2010, Kuala Lumpur, Malaysia, 110-117.
[6] A. Flores-Franulic, H. Román-Flores, A Chebyshev type inequality for fuzzy integrals, Appl. Math. Comput, 190 (2007) 1178-1184.
[7] A. Kandel, W.J. Byatt, Fuzzy sets, fuzzy algebra, and fuzzy statistics, Proceedings of the IEEE 66(1978) 1619-16390.
[8] E.P. Klement, R. Mesiar, E. Pap, Triangular norms, in: Trends in Logic,in:Studia Logica Library,Vol.8, Kluwer Academic Publishers. Dordrecht,2000.
[9] E.P. Klement, R. Mesiar, E. Pap, Integration with respect to decomposable measures, based on a conditionally distributive semiring on the unit interval, International Journal of Uncertainty Fuzziness and Knowledge-based Systems, 8 (2000) 701-717.
[10] R. Mesiar, Choquet-like integrals, Journal of Mathematical Analysis and Applications 194(1995) 477-488.
[11] R. Mesiar, Y. Ouyang, General Chebyshev type inequalities for Sugeno integrals, Fuzzy Sets and Systems, 160 (2009) 58-64.
[12] T. Murofushi, M. Sugeno, Fuzzy t-conorm integral with respect to fuzzy measures: Generalization of Sugeno integral and Choquet integral, Fuzzy Sets and Systems, 42 (1991) 57-71.
[13] Y. Ouyang, J. Fang, L. Wang, Fuzzy Chebyshev type inequality, International Journal of Approximate Reasoning, 48(2008) 829-835.
[14] Y. Ouyang, R. Mesiar, On the Chebyshev type inequality for seminormed, fuzzy integral, Appllied Mathematics Letters, 22 (2009) 1810-1815.
[15] Y. Ouyang, R. Mesiar, H. Agahi, An inequality related to Minkowski type for Sugeno integrals, Information Sciences, 180 (2010) 2793-2801.
[16] E. Pap, g-calculus, Univ. Novom Sadu zb. Rad. Prirod. Mat. Fak. Ser. Mat. 23 (10 (1993) 145-156.
[17] E. Pap, Null-additive Set Functions, Kluwer, Dordrecht, 1995.
[18] D. Ralescu, G. Adams, The fuzzy integral, J. Math. Anal. Appl, 75 (1980) 562-570.
[19] H. Román-Flores, A. Flores-Franulivc, R. Bassanezi and M. Rojas-Medar, On the level-continuity of fuzzy integrals, Fuzzy Sets and Systems, 80 (1996) 339-344.
[20] H. Román-Flores, Y. Chalco-Cano, H-continuity of fuzzy measures and set defuzzifincation, Fuzzy Sets and Systems, 157(2006)230-242.
[21] H. Román-Flores, Y. Chalco-Cano, Sugeno integral and geometric inequalities, International Journal of Uncertainty Fuzziness and Knowledge-based Systems, 15 (2007) 1-11.
[22] H. Román-Flores, A. Flores-Franulivc, Y. Chalco-Cano, The fuzzy integral for monotone functions, Applied Mathematics and Computation 185 (2007) 492-498.
[23] H. Román-Flores, A. Flores-Franulivc, Y. Chalco-Cano, A Jensen type inequality for fuzzy integrals, Information Sciences 177 (2007) 3192-3201.
[24] H. Román-Flores, A. Flores-Franulivc, Y. Chalco-cano, A convolution type inequality for fuzzy integrals, Applied Mathematics and Computation, 195 (2008) 94-99.
[25] H. Román-Flores, A. Flores-Franulivc, Y. Chalco-cano, A note on fuzzy integral inequality of Stolarsky type, Applied Mathematics and Computation, 196 (2008) 55-59.
[26] H. Román-Flores, A. Flores-Franulivc, Y. Chalco-cano, A Hardy type inequality for fuzzy integrals, Applied Mathematics and Computation, 204 (2008) 178-183.
[27] F. Suárez Garcacuteía, P. Gil Alvarez, Two families of fuzzy integrals, Fuzzy Sets and Systems, 18 (1986)
67-81.
[28] M. Sugeno, Theory of fuzzy integrals and its applications, Ph.D. thesis. Tokyo Institute of Technology, 1974.
[29] M. Sugeno, T. Murofushi, Pseudo-additive measures and integrals, Journal of Mathematical Analysis and Applications, 122 (1987) 197-222.
[30] S. Weber, Measures of fuzzy sets and measures of fuzziness, Fuzzy Sets and Systems, 13 (1984) 247-271.
[31] S. Weber, perp-decomposable measures and integrals for Archimedean t-conorms perp, Journal of Mathematical Analysis and Applications, 101 (1984) 114-138.
[32] Z. Wang, G.J. Klir, Fuzzy Measure Theory, Plenum Press, New York, 1992.
[33] C. Wu, S. Wang, M. Ma, Generalized fuzzy integrals: part I. Fundamental concepts, Fuzzy Sets and Systems, 57 (1993) 219-226.
[2] H. Agahi, R. Mesiar, Y. Ouyang, General Minkowski type inequalities for Sugeno integrals, Fuzzy Sets and Systems, 161 (2010) 708-715.
[3] H. Agahi, M. A. Yaghoobi, A Minkowski type inequality for fuzzy integrals, Journal of Uncertain Systems, Vol. 4, No. 3, (2010) 187-194.
[4] B. Daraby, Investigation of a Stolarsky type inequality for integrals in Pseudo-analysis, Fractional Calculus and Applied Analysis, Vol. 13, No. 4 (2010) 1-7.
[5] B. Daraby, Stplarsky type inequality for seminormed fuzzy integral., 30 November - 3 December 2010, Kuala Lumpur, Malaysia, 110-117.
[6] A. Flores-Franulic, H. Román-Flores, A Chebyshev type inequality for fuzzy integrals, Appl. Math. Comput, 190 (2007) 1178-1184.
[7] A. Kandel, W.J. Byatt, Fuzzy sets, fuzzy algebra, and fuzzy statistics, Proceedings of the IEEE 66(1978) 1619-16390.
[8] E.P. Klement, R. Mesiar, E. Pap, Triangular norms, in: Trends in Logic,in:Studia Logica Library,Vol.8, Kluwer Academic Publishers. Dordrecht,2000.
[9] E.P. Klement, R. Mesiar, E. Pap, Integration with respect to decomposable measures, based on a conditionally distributive semiring on the unit interval, International Journal of Uncertainty Fuzziness and Knowledge-based Systems, 8 (2000) 701-717.
[10] R. Mesiar, Choquet-like integrals, Journal of Mathematical Analysis and Applications 194(1995) 477-488.
[11] R. Mesiar, Y. Ouyang, General Chebyshev type inequalities for Sugeno integrals, Fuzzy Sets and Systems, 160 (2009) 58-64.
[12] T. Murofushi, M. Sugeno, Fuzzy t-conorm integral with respect to fuzzy measures: Generalization of Sugeno integral and Choquet integral, Fuzzy Sets and Systems, 42 (1991) 57-71.
[13] Y. Ouyang, J. Fang, L. Wang, Fuzzy Chebyshev type inequality, International Journal of Approximate Reasoning, 48(2008) 829-835.
[14] Y. Ouyang, R. Mesiar, On the Chebyshev type inequality for seminormed, fuzzy integral, Appllied Mathematics Letters, 22 (2009) 1810-1815.
[15] Y. Ouyang, R. Mesiar, H. Agahi, An inequality related to Minkowski type for Sugeno integrals, Information Sciences, 180 (2010) 2793-2801.
[16] E. Pap, g-calculus, Univ. Novom Sadu zb. Rad. Prirod. Mat. Fak. Ser. Mat. 23 (10 (1993) 145-156.
[17] E. Pap, Null-additive Set Functions, Kluwer, Dordrecht, 1995.
[18] D. Ralescu, G. Adams, The fuzzy integral, J. Math. Anal. Appl, 75 (1980) 562-570.
[19] H. Román-Flores, A. Flores-Franulivc, R. Bassanezi and M. Rojas-Medar, On the level-continuity of fuzzy integrals, Fuzzy Sets and Systems, 80 (1996) 339-344.
[20] H. Román-Flores, Y. Chalco-Cano, H-continuity of fuzzy measures and set defuzzifincation, Fuzzy Sets and Systems, 157(2006)230-242.
[21] H. Román-Flores, Y. Chalco-Cano, Sugeno integral and geometric inequalities, International Journal of Uncertainty Fuzziness and Knowledge-based Systems, 15 (2007) 1-11.
[22] H. Román-Flores, A. Flores-Franulivc, Y. Chalco-Cano, The fuzzy integral for monotone functions, Applied Mathematics and Computation 185 (2007) 492-498.
[23] H. Román-Flores, A. Flores-Franulivc, Y. Chalco-Cano, A Jensen type inequality for fuzzy integrals, Information Sciences 177 (2007) 3192-3201.
[24] H. Román-Flores, A. Flores-Franulivc, Y. Chalco-cano, A convolution type inequality for fuzzy integrals, Applied Mathematics and Computation, 195 (2008) 94-99.
[25] H. Román-Flores, A. Flores-Franulivc, Y. Chalco-cano, A note on fuzzy integral inequality of Stolarsky type, Applied Mathematics and Computation, 196 (2008) 55-59.
[26] H. Román-Flores, A. Flores-Franulivc, Y. Chalco-cano, A Hardy type inequality for fuzzy integrals, Applied Mathematics and Computation, 204 (2008) 178-183.
[27] F. Suárez Garcacuteía, P. Gil Alvarez, Two families of fuzzy integrals, Fuzzy Sets and Systems, 18 (1986)
67-81.
[28] M. Sugeno, Theory of fuzzy integrals and its applications, Ph.D. thesis. Tokyo Institute of Technology, 1974.
[29] M. Sugeno, T. Murofushi, Pseudo-additive measures and integrals, Journal of Mathematical Analysis and Applications, 122 (1987) 197-222.
[30] S. Weber, Measures of fuzzy sets and measures of fuzziness, Fuzzy Sets and Systems, 13 (1984) 247-271.
[31] S. Weber, perp-decomposable measures and integrals for Archimedean t-conorms perp, Journal of Mathematical Analysis and Applications, 101 (1984) 114-138.
[32] Z. Wang, G.J. Klir, Fuzzy Measure Theory, Plenum Press, New York, 1992.
[33] C. Wu, S. Wang, M. Ma, Generalized fuzzy integrals: part I. Fundamental concepts, Fuzzy Sets and Systems, 57 (1993) 219-226.
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