Skip to main content
Top
Published in: Soft Computing 7/2016

28-10-2015 | Foundations

On Carlson’s inequality for Sugeno and Choquet integrals

Authors: Michał Boczek, Marek Kaluszka

Published in: Soft Computing | Issue 7/2016

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

We present a Carlson type inequality for the Sugeno integral and a much wider class of functions than the comonotone functions. We also provide three Carlson type inequalities for the Choquet integral. Our inequalities generalize many known results.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Literature
go back to reference Agahi H, Yaghoobi MA (2010) General Hardy type inequality for seminormed fuzzy integrals. Appl Math Comput 216:1972–1977MathSciNetMATH Agahi H, Yaghoobi MA (2010) General Hardy type inequality for seminormed fuzzy integrals. Appl Math Comput 216:1972–1977MathSciNetMATH
go back to reference Agahi H, Mesiar R, Ouyang Y (2012) On some advanced type inequalities for Sugeno integral and T-(S-)evaluators. Inf Sci 190:64–75MathSciNetCrossRefMATH Agahi H, Mesiar R, Ouyang Y (2012) On some advanced type inequalities for Sugeno integral and T-(S-)evaluators. Inf Sci 190:64–75MathSciNetCrossRefMATH
go back to reference Barza S, Peoari J, Persson L-E (1998) Carlson type inequalities. J Inequal Appl 2:121–135 Barza S, Peoari J, Persson L-E (1998) Carlson type inequalities. J Inequal Appl 2:121–135
go back to reference Daraby B, Arabi L (2013) Related Fritz Carlson type inequalities for Sugeno integrals. Soft Comput 17:1745–1750CrossRefMATH Daraby B, Arabi L (2013) Related Fritz Carlson type inequalities for Sugeno integrals. Soft Comput 17:1745–1750CrossRefMATH
go back to reference Denneberg D (1994) Non-additive measure and integral. Kluwer Academic Publishers, DordrechtCrossRefMATH Denneberg D (1994) Non-additive measure and integral. Kluwer Academic Publishers, DordrechtCrossRefMATH
go back to reference Flores-Franulič A, Román-Flores H (2007) A Chebyshev type inequality for fuzzy integrals. Appl Math Comput 190:1178–1184MathSciNetMATH Flores-Franulič A, Román-Flores H (2007) A Chebyshev type inequality for fuzzy integrals. Appl Math Comput 190:1178–1184MathSciNetMATH
go back to reference Girotto B, Holzer S (2012) Chebyshev and Jensen inequalities for Choquet integral. Math Pannonica 23:267–275MathSciNetMATH Girotto B, Holzer S (2012) Chebyshev and Jensen inequalities for Choquet integral. Math Pannonica 23:267–275MathSciNetMATH
go back to reference Grabisch M, Labreuche C (2010) A decade of application of the Choquet and Sugeno integrals in multi-criteria decision aid. Ann Oper Res 175(1):247–286MathSciNetCrossRefMATH Grabisch M, Labreuche C (2010) A decade of application of the Choquet and Sugeno integrals in multi-criteria decision aid. Ann Oper Res 175(1):247–286MathSciNetCrossRefMATH
go back to reference Kaluszka M, Okolewski A, Boczek M (2014a) On Chebyshev type inequalities for generalized Sugeno integrals. Fuzzy Sets Syst 244:51–62 Kaluszka M, Okolewski A, Boczek M (2014a) On Chebyshev type inequalities for generalized Sugeno integrals. Fuzzy Sets Syst 244:51–62
go back to reference Kaluszka M, Okolewski A, Boczek M (2014b) On the Jensen type inequality for generalized Sugeno integral. Inf Sci 266:140–147 Kaluszka M, Okolewski A, Boczek M (2014b) On the Jensen type inequality for generalized Sugeno integral. Inf Sci 266:140–147
go back to reference Klement EP, Mesiar R, Pap E (2010) A universal integral as common frame for Choquet and Sugeno integral. IEEE Trans Fuzzy Syst 18:178–187CrossRef Klement EP, Mesiar R, Pap E (2010) A universal integral as common frame for Choquet and Sugeno integral. IEEE Trans Fuzzy Syst 18:178–187CrossRef
go back to reference Mesiar R, Li J, Pap E (2010) The Choquet integral as Lebesgue integral and related inequalities. Kybernetika 46:1098–1107MathSciNetMATH Mesiar R, Li J, Pap E (2010) The Choquet integral as Lebesgue integral and related inequalities. Kybernetika 46:1098–1107MathSciNetMATH
go back to reference Mitrinović DS, Pečarić JE, Fink AM (1991) Inequalities involving functions and their integrals and derivatives. Kluwer, DordrechtCrossRefMATH Mitrinović DS, Pečarić JE, Fink AM (1991) Inequalities involving functions and their integrals and derivatives. Kluwer, DordrechtCrossRefMATH
go back to reference Murofushi T, Sugeno M (1991) A theory of fuzzy measures: representations, the Choquet integral and null sets. J Math Anal Appl 159:532–549MathSciNetCrossRefMATH Murofushi T, Sugeno M (1991) A theory of fuzzy measures: representations, the Choquet integral and null sets. J Math Anal Appl 159:532–549MathSciNetCrossRefMATH
go back to reference Niculescu C, Persson L-E (2006) Convex functions and their applications. A contemporary approach. Springer, New YorkCrossRefMATH Niculescu C, Persson L-E (2006) Convex functions and their applications. A contemporary approach. Springer, New YorkCrossRefMATH
go back to reference Ouyang Y (2015) On Carlson inequality for the Choquet integral (personal communication) Ouyang Y (2015) On Carlson inequality for the Choquet integral (personal communication)
go back to reference Pap E (1995) Null-additive set functions. Kluwer, DordrechtMATH Pap E (1995) Null-additive set functions. Kluwer, DordrechtMATH
go back to reference Suárez García F, Gil Álvarez P (1986) Two families of fuzzy integrals. Fuzzy Sets Syst 18:67–81 Suárez García F, Gil Álvarez P (1986) Two families of fuzzy integrals. Fuzzy Sets Syst 18:67–81
go back to reference Sugeno M (1974) Theory of fuzzy integrals and its applications. Ph.D. dissertation, Tokyo Institute of Technology Sugeno M (1974) Theory of fuzzy integrals and its applications. Ph.D. dissertation, Tokyo Institute of Technology
go back to reference Tang YL, Ouyang Y (2012) On the Carlson inequality for the Choquet integral. J Huzhou Teach Coll 2:21–25 (in Chinese) Tang YL, Ouyang Y (2012) On the Carlson inequality for the Choquet integral. J Huzhou Teach Coll 2:21–25 (in Chinese)
go back to reference Wang X, Bai C (2011) General Fritz Carlson’s type inequality for Sugeno integrals. Hindawi Publ Corp J Inequal Appl Article ID 761430, 9 pages Wang X, Bai C (2011) General Fritz Carlson’s type inequality for Sugeno integrals. Hindawi Publ Corp J Inequal Appl Article ID 761430, 9 pages
go back to reference Zhao X, Zhang Q (2011) Hölder type inequality and Jensen type inequality for Choquet integral. In: Wang Y, Li T (eds) Knowledge engineering and management. AISC 123, pp 219–224 Zhao X, Zhang Q (2011) Hölder type inequality and Jensen type inequality for Choquet integral. In: Wang Y, Li T (eds) Knowledge engineering and management. AISC 123, pp 219–224
Metadata
Title
On Carlson’s inequality for Sugeno and Choquet integrals
Authors
Michał Boczek
Marek Kaluszka
Publication date
28-10-2015
Publisher
Springer Berlin Heidelberg
Published in
Soft Computing / Issue 7/2016
Print ISSN: 1432-7643
Electronic ISSN: 1433-7479
DOI
https://doi.org/10.1007/s00500-015-1909-9

Other articles of this Issue 7/2016

Soft Computing 7/2016 Go to the issue

Premium Partner