Abstract
In this paper, we prove some Milne type inequalities for interval-valued functions and, along with it, we explore some connections with other inequalities. More precisely, using the Aumann integral and the Kulisch–Miranker order and including-order on the space of real and compact intervals, we establish some Milne type inequalities for interval-valued functions. Also, using different orders, we obtain some connections with Chebyshev, Cauchy–Schwarz, and Hölder inequality. Finally, some new ideas and results based on submodular measures are explored as well as some examples and applications are presented for illustrating our results.
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References
Agahi H, Román-Flores H, Flores-Franulič A (2011) General Barnes-Godunova-Levin type inequalities for Sugeno integral. Inf Sci 181:1072–1079
Agahi H (2020) A refined Hölder’s inequality for Choquet expectation by Cauchy-Schwarz’s inequality. Inf Sci 512:929–934
Aubin JP, Cellina A (1984) Differential inclusions. Springer, New York
Aubin JP, Franskowska H (1990) Set-valued analysis. Birkhäuser, Boston
Aubin JP, Franskowska H (2000) Introduction: set-valued analysis in control theory. Set Value Anal 8:1–9
Aumann RJ (1965) Integrals of set-valued functions. J Math Anal Appl 12:1–12
Chalco-Cano Y, Flores-Franulič A, Román-Flores H (2012) Ostrowski type inequalities for inteval-valued functions using generalized Hukuhara derivative. Comput Appl Math 31:457–472
Chateauneuf A, Cohen M, Kast R (1997) Comonotone random variables in economics. A review of some results, GREQAM, Working Paper 97A07
Costa TM, Román-Flores H (2017) Some integral inequalities for fuzzy-interval-valued functions. Inf Sci 420:110–125
Costa TM, Román-Flores H (2019a) Gauss-type integral inequalities for interval and fuzzy-interval-valued functions. Comput Appl Math 38:Article No 58
Costa TM, Román-Flores H, Chalco-Cano Y (2019b) Opial-type inequalities for interval-valued functions. Fuzzy Sets Syst 358:48–63
Costa TM, Chalco-Cano Y, Román-Flores H (2020) Wirtinger-type integral inequalities for interval-valued functions. Fuzzy Sets Syst 396:102–114
Denneberg D (1994) Non-additive measure and integral, 1st edn. Springer, Amsterdam, p 1994
Flores-Franulič A, Román-Flores H (2007) A Chebyshev type inequality for fuzzy integrals. Appl Math Comput 190:1178–1184
Flores-Franulič A, Román-Flores H, Chalco-Cano Y (2008) A note on fuzzy integral inequality of Stolarsky type. Appl Math Comput 196:55–59
Flores-Franulič A, Román-Flores H, Chalco-Cano Y (2009) Markov type inequalities for fuzzy integrals. Appl Math Comput 207:242–247
Girotto B, Holder S (2011) Chebyshev type inequality for Choquet integral and comonotonicity. Int J Approx Reason 52:1118–1123
Kulisch U, Miranker W (1981) Computer arithmetic in theory and practice. Academic Press, New York
Markov S (1979) Calculus for interval functions of a real variable. Computing 22:325–337
Milne EA (1925) Note on Rosseland’s integral for the stellar absorption coefficient. Monthly Notices R Astron Soc 85:979–984
Mitrinović DS, Pečarić JE, Fink AM (1993) Classical and new inequalities in analysis. Mathematics and its applications. Kluwer Academic Publishers Group, Dordrecht
Moore RE (1966) Interval analysis. Prince-Hall, Englewood Cliffs (NJ)
Moore RE (1979) Method and application of interval analysis. SIAM, New York
Moore RE (1985) Computational functional analysis. Ellis Horwood Limited, England
Moore RE, Kearfott RB, Cloud MJ (2009) Introduction to interval analysis. SIAM, USA
Pap E (1995) Null-additive set functions. Kluwer, Dordrecht
Rokne JG (2001) Interval arithmetic and interval analysis: an introduction. In Granular Computing: An Emerging Paradigm, W. Pedrycz Ed.- Heldelberg; New York; Physica-Verlag, pp 1–22
Román-Flores H, Chalco-Cano Y (2006) H-continuity of fuzzy measures and set defuzzification. Fuzzy Sets Syst 157:230–242
Román-Flores H, Flores-Franulič A, Chalco-Cano Y (2007a) The fuzzy integral for monotone functions. Appl Math Comput 185:492–498
Román-Flores H, Flores-Franulič A, Chalco-Cano Y (2007b) A Jensen type inequality for fuzzy integrals. Inf Sci 177:3192–3201
Román-Flores H, Flores-Franulič A, Chalco-Cano Y (2008a) A convolution type inequality for fuzzy integrals. Appl Math Comput 19:94–99
Román-Flores H, Flores-Franulič A, Chalco-Cano Y (2008b) A Hardy-type inequality for fuzzy integrals. Appl Math Comput 204:178–183
Román-Flores H, Flores-Franulič A, Chalco-Cano Y, Dan R (2013) A two-dimensional Hardy type inequality for fuzzy integrals. Int J Uncertain Fuzzy Knowl Based Syst 21:165–173
Román-Flores H, Chalco-Cano Y, Lodwick W (2018) Some integral inequalities for interval-valued functions. Comput Appl Math 37:1306–1318
Román-Flores H, Flores-Franulič A, Aguirre-Cipe I, Martínez-Romero M (2020) A Sugeno integral inequality of Carleman–Knopp type and some refinements. Fuzzy Sets Syst 396:72–81
Wang Z, Klir G (2009) Generalized measure theory. Springer, New York
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Communicated by Rosana Sueli da Motta Jafelice.
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This work was supported in part by Conicyt-Chile through Projects Fondecyt 1190142.
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Román-Flores, H., Ayala, V. & Flores-Franulič, A. Milne type inequality and interval orders. Comp. Appl. Math. 40, 130 (2021). https://doi.org/10.1007/s40314-021-01500-y
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DOI: https://doi.org/10.1007/s40314-021-01500-y