A review on some fuzzy integral inequalities. (English) Zbl 1513.28021
Summary: In this paper, we introduce fuzzy measure and fuzzy integral concepts and express some of the fuzzy integral properties. The main purpose of this article is to reviewing of some important mathematical inequalities that have many applications in modeling mathematical problems. Firstly, we prove the related Gauss-Winkler type inequality for fuzzy integrals. Indeed, we prove fuzzy version provided by D. H. Hong. Another the famous mathematical inequality is Minkowski’s inequality. It is an important inequality from both mathematical and application points of view. Here, we state a Minkowski type inequality for fuzzy integrals. The established results are based on the classical Minkowski’s inequality for integrals. In the continue, we showed that by an example the classical Prékopa-Leindler type inequality is not valid for the Sugeno integral. We proved one version of the Prékopa-Leindler type inequality by adding concave fuzzy measure and quasi-concave fuzzy measure assumptions for the Sugeno integral with different proofs. Also, we obtained a derivation version of the Prékopa-Leindler inequality and illustrated all of the main results by examples. Finally, we investigate the Thunsdorff’s inequality for Sugeno integral. By an example, we show that the classical form of this inequality does not hold for the Sugeno integral. Then, by reviewing the initial conditions, we prove two main theorems for this inequality. By checking the special case of the aforementioned Thunsdorff’s inequality, we prove Frank-Pick type inequality for the Sugeno integral and illustrate it by an example.
Keywords:
set-valued functios; Sugeno integral; Gauss-Winkler inequality; Minkowski’s inequality; Prékopa-Leindler’s inequality; Thunsdorff’s inequality; Frank-Pick’s inequalityReferences:
| [1] | H. Agahi and M. A. Yaghoobi, A Minkowski type inequality for fuzzy integrals, Journal of Uncertain Systems, 4 (3) (2010), pp. 187-194. |
| [2] | H. Agahi, Y. Ouyang, R. Mesiar, E. Pap and M. Strboja, Holder and Minkowski type inequalities for pseudo-integral, Appl. Math. Comput., 217 (2011), pp. 8630-8639. · Zbl 1217.26039 |
| [3] | R.P. Agarwal and S.S. Dragomir, An application of Hayashiâs inequality for differentiable functions, Comput. Math. Appl., 32 (1996), pp. 95-99. · Zbl 0874.26017 |
| [4] | N. Balakrishnan and T. Rychlik, Evaluating expectation sof L-statistics by the Steffensen inequality, Metrika, 63(3) (2006), pp. 371-384. · Zbl 1095.62058 |
| [5] | L. Bougoffa, On Minkowski and Hardy integral inequalities, Journal of Inequalities in Pure and Applied Mathematics, 7(2) (2006), article 60. · Zbl 1132.26007 |
| [6] | P.S. Bullen, A dictionary of inequalities, Addison Wesley Longman Inc, (1998). · Zbl 0934.26003 |
| [7] | J. Caballero and K. Sadarangani, Fritz Carlsonas inequality for fuzzy integrals, Comput. Math. Appl., 59(8) (2010), pp. 2763-2767. · Zbl 1193.28015 |
| [8] | T.Y. Chen, H.L. Chang and G.H. Tzeng, Using fuzzy measures and habitual domains to analyze the public attitude and apply to the gas taxi policy, Eur. J. Oper. Res., 137 (2002), pp. 145-161. · Zbl 1003.90514 |
| [9] | D. Cordero-Erausquin, R.J. Mc-Cann and M. Schmuckenschlaager, Prekopa-Leindler type inequalities on Riemannian manifolds, Jacobi fields and optimal transport, Annals de la Faculte des sciences de Toulouse, XV (4) (2006), pp. 613-635. · Zbl 1125.58007 |
| [10] | B. Daraby, Investigation of a Stolarsky type inequality for integrals in pseudo-analysisFractional, Fract. Calc. Appl. Anal., 13 (5) (2010), pp. 467-473. · Zbl 1245.28013 |
| [11] | B. Daraby, Generalization of the Stolarsky type inequality for pseudo-integrals, Fuzzy Sets Syst., 194 (1) (2012), pp. 90-96. · Zbl 1253.28011 |
| [12] | B. Daraby, Generalization of the Stolarsky type inequality for pseudo-integrals, Fuzzy Sets Syst., 194 (2012), pp. 90-96. · Zbl 1253.28011 |
| [13] | B. Daraby, Markov type integral inequality for Pseudo-integrals, Casp. J. Appl. Math. Ecol. Econ., 1 (1) 2013, pp. 13-23. · Zbl 1474.28026 |
| [14] | B. Daraby and L. Arabi, Related Fritz Carlson type inequality for Sugeno integrals, Soft Comput., 17 (2013), pp. 1745-1750. · Zbl 1326.28016 |
| [15] | B. Daraby and F. Ghadimi, General Minkowsky type and related inequalities for seminormed fuzzy integrals, Sahand Commun. Math. Anal., 1(1) (2014), pp. 9-20. · Zbl 1317.26023 |
| [16] | B Daraby, A. Shafiloo and A. Rahimi, Geberalizations of the Feng Qi type inequality for Pseudo-integral, Gazi University Journal of Science, 28(4) (2015), pp. 695-702. |
| [17] | B. Daraby, F. Rostampour and A. Rahimi, Hardy’s Type Inequality For Pseudo-Integrals, Acta Univ. Apulensis, Math. Inform., 42 (2015), pp. 53-65 · Zbl 1374.26049 |
| [18] | B. Daraby, H. Ghazanfary Asll and I. Sadeqi, Favard’s inequality for seminormed fuzzy integral and semiconormed fuzzy integral, Mathematica, 58 (81) (2016), pp. 39-50. · Zbl 1389.26066 |
| [19] | B. Daraby, A Convolution Type Inequality For pseudo-Integrals, Acta Univ. Apulensis, Math. Inform., 48 (2016), pp. 27-35. · Zbl 1413.26054 |
| [20] | B. Daraby, Results Of The Chebyshev Type Inequality For Pseudo-Integral, Sahand Commun. Math. Anal., 4 (1) (2016), [pp.] 91-100. · Zbl 1413.26055 |
| [21] | B. Daraby and A. Rahimi, Jensen type inequality for seminormed fuzzy integrals, Acta Univ. Apulensis, Math. Inform., 46 (2016), pp. 1-8. · Zbl 1413.26056 |
| [22] | B. Daraby, H. Ghazanfary Asll and I. Sadeqi, General related inequalities to Carlson-type inequality for the Sugeno integral, Appl. Math. Comput., 305 (2017), pp. 323-329. · Zbl 1411.28009 |
| [23] | B. Daraby, Generalizations of the Well-Known Chebyshev Type Inequalities for Pseudo-Integrals, Gen. Math. Notes, 38 (1) (2017), pp. 32-45. |
| [24] | B. Daraby, H. Ghazanfary Asll and I. Sadeqi, General related inequalities to Carlson-type inequality for the Sugeno integral, Appl. Math. Comput., 305 (15) (2017), pp. 323-329. · Zbl 1411.28009 |
| [25] | B. Daraby, A, Shafiloo and A. rahimi, Carlson Type Inequality For Choquet-Like Expectation, Acta Univ. Apulensis, Math. Inform., 49 (2017), pp. 23-36. · Zbl 1413.30044 |
| [26] | B. Daraby, H. Ghazanfary Asll and I. Sadeqi, Gronwall’s Inequality For Pseudo-Integral, An. Univ. Oradea, Fasc. Mat., XXIV (1) (2017), pp. 67-74. · Zbl 1389.35016 |
| [27] | B. Daraby, F. Rostampour and A. Rahimi, Minkowski type inequality for fuzzy and pseudo-integrals, Tbil. Math. J., 10 (2) (2017), pp. 243-258. |
| [28] | B. Daraby, A. Shafiloo and A. Rahimi, General Lyapunov type inequality for Sugeno integral, J. Adv. Math. Stud., 11 (1) (2018), pp. 37-46. · Zbl 1401.26035 |
| [29] | B. Daraby, F. Rostampour and A. Rahimi, Minkowski type inequality for fuzzy and pseudo-integrals, Tibilis Mathematical Journal., 10 (4) (2017), pp. 159-174. · Zbl 1388.26014 |
| [30] | B. Daraby, H. Ghazanfary Asll and I. Sadeqi, General related inequalities to Carlson-type inequality for the Sugeno integral, Appl. Math. Comput., 305 (15) (2017), pp. 323-329. · Zbl 1411.28009 |
| [31] | B. Daraby, General Related Jensen type Inequalities for fuzzy integrals, TWMS J. Pure Appl. Math., 8 (1) (2018), pp. 1-7. · Zbl 1397.26010 |
| [32] | B. Daraby, H. Ghazanfary Asll and I. Sadeqi, Favard’s inequality for pseudo-integral, Asian-Eur. J. Math., 11 (1) (2018) · Zbl 1391.35015 |
| [33] | B. Daraby, F. Rostampour, A.R. Khodadadi and A. Rahimi, Related Gauss-Winkler Type Inequality for Fuzzy and Pseudo-Integrals, Thai J. Math., 19 (2) (2021), pp. 713-724. · Zbl 1477.28013 |
| [34] | B. Daraby, R. Mesiar, F. Rostampour and A. Rahimi, Related Thunsdorff type and FrankPick type inequalities for Sugeno integral, Appl. Math. Comput., 414 (2022). · Zbl 1510.26014 |
| [35] | B. Daraby, R. Mesiar, F. Rostampour and A. Rahimi, Related Thunsdorff type and Frank-P-ck type inequalities for Sugeno integral, Appl. Math. Comput., 414: 126683 (2022). · Zbl 1510.26014 |
| [36] | B. Daraby, F. Rostampour, A.R. Khodadadi, A. Rahimi and R. Mesiar, Polya-Knopp and Hardy-Knopp type inequalities for Sugeno integral, arXiv:1910.03812v1. |
| [37] | A. Flores-Franulic and H. Roman-Flores, A Chebyshev type inequality for fuzzy integrals, Appl. Math. Comput., 190 (2007), pp. 1178-1184. · Zbl 1129.26021 |
| [38] | A. Flores-Franulic, H. Roman-Flores and Y. Chalco-Cano, A note on fuzzy integral inequality of Stolarsky type, Appl. Math. Comput., 196 (2008), pp. 55-59. · Zbl 1134.26007 |
| [39] | L. Gajek and A. Okolewski, Steffensen type inequalities for order and record statistics, Ann. Univ. Mariae Curie-Skaodowska Lublin-Polonia, 16 (1997), pp. 41-59. · Zbl 0913.62049 |
| [40] | R.J. Gardner, The Brunn-Minkowski inequality, Bull. Am. Math. Soc., 39 (2002), pp. 355-405. · Zbl 1019.26008 |
| [41] | I. Gentil, From the Prekopa-Leindler inequality to modified logarithmic Sobolev inequality, Ann. Fac. Sci. Toulouse, Math., 17 (2) (2008), pp. 291-308. · Zbl 1175.26036 |
| [42] | D.H. Hong, Gauss-Winikler inequality for Sugeno integrals, Int. J. Pure Appl. Math., 116 (2) (2017), pp. 479-487. |
| [43] | J.Y. Lu , K.S. Wu and J.C. Lin, Fast full search in motion estimation by hierarchical use of Minkowski’s inequality, Pattern Recognition, 31 (1998), pp. 945-952. |
| [44] | R. Mesiar and Y. Ouyang, General Chebyshev type inequalities for Sugeno integrals, Fuzzy Sets Syst., 160 (2009), pp. 58-64. · Zbl 1183.28035 |
| [45] | H. Minkowski, Geometrie der Zahlen, Teubner, Leipzig, 1910. · JFM 41.0239.03 |
| [46] | Y. Ouyang, J. Fang and L. Wang, Fuzzy Chebyshev type inequality, Internatinal Journal of Approximate Reasoning, 48 (2008), pp. 829-835. · Zbl 1185.28025 |
| [47] | U.M. Ozkan, M.Z. Sarikaya and H. Yildirim, Extensions of certain integral inequalities on time scales, Appl. Math. Lett., 21 (2008), pp. 993-1000. · Zbl 1168.26316 |
| [48] | E. Pap, Null-additive Set Functions, Kluwer, Dordrecht, 1995. · Zbl 0856.28001 |
| [49] | A. Prekopa, Stochastic Programming, Kluwer, Dordretch, 1995. · Zbl 0863.90116 |
| [50] | D. Ralescu and G. Adams, The fuzzy integral, J. Appl. Math. Anal. Appl., 75 (1980), pp. 562-570. · Zbl 0438.28007 |
| [51] | H. Roman-Flores, A. Flores-Franulic and Y. Chalco-Cano, The fuzzy integral for monotone functions, Appl. Math. Comput., 185 (2007), pp. 492-498. · Zbl 1116.26024 |
| [52] | H. Roman-Flores and Y. Chalco-Cano, Sugeno integral and geometric inequalities, International Journal of Uncertainity, Fuzziness and Knowledge-Based Systtem, 15 (2007), pp. 1-11. · Zbl 1118.28012 |
| [53] | H. Roman-Flores, A. Flores-Franulic and Y. Chalco-Cano, A Jensen type inequality for fuzzy integrals, Inf. Sci., 177 (2007), pp. 3192-3201. · Zbl 1127.28013 |
| [54] | H. Roman-Flores, A. Flores-Franulic and Y. Chalco-Cano, A convolution type inequality for fuzzy integrals, Appl. Math. Comput., 195 (2008), pp. 94-99. · Zbl 1149.26033 |
| [55] | H. Roman-Flores, A. Flores-Franulic and Y. Chalco-Cano, A note on fuzzy integral inequality of Stolarsky type, Appl. Math. Comput., 196 (2008), pp. 55-59. · Zbl 1134.26007 |
| [56] | H. Roman-Flores, A. Flores-Franulic and Y. Chalco-Cano, A Hardy-type inequality for fuzzy integrals, Appl. Math. Comput., 204 (2008), pp. 178-183. · Zbl 1168.26317 |
| [57] | W. Rudin, Principles of Mathematical Analysis, 3rd Edition, McGraw-Hill, New York, 1976. · Zbl 0346.26002 |
| [58] | W. Rudin, Real and Complex Analysis, 3rd Edition, McGraw-Hill, New York, 1987. · Zbl 0925.00005 |
| [59] | H. Roman-Flores, A. Flores-Franulic and Y. Chalco-Cano, A Hardy-type inequality for fuzzy integrals, Appl. Math. Comput., 204 (2008), pp. 178-183. · Zbl 1168.26317 |
| [60] | I. Sadeqi, H. Ghazanfary Asll and B. Daraby, Gauss type inequality for Sugeno integral, J. Adv. Math. Stud., 10 (2) (2017), pp. 167-173. · Zbl 1371.28037 |
| [61] | R. Srivastava, Some families of integral, trigonometric and other related inequalities, Appl. Math. Inf. Sci., 5 (2011), pp. 342-360. |
| [62] | M. Sugeno, Theory of fuzzy integrals and its applications, Ph.D. thesis, Tokyo Institute of Technology, 1974. |
| [63] | Z. Wang and G. Klir, Fuzzy Measure Theory, Plenum Press, New York, 1992. · Zbl 0812.28010 |
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