Document Type : Review Paper

Author

Department of Mathematics, University of Maragheh, P. O. Box 55181-83111, Maragheh, Iran.

Abstract

In this paper, we express and prove  Stolarsky, Feng Qi and Markov type inequalities for two classes of pseudo-integrals. One of them concerning the pseudo-integrals based on a function reduces on the g-integral where pseudo-operations are defined by a monotone and continuous function $g$. The other one concerns the pseudo-integrals based on a semiring $( [a, b], \max, \odot )$, where $\odot$ is generated.  The integral  inequalities are  appling in multivariate approximation theory and probability theory and etc.

Keywords

[1] H. Agahi, R. Mesiar and Y. Ouyang, New general extensions of Chebyshev type inequalities for Sugeno integrals,     Int. J. Approx. Reasoning, 51 (2009), pp. 135-140.
[2] H. Agahi, R. Mesiar and Y. Ouyang, General Minkowski type inequalities for Sugeno integrals, Fuzzy Sets Syst., 161 (2010), pp. 708-715.
[3] H. Agahi, R. Mesiar and Y. Ouyang, Chebyshev type inequalities for pseudo-integrals, Nonlinear Analysis, 72 (2010), pp. 2737-2743.
[4] H. Agahi and M.A. Yaghoobi, A Feng Qi type inequality for Sugeno integral, Fuzzy Inf. Eng., 3 (2010), pp. 293-304.
[5] G. Anastassiou, Chebyshev Gruss type inequalities via Euler type and Fink identities, Math. Comput. Modelling, 45 (2007), pp. 1189-1200.
[6] L. Bougoffa, On Minkowski and Hardy integral inequalities, JIPAM, J. Inequal. Pure Appl. Math., 7 (2) (2006), Article 60.
[7] P.S. Bullen, A Dictionary of Inequalities, Addison Wesley Longman Limited, 1998.
[8] J. Caballero and K. Sadaragani, Chebyshev inequality for Sugeno integrals, Fuzzy Sets Syst., 161 (2010), pp. 1480-1487.
[9] 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.
[10] B. Daraby, Investigation of a Stolarsky type inequality for integrals in pseudo-analysis, Fractional Calculus & Applied Analysis, 13 (5) (2010), pp. 467-473.
[11] B. Daraby, Generalization of the Stolarsky type inequality for pseudo-integrals, Fuzzy Sets Syst., 194 (2012), pp. 90-96.
[12] B. Daraby and L. Arabi, Related Fritz Carlson type inequality for sugeno integrals, Soft Comput., 17 (10) (2013), pp. 1745-1750.
[13] B. Daraby, Markov type integral inequality for pseudo-integrals, Casp. J. Appl. Math. Ecol. Econ., 1 (1) (2013), pp. 13-23.
[14] 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.
[15] B. Daraby, F. Rostampour and A. Rahimi, Hardy's type tnequality for pseudo-integrals, Acta Univ. Apulensis, Math. Inform., 42 (2015), pp. 53-65.
[16] B. Daraby, A convolution type inequality for pseudo-integrals, Acta Univ. Apulensis, Math. Inform., 48 (2016), pp. 27-35.
[17] B. Daraby, Results Of The Chebyshev Type Inequality For Pseudo-Integral,     Sahand Commun. Math. Anal., 4 (1) (2016), pp. 91-100.
[18] B. Daraby, H.G. Asll and I. Sadeqi, General related inequalities to Carlson-type inequality for the Sugeno integral, Appl. Math. Comput., 305 (2017), pp. 323-329.
[19] B. Daraby, Generalizations of the Well-Known Chebyshev type inequalities for pseudo-integrals, Gen. Math. Notes, 38 (1) (2017), pp. 32-45.
[20] B. Daraby, F. Rostampour and A. Rahimi, Minkowski type inequality for fuzzy and pseudo-integrals, Tbil. Math. J., 10 (4) (2017), pp. 159-174.
[21] B. Daraby, A, Shafiloo and A. rahimi, Carlson type inequality for Choquet-like expectation, Acta Univ. Apulensis, Math. Inform., 49 (2017), pp. 23-36.
[22] 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.
[23] B. Daraby, H. Ghazanfary Asll and I. Sadeqi, Favard's inequality for pseudo-integral, Asian-Eur. J. Math., 11 (1) 2018, 1850015.
[24] 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), 713-724.
[25] B. Daraby, R. Mesiar, F. Rostampour and A. Rahimi Related Thunsdorff type and Frank-Pick type inequalities for Sugeno integral, Appl. Math. Comput., 414 (2022), 126683.
[26] A. Flores-Franulic and H. Roman-Flores, A Chebyshev type inequality for fuzzy integrals, Appl. Math. Comput., 190 (2007), pp. 1178-1184.
[27] 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.
[28] A. Flores-Franulic, H. Roman-Flores and Y. Chalco-Cano, A convolution type inequality for fuzzy integrals, Appl. Math. Comput., 195 (2008),94-99.
 
[29] A. Flores-Franulic, H. Roman-Flores and Y. Chalco-Cano, Markov type inequalities for fuzzy integrals, Appl. Math. Comput., 207 (2009), pp. 242-247.
[30] A. Flores-Franulic, H. Roman-Flores and Y. Chalco-Cano, Markov type inequalities for fuzzy integrals, Appl. Math. Comput., 207 (2009), pp. 242-247.
[31] D.H. Hong, A sharp Hardy-type inequality of Sugeno integrals, Appl. Math. Comput., 217 (2010), 437-440.
[32] A. Kolmogorov and S. Fomin, Elements of the Theory of Functions and Functional Analysis, Dover, New York, 1999.
[33] S.G. Krantz, Jensen's Inequality, $sharp$ 9.1.3 in Handbook of Complex Variables, Boston, MA: Birkhauser, 119, 1999.
[34] W. Kuich, Semirings, Automata, Languages, Springer-Verlag, Berlin, 1986.
[35] Y. Liu and M. Luo, Fuzzy topology,     Adv. Fuzzy Syst., Appl. Theory, 9, World Scientific, Singapore, 1997.
[36] 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.
[37] R. Mesiar and E. Pap, Idempotent integral as limit of g-integrals, Fuzzy Sets Syst., 102 (1999), pp. 385-392.
[38] R. Mesiar and Y. Ouyang, General Chebyshev type inequalities for Sugeno integrals, Fuzzy Sets Syst., 160 (2009), pp. 58-64.
[39] H. Minkowski, Geometrie der Zahlen, Teubner, Leipzig, 1910.
[40] Y. Ouyang, J. Fang and L. Wang, Fuzzy Chebyshev type inequality, Int. J. Approx. Reasoning, 48 (2008), pp. 829-835.
[41] 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.
[42] E. Pap, An integral generated by decomposable measure, Univ. Novom Sadu Zb. Rad. Prirod. -Mat. Fak. Ser. Mat., 20 (1) (1990), pp. 135-144.
[43] E. Pap, $ g $-calculus, Univ. Novom Sadu Zb. Rad. Prirod.-Mat. Fak. Ser. Mat., 23 (1) (1993), pp. 145-156.
[44] E. Pap, Null-additive Set Functions, Kluwer, Dordrecht, 1995.
[45] E. Pap, N. Ralevic, Pseudo-Laplace transform, Nonlinear Analysis, 33 (1998), pp. 553-560.
[46] E. Pap, Pseudo-additive measures and their applications, in: E. Pap (Ed.), Handbook of Measure Theory, Elsevier, Amsterdam, 2002, pp. 1403-1465.
[47] E. Pap, Generalized real analysis and its applications, Int. J. Approx. Reasoning, 47 (2008), pp. 368-386.
[48] E. Pap and M. Strboja, Generalization of the Jensen inequality for pseudo-integral, Inf. Sci., 180 (2010), pp. 543-548.
[49] F. Qi, Several integral inequalities, J. Inequal. Pure Appl. Math., 1 (2) (2000), Article 19.
[50] H. Roman-Flores, A. Flores-Franulic and Y. Chalco-Cano, The fuzzy integral for monotone functions, Appl. Math. Comput., 185 (2007), pp. 492-498.
[51] H. Roman-Flores, A. Flores-Franulic and Y. Chalco-Cano, A Jensen type inequality for fuzzy integrals, Inf. Sci., 177 (2007), pp. 3192-3201.
[52] H. Roman-Flores, H.Y. Chalco-Cano, Sugeno integral and geometric inequalities, International Journal of Uncertainity, Fuzziness and Knowledge-Based Systtem, 15 (2007), pp. 1-11.
[53] 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.
[54] 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.
[55] 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.
[56] K.B. Stolarsky, From Wythoff,s Nim to Chebyshev,s inequality, Am. Math. Mon., 98 (1991), pp. 889-900.
[57] M. Sugeno and T. Murofushi, Pseudo-additive measures and integrals, J. Appl. Math. Anal. Appl., 122 (1987), pp. 197-222.
[58] Z. Wang and G.J. Klir, Fuzzy Measure Theory, Plenum Press, New York, 1992.
[59] L. Wu, J. Sun, X. Ye and L. Zhu, Holder type inequality for Sugeno integral, Fuzzy Sets Syst., 161 (2010), pp. 2337-2347.
[60] K.W. Yu. and F. Qi, A short note on an integral inequality, RGMIA Res. Rep. Coll., 4 (1) (2001), pp. 23-25
[61] L.A. Zadeh, Is there a need for fuzzy logic?, Inf. Sci., 178 (2008), 2751-2779.
[62] L.A. Zadeh, Fuzzy sets, Inf. Control, 8 (1965), pp. 338-353.