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Fractional Ostrowski-Sugeno Fuzzy univariate inequalities

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  • Here we present fractional univariate Ostrowski-Sugeno Fuzzy type inequalities. These are of Ostrowski-like inequalities in the setting of Sugeno fuzzy integral and its special-particular properties. In a fractional environment, they give tight upper bounds to the deviation of a function from its Sugeno-fuzzy averages. The fractional derivatives we use are of Canavati and Caputo types. This work is greatly inspired by [8], [1] and [2].

    Mathematics Subject Classification: Primary: 26A33, 26D07, 26D10, 26D15, 41A44; Secondary: 26A24, 26D20, 28A25.


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  • [1] G. A. Anastassiou, Fractional Differentiation Inequalities, Springer, Dordrecht, 2009. doi: 10.1007/978-0-387-98128-4.
    [2] G. A. Anastassiou, Advances on Fractional Inequalities, SpringerBriefs in Mathematics, Springer, New York, 2011. doi: 10.1007/978-1-4614-0703-4.
    [3] G. A. Anastassiou, Intelligent Mathematics: Computational Analysis, Intelligent Systems Reference Library, 5. Springer-Verlag, Berlin, 2011. doi: 10.1007/978-3-642-17098-0.
    [4] G. A. Anastassiou, Intelligent Comparisons: Analytic Inequalities, Studies in Computational Intelligence, 609. Springer, Cham, 2016. doi: 10.1007/978-3-319-21121-3.
    [5] M. Boczek and M. Kaluszka, On the Minkowaki-Hölder type inequalities for generalized Sugeno integrals with an application, Kybernetika (Prague), 52 (2016), 329-347.  doi: 10.14736/kyb-2016-3-0329.
    [6] J. A. Canavati, The Riemann-Liouville integral, Nieuw Arch. Wisk., 5 (1987), 53-75. 
    [7] K. Diethelm, The Analysis of Fractional Differential Equations, An application-oriented exposition using differential operators of Caputo type, Lecture Notes in Mathematics, 2004. Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-14574-2.
    [8] A. Ostrowski, Über die Absolutabweichung einer differentiebaren Funktion von ihrem Integralmittelwert, (German) Comment. Math. Helv., 10 (1938), 226-227.  doi: 10.1007/BF01214290.
    [9] E. Pap, Null-Additive Set functions, Mathematics and its Applications, 337, Kluwer Academic Publishers Group, Dordrecht; Ister Science, Bratislava, 1995.
    [10] D. Ralescu and G. Adams, The fuzzy integral, J. Math. Anal. Appl., 75 (1980), 562-570.  doi: 10.1016/0022-247X(80)90101-8.
    [11] M. Sugeno, Theory of Fuzzy Integrals and Its Applications[J], PhD thesis, Tokyo Institute of Technology, 1974.
    [12] Z. Wang and  G. J. KlirFuzzy Measure Theory, Plenum Press, New York, 1992.  doi: 10.1007/978-1-4757-5303-5.
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