Consensus of classical and fractional inequalities having congruity on time scale calculus

Sahir, Muhammad Jibril Shahab

doi:10.2478/gm-2019-0006

Abstract

In this paper, we find accordance of some classical inequalities and fractional dynamic inequalities. We find inequalities such as Radon’s inequality, Bergström’s inequality, Rogers-Hölder’s inequality, Cauchy-Schwarz’s inequality, the weighted power mean inequality and Schlömilch’s inequality in generalized and extended form by using the Riemann-Liouville fractional integrals on time scales.

References

[1] J. Aczél, E. F. Beckenbach, On Hölder’s inequality, in “General inequalities 2” (Proc. Second Internat. Conf. Oberwolfach, 1978), 145-150, Birkhäuser, Basel/Stuttgart, 1980.10.1007/978-3-0348-6324-7_15
Search in Google Scholar Back to article
[2] R. P. Agarwal, D. O’Regan, S. H. Saker, Dynamic inequalities on time scales, Springer International Publishing, Cham, Switzerland, 2014.10.1007/978-3-319-11002-8
Search in Google Scholar Back to article
[3] G. A. Anastassiou, Principles of delta fractional calculus on time scales and inequalities, Mathematical and Computer Modelling, vol. 52, no. 3-4, 2010, 556-566.10.1016/j.mcm.2010.03.055
Search in Google Scholar Back to article
[4] G. A. Anastassiou, Foundations of nabla fractional calculus on time scales and inequalities, Computers & Mathematics with Applications, vol. 59, no. 12, 2010, 3750-3762.10.1016/j.camwa.2010.03.072
Search in Google Scholar Back to article
[5] G. A. Anastassiou, Integral operator inequalities on time scales, International Journal of Difference Equations, vol. 7, no. 2, 2012, 111-137.
Search in Google Scholar Back to article
[6] D. Anderson, J. Bullock, L. Erbe, A. Peterson, H. Tran, Nabla dynamic equations on time scales, Pan-American Mathematical Journal, vol. 13, no. 1, 2003, 1-48.10.1007/978-0-8176-8230-9_3
Search in Google Scholar Back to article
[7] E. F. Beckenbach, R. Bellman, Inequalities, Springer, Berlin, Göttingen and Heidelberg, 1961.10.1007/978-3-642-64971-4
Search in Google Scholar Back to article
[8] R. Bellman, Notes on matrix theory-IV (An inequality due to Bergström), Amer. Math. Monthly, vol. 62, 1955, 172-173.10.2307/2306621
Search in Google Scholar Back to article
[9] H. Bergström, A triangle inequality for matrices, Den Elfte Skandinaviske Matematikerkongress, 1949, Trondheim, Johan Grundt Tanums Forlag, Oslo, 1952, 264-267.
Search in Google Scholar Back to article
[10] M. Bohner, A. Peterson, Dynamic equations on time scales, Birkhäuser Boston, Inc., Boston, MA, 2001.10.1007/978-1-4612-0201-1
Search in Google Scholar Back to article
[11] M. Bohner, A. Peterson, Advances in dynamic equations on time scales, Birkhäuser Boston, Boston, MA, 2003.10.1007/978-0-8176-8230-9
Search in Google Scholar Back to article
[12] M. Bohner, H. Luo, Singular second-order multipoint dynamic boundary value problems with mixed derivatives, Advances in Difference Equations, 2006, 1-15.10.1155/ADE/2006/54989
Search in Google Scholar Back to article
[13] Z. Dahmani, About some integral inequalities using Riemann-Liouville integrals, General Mathematics, vol. 20, no. 4, 2012, 63-69.
Search in Google Scholar Back to article
[14] C. Dinu, Convex functions on time scales, Annals of the University of Craiova, Math. Comp. Sci. Ser., vol. 35, 2008, 87-96.
Search in Google Scholar Back to article
[15] D. M. Bătineţu-Giurgiu, O. T. Pop, A generalization of Radon’s inequality, Creative Math. & Inf., vol. 19, no. 2, 2010, 116-121.
Search in Google Scholar Back to article
[16] G. H. Hardy, J. E. Littlewood, G. Pölya, Inequalities, 2nd Ed., Cambridge, University Press, 1952.
Search in Google Scholar Back to article
[17] S. Hilger, Ein Maβkettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten, Ph.D. thesis, Universität Würzburg, 1988.
Search in Google Scholar Back to article
[18] C. H. Hong, C. C. Yeh, Rogers-Hölder’s inequality on time scales, International Journal of Pure and Applied Mathematics, vol. 29, no. 3, 2006, 289-309.
Search in Google Scholar Back to article
[19] D. S. Mitrinović, Analytic inequalities, Springer-Verlag, Berlin, 1970.10.1007/978-3-642-99970-3
Search in Google Scholar Back to article
[20] J. Radon, Theorie und Anwendungen der absolut additiven Mengenfunktionen, Sitzungsber. Acad. Wissen. Wien, vol. 122, 1295-1438, 1913.
Search in Google Scholar Back to article
[21] M. J. S. Sahir, Dynamic inequalities for convex functions harmonized on time scales, Journal of Applied Mathematics and Physics, vol. 5, 2017, 2360-2370.10.4236/jamp.2017.512193
Search in Google Scholar Back to article
[22] M. J. S. Sahir, Fractional dynamic inequalities harmonized on time scales, Cogent Mathematics & Statistics, vol. 5, 2018, 1-7.10.1080/23311835.2018.1438030
Search in Google Scholar Back to article

Consensus of classical and fractional inequalities having congruity on time scale calculus

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