Hardy Type Inequalities for Conformable Fractional Integrals

Abstract

The main target addressed in this article are presenting Hardy type inequalities for Katugampola conformable fractional integral. In accordance with this purpose we try to use more general type of function in order to make a generalization. Thus our results cover the previous published studies for Hardy type inequalities.

Keywords

• Hardy inequality,Hölder's inequality

References

1. [1] T. Abdeljawad, On conformable fractional calculus, Journal of Computational and Applied Mathematics 279 (2015) 57-66.
2. [2] D. R. Anderson and D. J. Ulness, Results for conformable differential equations, preprint, 2016.
3. [3] A. Atangana, D. Baleanu, and A. Alsaedi, New properties of conformable derivative, Open Math. 2015; 13: 889-898.
4. [4] G. H. Hardy, Note on a theorem of Hilbert, Math. Z. 6 (1920), 314-317.
5. [5] G. H. Hardy, Notes on some points in the integral calculus, LX. An inequality between integrals, Messenger of Math. 54, (1925), 150-156.
6. [6] G. H. Hardy, Notes on some points in the integral calculus, LXIV. Further inequalities between integrals. Messenger of Math. 57 (1928), 12-16.
7. [7] M. Izumi, S. Izumi and G. Peterson, On Hardy’s Inequality and its Generalization, Tohoku Math .J. ; 21, (1999), 601-613.
8. [8] O. S. Iyiola and E. R. Nwaeze, Some new results on the new conformable fractional calculus with application using D’Alambert approach, Progr. Fract. Differ. Appl., 2(2), 115-122, 2016.

9. [9] U. Katugampola, A new fractional derivative with classical properties, ArXiv:1410.6535v2.
10. [10] R. Khalil, M. Al horani, A. Yousef and M. Sababheh, A new definition of fractional derivative, Journal of Computational Apllied Mathematics, 264 (2014), 65-70.
11. [11] A. A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier B.V., Amsterdam, Netherlands, 2006.
12. [12] A. Kufner, L. Maligranda and L.E. Persson, The Hardy Inequality- About its History and Some Related Results, Vydavatelsky Servis Publishing House, Pilsen, 2007.
13. [13] A. Kufner and L.E. Persson,Weighted Inequalities of Hardy Type, World Scientific Publishing Co., Singapore, 2003.
14. [14] A. Moazzena, R. Lashkaripour, Some new extensions of Hardy‘s inequality, Int. J. Nonlinear Anal. Appl. 5 (2014) No. 1, 98-109.
15. [15] J. A. Oguntuase, Remark on an Integral Inequality of the Hardy type, Krag. J. Math.32, 2009, 133-138.
16. [16] J. A. Oguntuase, On Hardy’s integral inequality, Proceedings of the Jangjeon Mathematical Society, Vol. 3, 2001, 37-44.
17. [17] S. H. Saker, D. O’Regan, M. R. Kenawy, R. P. Agarwal, Fractional Hardy Type Inequalities via Conformable Calculus, Memoirs on Differential Equations and Mathematical Physics, Volume 73, 2018, 131-140.
18. [18] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordonand Breach, Yverdon et alibi, 1993.
19. [19] M. Z. Sarikaya and H. Yildirim, Some Hardy-type integral inequalities, J. Ineq. Pure .Appl. Math, 7(5), Art 178 (2006).