We use the definition of a fractional integral, recently proposed by Katugampola, to establish a generalization of the reverse Minkowski's inequality. We show two new theorems associated with this inequality, as well as state and show other inequalities related to this fractional operator.
Citation: J. Vanterler da C. Sousa, E. Capelas de Oliveira. The Minkowski’s inequality by means of a generalized fractional integral[J]. AIMS Mathematics, 2018, 3(1): 131-147. doi: 10.3934/Math.2018.1.131
We use the definition of a fractional integral, recently proposed by Katugampola, to establish a generalization of the reverse Minkowski's inequality. We show two new theorems associated with this inequality, as well as state and show other inequalities related to this fractional operator.
| [1] | C. Bandle, A. Gilányi, L. Losonczi, et al. Inequalities and Applications: Conference on Inequalities and Applications Noszvaj (Hungary), vol. 157, Springer Science & Business Media, 2008. |
| [2] | D. Bainov, P. Simeonov, Integral Inequalities and Applications, vol. 57, Springer Science & Business Media, 2013. |
| [3] | A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204, North-Holland Mathematics Studies, Elsevier, Amsterdam, 2006. |
| [4] | O. Hutník, On Hadamard type inequalities for generalized weighted quasi-arithmetic means, Journal of Inequalities in Pure and Applied Mathematics, 7 (2006), 1–10. |
| [5] | E. Set, M. Özdemir, S. Dragomir, On the Hermite-Hadamard inequality and other integral inequalities involving two functions, J. Inequal. Appl., 2010 (2010), 148102. |
| [6] | W. Szeligowska, M. Kaluszka, On Jensen's inequality for generalized Choquet integral with an application to risk aversion, eprint arXiv: 1609. 00554,2016. |
| [7] | L. Bougoffa, On Minkowski and Hardy integral inequalities, Journal of Inequalities in Pure and Applied Mathematics, 7 (2006), 1–3. |
| [8] | R. A. Adams, J. J. F. Fournier, Sobolev spaces, vol. 140, Academic press, 2003. |
| [9] | D. Idczak, S. Walczak, Fractional Sobolev spaces via Riemann-Liouville derivatives, J. Funct. Space. Appl., 2013 (2013), 1–15. |
| [10] | T. Krainer, B. W. Schulze, Weighted Sobolev spaces, vol. 138, Springer, 1985. |
| [11] | V. Gol'dshtein, A. Ukhlov, Weighted Sobolev spaces and embedding theorems, T. Am. Math. Soc., 361 (2009), 3829–3850. |
| [12] | O. Hutnök, Some integral inequalities of H¨older and Minkowski type, Colloq. Math-Warsaw, 108 (2007), 247–261. |
| [13] | P. R. Beesack, Hardys inequality and its extensions, Pac. J. Math., 11 (1961), 39–61. |
| [14] | C. O. Imoru, New generalizations of Hardy's integral inequality, J. Math. Anal. Appl., 241 (1987), 73–82. |
| [15] | R. Herrmann, Fractional calculus: An Introduction for Physicists, World Scientific Publishing Company, Singapore, 2001. |
| [16] | I. Podlubny, Fractional Differential Equation, vol. 198, Mathematics in Science and Engineering, Academic Press, San Diego, 1999. |
| [17] | U. N. Katugampola, A new approach to generalized fractional derivatives, Bull. Math. Anal. Appl., 6 (2014), 1–15. |
| [18] | J. Vanterler da C. Sousa, E. Capelas de Oliveira, On the ψ-Hilfer fractional derivative, Commun. Nonlinear Sci., 60 (2018), 72–91. |
| [19] | R. Khalil, M. Al Horani, A. Yousef, et al. A new definition of fractional derivative, J. Comput. Appl. Math., 264 (2014), 65–70. |
| [20] | T. Abdeljawad, On conformable fractional calculus, J. Comput. Appl. Math., 279 (2015), 57–66. |
| [21] | M. Bohner, T. Matthews, The Grüss inequality on time scales, Commun. Math. Anal., 3 (2007), 1–8. |
| [22] | D. R. Anderson, Taylors Formula and Integral Inequalities for Conformable Fractional Derivatives, Contr. Math. Eng., 2014 (2016), 25–43. |
| [23] | M. E. Özdemir, M. Avci, H. Kavurmaci, Hermite-Hadamard-type inequalities via (α, m)-convexity, Comput. Math. Appl., 61 (2011), 2614–2620. |
| [24] | F. Chen, Extensions of the Herminte–Hadamard inequality for convex functions via fractional integrals, J. Math. Inequal., 10 (2016), 75–81. |
| [25] | J. Wang, X. Li, M. Fekan, et al. Hermite-Hadamard-type inequalities for Riemann-Liouville fractional integrals via two kinds of convexity, Appl. Anal., 92 (2013), 2241–2253. |
| [26] | H. Chen, U. N. Katugampola, Hermite-Hadamard and Hermite-Hadamard-Fejér type inequalities for generalized fractional integrals, J. Math. Anal. Appl., 446 (2017), 1274–1291. |
| [27] | I. İşcan, On generalization of different type inequalities for harmonically quasi-convex functions via fractional integrals, Appl. Math. Comput., 275 (2016), 287–298. |
| [28] | I. İşcan, S. Wu, Hermite-Hadamard type inequalities for harmonically convex functions via fractional integrals, Appl. Math. Comput., 238 (2014), 237–244. |
| [29] | F. Chen, Extensions of the Hermite-Hadamard inequality for harmonically convex functions via fractional integrals, Appl. Math. Comput., 268 (2015), 121–128. |
| [30] | H. Shiow-Ru, Y. Shu-Ying and T. Kuei-Lin, Refinements and similar extensions of Hermite-Hadamard inequality for fractional integrals and their applications, Appl. Math. Comput., 249 (2014), 103–113. |
| [31] | M. Z. Sarikaya, H. Budak, New inequalities of Opial type for conformable fractional integrals, Turk. J. Math., 41 (2017), 1164–1173. |
| [32] | J. Vanterler da C. Sousa and E. Capelas de Oliveira, A Gronwall inequality and the Cauchy-type problem by means of ψ-Hilfer operator, arXiv: 1709. 03634,2017. |
| [33] | U. N. Katugampola, New fractional integral unifying six existing fractional integrals, arXiv. org/abs/1612. 08596,2016. |
| [34] | E. Set, M. Özdemir, S. Dragomir, On the Hermite-Hadamard Inequality and Other Integral Inequalities Involving Two Functions, J. Inequal. Appl., 2010 (2010), 148102. |
| [35] | Z. Dahmani, On Minkowski and Hermite-Hadamard integral inequalities via fractional integration, Ann. Funct. Anal., 1 (2010), 51–58. |
| [36] | V. L. Chinchane, D. B. Pachpatte, New fractional inequalities via Hadamard fractional integral, Internat. J. Functional Analysis, Operator Theory and Application, 5 (2013), 165–176. |
| [37] | S. S. Dragomir, Hermite-Hadamards type inequalities for operator convex functions, Appl. Math. Comput., 218 (2011), 766–772. |
| [38] | H. Yildirim, Z. Kirtay, Ostrowski inequality for generalized fractional integral and related inequalities, Malaya Journal of Matematik, 2 (2014), 322–329. |
| [39] | R. F. Camargo, E. Capelas de Oliveira, Fractional Calculus (In Portuguese), Editora Livraria da Física, São Paulo, 2015. |
| [40] | S. Taf, K. Brahim, Some new results using Hadamard fractional integral, Int. J. Nonlinear Anal. Appl., 7 (2015), 103–109. |
| [41] | V. L. Chinchane, D. B. Pachpatte, New fractional inequalities involving Saigo fractional integral operator, Math. Sci. Lett., 3 (2014), 133–139. |
| [42] | V. L. Chinchane, New approach to Minkowski's fractional inequalities using generalized kfractional integral operator, arXiv: 1702. 05234,2017. |
| [43] | W. T. Sulaiman, Reverses of Minkowski's, Hölders, and Hardys integral inequalities, Int. J. Mod. Math. Sci., 1 (2012), 14–24. |
| [44] | B. Sroysang, More on Reverses of Minkowski's Integral Inequality, Math. Aeterna, 3 (2013), 597–600. |
| [45] | E. Kreyszig, Introductory Functional Analysis with Applications, vol. 1, Wiley, New York, 1989. |
| [46] | J. Vanterler da C. Sousa, D. S. Oliveira, E. Capelas de Oliveira, Grüss-type inequality by mean of a fractional integral, arXiv: 1705. 00965,2017. |
| [47] | J. Vanterler da C. Sousa, E. Capelas de Oliveira, A new truncated M-fractional derivative unifying some fractional derivatives with classical properties, International Journal of Analysis and Applications, 16 (2018), 83–96. |
J. Vanterler da C. Sousa, E. Capelas de Oliveira. The Minkowski’s inequality by means of a generalized fractional integral[J]. AIMS Mathematics, 2018, 3(1): 131-147. doi: 10.3934/Math.2018.1.131
DownLoad: