New extensions of Chebyshev type inequalities using generalized Katugampola integrals via Polya-Szegö inequality
A number of Chebyshev type inequalities involving various fractional integral operators have, recently, been presented. In this work, motivated essentially by the earlier works and their applications in diverse research subjects, we establish some new Polya-Szego inequality involving generalized Katugampola fractional integral operator and use them to prove some new fractional Chebyshev type inequalities which are extensions of the results in the paper: [On Polya-Szego and Chebyshev type inequalities involving the Riemann-Liouville fractional integral operators, J. Math. Inequal, 10(2) (2016)].
[1] Ceby~sev, P.L. (1982). Sur les expressions approxima- tives des integrales definies par les autres prises entre les memes limites. Proc. Math. Soc. Charkov, 2, 93-98.
[2] Belarbi, S. and Dahmani, Z. (2009). On some new fractional integral inequalities. J. Ineq. Pure Appl. Math., 10(3) Article 86, 5 pages.
[3] Dahmani, Z., Mechouar, O. and Brahami, S. (2011). Certain inequalities related to the Chebyshevs func- tional involving a type Riemann-Liouville operator. Bull. of Math. Anal. and Appl., 3(4), 38-44.
[4] Dahmani, Z. (2010). New inequalities in fractional in- tegrals. Int. J. of Nonlinear Sci., 9(4), 493-497.
[5] Set, E., Akdemir, O.A. and Mumcu, I(˙) . Chebyshev type inequalities for conformable fractional integrals. Submitted.
[6] O(¨)zdemir, M.E., Set, E., Akdemir, O.A. and Sarlkaya,M.Z. (2015). Some new Chebyshev type inequalities for functions whose derivatives belongs to Lp spaces. Afr. Mat., 26, 1609-1619.
[7] Polya, G. and Szeg¨o, G. (1925). Aufgaben und Lehrsatze aus der Analysis, Band 1, Die Grundlehren der mathmatischen Wissenschaften 19, J. Springer, Berlin.
[8] Dragomir, S.S. and Diamond, N.T. (2003). Integral inequalities of Gr¨uss type via P/olya-Szeg¨o and Shisha- Mond results. East Asian Math. J., 19, 27-39.
[9] Srivastava, H.M. and Choi, J. (2012). Zeta and q-Zeta Functions and Associated Series and Integrals, Else- vier Science Publishers, Amsterdam, London and New York.
[10] Kilbas, A.A., Srivastava, H.M. and Trujillo, J.J.(2006). Theory and applications of fractional diferen- tial equations, North-Holland Mathematical Studies, Vol.204, Elsevier (North-Holland) Science Publishers, Amsterdam, London and New York.
[11] Podlubny, I. (1999). Fractional diferential equations. An introduction to fractional derivatives, fractional diferential equations, to methods of their Solution and some of their applications, Academic Press, San Diego, CA.
[12] Katugampola, U.N. (2014). A new approach to gener- alized fractional derivatives. Bull. Math. Anal. Appl., 6, 1-15.
[13] Katugampola, U.N. (2016). New fractional in- tegral unifying six existing fractional integrals. arXiv:1612.08596v1 [math.CA].
[14] Sousa, J. and Oliviera, E.C. The Minkowski’s inequal- ity by means of a generalized fractional integral. arXiv preprint arXiv:1705.07191.
[15] Sousa, J., Oliveira, D.S and Oliveira, E.C. Gr¨uss- type inequality by mean of a fractional integral. arXiv preprint arXiv:1705.00965.
[16] Ntouyas, S.K., Agarwal, P. and Tariboon, J. (2016). On P/olya-Szeg¨o and Chebyshev type inequalities in- volving the Riemann-Liouville fractional integral op- erators. J. Math. Inequal, 10(2), 491-504.