The Minkowski's inequalities utilizing newly defined generalized fractional integral operators

Year 2019, Volume: 68 Issue: 1, 686 - 701, 01.02.2019

Fuat Usta , Hüseyin Budak , Fatma Ertugral Mehmet Zeki Sarıkaya

https://doi.org/10.31801/cfsuasmas.463983

Cited By: 8

https://izlik.org/JA67BX24MD

Abstract

Motivated by the recent generalized fractional integral operators proposed by Tunc et. al. <cite>tunc</cite>, we establish a generalization of the reverse Minkowski's inequalities. Within this context, we provide new upper bounds of inequalities utilizing generalized fractional integral operators and show and state other inequalities related to this fractional integral operator.

Keywords

Fractional integral operators , Hermite-Hadamard inequality , midpoint inequality , convex function

References

• R. P. Agarwal, M.-J. Luo and R. K. Raina, On Ostrowski type inequalities, Fasciculi Mathematici, 24, De Gruyter, (2016) doi:10.1515/fascmath-2016-0001.
• P. Agarwal and J. E. Restrepo, An extension by means of ω-weighted classes of the generalized Riemann-Liouville k-fractional integral inequalities, (pending).
• A. Akkurt, M. E. Yıldırım, and H. Yıldırım, On some integral inequalities for (k,h)-Riemann-Liouville fractional integral, New Trends in Mathematical Science, 4 (2016), no. 2, 138--138.
• L. Bougoffa, On Minkowski and Hardy integral inequality, J. Inequal. Pure and Appl. Math. 7 (2006).
• V.L. Chinchane, New approach to Minkowski fractional inequalities using generalized kfractional integral operator, arXiv:1702.05234v1 [math.CA].
• V.L. Chinchane and D. B. Pachpatte, New fractional inequalities via Hadamard fractional integral, Internat. J. Functional Analyisis, Operator Theory and Application, 5(3)(2013), 165-176.
• Z. Dahmani, On Minkowski and Hermite-Hadamard integral inequalities via fractional integration, Ann. Funct. Anal., 1(1)(2010), 51-58.
• R. Diaz and E. Pariguan, On hypergeometric functions and Pochhammer k symbol, Divulg.Math, 15, (2007), 179- 192.
• U. Katugampola, New approach to a generalized fractional integral, Applied Mathematics and Computation (2011).xxxxxxxxxxx
• U. Katugampola, On Generalized Fractional Integrals and Derivatives, Ph.D. Dissertation, Southern Illinois University, Carbondale, August, 2011.
• R. Khalil, M. Al horani, A. Yousef, M. Sababheh, A new definition of fractional derivative, Journal of Computational Apllied Mathematics, 264 (2014), 65-70.
• A. A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Diferential Equations, Elsevier B.V., Amsterdam, Netherlands, 2006.
• V. Kiryakova, On two Saigos fractional integral operator in the class of univalent functions, Fract. Calc. Appl. Anal., 9(2)(2006), 159-176.
• H. Kober, On fractional integrals and derivatives, The Quarterly Journal of Mathematics (Oxford Series), (1940). xxxxxxxxxxxxxxx
• S. Mubeen and G. M. Habibullah, k-Fractional integrals and applications, International Journal of Contemporary Mathematical Sciences, 7(2012), 89--94.
• E. Set, M. Tomar, M. Sarikaya, On generalized Gruss type inequalities for k-fractional integrals, Appl. Math. Comput. (2015) xxxxxxxxxxxxxxx
• M. Saigo, A remark on integral operators involving the Gauss hypergeometric function, Rep. College General Ed., Kyushu Univ., 11 (1978), 135-143.
• M. Sarikaya, Z. Dahmani, M. Kiris, F. Ahmad, (k; s)-Riemann-Liouville fractional integral and applications, Hacet. J. Math. Stat., (2016) xxxxxxxxxxxxxxxxx
• B. Sroysang, More on Reverses of Minkowskis Integral Inequality, Mathematica Aeterna, Vol. 3, (2013), no. 7, 597-600.
• W. T. Sulaiman, Reverses of Minkowski's, Hölder's, and Hardy's integral inequalities, Int. J. Mod. Math. Sci., (2012), 1(1), 1424.
• R.K. Raina, On generalized Wright's hypergeometric functions and fractional calculus operators, East Asian Math. J., 21(2) (2005), 191-203.
• T. Tunç, H. Budak, F. Usta, M. Z. Sarikaya, On new generalized fractional integral operators and related fractional inequalities, ResearchGate Article, Available online at: https://www.researchgate.net/publication/313650587.