Chebyshev inequality on conformable derivative

Year 2021, Volume: 70 Issue: 2, 900 - 909, 31.12.2021

Aysun Selçuk Kızılsu , Ayşe Feza Güvenilir

https://doi.org/10.31801/cfsuasmas.773392

Abstract

Integral inequalities are very important in applied sciences. Chebyshev's integral inequality is widely used in applied mathematics. First of all, some necessary definitions and results regarding conformable derivative are given in this article. Then we give Chebyshev inequality for simultaneously positive (or negative) functions using the conformable fractional derivative. We used the Gronwall inequality to prove our results, unlike other studies in the literature.

Keywords

Conformable Derivative, conformable integral, Chebyshev inequality

References

• Abdeljawad, T., On conformable fractional calculus, Journal of Computational and Applied Mathematics, 279 (2015), 57-66. https://doi:101016/j.cam.2014.10.016
• Anderson, D. R., Ulness, D. J., Newly de…ned conformable derivatives, Advances in Dynamical Systems and Applications, 10(2) (2015), 109-137.
• Beckenbach, E. F., Bellman, R., Inequalities, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1961.
• Belarbi, S., Dahmani, Z., On some new fractional integral inequalities, Journal of Inequalities in Pure and Applied Mathematics, 10(3) (2009), Article 86, 5 pp.
• Butt, S. I., Umar, M., Rashid, S., Akdemir, A. O., Chu, Y., New Hermite-Jensen-Mercer-type inequalities via k-fractional integrals, Advances in Difference Equations, (2020), Article 635, 24 pp. https://doi:10.1186/s13662-020-03093-y
• Chebyshev, P. L., Sur les expressions approximatives des integrales definies par les autres prises entre les mêmes limites, Proc. Math. Soc. Charkov, 2 (1882), 93-98.
• Chen, S., Rashid, S., Hammouch, Z., Noor, M. A., Ashraf, R., Chu, Y., Integral inequalities via Raina's fractional integrals operator with respect to a monotone function, Advances in Difference Equations, Volume 2020 (2020), Article 647, 20 pp. https://doi:10.1186/s13662-020-03108-8
• Diethelm, K., The Analysis of Fractional Di¤erential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type, Springer-Verlag, Berlin, 2010.
• Fink, A. M., An essay on the history of inequalities, Journal of Mathematical Analysis and Applications, 249(1) (2000), 118-134. https://doi:10.1006/jmaa.2000.6934
• Hardy G. H., Littlewood J. E., Polya, G., Inequalities, Cambridge University Press, Cambridge, 1952.
• Katugampola, U. N., A new fractional derivative with classical properties, (2014), 8 pp. arXiv:1410.6535v2
• Khalil, R., Al Horani, M., Yousef, A., Sababheh, M., A new definition of fractional derivative, Journal of Computational and Applied Mathematics, 264 (2014), 65-70. https://doi:10.1016/j.cam.2014.01.002
• Khan, Z. A., Rashid, S., Ashraf, R., Baleanu, D., Chu, Y., Generalized trapezium-type inequalities in the settings of fractal sets for functions having generalized convexity property, Advances in Difference Equations, Volume 2020 (2020), Article 657, 24 pp. https://doi:10.1186/s13662-020-03121-x
• Miller, K. S., Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993.
• Mitrinovic, D. S., Analytic Inequalities. Springer-Verlag, Berlin-Heidelberg-New York, 1970.
• Musraini, M., Efendi, R., Lily, E., Hidayah, P., Classical properties on conformable fractional calculus, Pure and Applied Mathematics Journal, 8(5) (2019), 83-87. https://doi:10.11648/j.pamj.20190805.11
• Podlubny, I., Fractional Differential Equations, Academic Press, San Diego, 1999.
• Rashid, S., Jarad, F., Noor, M. A., Kalsoom, H., Chu, Y., Inequalities by means of generalized proportional fractional integral operators with respect to another function, Mathematics, 7(12) (2019), Article 1225, 18 pp. https://doi:10.3390/math7121225
• Rashid, S., Ahmad, H., Khalid, A., Chu, Y., On discrete fractional integral inequalities for a class of functions, Hindawi, Volume 2020 (2020), Article ID 8845867, 13 pp. https://doi:10.1155/2020/8845867
• Samko, S. G., Kilbas, A. A., Marichev, O. I., Fractional Integrals and Derivatives, Gordon and Breach Science Publishers, Yverdon, 1993.
• Set E., Akdemir A. O., Mumcu I., Chebyshev type inequalities for conformable fractional integrals, Miskolc Mathematical Notes, 20(2) (2019), 1227-1236. https://doi:10.18514/MMN.2019.2766