AIMS Mathematics, 2020, 5(4): 3573-3583. doi: 10.3934/math.2020232

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Chebyshev type inequalities involving extended generalized fractional integral operators

1 Department of Mathematics, Faculty of Science and Arts, Ordu University, Ordu, Turkey
2 Department of Mathematics Education, Education Faculty, Bursa Uludağ University, Görükle Campus, Bursa, Turkey

In this paper, mainly by using the extended generalized fractional integral operator that involve a further extension of Mittag-Leffler function in the kernel, we obtain several fractional Chebyshev type integral inequalities. So, results of Dahmani et al. from [4] are generalized. Also, it is point out that new results are obtained for different fractional integral operators with the help of special selection of parameters.
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1. M. Andric, G. Farid, J. Pečarić, A further extension of Mittag-Leffler function, Fract. Calc. Appl. Anal., 21 (2018), 1377-1395.    

2. S. Belarbi, Z. Dahmani, On some new fractional integral inequalities, JIPAM, 10 (2009), 1-12.

3. Z. Dahmani, About some integral inequalities using Riemann-Liouville integrals, General Mathematics, 20 (2012), 63-69.

4. Z. Dahmani, O. Mechouar, S. Brahami, Certain inequalities related to the Chebyshev's functional involving a Riemann-Liouville operator, Bull. Math. Anal. Appl., 3 (2011), 38-44.

5. Z. Dahmani, New inequalities in fractional integrals, Int. J. Nonlinear Sci., 9 (2010), 493-497.

6. Z. Dahmani, Some results associated with fractional integrals involving the extended Chebyshev functional, Acta Universitatis Apulansis, 27 (2011), 217-224.

7. J. Daiya, J. Ram, R. K. Saxena, New fractional integral inequalities associated with Pathway operator, Acta Comment. Univ. Tartu. Math., 19 (2015), 121-126.

8. S. M. Kang, G. Farid, W. Nazeer, et al. Hadamard and Fejér-Hadamard inequalities for extended generalized fractional integrals involving special functions, J. Ineq. Appl., 2018 (2018), 119.

9. S. M. Kang, G. Farid, W. Nazeer, et al. (h-m)-convex functions and associated fractional Hadamard and Fejér-Hadamard inequalities via an extended generalized Mittag-Leffler function, J. Ineq. Appl., 2019 (2019), 78.

10. C. P. Niculescu, I. Roventa, An extention of Chebyshev's algebric inequality, Math. Reports, 15 (2013), 91-95.

11. M. E. Özdemir, E. Set, A. O. Akdemir, et al. Some new Chebyshev type inequalities for functions whose derivatives belongs to spaces, Afrika Matematika, 26 (2015), 1609-1619.    

12. B. G. Pachpatte, A note on Chebyshev-Grüss type inequalities for diferential functions, Tamsui Oxford Journal of Mathematical Sciences, 22 (2006), 29-36.

13. T. R. Prabhakar, A singular integral equation with generalized Mittag-Leffler function in the kernel, Yokohama Math. J., 19 (1971), 7-15.

14. S. D. Purohit, S. L. Kalla, Certain inequalities related to the Chebyshev's functional involving Erdelyi-Kober operators, Scientia Mathematical Sciences, 25 (2014), 55-63

15. G. Rahman, D. Baleanu, M. A. Qurashi, et al. The extended Mittag-Leffler function via fractional calculus, J. Nonlinear Sci. Appl., 10 (2017), 4244-4253.    

16. T. O. Salim, A. W. Faraj, A generalization of Mittag-Leffler function and integral operator associated with fractional calculus, J. Fract. Calc. Appl., 3 (2012), 1-13.    

17. S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional Integral and Derivatives: Theory and Applications, Gordon and Breach, 1993.

18. M. Z. Sarıkaya, N. Aktan, H. Yıldırım, On weighted Chebyshev-Grüss like inequalities on time scales, J. Math. Ineq., 2 (2008), 185-195.

19. M. Z. Sarıkaya, A. Saglam, H. Yıldırım, On generalization of Chebyshev type inequalities, Iranian J. Math. Sci. Inform., 5 (2010), 41-48.

20. M. Z. Sarıkaya, M. E. Kiriş, On Ostrowski type inequalities and Chebyshev type inequalities with applications, Filomat, 29 (2015), 123-130.

21. E. Set, M. Z. Sarıkaya, F. Ahmad, A generalization of Chebyshev type inequalities for first differentiable mappings, Miskolc Mathematical Notes, 12 (2011), 245-253.    

22. E. Set, Z. Dahmani and İ. Mumcu, New extensions of Chebyshev type inequalities using generalized Katugampola integrals via Polya-Szegö inequality, An International Journal of Optimization and Control: Theories Applications, 8 (2018), 137-144.

23. E. Set, J. Choi, İ. Mumcu, Chebyshev type inequalities involving generalized Katugampola fractional integral operators, Tamkang J. Math., 50 (2019), 381-390.    

24. E. Set, A. O. Akdemir, İ. Mumcu, Chebyshev type inequalities for conformable fractional integrals, Miskolc Mathematical Notes, 20 (2019).

25. E. Set, İ. Mumcu, S. Demirbaş, Chebyshev type inequalities involving new conformable fractional integral operators, RACSAM, 113 (2018), 2253-2259.

26. H. M. Srivastava, Z. Tomovski, Fractional calculus with an integral operator containing a generalized Mittag-Leffler function in the kernel, Appl. Math. Comput., 211 (2009), 198-210.

27. S. Ullah, G. Farid, K. A. Khan, et al. Generalized fractional inequalities for quasi-convex functions, Adv. Difference Equ., 2019 (2019), 1-16.    

28. F. Usta, H. Budak, M. Z. Sarıkaya, On Chebyshev Type Inequalities for Fractional Integral Operators, AIP Conference Proceedings, 1833 (2017), 020045.

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