AIMS Mathematics, 2020, 5(5): 4781-4792. doi: 10.3934/math.2020306.

Research article

Export file:

Format

• RIS(for EndNote,Reference Manager,ProCite)
• BibTex
• Text

Content

• Citation Only
• Citation and Abstract

Derivation of bounds of integral operators via convex functions

1 Department of Mathematics and Physics, Nanjing Institute of Technology, Nanjing, Jiangsu, China
2 Department of Mathematics, COMSATS University Islamabad, Attock Campus, Pakistan
3 Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology (VIT) University, Vellore 632 014, Tamil Nadu, India
4 Department of Mathematics, COMSATS University Islamabad, Attock Campus, Pakistan
5 Department of Mathematics, Air Univeristy, Islamabad, Pakistan

## Abstract    Full Text(HTML)    Figure/Table    Related pages

Convex functions play a vital role in the derivation of inequalities. In this paper these functions are used to obtain certain bounds of a unified integral operator. A Hadamard inequality for these operators is established. Further bounds of several kinds of fractional and conformable integral operators are deduced in particular.
Figure/Table
Supplementary
Article Metrics

Citation: Xinghua You, Ghulam Farid, Lakshmi Narayan Mishra, Kahkashan Mahreen, Saleem Ullah. Derivation of bounds of integral operators via convex functions. AIMS Mathematics, 2020, 5(5): 4781-4792. doi: 10.3934/math.2020306

References

• 1. A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, New York-London, 2006.
• 2. Y. C. Kwun, G. Farid, W. Nazeer, et al. Generalized Riemann-Liouville k-fractional integrals associated with Ostrowski type inequalities and error bounds of Hadamard inequalities, IEEE Access, 6 (2018), 64946-64953.
• 3. S. Mubeen, A. Rehman, A note on k-Gamma function and Pochhammer k-symbol, J. Math. Sci., 6 (2014), 93-107.
• 4. M. Arshad, J. Choi, S. Mubeen, et al. A new extension of Mittag-Leffler function, Commun. Korean Math. Soc., 33 (2018), 549-560.
• 5. H. J. Haubold, A. M. Mathai, R. K. Saxena, Mittag-Leffler functions and their applications, J. Appl. Math., 2011 (2011), 1-51.
• 6. G. Rahman, D. Baleanu, M. A. Qurashi, et al. The extended Mittag-Leffler function via fractional calculus, J. Nonlinear Sci. Appl., 10 (2013), 4244-4253.
• 7. T. R. Prabhakar, A singular integral equation with a generalized Mittag-Leffler function in the kernel, Yokohama Math. J., 19 (1971), 7-15.
• 8. M. Andrić, G. Farid, J. Pečarić, A further extension of Mittag-Leffler function, Fract. Calc. Appl. Anal., 21 (2018), 1377-1395.
• 9. G. Farid, A unified integral operator and its consequences, Open J. Math. Anal., 4 (2020), 1-7.
• 10. S. Mubeen, G. M. Habibullah, k-fractional integrals and applications, Int. J. Contemp. Math., 7 (2012), 89-94.
• 11. 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.
• 12. T. U. Khan, M. A. Khan, Generalized conformable fractional operators, J. Comput. Appl. Math., 346 (2019), 378-389.
• 13. S. Habib, S. Mubeen, M. N. Naeem, Chebyshev type integral inequalities for generalized kfractional conformable integrals, J. Inequal. Spec. Funct., 9 (2018), 53-65.
• 14. M. Z. Sarikaya, M. Dahmani, M. E. Kiris, et al. (k, s)-Riemann-Liouville fractional integral and applications, Hacet. J. Math. Stat., 45 (2016), 77-89.
• 15. F. Jarad, E. Ugurlu, T. Abdeljawad, et al. On a new class of fractional operators, Adv. Differ. Equ., 2017 (2017), 1-16.
• 16. T. Tunc, H. Budak, F. Usta, et al. On new generalized fractional integral operators and related fractional inequalities, Available from: https://www.researchgate.net/publication/313650587.
• 17. S. S. Dragomir, Inequalities of Jensens type for generalized k-g-fractional integrals of functions for which the composite fg-1 is convex, Fract. Differ. Calc., 8 (2018), 127-150.
• 18. T. O. Salim, A. W. Faraj, A generalization of Mittag-Leffler function and integral operator associated with integral calculus, J. Fract. Calc. Appl., 3 (2012), 1-13.
• 19. H. M. Srivastava, Z. Tomovski, Fractional calculus with an integral operator containing generalized Mittag-Leffler function in the kernel, Appl. Math. Comput., 211 (2009), 198-210.
• 20. G. Farid, Existence of an integral operator and its consequences in fractional and conformable integrals, Open J. Math. Sci., 3 (2019), 210-216.
• 21. A. W. Roberts, D. E. Varberg, Convex Functions, Acadamic press, New York and London, 1993.
• 22. G. Farid, Some Riemann-Liouville fractional integrals inequalities for convex function, J. Anal., 27 (2019), 1095-1102.