Bounds of Generalized Proportional Fractional Integrals in General Form via Convex Functions and Their Applications
Abstract
:1. Introduction
2. Preliminaries
- i.
- ii.
- if we take , and , we get the left and right sided Katugampola fractional integral operators,
- iii.
- if we take , then it reduces to the general form of Riemann–Liouville fractional integral given in [52],
- iv.
- v.
- if we take and (where , and ), then it reduces to the generalized fractional conformable integrals given in [53].
3. Main Results
4. Applications
5. Concluding Remarks
Author Contributions
Funding
Conflicts of Interest
References
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Rahman, G.; Abdeljawad, T.; Jarad, F.; Nisar, K.S. Bounds of Generalized Proportional Fractional Integrals in General Form via Convex Functions and Their Applications. Mathematics 2020, 8, 113. https://doi.org/10.3390/math8010113
Rahman G, Abdeljawad T, Jarad F, Nisar KS. Bounds of Generalized Proportional Fractional Integrals in General Form via Convex Functions and Their Applications. Mathematics. 2020; 8(1):113. https://doi.org/10.3390/math8010113
Chicago/Turabian StyleRahman, Gauhar, Thabet Abdeljawad, Fahd Jarad, and Kottakkaran Sooppy Nisar. 2020. "Bounds of Generalized Proportional Fractional Integrals in General Form via Convex Functions and Their Applications" Mathematics 8, no. 1: 113. https://doi.org/10.3390/math8010113
APA StyleRahman, G., Abdeljawad, T., Jarad, F., & Nisar, K. S. (2020). Bounds of Generalized Proportional Fractional Integrals in General Form via Convex Functions and Their Applications. Mathematics, 8(1), 113. https://doi.org/10.3390/math8010113