On Caputo Fractional Derivatives via Exponential s-Convex Functions

Abstract

In this paper, we establish several new integral inequalities including Caputo fractional integrals for exponential s-convex functions. By using convexity for exponential s-convex functions of any integer order differentiable function some novel results are given.

In this paper, we establish several new integral inequalities including Caputo fractional integrals for exponential s-convex functions. By using convexity for exponential s-convex functions of any integer order differentiable function some novel results are given.

Keywords

• convex function
• exponential s-convex functions
• Caputo-Fractional derivatives

References

1. A. A. Kilbas, H.M. Srivastava, J. J. Trujillo, Theory and applications of Fractional Differential Equations,North-Holland Mathematics Studies, Elsevier, New York-London, 2006.
2. C. P. Li and W. H. Deng, Remarks on fractional derivatives, Appl. Math. Comput., 187(2) (2007), 777-784.
3. G. Farid, S. Naqvi and A. U. Rehman, A version of the Hadamard inequality for Caputo fractional derivatives and related results, RGMIA Research Report Collection, 20 (2017).
4. G. Farid,On Caputo Fractional Derivatives Via Convexity,Kragujevac Journal Of Mthematics, 44(3),(2020), 393-399.
5. G.H. Hardy, J.E. Littlewood, G. Polya, Inequalities,Cambridge, UK : Cambridge University Press, Cambridge mathematical library, 1952.
6. H.Hudzik, L. Maligranda,Some remarks on s-convex functions,Aequ. Math. 48(1), 1994, 100-111.
7. L. N. Mishra, Q. U. Ain, G. Farid and A. U. Rehman, k−fractional integral inequalities for (h,m)− convex functions via Caputo k− fractional derivatives, The Korean Journal of Mathematics, 27(2), (2019), 357-374.
8. M. Caputo,Linear model of dissipation whose Q is almost frequency independent Geophysical Journal International, 13(5), 1967, 529-539.

9. M.U. Awan, M.A. Noor, K.I. Noor,Hermite-Hadamard inequalities for exponentially convex functions,Appl. Math. Inf. Sci. 12(2), 2018, 405-409.
10. N. Mehreen, M. Anwar,Hermite-Hadamard type inequalities fo exponentially p-convex functions and exponentialli sconvex functions in the second sense with applications,J Inequal. Appl. 92, 2019.
11. S. Naqvi, G. Farid and B. Tariq, Caputo fractional integral inequalities via m convex function,RGMIA Research Report Collection, (2014).
12. S. Miller, B. Ross,An introduction to fractional calculus and fractional differential equations,John Wiley And Sons, Usa., (2), 1993.