Korean J. Math.  Vol 27, No 2 (2019)  pp.357-374
DOI: https://doi.org/10.11568/kjm.2019.27.2.357

$k-$fractional integral inequalities for $(h-m)-$convex functions via Caputo $k-$fractional derivatives

Lakshmi Narayan Mishra, Qurat Ul Ain, Ghulam Farid, Atiq Ur Rehman


In this paper, first we obtain some inequalities of Hadamard type for $(h-m)-$convex functions via Caputo $k-$fractional derivatives. Secondly, two integral identities including the $(n+1)$ and $(n+2)$ order derivatives of a given function via Caputo $k-$fractional derivatives have been established. Using these identities estimations of Hadamard type integral inequalities for the Caputo $k-$fractional derivatives have been proved.


Caputo fractional derivatives; Caputo $k-$fractional derivatives; $(h-m)-$convex functions

Subject classification

26B15, 26A51, 34L15.


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F. Chen, On Hermite-Hadamard type inequalities for Riemann-Liouville fractional integrals via two kinds of convexity, Chin. J. Math. (2014) Article ID 173293, pp 7. (Google Scholar)

G. Farid, A. Javed and A. U. Rehman, On Hadamard inequalities for n−times differentiable functions which are relative convex via Caputo k-fractional derivatives, Nonlinear Anal. Forum. 22 (2) (2017), 17–28. (Google Scholar)

G. Farid, A. U. Rehman and M. Zahra, On Hadamard inequalities for k- fractional integrals, Nonlinear Funct. Anal. Appl. 21 (3) (2016), 463–478. (Google Scholar)

R. Gorenflo and F. Mainardi, Fractional calculus: Integral and differential equations of fractional order, Springer Verlag, Wien, (1997). (Google Scholar)

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier, Amsterdam, (2006). (Google Scholar)

M. Lazarevi ́c, Advanced topics on applications of fractional calculus on control problems, System stability and modeling, WSEAS Press, Belgrade, Serbia, (2012). (Google Scholar)

S. Mubeen and G. M. Habibullah, k-Fractional integrals and applications, Int. J. Contemp. Math. Sci. 7 (2012), 89–94. (Google Scholar)

K. Oldham, J. Spanier, The fractional calculus theory and applications of differentiation and integration to arbitrary order, Academic Press, New York-London (1974). (Google Scholar)

M.E.O ̈zdemir, A.O.Akdemri and E.Set, On(h−m)−convexity and Hadamard- type inequalities, Transylv. J. Math. Mech. 8 (1) (2016), 51–58. (Google Scholar)

I. Podlubni, Fractional differential equations, Academic press, San Diego, (1999). (Google Scholar)


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