Riemann-Liouville Fractional Versions of Hadamard inequality for Strongly m-Convex Functions

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Ghulam Farid
Saira Bano Akbar
Laxmi Rathour
Lakshmi Narayan Mishra
Vishnu Narayan Mishra


This paper deals with Hadamard inequalities for strongly m-convex functions via Riemann-Liouville fractional integrals. These inequalities provide refinements of well known fractional integral inequalities for convex functions. Further, by applying an identity error estimations are obtained and compared with already known error estimations.

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  1. S.S. Dragomir, R.P. Agarwal, Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula, Appl. Math. Lett. 11 (1998), 91–95. https://doi.org/10.1016/S0893-9659(98)00086-X.
  2. S.S. Dragomir, C.E.M. Pearce, Selected topics on Hermite-Hadamard inequalities and applications, RGMIA Monographs, Victoria University, 2000. http://rgmia.org/monographs/hermite_hadamard.html.
  3. G. Farid, A. Ur. Rehman, B. Tariq, On Hadamard-type inequalities for m-convex functions via Riemann-Liouville fractional integrals, Stud. Univ. Babes,-Bolyai Math. 62 (2017), 141-150. https://doi.org/10.24193/subbmath.2017.2.01.
  4. G. Farid, A. Ur. Rehman, B. Tariq, A. Waheed, On Hadamard type inequalities for m-convex functions via fractional integrals, J. Inequal. Spec. Funct. 7 (2016), 150-167.
  5. U. Kirmaci, Inequalities for differentiable mappings and applications to special means of real numbers to midpoint formula, Appl. Math. Comput. 147 (2004), 137-146. https://doi.org/10.1016/S0096-3003(02)00657-4.
  6. T. Lara, N. Merentes, R. Quintero, E. Rosales, On Strongly m-convex functions, Math. Aeterna, 5 (2015), 521-535.
  7. B.T. Polyak, Existence theorems and convergence of minimizing sequences in extremum problems with restrictions, Soviet Math. Doklady, 7 (1966), 72-75.
  8. A.W. Roberts, D.E. Varberg, Convex functions, Academic Press, New York, 1973.
  9. M. Z. Sarikaya, E. S. H. Yaldiz, N. Başak, Hermite-Hadamard’s inequalities for fractional integrals and related fractional inequalities, Math. Comput. Model. 57 (2013), 2403-2407. https://doi.org/10.1016/j.mcm.2011.12.048.
  10. 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. https://doi.org/10.1016/j.amc.2009.01.055.
  11. M.Z. Sarikaya, H. Yildirim, On Hermit-Hadamard type inequalities for Rieman-Liouville fractional integrals, Miskolc Math. Notes, 17 (2017), 1049-1059.
  12. G. Toader, Some generalizations of the convexity, in: Proceedings of the Colloquium on Approximation and Optimization, Univ. Cluj-Napoca, Cluj-Napoca, (1985), 329-338.