Korean Journal of Mathematics

The Kangwon-Kyungki Mathematical Society (강원경기수학회)
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RIEMANN-LIOUVILLE FRACTIONAL VERSIONS OF HADAMARD INEQUALITY FOR STRONGLY (α, m)-CONVEX FUNCTIONS

  • Received : 2021.06.07
  • Accepted : 2021.10.22
  • Published : 2021.12.30
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Abstract

The refinement of an inequality provides better convergence of one quantity towards the other one. We have established the refinements of Hadamard inequalities for Riemann-Liouville fractional integrals via strongly (α, m)-convex functions. In particular, we obtain two refinements of the classical Hadamard inequality. By using some known integral identities we also give refinements of error bounds of some fractional Hadamard inequalities.

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References

  1. S.S.Dragomir and R.P.Agarwal, Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula, Appl. Math. Lett. 11 (5) (1998), 91-95. https://doi.org/10.1016/S0893-9659(98)00086-X
  2. S.S.Dragomir and C.E.M.Pearce, Selected topics on Hermite-Hadamard inequalities and applications, RGMIA Monographs. Victoria University, 2000.
  3. G.Farid, A.U.Rehman, B.Tariq and A.Waheed, On Hadamard type inequalities for m-convex functions via fractional integrals, J. Inequal. Spec. Funct. 7 (4) (2016), 150-167.
  4. G.Farid, A.U.Rehman and B.Tariq, On Hadamard-type inequalities for m-convex functions via Riemann-Liouville fractional integrals, Studia Univ. Babes-Bolyai, Math. 62 (2) (2017), 141-150. https://doi.org/10.24193/subbmath.2017.2.01
  5. Y.Dong, M.Saddiqa, S.Ullah and G.Farid, Study of fractional integral operators containing Mittag-Leffler functions via strongly (α, m)-convex functions, Math. Prob. Eng. 2021 (2021), 13 pages.
  6. G.Farid, A.U.Rehman and S.Mehmood, Hadamard and Fejer-Hadamard type integral inequalities for harmonically convex functions via an extended generalized Mittag-Leffler function, J. Math. Comput. Sci. 8 (5) (2018), 630-643.
  7. I.Iscan, Hermite-Hadamard-Fejer type inequalities for convex functions via fractional integrals, Stud. Univ. Babes-Bolyai Math. 60 (3) (2015), 355-366.
  8. Y.C.Kwun, G.Farid, S.B.Akbar and S.M.Kang, Rieman-Liouville fractional version of Hadamard inequality for strongly m-convex functions, Submitted.
  9. S.M.Kang, G.Farid, W.Nazeer and S.Mehmood, (h;m)-convex functions and associated fractional Hadamard and Fejer-Hadamard inequalities via an extended generalized Mittag-Leffler function, J. Inequal. Appl. 2019 (2019), 78 pages.
  10. A.A.Kilbas, H.M.Srivastava, J.J.Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies, 204, Elsevier, New York-London, 2006.
  11. U.S.Kirmaci, Inequalities for differentiable mappings and applications to special means of real numbers to midpoint formula, Appl. Math. Comput. 147 (5) (2004), 137-146.
  12. T.Lara, N.Merentes, R.Quintero and E.Rosales, On strongly m-convex functions, Mathematica Aeterna. 5 (3) (2015), 521-535.
  13. S.Mehmood and G.Farid Fractional Hadamard and Fejer-Hadamard inequalities for exponentially m-convex function, Stud. Univ. Babes-Bolyai Math.
  14. S.Mehmood, G.Farid, K.A.Khan and M.Yussouf, New Hadamard and Fejer-Hadamard fractional inequalities for exponentially m-convex function, Eng. Appl. Sci. Lett. 3 (1) (2020), 45-55. https://doi.org/10.30538/psrp-easl2020.0034
  15. S.Mehmood, G.Farid, K.A.Khan and M.Yussouf, New fractional Hadamard and Fejer-Hadamard inequalities associated with exponentially (h - m)-convex function, Eng. Appl. Sci. Lett. 3 (2) (2020), 9-18.
  16. E.Mittal, S.Joshi, R.M.Pandey and V.N.Mishra, Fractional integral and integral transform formulae using generalized Appell hypergeometric function, Nonlinear Science Letters A. 8, (2) (2017), 221-227.
  17. V.N.Mishra, D.L.Suthar and S.D.Purohit, Marichev-Saigo-Maeda Fractional Calculus Operators, Srivastava Polynomials and Generalized Mittag-Leffler Function, Cogent Math. 4 (2017), 1-11. https://doi.org/10.1080/2331205X.2017.1328030
  18. V.G.Mihesan, A generalization of the convexity, Seminar on Functional Equations, Approx. and Convex, Cluj-Napoca, Romania, 1993.
  19. J.Mako and A.Hazy, On strongly convex functions, Carpathian J. Math. 32 (1) (2016), 87-95. https://doi.org/10.37193/CJM.2016.01.09
  20. A.F.Nikiforov and V.B.Uvarov, Special Functions of Mathematical Physics, Springer Basel AG, 1988.
  21. K.Nikodem and Z.Pales, Characterizations of inner product spaces by strongly convex functions, Banach J. Math. Anal. 5 (1) (2011), 83-87. https://doi.org/10.15352/bjma/1313362982
  22. C.P.Niculescu and L.E.Persson, Old and new on the Hermite-Hadamard inequality, Real Anal. Exch. 29 (2) (2003/2004), 663-685. https://doi.org/10.14321/realanalexch.29.2.0663
  23. B.T.Polyak, Existence theorems and convergence of minimizing sequences in extremum problems with restrictions, Soviet Mathematics Doklady. 7 (1966), 72-75.
  24. A.W.Roberts and D.E.Varberg, Convex Functions, Academic Press, New York, 1973.
  25. Z.Retkes, An extension of the Hermite-Hadamard Inequality, Acta Sci. Math. (Szeged), 74 (1) (2008), 95-106.
  26. M.Z.Sarikaya, E.Set, H.Yaldiz and N.Basak, Hermite-Hadamard's inequalities for fractional integrals and related fractional inequalities, Math. comput. model. 57 (9) (2013), 2403-2407. https://doi.org/10.1016/j.mcm.2011.12.048
  27. M.Z.Sarikaya, and H.Yildirim, On Hermite-Hadamard type inequalities for Riemann-Liouville fractional integrals, Miskolc Math. Notes. 17 (2) (2017), 1049-1059. https://doi.org/10.18514/MMN.2017.1197
  28. G.Toader, Some generalizations of the convexity, Proceedings of the Colloquium on Approximation and Optimization, Univ. Cluj-Napoca, Cluj-Napoca, (1985), 329-338.
  29. J.P.Vial, Strong convexity of sets and functions, J. Math. Econ. 9 (1-2) (1982), 187-205. https://doi.org/10.1016/0304-4068(82)90026-X