Korean Journal of Mathematics

The Kangwon-Kyungki Mathematical Society (강원경기수학회)
권호 표지
DOI QR코드

DOI QR Code

FRACTIONAL VERSIONS OF HADAMARD INEQUALITIES FOR STRONGLY (s, m)-CONVEX FUNCTIONS VIA CAPUTO FRACTIONAL DERIVATIVES

  • Received : 2022.08.24
  • Accepted : 2023.03.22
  • Published : 2023.03.30
Download PDF
⟨ Previous Next ⟩

Abstract

We aim in this article to establish variants of the Hadamard inequality for Caputo fractional derivatives. We present the Hadamard inequality for strongly (s, m)-convex functions which will provide refinements as well as generalizations of several such inequalities already exist in the literature. The error bounds of these inequalities are also given by applying some known identities. Moreover, various associated results are deduced.

Keywords

References

  1. P. Agarwal, M. Jleli and M. Tomar, Certain Hermite-Hadamard type inequalities via generalized k-fractional integrals, J. Inequal. Appl, 2017 (2017), 2017:55.
  2. M. Bracamonte, J. Gimenez and M. Vivas-Cortez, Hermite-Hadamard-Fejer type inequalities for strongly (s, m)-convex functions with modulus c, in second sense, Appl. Math. Inf. Sci. 10 (2016), 2045-2053. https://doi.org/10.18576/amis/100606
  3. 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.
  4. Y. Dong, M. Zeb, G. Farid and S. Bibi, Hadamard inequalities for strongly b(α, m)-convex functions via Caputo fractional derivatives, J. Mathematics. 2021 (2021), 16 pages.
  5. S. S. Dragomir and C. E. M. Pearce, Selected topics on Hermite-Hadamard inequalities and applications, Mathematics Preprint Archive 1 (2003), 463-817.
  6. G. Farid, S. B. Akbar, S. U. Rehman and J. Pecaric, Boundedness of fractional integral operators containing Mittag-Leffler functions via (s, m)-convexity, AIMS. Math. 5 (2020), 966-978. https://doi.org/10.3934/math.2020067
  7. G. Farid, A. U. Rehman, S. Bibi and Y. M. Chu, Refinements of two fractional versions of Hadamard inequalities for Caputo fractional derivatives and related results, Open J. Math. Sci. 5 (2021), 1-10. https://doi.org/10.30538/oms2021.0139
  8. X. Feng, B. Feng, G. Farid, S. Bibi, Q. Xiaoyan and Z. Wu, Caputo fractional derivative Hadamard inequalities for strongly m-convex functions, J. Funct. Spac. 2021 (2021), 11 pages.
  9. G. Farid, A. Javed and S. Naqvi, Hadamard and Fejer Hadamard inequalities and related results via Caputo fractional derivatives, Bull. Math. Anal. Appl. 9 (3) (2017), 16-30.
  10. G. Farid, A. Javed and A. U. Rehman, Fractional integral inequalities of Hadamard type for m-convex functions via Caputo k-fractional derivatives, J. Fract. Calculus. Appl. 10 (1) (2019), 120-134.
  11. U. Ishtaiq, K. Javed, F. Uddin, M. Sen, K. Ahmed and M. U. Ali, Fixed point results in orthogonal Neutrosophic Metric spaces, Complexity. 2021 (2021), 18 pages.
  12. K. Javed, F. Uddin, H. Aydi, M. Arshad, U. Ishtiaq and H. Alsamir, On Fuzzy b-Metric-like spaces, J. Funct. Spaces. 2021 (2021), 9 pages.
  13. M. A. Khan, F. Alam and S. Z. Ullah, Majorization type inequalities for strongly convex functions, Turkish J. Ineq. 3 (2) (2019), 62-78. https://doi.org/10.1186/s13660-019-1964-3
  14. S. M. Kang, G. Farid, W. Nazeer and S. Naqvi, A version of the Hadamard inequality for Caputo fractional derivatives and related results, J. Comput. Anal. Appl. 27 (6) (2019), 962-972. https://doi.org/10.3390/math6070122
  15. A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier, Amsterdam, (2006).
  16. J. Mako and A. Hezy, On strongly convex functions, Carpathian J. Math. 32 (1) (2016), 87-95. https://doi.org/10.37193/CJM.2016.01.09
  17. T. Lara, N. Merentes, R. Quintero and E. Rosales, On Strongly m-Convex Functions, Mathematica Aeterna. 5 (3) (2015), 521-535.
  18. N. Merentes and K. Nikodem, Remarks on strongly convex functions, Aequationes Math. 80 (2010), 193-199. https://doi.org/10.1007/s00010-010-0043-0
  19. K. Nikodem, Strongly convex functions and related classes of functions, Handbook of Functional Equations: Functional Inequalities, 365-40.
  20. 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
  21. M. E. Ozdemir, A. O. Akdemri and E. Set, On (h - m)-convexity and Hadamard-type inequalities, Transylv. J. Math. Mech. 8 (1) (2016), 51-58.
  22. B. T. Polyak, Existence theorems and convergence of minimizing sequences in extremum problems with restrictions, Soviet Mathematics Doklady 7 (1966), 72-75.
  23. M. Z. Sarikaya, E. Set, H. Yaldiz and N. Ba,sak, Hermite-Hadamard's inequalities for fractional integrals and related fractional inequalities, Math. Comput. Model 57 (9-10) (2013), 2403-2407. https://doi.org/10.1016/j.mcm.2011.12.048
  24. G. H. Toader, Some generalisations of the convexity, Proc. Colloq. Approx. Optim, Cluj-Napoca (Romania), 1984, 329-338.
  25. 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