DOI QR코드

DOI QR Code

SOME INTEGRAL INEQUALITIES IN THE FRAMEWORK OF GENERALIZED K-PROPORTIONAL FRACTIONAL INTEGRAL OPERATORS WITH GENERAL KERNEL

  • Valdes, Juan E. Napoles (UNNE, FaCENA and UTN-FRRE)
  • Received : 2021.03.22
  • Accepted : 2021.09.08
  • Published : 2021.12.25

Abstract

In this article, using the concept proposed reciently by the author, of a Generalized k-Proportional Fractional Integral Operators with General Kernel, new integral inequalities are obtained for convex functions. It is shown that several known results are particular cases of the proposed inequalities and in the end new directions of work are provided.

Keywords

References

  1. B. Ahmad, A. Alsaedi, M. Kirane, B. T. Toberek, Hermite-Hadamard, Hermite-Hadamard-Fejer, Dragomir-Agarwal and Pachpatte type inequalities for convex functions via new fractional integrals, ArXiv: 1701.00092 https://doi.org/10.1016/j.cam.2018.12.030
  2. M. A. Ali, J. E. Napoles, A. Kashuri, Z. Zhang, Fractional non conformable Hermite-Hadamard inequalities for generalized ϕ-convex functions, Fasciculi Mathematici, Nr 64 2020, 5-16 DOI: 10.21008/j.0044-4413.2020.0007
  3. A. Atangana, Derivative with a New Parameter Theory, Methods and Applications, Academic Press, 2016.
  4. D. Baleanu, COMMENTS ON: Ortigueira M., Martynyuk V., Fedula M., Machado J.A.T., The failure of certain fractional calculus operators in two physical models, in Fract. Calc. Appl. Anal. 22(2)(2019), Fract. Calc. Appl. Anal., Volume 23: Issue 1, DOI: https://doi.org/10.1515/fca-2020-0012.
  5. D. Baleanu, A. Fernandez, On fractional operators and their classifications, Mathematics, 7(830)(2019).
  6. S. Bermudo, P. Korus, Juan E. Napoles, On q-Hermite-Hadamard inequalities for general convex functions, Acta Math. Hungar. 162, 364-374 (2020). https://doi.org/10.1007/s10474-020-01025-6
  7. V. L. Chinchane, D. B. Pachpatte, On new fractional integral inequalities involving convex functions using Hadamard fractional integral, Bull. Allahabad Math. Soc., 2016, 31(2), 183-192.
  8. Z. Dahmani, A note on some new fractional results involving convex functions, Acta Math. Univ. Comenianae, 81(2)(2012), 241-246.
  9. O. M. Duarte, Fractional Calculus for Scientists and Engineers, Dordrecht Heidelberg London New York: Springer, 2011.
  10. J. D. Galeano, J. E. Napoles, E. Perez, A note on some integral inequalities in a generalized framework, Int. J. Appl. Math. Stat.; Vol. 60; Issue No. 1; Year 2021, 45-52.
  11. J. D. Galeano, J. E. Napoles, E. Perez, Concerning to the generalized Hermite-Hadamard integral inequality, submitted.
  12. J. D. Galeano, J. E. Napoles, E. Perez, On a general formulation of the fractional operator Riemann-Liouville and related inequalities, submitted.
  13. R. Gorenflo, F. Mainardi, Fractals and fractional calculus in continuum mechanics, 223-276, Springer, Vienna, 1997.
  14. J. Hadamard, Essai sur l'etude des fonctions donnees par leur developpement de Taylor, J. Math. Pures Appl. (4) 8, 101-186 (1892)
  15. J. Hadamard, Etude sur les proprietes des fonctions enti'eres et en particulier d'une fonction consideree par Riemann, J. Math. Pures Appl. 58, 171-215 (1893)
  16. E. Kacar, Z. Kacar, H. Yildirim, Integral inequalities for Riemann-Liouville fractional integrals of a function with respect to another function, Iran. J. Math. Sci. Inform. 2018, 13, 1-13. https://doi.org/10.22457/jmi.v13a1
  17. U. N. Katugampola, New Approach Generalized Fractional Integral, Applied Math and Comp. 218(2011),860-865. https://doi.org/10.1016/j.amc.2011.03.062
  18. A. A. Kilbas, O. I. Marichev, S. G. Samko, Fractional Integrals and Derivatives, Theory and Applications, Gordon & Breach, Switzerland (1993).
  19. A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Amsterdam, Netherlands, Elsevier, Febrary 2006.
  20. P. Korus, L. M. Lugo, J. E. Napoles Valdes, Integral inequalities in a generalized context, Studia Scientiarum Mathematicarum Hungarica 57 (3), 312-320 (2020) https://doi.org/10.1556/012.2020.57.3.1464
  21. W. J. Liu, Q. A. Ngo, V. N. Huy, Several interesting integral inequalities, Journal of Math. Inequal., 3(2) (2009), 201-212.
  22. F. Mainardi, Fractional Calculus and Waves in Linear Viscoelsticity, Ed. Imperial College Press, 2010.
  23. C. A. Monje, Y. Chen, B. M. Vinagre, D. Xue, V. Feliu-Batle, Fractional Order Systems and Controls, Fundamentals and Applications, Londres: Springer-Verlag London Limited, 2010.
  24. S. Mubeen, G. M. Habibullah, k-Fractional Integrals and Application, Int. J. Contemp. Math. Sciences, Vol. 7, 2012, no. 2, 89 - 94
  25. J. E. Napoles V., A Generalized k-Proportional Fractional Integral Operators with General Kernel, submited
  26. J. E. Napoles V., Hermite-Hadamard inequality in generalized context, VI COLLOQUIUM ON APPLIED MATHEMATICS and II INTERNATIONAL MEETING OF APPLIED MATHEMATICS, UNIMILITAR, BOGOTA, COLOMBIA, NOVEMBER 11-13, 2020
  27. J. E. Napoles V., New generalized fractional integral inequalities of Hermite-Hadamard type for harmonically convex functions, XVI International Meeting of Mathematics, Barranquilla, Colombia NOVEMBER 17-20, 2020
  28. J. E. Napoles Valdes, F. Rabossi, A. D. Samaniego, Convex functions: Ariadne's thread or Charlotte's spiderweb?, Advanced Mathematical Models & Applications Vol.5, No.2, 2020, 176-191 https://doi.org/10.11648/j.mma.20200503.16
  29. J. E. Napoles Valdes, J. M. Rodriguez, J. M. Sigarreta, New Hermite-Hadamard Type Inequalities Involving Non-Conformable Integral Operators, Symmetry 2019, 11, 1108; doi:10.3390/sym11091108
  30. T. F. Nonnenmacher, R. Metzler, Applications of Fractional Calculus Ideas to Biology, World Scientific, 1998.
  31. K. Oldham, J. Spanier, Applications of Differentiation and Integration to Arbitrary Order, Volume 111, Elsevier Science, 1974.
  32. I. Podlubny, Fractional differential equations: An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Mathematics in science and engineering; v. 198, San Diego: Academic Press, 1999.
  33. G. Rahman, T. Abdeljawad, F. Jarad, A. Khan, K. S. Nisar, Certain inequalities via generalized proportional Hadamard fractional integral operators, Adv. Diff. Eqs. 2019, 2019, Article ID 454, 10 pages.
  34. S. Rashid, Z. Hammouch, F. Jarad, Y. M. Chu, New Estimates of Integral Inequalities via Generalized Proportional Fractional Integral Operator with Respect to Another Function, Fractals, doi: 10.1142/S0218348X20400277
  35. S. Rashid, F. Jarad, M. A. Noor, H. Kalsoom, Y. M. Chu, Inequalities by Means of Generalized Proportional Fractional Integral Operators with Respect to Another Function, Mathematics 2020, 7, 1225; doi:10.3390/math7121225
  36. S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional Integrals and Derivatives, Gordon & Breach Science, Yverdon, 1993.
  37. V. E. Tarasov, Fractional Dynamics; Applications of the Fractional Calculus to Dynamics of Particles, fields and Media, Dordrecht Heidelberg London New York: Springer, 2010.
  38. H. U. Rehman, M. Darus, J. Salah, A Note on Caputo's Derivative Operator Interpretation in Economy, Journal of Applied Mathematics, 2018, Article ID 1260240, 7 pages https://doi.org/10.1155/2018/1260240
  39. S. Umarov, S. Steinberg, Variable order differential equations with piecewise constant order-function and diffusion with changing modes, Z. Anal. Anwend. 28 (4) (2009) 431-450.
  40. H. Yildirim, Z. Kirtay, Ostrowski inequality for generalized fractional integral and related inequalities, Malaya J. Mat. 2014, 2, 322-329.