Honam Mathematical Journal (호남수학학술지)

The Honam Mathematical Society (호남수학회)
권호 표지
DOI QR코드

DOI QR Code

PERTURBED FRACTIONAL NEWTON-TYPE INEQUALITIES BY TWICE DIFFERENTIABLE FUNCTIONS

  • Fatih Hezenci (Department of Mathematics, Faculty of Science and Arts, Duzce University) ;
  • Hasan Kara (Department of Mathematics, Faculty of Science and Arts, Duzce University) ;
  • Huseyin Budak (Department of Mathematics, Faculty of Science and Arts, Duzce University)
  • Received : 2022.10.03
  • Accepted : 2023.01.25
  • Published : 2023.06.01
Download PDF
⟨ Previous Next ⟩

Abstract

In the present paper, we establish some perturbed Newton-type inequalities in the case of twice differentiable convex functions. These inequalities are established by using the well-known Riemann-Liouville fractional integrals. With the aid of special cases of our main results, we also give some previously obtained Newton-type inequalities.

Keywords

References

  1. M. W. Alomari, A companion of Dragomir's generalization of Ostrowski's inequality and applications in numerical integration, Ukranianian Math. J. 64 (2012), 435-450.  https://doi.org/10.1007/s11253-012-0661-x
  2. P. Agarwal, Some inequalities involving Hadamard-type k-fractional integral operators, Math. Methods Appl. Sci. 40 (2017), no. 11, 3882-3891.  https://doi.org/10.1002/mma.4270
  3. P. Agarwal, M. Jleli, and M. Tomar, Certain Hermite-Hadamard type inequalities via generalized k -fractional integrals, J. Inequal. Appl. 2017 (2017), no. 1, 1-10.  https://doi.org/10.1186/s13660-016-1272-0
  4. M. U. Awan, S. Talib, Y. M. Chu, M. A. Noor, and K. I. Noor, Some new refinements of Hermite-Hadamard-type inequalities involving Riemann-Liouville fractional integrals and applications, Math. Probl. Eng. 2020 (2020), 3051920. 
  5. H. Budak, F. Hezenci, and H. Kara, On parametrized inequalities of Ostrowski and Simpson type for convex functions via generalized fractional integral, Math. Methods Appl. Sci., 44 (2021), no. 17, 12522-12536.  https://doi.org/10.1002/mma.7558
  6. H. Budak, F. Hezenci, and H. Kara, On generalized Ostrowski, Simpson and Trapezoidal type inequalities for co-ordinated convex functions via generalized fractional integrals, Adv. Difference Equ. 2021 (2021), no.1, 1-32.  https://doi.org/10.1186/s13662-020-03162-2
  7. S. Erden, S. Iftikhar, P. Kumam, and M. U. Awan, Some Newton like inequalities with applications,RACSAM Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. 114 (2020), no. 4, 1-13.  https://doi.org/10.1007/s13398-019-00732-2
  8. S. Gao and W. Shi, On new inequalities of Newton-type for functions whose second derivatives absolute values are convex, Int. J. Pure Appl. Math. 74 (2012), no. 1, 33-41. 
  9. R. Gorenflo and F. Mainardi, Fractional calculus: Integral and differential equations of fractional order, Springer Verlag, Wien, 1997. 
  10. F. Hezenci, H. Budak, and H. Kara, New version of Fractional Simpson type inequalities for twice differentiable functions, Adv. Difference Equ. 2021 (2021), no. 1, 1-10.  https://doi.org/10.1186/s13662-020-03162-2
  11. F. Hezenci, H. Budak, and P. Kosem, On New version of Newton's inequalities for Riemann-Liouville fractional integrals, Rocky Mountain J. Math., in press. 
  12. S. Iftikhar, P. Kumam, and S. Erden, Newton's-type integral inequalities via local fractional integrals, Fractals 2020 (2020), no. 3, 2050037. 
  13. S. Iftikhar, S.Erden, P. Kumam, and M. U. Awan, Local fractional Newton's inequalities involving generalized harmonic convex functions, Adv. Difference Equ. 2020 (2020), no. 1, 1-14.  https://doi.org/10.1186/s13662-019-2438-0
  14. M. A. Khan, A. Iqbal, M. Suleman and Y. M. Chu, Hermite-Hadamard type inequalities for fractional integrals via Green's function, J. Inequal. Appl. 2018 (2018), 1-15.  https://doi.org/10.1186/s13660-017-1594-6
  15. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier, Amsterdam, 2006. 
  16. M. A. Noor, K. I. Noor, and S. Iftikhar, Some Newton's type inequalities for harmonic convex functions, J. Adv. Math. Stud. 9 (2016), no. 1, 7-16.  https://doi.org/10.2298/FIL1609435N
  17. M. A. Noor, K. I. Noor, and S. Iftikhar, Newton inequalities for p-harmonic convex functions, Honam Math. J. 40 (2018), no. 2, 239-250. 
  18. J. Park, On some integral inequalities for twice differentiable quasi-convex and convex functions via fractional integrals, Appl. Math. Sci. 9 (2015), no. 62, 3057-3069.  https://doi.org/10.12988/ams.2015.53248
  19. J. E. Pecaric, F. Proschan, and Y. L. Tong, Convex Functions, Partial Orderings and Statistical Applications, Academic Press, Boston, 1992. 
  20. C. Peng, C. Zhou, and T. S. Du, Riemann-Liouville fractional Simpson's inequalities through generalized (m, h1, h2)-preinvexity, Ital. J. Pure Appl. Math. 38 (2017), 345-367. 
  21. M. Z. Sarikaya and H. Yildirim, On Hermite-Hadamard type inequalities for Riemann-Liouville fractional integrals, Miskolc Math. Notes 17 (2016), 1049-1059.  https://doi.org/10.18514/MMN.2017.1197
  22. 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 (2013), 2403-2407.  https://doi.org/10.1016/j.mcm.2011.12.048
  23. M. Z. Sarikaya and F. Ertugral, On the generalized Hermite-Hadamard inequalities, Ann. Univ. Craiova Math. 47 (2020), 193-213. 
  24. T. Sitthiwirattham, K. Nonlaopon, M. A. Ali, and H. Budak, Riemann-Liouville fractional Newton's type inequalities for differentiable convex functions, Fractal Fract. 6 (2022) no. 3, 175. 
  25. J. Tariboon, S. K. Ntouyas, and P. Agarwal, New concepts of fractional quantum calculus and applications to impulsive fractional q-difference equations, Adv. Difference Equ. 2015 (2015), no. 1, 1-19.  https://doi.org/10.1186/s13662-014-0348-8
  26. M. Tunc, On new inequalities for h -convex functions via Riemann-Liouville fractional integration, Filomat 27 (2013), no. 4, 559-565.  https://doi.org/10.2298/FIL1304559T
  27. X. You, F. Hezenci, H. Budak, and H. Kara, New Simpson type inequalities for twice differentiable functions via generalized fractional integrals, AIMS Mathematics 7 (2021), no. 3, 3959-3971.