Korean Journal of Mathematics

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

DOI QR Code

NOTE ON NEWTON-TYPE INEQUALITIES INVOLVING TEMPERED FRACTIONAL INTEGRALS

  • Fatih Hezenci (Department of Mathematics, Faculty of Science and Arts, Duzce University) ;
  • Huseyin Budak (Department of Mathematics, Faculty of Science and Arts, Duzce University)
  • Received : 2024.02.04
  • Accepted : 2024.05.15
  • Published : 2024.06.30
Download PDF
⟨ Previous Next ⟩

Abstract

We propose a new method of investigation of an integral equality associated with tempered fractional integrals. In addition to this, several Newton-type inequalities are considered for differentiable convex functions by taking the modulus of the newly established identity. Moreover, we establish some Newton-type inequalities with the help of Hölder and power-mean inequality. Furthermore, several new results are presented by using special choices of obtained inequalities.

Keywords

References

  1. Ali, M.A.; Budak, H.; Zhang, Z. A new extension of quantum Simpson's and quantum Newton's type inequalities for quantum differentiable convex functions, Math. Methods Appl. Sci. 45 (4) (2022), 1845-1863. https://doi.org/10.1002/mma.7889
  2. Buschman, R.G. Decomposition of an integral operator by use of Mikusinski calculus, SIAM J. Math. Anal. 3 (1972), 83-85. https://doi.org/10.1137/0503010
  3. Erden, S.; Iftikhar, S.; Kumam, P.; Awan, M. U. Some Newton's like inequalities with applications, Rev. R. Acad. Cienc. Exactas F'is. Nat. Ser. A Mat. RACSAM, 114 (4) (2020), Paper No. 195. https://doi.org/10.1007/s13398-020-00926-z
  4. Erden, S.; Iftikhar, S.; Kumam, P.; Thounthong, P. On error estimations of Simpson's second type quadrature formula, Math. Methods Appl. Sci. 2020 (2020), 1-13. https://doi.org/10.1002/mma.7019
  5. Hezenci, F.; Budak, H.; Kosem, P. A new version of Newton's inequalities for Riemann-Liouville fractional integrals, Rocky Mountain J. Math. 53 (1) (2023), 49-64. https://doi.org/10.1216/rmj.2023.53.49
  6. Gao, S.; Shi, W. On new inequalities of Newton's type for functions whose second derivatives absolute values are convex, Int. J. Pure Appl. Math. 74 (1) (2012), 33-41.
  7. Gorenflo R.; Mainardi F. Fractional calculus: Integral and differential equations of fractional order, Springer Verlag, Wien, 1997. https://doi.org/10.1007/978-3-7091-2664-6_5
  8. Iftikhar, S.; Erden, S.; Ali, M.A.; Baili, J.; Ahmad, H. Simpson's second-type inequalities for coordinated convex functions and applications for cubature formulas, Fractal Fract. 6 (1) (2022), 33. https://doi.org/10.3390/fractalfract6010033
  9. Iftikhar, S.; Erden, S.; Kumam, P.; Awan, M. U. Local fractional Newton's inequalities involving generalized harmonic convex functions, Adv. Difference Equ. 2020 (1) (2020), Paper No. 185. https://doi.org/10.1186/s13662-020-02637-6
  10. Iftikhar, S.; Kumam, P.; Erden, S. Newton's-type integral inequalities via local fractional integrals, Fractals 28 (3) (2020), Article ID 2050037. https://doi.org/10.1142/S0218348X20500371
  11. Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J. Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204, Elsevier Sci. B.V., Amsterdam, 2006.
  12. Li, C.; Deng W.; Zhao, L. Well-posedness and numerical algorithm for the tempered fractional ordinary differential equations, Discrete Contin. Dyn. Syst. Ser. B 24 (2019), 1989-2015. https://doi.org/10.3934/dcdsb.2019026
  13. Luangboon, W.; Nonlaopon, K.; Tariboon, J.; Ntouyas, S.K. Simpson-and Newton-type inequalities for convex functions via (p, q)-calculus, Mathematics 9 (12) (2021), 1338. https://doi.org/10.3390/math9121338
  14. Meerschaert, M.M.; Sabzikar, F.; Chen, J., Tempered fractional calculus, J. Comput. Phys. 293 (2015), 14-28. https://doi.org/10.1016/j.jcp.2014.04.024
  15. Meerschaert, M.M.; Sikorskii, A. Stochastic Models for Fractional Calculus, De Gruyter Studies in Mathematics, 43 , Walter de Gruyter and Co., Berlin, 2012. https://doi.org/10.1515/9783110258165
  16. Mohammed, P.O.; Sarikaya, M.Z.; Baleanu, D. On the generalized Hermite-Hadamard inequalities via the tempered fractional integrals, Symmetry 12 (4) (2020), 595. https://doi.org/10.3390/sym12040595
  17. Noor, M.A.; Noor, K.I.; Awan, M.U. Some Newton's type inequalities for geometrically relative convex functions, Malays. J. Math. Sci. 9 (3) (2015), 491-502.
  18. Noor, M. A.; Noor, K.I.; Iftikhar, S. Some Newton's type inequalities for harmonic convex functions, J. Adv. Math. Stud. 9 (1) (2016), 07-16. https://doi.org/10.2298/FIL1609435N
  19. Noor,M.A.; Noor, K.I.; Iftikhar, S. Newton inequalities for p-harmonic convex functions, Honam Math. J. 40 (2) (2018), 239-250. https://dx.doi.org/10.5831/HMJ.2018.40.2.239
  20. Pecaric J.E.; Proschan F.; Tong Y.L. Convex Functions, Partial Orderings and Statistical Applications, Academic Press, Boston, 1992.
  21. Samko, S.; Kilbas, A.; Marichev, O. Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, London, 1993.
  22. Sarikaya, M.Z.; Set, E.; Yaldiz, H.; Basak, N. 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. Sarikaya, M.Z.; Yildirim, H. On Hermite-Hadamard type inequalities for Riemann-Liouville fractional integrals, Miskolc Math. Notes 17 (2017), 1049-1059. https://doi.org/10.18514/MMN.2017.1197
  24. Sitthiwirattham, T.; Nonlaopon, K.; Ali, M.A.; Budak, H. Riemann-Liouville fractional Newton's type inequalities for differentiable convex functions, Fractal Fract. 6 (3) (2022), 175. https://doi.org/10.3390/fractalfract6030175
  25. Srivastava, H.M.; Buschman, R.G. Convolution Integral Equations with Special Function Kernels, John Wiley & Sons, New York, 1977.