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New Midpoint-type Inequalities of Hermite-Hadamard Inequality with Tempered Fractional Integrals

Year 2023, Volume: 44 Issue: 4, 758 - 767, 28.12.2023

Tuba Tunç , Ayşe Nur Altunok

https://doi.org/10.17776/csj.1320515

Abstract

In this research, we get some midpoint type inequalities of Hermite-Hadamard inequality via tempered fractional integrals. For this, we first obtain an identity. After that, using this identity and with the help of modulus function, Hölder inequality, power mean inequality, ongoing research and the papers mentioned, we have reached our intended midpoint type inequalities. Also, we give the special cases of our results. We see that our special results give earlier works.

Keywords

Hermite-Hadamard-type inequalities Midpoint-type inequalities Convex functions Riemann-Liouville fractional integrals Tempered fractional integrals.

References

  • [1] Dragomir S.S., Pearce C.E.M., Selected topics on the Hermite-Hadamard inequalities and applications, RGMIA Monographs, Victoria University.
  • [2] Hadamard J., Etude sur les proprietes des fonctions entieres et en particulier d'une fonction considree par Riemann, Journal de Math´ematiques Pures et Appliqu´ees, 58 (1893) 171-215.
  • [3] Metzler R., Klafter J., The random walk's guide to anomalous diffusion: a fractional dynamics approach, Physics Reports, 339 (2000) 1-77.
  • [4] Mohammed P.O., Brevik I., A new version of the Hermite-Hadamard inequality for Riemann--Liouville fractional integrals, Symmetry, 12(4) (2020), 1-11.
  • [5] Nonlaopon K., Awan M.U., Javed M.Z., Budak H., Noor M.A., Some q-fractional estimates of trapezoid like inequalities involving Raina’s function, Fractal and Fractional, 6(4) (2022) 1-19.
  • [6] Tomovski Z., Generalized Cauchy type problems for nonlinear fractional differential equations with composite fractional derivative operator, Nonl. Anal., 75(7) (2012) 3364-3384.
  • [7] Podlubny I., Fractional differential equations, Academic Press, San Diego, (1999).
  • [8] Samko S., Kilbas A., Marichev O., Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, London, (1993).
  • [9] Hilfer R., Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.
  • [10] Bin-Mohsin B., Awan M.U., Javed M.Z., Khan A.G., Budak H., Mihai M.V., Noor M.A., Generalized AB-fractional operator inclusions of Hermite–Hadamard’s type via fractional integration, Symmetry, 15(5) (2023) 1-21.
  • [11] Budak H., Kılınç Yıldırım S., Sarıkaya M.Z., Yıldırım H., Some parameterized Simpson-, midpoint- and trapezoid-type inequalities for generalized fractional integrals, J. Inequal. Appl., 2022(1) (2022) 1-23.
  • [12] Ertuğral F., Sarikaya M.Z., Budak H., On Hermite-Hadamard type inequalities associated with the generalized fractional integrals, Filomat, 36(12) (2022) 3983-3995.
  • [13] Jarad F., Abdeljawad T., Baleanu D., On the generalized fractional derivatives and their Caputo modification, J. Nonlinear Sci. Appl., 10(5) (2017) 2607-2619.
  • [14] Kirmaci U.S., Özdemir M.E., On some inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula, Appl. Math. Comput., 153(2) (2004) 361-368.
  • [15] 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).
  • [16] Dragomir S.S., Agarwal R.P., 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.
  • [17] Kirmaci U.S, Inequalities for differentiable mappings and applications to special means of real numbers to midpoint formula, Appl. Math. Comput., 147 (5) (2004) 137-146.
  • [18] Sarikaya M.Z., Set E., Yaldiz H., Basak N., Hermite-Hadamard's inequalities for fractional integrals and related fractional inequalities, Math. Comput. Model., 57(9-10) (2013) 2403-2407.
  • [19] Iqbal M., Iqbal B.M., Nazeer K., Generalization of inequalities analogous to Hermite-Hadamard inequality via fractional integrals, Bull. Korean Math. Soc., 52(3) (2015) 707-716.
  • [20]Budak H., Ertugral F., Sarikaya M.Z., New generalization of Hermite-Hadamard type inequalities via generalized fractional integrals, An. Univ. Craiova Ser. Mat. Inform., 47(2) (2020) 369-386.
  • [21] Mohammed P.O., Sarikaya M.Z., Baleanu D., On the Generalized Hermite--Hadamard Inequalities via the Tempered Fractional Integrals, Symmetry, 12(4) (2020) 1-17.
  • [22] Chaudhry M.A., Zubair S.M., Generalized incomplete gamma functions with applications, J. Comput. Appl. Math., 55 (1994) 99-124.
  • [23]Li C., Deng W. , Zhao L., Well-posedness and numerical algorithm for the tempered fractional ordinary differential equations, Discret. Cont. Dyn-B, 24 (2019) 1989-2015.
  • [24]Meerschaert M.M., Sabzikar F., Chen J., Tempered fractional calculus, J. Comput. Phys., 293 (2015) 14-28.
  • [25]Buschman R. G., Decomposition of an integral operator by use of Mikusinski calculus, SIAM J. Math. Anal., 3 (1972) 83-85.
  • [26]Meerschaert M.M., Sikorskii A., Stochastic Models for Fractional Calculus, De Gruyter Studies in Mathematics Vol. 43, (2012).
  • [27] Srivastava H.M., Buschman R.G., Convolution Integral Equations with Special Function Kernels, John Wiley and Sons, New York, (1977).
  • [28]Sarikaya M.Z., Yildirim H., On Hermite-Hadamard type inequalities for Riemann-Liouville fractional integrals, Miskolc Math. Notes, 17(2) (2017) 1049-1059.

Year 2023, Volume: 44 Issue: 4, 758 - 767, 28.12.2023

Tuba Tunç , Ayşe Nur Altunok

https://doi.org/10.17776/csj.1320515

Abstract

References

  • [1] Dragomir S.S., Pearce C.E.M., Selected topics on the Hermite-Hadamard inequalities and applications, RGMIA Monographs, Victoria University.
  • [2] Hadamard J., Etude sur les proprietes des fonctions entieres et en particulier d'une fonction considree par Riemann, Journal de Math´ematiques Pures et Appliqu´ees, 58 (1893) 171-215.
  • [3] Metzler R., Klafter J., The random walk's guide to anomalous diffusion: a fractional dynamics approach, Physics Reports, 339 (2000) 1-77.
  • [4] Mohammed P.O., Brevik I., A new version of the Hermite-Hadamard inequality for Riemann--Liouville fractional integrals, Symmetry, 12(4) (2020), 1-11.
  • [5] Nonlaopon K., Awan M.U., Javed M.Z., Budak H., Noor M.A., Some q-fractional estimates of trapezoid like inequalities involving Raina’s function, Fractal and Fractional, 6(4) (2022) 1-19.
  • [6] Tomovski Z., Generalized Cauchy type problems for nonlinear fractional differential equations with composite fractional derivative operator, Nonl. Anal., 75(7) (2012) 3364-3384.
  • [7] Podlubny I., Fractional differential equations, Academic Press, San Diego, (1999).
  • [8] Samko S., Kilbas A., Marichev O., Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, London, (1993).
  • [9] Hilfer R., Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.
  • [10] Bin-Mohsin B., Awan M.U., Javed M.Z., Khan A.G., Budak H., Mihai M.V., Noor M.A., Generalized AB-fractional operator inclusions of Hermite–Hadamard’s type via fractional integration, Symmetry, 15(5) (2023) 1-21.
  • [11] Budak H., Kılınç Yıldırım S., Sarıkaya M.Z., Yıldırım H., Some parameterized Simpson-, midpoint- and trapezoid-type inequalities for generalized fractional integrals, J. Inequal. Appl., 2022(1) (2022) 1-23.
  • [12] Ertuğral F., Sarikaya M.Z., Budak H., On Hermite-Hadamard type inequalities associated with the generalized fractional integrals, Filomat, 36(12) (2022) 3983-3995.
  • [13] Jarad F., Abdeljawad T., Baleanu D., On the generalized fractional derivatives and their Caputo modification, J. Nonlinear Sci. Appl., 10(5) (2017) 2607-2619.
  • [14] Kirmaci U.S., Özdemir M.E., On some inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula, Appl. Math. Comput., 153(2) (2004) 361-368.
  • [15] 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).
  • [16] Dragomir S.S., Agarwal R.P., 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.
  • [17] Kirmaci U.S, Inequalities for differentiable mappings and applications to special means of real numbers to midpoint formula, Appl. Math. Comput., 147 (5) (2004) 137-146.
  • [18] Sarikaya M.Z., Set E., Yaldiz H., Basak N., Hermite-Hadamard's inequalities for fractional integrals and related fractional inequalities, Math. Comput. Model., 57(9-10) (2013) 2403-2407.
  • [19] Iqbal M., Iqbal B.M., Nazeer K., Generalization of inequalities analogous to Hermite-Hadamard inequality via fractional integrals, Bull. Korean Math. Soc., 52(3) (2015) 707-716.
  • [20]Budak H., Ertugral F., Sarikaya M.Z., New generalization of Hermite-Hadamard type inequalities via generalized fractional integrals, An. Univ. Craiova Ser. Mat. Inform., 47(2) (2020) 369-386.
  • [21] Mohammed P.O., Sarikaya M.Z., Baleanu D., On the Generalized Hermite--Hadamard Inequalities via the Tempered Fractional Integrals, Symmetry, 12(4) (2020) 1-17.
  • [22] Chaudhry M.A., Zubair S.M., Generalized incomplete gamma functions with applications, J. Comput. Appl. Math., 55 (1994) 99-124.
  • [23]Li C., Deng W. , Zhao L., Well-posedness and numerical algorithm for the tempered fractional ordinary differential equations, Discret. Cont. Dyn-B, 24 (2019) 1989-2015.
  • [24]Meerschaert M.M., Sabzikar F., Chen J., Tempered fractional calculus, J. Comput. Phys., 293 (2015) 14-28.
  • [25]Buschman R. G., Decomposition of an integral operator by use of Mikusinski calculus, SIAM J. Math. Anal., 3 (1972) 83-85.
  • [26]Meerschaert M.M., Sikorskii A., Stochastic Models for Fractional Calculus, De Gruyter Studies in Mathematics Vol. 43, (2012).
  • [27] Srivastava H.M., Buschman R.G., Convolution Integral Equations with Special Function Kernels, John Wiley and Sons, New York, (1977).
  • [28]Sarikaya M.Z., Yildirim H., On Hermite-Hadamard type inequalities for Riemann-Liouville fractional integrals, Miskolc Math. Notes, 17(2) (2017) 1049-1059.
There are 28 citations in total.

Details

Primary Language English
Subjects Applied Mathematics (Other)
Journal Section Natural Sciences
Authors

Tuba Tunç 0000-0002-4155-9180

Ayşe Nur Altunok 0009-0002-6116-583X

Publication Date December 28, 2023
Submission Date June 27, 2023
Acceptance Date December 1, 2023
Published in Issue Year 2023Volume: 44 Issue: 4

Cite

APA Tunç, T., & Altunok, A. N. (2023). New Midpoint-type Inequalities of Hermite-Hadamard Inequality with Tempered Fractional Integrals. Cumhuriyet Science Journal, 44(4), 758-767. https://doi.org/10.17776/csj.1320515
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