Some k-Fractional Integral Inequalities for p-Convex Functions
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Authors: N. MEHREEN AND M. ANWAR
DOI: 10.46793/KgJMat2401.025M
Abstract:
In this paper, we use Riemann-Liouville k-fractional and k-fractional confomable integrals to prove Hermite-Hadamard inequality, an identity and Hermite-Hadamard type inequality for p-convex functions. Some special cases are also discussed. Our work is extensions of other related previous results.
Keywords:
Hermite-Hadamard inequality, p-convex function, Riemann-Liouville k-fractional integrals, k-fractional conformable integrals.
References:
[1] R. Diaz and E. Pariguan, On hypergeometric functions and Pochhammer k-symbol, Divulgaciones matemáticas 15(2) (2007), 179–192.
[2] S. S. Dragomir and R. P. Agarwal, Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula, Appl. Math. Lett. 11 (1998), 91–95. https://doi.org/10.1016/S0893-9659(98)00086-X.
[3] S. S. Dragomir and S. Fitzpatrick, The Hadamard’s inequality for s-convex functions in the second sense, Demonstratio Math. 32 (1999), 687–696.
[4] S. S. Dragomir, J. Pečarić and L. E. Persson, Some inequalities of Hadamard type, Soochow J. Math. 21 (1995), 335–341.
[5] G. Farid, A. U. Rehman and M. Zahra, On Hadamard’s inequalities for k-fractional integrals, Nonlinear Functional Analysis and Applications 21(3) (2016), 463–478. https://nfaa.kyungnam.ac.kr/journal-nfaa
[6] J. Hadamard, Étude sur les propriétés des fonctions entières et en particulier d’une fonction considérée par Riemann, J. Math. Pures Appl. (1893), 171–215.
[7] Ch. Hermite, Sur deux limites d’une inegrale ďenie, Mathesis 3 (1883), 82.
[8] I. Iscan, Hermite-Hadamard type inequalities for p-convex functions, International Journal of Analysis and Applications 11(2) (2016), 137–145. https://www.etamaths.com
[9] I. Iscan and S. Wu, Hermite-Hadamard type inequalities for harmonically convex functions via fractional integrals, Appl. Math. Comput. 237 (2014), 237–244. https://doi.org/10.1016/j.amc.2014.04.020
[10] F. Jarad, E. Ugurlu, T. Abdeljawad and D. Baleanu, On a new class of fractional operators, Adv. Diference Equ. 2017 (2017). https://doi.org/10.1186/s13662-017-1306-z
[11] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential equations, Elsevier, Amsterdam, 2006.
[12] U. S. Kirmaci, M. K. Bakula, M. E. Özdemir and J. Pečarić, Hadamard-type inequalities for s-convex functions, Appl. Math. Comput. 193 (2007), 26–35. https://doi.org/10.1016/j.amc.2007.03.030
[13] N. Mehreen and M. Anwar, Integral inequalities for some convex functions via generalized fractional integrals, J. Inequal. Appl. 2018 (2018). https://doi.org/10.1186/s13660-018-1807-7
[14] N. Mehreen and M. Anwar, Hermite-Hadamard type inequalities via exponentially p-convex functions and exponentially s-convex functions in second sense with applications, J. Inequal. Appl. 2019 (2019). https://doi.org/10.1186/s13660-019-2047-1
[15] N. Mehreen and M. Anwar, Hermite-Hadamard type inequalities via exponentially (p,h)-convex functions, IEEE Access 8 (2020), 37589–37595. https://doi.org/10.1109/ACCESS.2020.2975628
[16] N. Mehreen and M. Anwar, On some Hermite-Hadamard type inequalities for tgs-convex functions via generalized fractional integrals, Adv. Difference Equ. 2020 (2020). https://doi.org/10.1186/s13662-019-2457-x
[17] N. Mehreen and M. Anwar, Hermite-Hadamard and Hermite-Hadamard-Fejer type inequalities for p-convex functions via conformable fractional integrals, J. Inequal. Appl. 2020 (2020). https://doi.org/10.1186/s13660-020-02363-3
[18] N. Mehreen and M. Anwar, Hermite-Hadamard and Hermite-Hadamard-Fejer type inequalities for p-convex functions via new fractional conformable integral operators, J. Math. Comput. Sci. 19 (2019), 230–240.
[19] N. Mehreen and M. Anwar, Some inequalities via ψ-Riemann-Liouville fractional integrals, AIMS Mathematics 4(5) (2019), 1403–1415. https://doi.org/10.3934/math.2019.5.1403
[20] S. Mubeen and G. M. Habibullah, k-Fractional integrals and application, International Journal of Contemporary Mathematical Sciences 7(2) (2012), 89–94.
[21] K. S. Nisar and F. Qi, On solutions of fractional kinetic equations involving the generalized k-Bessel function, Note Mat. 37(2) (2017), 11–20. https://doi.org/10.1285/i15900932v37n2p11
[22] F. Qi, S. Habib, S. Mubeen and M. N. Naeem, Generalized k-fractional conformable integrals and related inequalities, AIMS Mathematics 4(3) (2019), 343–358. https://hal.archives-ouvertes.fr/hal-01788916
[23] M. Z. Sarikaya, E. Set, H. Yaldiz and N. Başak, 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
[24] E. Set, J. Choi and A. Gözpnar, Hermite-Hadamard type inequalities for new fractional conformable integral operators, AIP Conference Proceedings, 2018.
[25] E. Set, M. Z. Sarikaya, M. E. Özdemir and H. Yaldirm, The Hermite-Hadamard’s inequality for some convex functions via fractional integrals and related results, J. Appl. Math. Stat. Inform. 10(2) (2014), 69–83. https://doi.org/10.2478/jamsi-2014-0014
[26] T. Tunç, S. Sönmezoğlu and M. Z. Sarıkaya, On integral inequalities of Hermite-Hadamard type via Green function and applications, Appl. Appl. Math. 14(1) (2019), 452–462.