Korean J. Math. Vol. 27 No. 4 (2019) pp.843-860
DOI: https://doi.org/10.11568/kjm.2019.27.4.843

Some new estimates for exponentially $(\hbar,\mathfrak{m})$-convex functions via extended generalized fractional integral operators

Main Article Content

Saima Rashid
Muhammad Aslam Noor
Khalida Inayat Noor

Abstract

In the article, we present several new Hermite-Hadamard and Hermite-Hadamard-Fej\'{e}r type inequalities for the exponentially $(\hbar,\mathfrak{m})$-convex functions via an extended generalized Mittag-Leffler function. As applications, some variants for certain typ e of fractional integral operators are established and some remarkable special cases of our results are also have been obtained.



Article Details

References

[1] Alirezaei. G.; Mathar. R., On exponentially concave functions and their impact in information theory, J. inform. Theory. Appl, 2018, 9, 5, 265-274. Google Scholar

[2] M. U. Awan, M. A. Noor, and K. I. Noor, Hermite-Hadamard inequalities for exponentially convex functions, Appl. Math. Inf. Sci., 2018, 12(2), 405–409. Google Scholar

[3] M. U. Awan, M. A. Noor, M. V. Mihai, K. I. Noor, and A. G. Khan, Some new bounds for Simpson’s rule involving special functions via harmonic h-convexity, J. Nonlinear Sci. Appl., 2017, 10(4), 1755–1766. Google Scholar

[4] A. G. Azpeitia, Convex functions and the Hadamard inequality, Rev. Colombiana Mat., 1994, 28(1), 7–12. Google Scholar

[5] S. N. Bernstein, Sur les fonctions absolument monotones, Acta Math. 1929, 52, 1-66. Google Scholar

[6] G. Cristescu, M. A. Noor, K. I. Noor, and M. U. Awan, Some inequalities for functions having Orlicz-convexity, Appl. Math. Comput., 2016, 273, 226–236. Google Scholar

[7] M. R. Delavar and S. S. Dragomir, On -convexity, Math. Inequal. Appl., 2017, 20(1), 203–216. Google Scholar

[8] S. S. Dragomir, On some new inequalities of Hermite-Hadamard type for m-convex functions, Tamkang J. Math., 2002, 33(1), 55–65. Google Scholar

[9] S. S. Dragomir and I. Gomm, Some Hermite-Hadamard type inequalities for functions whose exponentials are convex, Stud. Univ. Babe¸s-Bolyai Math., 2015, 60(4), 527¨C-534. Google Scholar

[10] S. S. Dragomir and Gh. Toader, Some inequalities for m-convex functions, Studia Univ. Babe¸s-Bolyai Math., 1993, 38(1), 21–28. Google Scholar

[11] C. P. Niculescu, Convexity according to the geometric mean, Math. Inequal. Appl., 2010, 3(2), 155–167. Google Scholar

[12] J. Hadamard, ´Etude sur les propri´et´es des fonctions enti`eres et en particulier `dune fonction consid´er´ee par Riemann, J. Math. Pures Appl., 1893, 58, 171–215. Google Scholar

[13] ˙I. ˙ I¸scan, Hermite-Hadamard-Fej´er type inequalities for convex functions via fractional integrals, Stud. Univ. Babe¸s-Bolyai Math., 2015, 60(3), 355–366 Google Scholar

[14] A. A. Kilbas and A. A. Koroleva, Integral transform with the extended generalized Mittag-Leffler function, Math. Model. Anal., 2006, 11(2), 173–186. Google Scholar

[15] M. A. Latif, Estimates of Hermite-Hadamard inequality for twice differentiable harmonically-convex functions with applications, Punjab Univ. J. Math. 50, No. 1 (2018) 1-13. Google Scholar

[16] C. P. Niculescu, Convexity according to means, Math. Inequal. Appl., 2003, 6(4), 571–579. Google Scholar

[17] M. A. Noor, K. I. Noor, and S. Rashid, On exponentially r-convex functions and inequalities, Preprint. Google Scholar

[18] M. E. ¨ Ozdemir, M. Avcı, and E. Set, On some inequalities of Hermite-Hadamard type via m-convexity, Appl. Math. Lett., 2010, 23(9), 1065–1070. Google Scholar

[19] S. Pal and T. K. L. Wong., On exponentially concave functions and a new information geometry, Annals. Prob, 2018, 46(2), 1070-1113. Google Scholar

[20] T. P. Prabhakar, A singular integral equation with a generalized Mittag Leffler function in the kernel, Yokohama Math. J., 1971, 19, 7–15. Google Scholar

[21] G. Rahman, D. Baleanu, M. Al Qurashi, S. D. Purohit, S. Mubeen, and M. Arshad, The extended Mittag-Leffler function via fractional calculus, J. Nonlinear Sci. Appl., 2017, 10(8), 4244–4253. Google Scholar

[22] S. Rashid, M. A. Noor, and K. I. Noor, Fractional exponentially m-convex functions and inequalities, Inter. J. Anal. Appl. 17(3)(2019), 464-478. Google Scholar

[23] T. O. Salim and A. W. Faraj, A generalization of Mittag-Leffler function and integral operator associated with fractional calculus, J. Fract. Calc. Appl., 2012, 3(5), 1–13. Google Scholar

[24] E. Set, A. O. Akdemir, and I. Mumcu, Hermite-Hadamard’s inequality and its extensions for conformable fractional integrals of any order > 0 (submitted)(2019). Google Scholar

[25] H. M. Srivastava and ˇZ. Tomovski, Fractional calculus with an integral operator containing a generalized Mittag-Leffler function in the kernel, Appl. Math. Comput., 2009, 211(1), 198–210. Google Scholar

[26] Gh. Toader, Some generalizations of the convexity, in: Proceedings of the colloquium on approximation and optimization (Cluj-Napoca, 1985), 329–338, Univ. Cluj-Napoca, Cluj-Napoca, 1985. Google Scholar

[27] M. Tomer, E. Set and M. Z. Sarikaya, Hermite Hadamard type Reimann-Liouville fractional integral inequalities for convex funtions, AIP Conference Proceedings 1726,020035 (2016). Google Scholar

[28] S. Varoˇsanec, On h-convexity, J. Math. Anal. Appl., 2007, 326(1), 303-311. Google Scholar