Fractional Hadamard and Fejer-Hadamard inequalities for exponentially m-convex function

Sajid Mehmood, Ghulam Farid

Abstract


Fractional integral operators play a vital role in the advancement
of mathematical inequalities. The aim of this paper is to present the Hadamard
and the Fejer-Hadamard integral inequalities for generalized fractional inte-
gral operators containing Mittag-Leffler function. Exponentially m-convexity
is utilized to establish these inequalities. By fixing parameters involved in the
Mittag-Leffler function Hadamard and the Fejer-Hadamard integral inequali-
ties for various well known fractional integral operators can be obtained.


Keywords


Convex functions, Exponentially $m$-convex functions, Hadamard inequality, Fej\'{e}r-Hadamard inequality, Fractional integral operators, Mittag-Leffler function

Full Text:

PDF

References


G. Abbas, G. Farid, Hadamard and Fejer-Hadamard type inequalities for harmonically convex functions via generalized fractional integrals, J. Anal., 25(1) (2017), 107-119.

M. Andri´ c, G. Farid and J. Peˇ cari´ c, A further extension of Mittag-Leffler function, Fract. Calc. Appl. Anal., 21(5) (2018), 1377-1395.

T. Antczak, (p,r)-invex sets and functions, J. Math. Anal. Appl., 263 (2001), 355-379.

F. Chen, On Hermite-Hadamard type inequalities for Riemann-Liouville fractional integrals via two kinds of convexity, Chin. J. Math., 2014 (2014), pp 7.

H. Chen and U. N. Katugampola, Hermite-Hadamard-Fej´ er type inequalities for generalized fractional integrals, J. Math. Anal. Appl., 446 (2017), 1274-1291.

S. S. Dragomir and I. Gomm, Some Hermite-Hadamard type inequalities for functions whose exponentials are convex, Stud. Univ. Babes-Bolyai Math., 60(4)(2015), 527-534.

G. Farid, Hadamard and Fej´ er-Hadamard inequalities for generalized fractional integral involving special functions, Konuralp J. Math., 4(1) (2016), 108-113.

G. Farid, A Treatment of the Hadamard inequality due to m-convexity via generalized fractional integral, J. Fract. Calc. Appl., 9(1) (2018), 8-14.

G. Farid, A. U. Rehman and B. Tariq, On Hadamard-type inequalities for m−convex functions via Riemann-Liouville fractional integrals, Studia Univ. Babes-Bolyai, Math., 62(2) (2017), 141-150.

G. Farid, A. U. Rehman, S. Mehmood, Hadamard and Fej´ er-Hadamard type integral inequalities for harmonically convex functions via an extended generalized Mittag-Leffler function, J.Math. Comput. Sci., 8(5) (2018), 630-643.

G. Farid, K. A. Khan, N. Latif, A. U. Rehman and S. Mehmood, General fractional integral inequalities for convex and m-convex functions via an extended generalized Mittag-Leffler function, J. Inequal. Appl., 2018 (2018), 243 pp.

L. Fejer, Uberdie Fourierreihen II, Math Naturwiss Anz Ungar Akad Wiss, 24 (1906), 369-390.

S. M. Kang, G. Farid, W. Nazeer and S. Mehmood, (h,m)-convex functions and associated fractional Hadamard and Fej´ er-Hadamard inequalities via an extended generalized Mittag-Leffler function, J. Inequal. Appl., 2019 (2019), 78 pp.

G. M. Mittag-Leffler, Sur la nouvelle fonction E σ (x), Comptes Rendus Hebdomadaires des S´ eances de l’ Acad´ emie des Sciences Paris, 137 (1903), 554-558.

T. R. Prabhakar, A singular integral equation with a generalized Mittag-Leffler function in the kernel, Yokohama Math. J., 19 (1971), 7-15.

G. Rahman, D. Baleanu, M. A. Qurashi, S. D. Purohit, S. Mubeen and M. Arshad, The extended Mittag-Leffler function via fractional calculus, J. Nonlinear Sci. Appl., 10 (2017), 4244-4253.

S. Rashid, M. A. Noor and K. I. Noor, Fractional exponentially m-convex functions and inequalities, Int. J. Anal. Appl., 17(3) (2019), 464-478.

T. O. Salim and A. W. Faraj, A generalization of Mittag-Leffler function and integral operator associated with integral calculus, J. Frac. Calc. Appl., 3(5) (2012), 1-13.

M. Z. Sarikaya, E. Set, H. Yaldiz and N. Basak, Hermite- Hadamard inequalities for fractional integrals and related fractional inequalities, J. Math. Comput. Model, 57(9) (2013), 2403-2407.

A. K. Shukla and J. C. Prajapati, On a generalization of Mittag-Leffler function and its properties, J. Math. Anal. Appl., 336 (2007), 797-811.

H. M. Srivastava and Z. Tomovski, Fractional calculus with an integral operator containing generalized Mittag-Leffler function in the kernal, Appl. Math. Comput., 211(1) (2009), 198-210.

M. Z. Sarikaya and H. Yildirim, On Hermite-Hadamard type inequalities for Riemann-Liouville fractional integrals, Miskolc Math. Notes, 17(2) (2016), 1049-1059.




DOI: http://dx.doi.org/10.24193/subbmath.2021.4.03

Refbacks

  • There are currently no refbacks.