Document Type : Research Paper


1 Department of Mathematics, Govt. Graduate College Sahiwal, Pakistan.

2 Centre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University, Multan 60800, Pakistan.


In this paper, for generalised preinvex functions, new estimates of the Fej'{e}r-Hermite-Hadamard inequality on fractional sets $\mathbb{R}^{\rho }$ are given in this study. We demonstrated a fractional  integral inequalities based on Fej'{e}r-Hermite-Hadamard theory. We establish two new local fractional integral identities for differentiable functions. We construct several novel Fej'{e}r-Hermite-Hadamard-type inequalities for generalized convex function in local fractional calculus
contexts using these integral identities. We provide a few illustrations to highlight the uses of the obtained findings. Furthermore, we have also given a few examples of new inequalities in use.


Main Subjects

[1] G. Anastassiou, A. Kashuri and R. Liko, Local fractional integrals involving generalized strongly $m$-convex mappings, Arab J. Math., 8 (2019), pp. 95-107.
[2] T. Antczak, Mean value in invexity analysis, Nonlinear Anal., 60 (2005), pp. 1473-1484.
[3] A. Barani, A.G. ghazanfari and S.S Dragomir, Hermite Hadamard inequality for functions whose derivatives absolutes values are preinvex . J. Inequal. Appl., 2012 (2012), pp. 247.
[4] S.I. Butt, M. Nadeem and G. Farid, On Caputo fractional derivatives via exponential s-convex functions, Turkish Journal of Science, 5(2)020), pp. 140-146.
[5] T. Du, T. Wang, M.A. Adil and Y. Zhang, Certain integral inequalities considering generalized $m$-convexity on fractal sets and their applications, World Scientific Publishing Co., 27(7) (2019), pp. 17.
[6] A. Ekinci and M.E. Ozdemir, Some new integral inequalities via Riemann-Liouville integral operators, Appl. Comput. Math., 18(3) (2019).
[7] S. Erden, M.Z. Sarikaya and N. Celik, Some generalized inequalities involving local fractional integrals and their applications for random variables and numerical integration, J. Appl. Math. Stat. Inform., 12(2) (2016) pp. 49-65.
[8] L. Fejér, Uber die Fourierreihen, II. Math. Naturwiss Anz. Ungar. Akad. Wiss., 24 (1906) pp. 369-390.
[9] H. Kalsoom, M.A. Ali, M. Abbas, H. Budak and G. Murtaza, Generalized quantum Montgomery identity and Ostrowski type inequalities for preinvex functions, TWMS J. Pure Appl. Math., 13(1) (2022) pp. 72-90.
[10] S. Kizil and M.A. Ardic, Inequalities for strongly convex functions via Atangana-Baleanu Integral Operators, Turkish Journal of Science, 6(2) (2021) pp. 96-109.
[11] C. Luo, H. Wang and T. Du, Fejér-Hermite-Hadamard type inequalities involving generalized $h$-convexity on fractal sets and their applications, Chaos Solitons Fractals, 131 (2020) pp. 13.
[12] H. Mo, X. Sui and D. Yu, Generalized convex functions on fractal sets and two related inequalities, Abstr. Appl. Anal., 2014 (2020) p. 7.
[13] W. Sun, Some local fractional integral inequalities for generalized preinvex functions and applications to numerical quadrature, World Scientific Publishing Co., 27(5) (2019) pp. 14.
[14] W. Sun, Generalized preinvex functions and related Hermite-Hadamard type integral inequalities on fractal space, J. Zhejiang Univ., 46(5) (2019) pp. 543-549.
[15] W. Sun and L. Qiong, New inequalities of Hermite Hadamard type for generalized convex functions on fractal sets and its applications, J. Zhejiang Univ., 44(1) (2017) pp. 47-52.
[16] T. Weir and B. Mond, Preinvex functions in multiple objective optimization, J. Math. Anal. Appl., 136(1) (1998) pp. 29-38.
[17] X.J. Yang, Advanced local fractional calculus and its applications, WorldScience, New York, 2012.
[18] U.S. Kirmaci, Inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula, Appl. Math. Comp., 147(1) (2004) pp. 137-146.