Fractional trapezium type inequalities for twice differentiable preinvex functions and their applications

Artion Kashuri, Rozana Liko


Trapezoidal inequalities for functions of divers natures are useful in numerical computations. The authors have proved an identity for a generalized integral operator via twice differentiable preinvex function. By applying the established identity, the generalized trapezoidal type integral inequalities have been discovered. It is pointed out that the results of this research provide integral inequalities for almost all fractional integrals discovered in recent past decades. Various special cases have been identified. Some applications of presented results to special means have been analyzed. The ideas and techniques of this paper may stimulate further research.


Trapezium type integral inequalities; preinvexity; general fractional integrals.

Full Text:



Aslani, S.M., Delavar, M.R. and Vaezpour, S.M., Inequalities of Fejer type related to generalized convex functions with applications, Int. J. Anal. Appl., 16(1) (2018), pp. 38-49.

Chen, F.X. and Wu, S.H., Several complementary inequalities to inequalities of Hermite-Hadamard type for s-convex functions, J. Nonlinear Sci. Appl., 9(2) (2016), pp. 705-716.

Chu, Y.M., Khan, M.A., Khan, T.U. and Ali, T., Generalizations of Hermite-Hadamard type inequalities for MT-convex functions, J. Nonlinear Sci. Appl., 9(5) (2016), pp. 4305-4316.

Dahmani, Z., On Minkowski and Hermite-Hadamard integral inequalities via fractional integration, Ann. Funct. Anal., 1(1) (2010), pp. 51-58.

Delavar, M.R. and Dragomir, S.S., On eta-convexity, Math. Inequal. Appl., 20 (2017), pp. 203-216.

Delavar, M.R. and De La Sen, M. Some generalizations of Hermite-Hadamard type inequalities, Springer-Plus, 5(1661) (2016).

Dragomir, S.S. and Agarwal, R.P., Two inequalities for differentiable mappings and applications to special means of real numbers and trapezoidal formula, Appl. Math. Lett., 11(5) (1998), pp. 91-95.

Khan, M.A., Chu, Y.M., Kashuri, A., Liko, R. and Ali, G., New Hermite-Hadamard inequalities for conformable fractional integrals, J. Funct. Spaces, (2018), Article ID 6928130, pp. 9.

Khan, M.A., Khurshid, Y. and Ali, T., Hermite-Hadamard inequality for fractional integrals via eta-convex functions, Acta Math. Univ. Comenianae, 79(1) (2017), pp. 153-164.

Liu, W.J., Some Simpson type inequalities for h-convex and (alpha,m)-convex functions, J. Comput. Anal. Appl., 16(5) (2014), pp. 1005-1012.

Liu, W., Wen, W. and Park, J., Hermite-Hadamard type inequalities for MT-convex functions via classical integrals and fractional integrals, J. Nonlinear Sci. Appl., 9 (2016), pp. 766-777.

Mihai, M.V., Some Hermite-Hadamard type inequalities via Riemann-Liouville fractional calculus, Tamkang J. Math, 44(4) (2013), pp. 411-416.

Mubeen, S. and Habibullah, G.M., k-Fractional integrals and applications, Int. J. Contemp. Math. Sci., 7 (2012), pp. 89-94.

Noor, M.A., Noor, K.I., Awan, M.U. and Khan, S., Hermite-Hadamard inequalities for s-Godunova-Levin preinvex functions, J. Adv. Math. Stud., 7(2) (2014), pp. 12-19.

Omotoyinbo, O. and Mogbodemu, A., Some new Hermite-Hadamard integral inequalities for convex functions, Int. J. Sci. Innovation Tech., 1(1) (2014), pp. 1-12.

Ozdemir, M.E., Dragomir, S.S. and Yildiz, C., The Hadamard's inequality for convex function via fractional integrals, Acta Mathematica Scientia, 33(5) (2013), pp. 153-164.

Sarikaya, M.Z. and Yildirim, H., On generalization of the Riesz potential, Indian Jour. of Math. and Mathematical Sci., 3(2), (2007), pp. 231-235.

Set, E., Noor, M.A., Awan, M.U. and Gozpinar, A., Generalized Hermite-Hadamard type inequalities involving fractional integral operators, J. Inequal. Appl., 169 (2017), pp. 1-10.

Wang, H., Du, T.S. and Zhang, Y., k-fractional integral trapezium-like inequalities through (h,m)-convex

and (alpha,m)-convex mappings, J. Inequal. Appl., 2017(311) (2017), pp. 20.

Weir, T. and Mond, B., Preinvex functions in multiple objective optimization, J. Math. Anal. Appl., 136 (1988), pp. 29-38.

Zhang, X.M., Chu, Y.M. and Zhang, X.H., The Hermite-Hadamard type inequality of GA-convex functions and its applications, J. Inequal. Appl., (2010), Article ID 507560, pp. 11.

Zhang, Y., Du, T.S., Wang, H., Shen, Y.J. and Kashuri, A., Extensions of different type parameterized inequalities for generalized (m,h)-preinvex mappings via k-fractional integrals, J. Inequal. Appl., 2018(49) (2018), pp. 30.



  • There are currently no refbacks.

Copyright (c) 2020 Artion Kashuri, Rozana Liko

Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.


   ithe_170     crossref_284         ind_131_43_x_117_117  Scopus  EBSCO_Host    ULAKBIM     ZBMATH more...