On New Fractional Hermite-Hadamard Type Inequalities for $(\alpha^{*},m)$-Convex Functions

Abstract

The aim of the present paper is to investigate some new Hermite-Hadamard type integral inequalities for $(\alpha^{*},m)$-convex functions via Riemann-Liouville fractional integrals.

Keywords

• Fractional Hermite-Hadamard inequalities,$m$-convex function

References

1. [1] R. Bai, F. Qi, B. Xi, Hermite-Hadamard type inequalities for the m- and (a;m)-logarithmically convex functions, Filomat, 27 (2013), 1-7.
2. [2] M.K. Bakula, M.E. Özdemir and J. Pecaric, Hadamard type inequalities for m-convex and (a;m)-convex functions, J.Ineq. Pure Appl. Math., 9(2008), Article 96, [ONL˙INE:http://jipam.vu.edu.au].
3. [3] S. Belarbi, Z. Dahmani, On some new fractional integral inequalities, J. Ineq. Pure Appl. Math., 10(3) (2009), Art.86.
4. [4] G. Cristescu, L. Lupsa, Non-connected Convexities abd Applications, Kluwer Academic Publishers, Dordrecht, Holland, (2002).
5. [5] Z. Dahmani, New inequalities in fractional integrals, Int. J. Nonlinear Sci., 9(4) (2010),493-497.
6. [6] Z. Gao, M. Li, J. Wang, On some fractional Hermite-Hadamard inequalities via s-convex and s-Godunova-Levin functions and their applications, Bol. Soc. Mat. Mex., DOI 10.1007/s40590-016-0087-9.
7. [7] E.K. Godunova and V.I. Levin, Neravenstva dlja funckcii sirokogo klassa, soderzascego vypuklye, monotonnye inekotorye drugie vidy funkii, Vycislitel. Mat. i. Fiz. Mezvuzov. Sb. Nauc. Trudov, MGPI, Moskova., (1985),138-142.
8. [8] I. Iscan, Hermite-Hadamard type inequalities for harmonically convex functions, Hacettepe J. Math. Statis., 43(6) (2014),935-942.

9. [9] A. A. Kilbas, M. H. Srivastava , J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science B. V, Amsterdam(2006).
10. [10] V.G. Mihes¸an, A generalization of the convexity, Seminar on Functional Equations, Approx. and Convex., Cluj-Napoca, (Romania) (1993).
11. [11] S. Miller, B. Ross, An introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley. Soons. USA., (1993),2.
12. [12] M. A. Noor, K.I. Noor, M.U. Awan, Geometrically relative convex functions, Appl. Math. Infor. Sci., 8(2) (2014),607-616.
13. [13] M. E. Özdemir, H. Kavurmacı, E. Set, Ostrowski’s type inequalities for (a;m)-convexity functions, Kyungpook Math. J., 50 (2010), 371-378.
14. [14] M. E. Özdemir, E. Set, M.Z. Sarıkaya, Some new Hadamard type inequalities for co-ortinated m-convex and (a;m)-convex functions, Hacettepe Journal of Mathematics and Statistics 40 (2) (2011), 219 – 229.
15. [15] M. Z. Sarıkaya, E. Set, H. Yaldız, N. Basak, Hermite-Hadamard’s inequalities for fractional integrals and related fractional inequalities, Math. Comput. Model., 57 (2013),2403-2407.
16. [16] E. Set, M. E. Özdemir, M.Z. Sarıkaya, Inequalities of Hermite-Hadamard’s type for functions whose second derivatives absolute values are m-convex, AIP Conference Proceedings, 1309(1) (2010),861-873.
17. [17] G. Toader, Some generalizations of the convexity, Proceedings of The Colloquium On Appraximation and Optimization, Univ. Cluj-Napoca, (1984),329- 338.
18. [18] S. Varosanec, On h-convexity , J. Math. Anal. Appl., 326 (2007),303-311.
19. [19] J. Wang, J. Deng, M. Feˇckan, Hermite-Hadamard type inequalities for r-convex functions via Riemann-Liouville fractional integrals, Ukr. Math. J., 65 (2013),193-211.
20. [20] J. Wang, X. Li, M. Feckan, Y. Zhou, Hermite-Hadamard type inequalities for Riemann-Liouville fractional integrals via two kinds of convexity Appl. Anal. Int. J., 92 (2013),2241-2253.