HERMITE-HADAMARD'S INEQUALITIES FOR PREQUASIINVEX FUNCTIONS VIA FRACTIONAL INTEGRALS

Abstract

In this paper, we extend some estimates of the right hand side ofHermite-Hadamard type inequality for prequasiinvex functions via fractionalintegrals

Keywords

• Hermite-Hadamard type inequalities,prequasiinvex function,fractional integral

References

1. T. Antczak, Mean value in invexity analysis, Nonlinear Analysis, 60 (2005) 1471-1484.
2. M. Alomari, M. Darus and U.S. Kırmacı, Refinements of Hadamard-type inequalities for quasi-convex functions with applications to trapezoidal formula and to special means, Comp.and Math. with Applications, 59 (2010), 225-232.
3. M.K. Bakula, M.E. Ozdemir and J. Peˇcari´c, Hadamard type inequalities for m-convex and (α, m)-convex functions, J. Inequal. Pure Appl. Math. 9 (2008) Article 96. [Online: http://jipam.vu.edu.au].
4. A. Barani, A.G. Ghazanfari and S.S. Dragomir, Hermite-Hadamard inequality through pre- quasiinvex functions, RGMIA Res. Rep. Coll., 14 (2011), Article 48.
5. S.S. Dragomir and C.E.M. Pearce, Selected Topics on Hermite-Hadamard Inequalities and Applications, RGMIA Monographs,Victoria University, 2000.
6. M.A. Hanson, On sufficiency of the Kuhn-Tucker conditions, J. Math. Anal. Appl., 80 (1981) 545-550.
7. D.A. Ion, Some estimates on the Hermite-Hadamard inequalities through quasi-convex func- tions, Annals of University of Craiova, Math. Comp. Sci. Ser., 34 (2007), 82-87.
8. I. Iscan, Hermite-Hadamard’s inequalities for preinvex functions via fractional integrals and related fractional inequalities, arXiv:1204.0272, submitted.

9. S.R.Mohan and S.K. Neogy, On invex sets and preinvex functions, J. Math. Anal. Appl., 189 (1995), 901-908. [10] M. Aslam Noor, Some new classes of nonconvex functionss, Nonl. Funct. Anal. Appl., 11 (2006), 165-171. [11] M. Aslam Noor, On Hadamard integral inequalities invoving two log-preinvex functions, J. Inequal. Pure Appl. Math., 8 (2007), No. 3, 1-6, Article 75.
10. M. Aslam Noor, Hadamard integral inequalities for product of two preinvex function, Nonl. anal. Forum, 14 (2009), 167-173.
11. M.E. ¨Ozdemir and C¸ . Yıldız, The Hadamard’s inequality for quasi-convex functions via frac- tional integrals, RGMIA Res. Rep. Coll., 14 (2011), Article 101.
12. J. Park, Simpson-like and Hermite-Hadamard-like type integral inequalities for twice differ- entiable preinvex functions, Int. Journal of Pure and Appl. Math.,79 (4) (2012), 623-640.
13. R. Pini, Invexity and generalized Convexity, Optimization, 22 (1991) 513-525.
14. M.Z. Sarıkaya and H. Ogunmez, On new inequalities via Riemann-Liouville fractional integration, Abstract and Applied Analysis, 2012 (2012), Article ID 428983, 10 pages, doi:10.1155/2012/428983.
15. M.Z. Sarıkaya, E. Set and M.E. ¨Ozdemir, On some new inequalities of Hadamard type involv- ing h-convex functions, Acta Nath. Univ. Comenianae vol. LXXIX, 2 (2010), pp. 265-272.
16. M.Z. Sarıkaya, E. Set, H. Yaldız and N. Ba¸sak, Hermite-Hadamard’s inequalities for frac- tional integrals and related fractional inequalities, Mathematical and Computer Modelling, DOI:10.1016/j.mcm.2011.12.048.
17. E. Set, New inequalities of Ostrowski type for mapping whose derivatives are s-convex in the second sense via fractional integrals, Computers and Math. with Appl., 63 (2012) 1147-1154. [20] T. Weir, and B. Mond, Preinvex functions in multiple objective optimization, Journal of Mathematical Analysis and Applications, 136, (1998) 29-38.
18. X.M. Yang and D. Li, On properties of preinvex functions, J. Math. Anal. Appl. 256 (2001) 229-241.
19. X.M. Yang, X.Q. Yang and K.L. Teo, Characterizations and applications of prequasiinvex functions, properties of preinvex functions, J. Optim. Theo. Appl., 110 (2001) 645-668.
20. Giresun University, Science and Art Faculty, Department of Mathematics, Giresun- TURKEY
21. E-mail address: [e-mail]