On two times differentiable preinvex and prequasiinvex functions

Abstract

The main goal of this paper is to establish a new identity for functions defined on an open invex subset of real numbers. By using this identity, the Hölder integral inequality and power mean integral inequality, we introduce some new type integral inequalities for functions whose powers of second derivatives in absolute values are preinvex and prequasiinvex.

Keywords

• Invex set,preinvex and prequasiinvex,Hölder integral inequality

References

1. Antczak, T., Mean value in invexity analysis, Nonl. Anal., 60, (2005), 1473-1484. Barani, A, Ghazanfari, A.G., Dragomir, S.S., Hermite-Hadamard inequality through prequasiinvex functions, RGMIA Research Report Collection 14, Article 48, (2011), 7 pp.
2. Barani, A., Ghazanfari A.G., Dragomir, S.S., Hermite-Hadamard inequality for functions whose derivatives absolute values are preinvex, J. Inequal. Appl. (2012), 247.
3. Dragomir, S.S. and Pearce, C.E.M., Selected Topics on Hermite-Hadamard Inequalities and Applications, RGMIA Monographs, Victoria University, 2000.
4. Hadamard, J., Étude sur les propriétés des fonctions entières et en particulier d'une fonction considerée par Riemann, J. Math Pures Appl. 58, (1893), 171--215.
5. Ion, D.A., Some estimates on the Hermite-Hadamard inequality through quasi-convex functions, Annals of University of Craiova, Math. Comp. Sci. Ser, Volume 34, (2007), 82--87.
6. Israel, A.B., and Mond, B., What is invexity? J. Aust. Math. Soc. Ser. B 28(1), (1986), 1-9.
7. İşcan, İ., Set, E. and Özdemir, M.E., On new general integral inequalities for s-convex functions, Applied Mathematics and Computation 246, (2014), 306-315.
8. İşcan, İ., Ostrowski type inequalities for functions whose derivatives are preinvex, Bulletin of the Iranian Mathematical Society 40 (2), (2014), 373-386.

9. İşcan İ., Kadakal, H. and Kadakal, M., Some New Integral Inequalities for n- Times Differentiable Quasi-Convex Functions, Sigma Journal of Engineering and Natural Sciences, 35 (3), (2017) 363-368.
10. Latif, M.A. and Dragomir, S.S., Some Hermite-Hadamard type inequalities for functions whose partial derivatives in absloute value are preinvex on the co-oordinates, Facta Universitatis (NIŠ) Ser. Math. Inform, Vol. 28, No 3, (2013), 257-270.
11. Maden, S., Kadakal, H., Kadakal, M. and İşcan İ., Some new integral inequalities for n-times differentiable convex functions, J. Nonlinear Sci. Appl., 10 (12), (2017), 6141-6148.
12. Matloka, M., On some new inequalities for differentiable (h₁; h₂)-preinvex functions on the co-ordinates, Mathematics and Statistics, 2(1), (2014), 6-14.
13. Mohan, S.R., Neogy, S.K., On invex sets and preinvex functions, J. Math. Anal. Appl., 189, (1995), 901-908.
14. Noor, M.A., Hermite-Hadamard integral inequalities for log-preinvex functions, J. Math. Anal. Approx. Theory, 2, (2007), 126-131.
15. Noor, M.A., Invex equilibrium problems. J. Math. Anal. Appl., 302, (2005), 463-475.
16. Noor, M.A., Variational-like inequalities. Optimization, 30, (1994), 323-330.
17. Pečarić, J.E., Porschan, F. and Tong, Y.L., Convex Functions, Partial Orderings, and Statistical Applications, Academic Press Inc., 1992.
18. Pini, R., Invexity and generalized convexity. Optimization, 22, (1991), 513-525.
19. Weir, T., and Mond, B., Preinvex functions in multiple objective optimization, J Math Anal Appl., 136, (1998), 29-38.
20. Yang, X.M., Yang X.Q., Teo, K.L., Generalized invexity and generalized invariant monotonicity, J. Optim. Theory. Appl., 117, (2003), 607-625.
21. Yang, X.M., and Li, D., On properties of preinvex functions. J. Math. Anal. Appl., 256, (2001), 229-241.