On new general integral inequalities for h-convex function

Abstract

In this paper, we derive new estimates for the remainder term of the midpoint, trapezoid, and Simpson formulae for functions whose derivatives in absolute value at certain power are -convex and -concave by using power mean inequality, Hölder inequality and some other integral inequalities. Some applications to special means of real numbers are also given.

Keywords

• h-convex function,h-concave function,Simpson’s inequality

References

1. M.W. Alomari, M. Darus, U.S. Kirmaci, Some inequalities of Hermite-Hadamard type for s-convex functions, Acta Math. Sci. 31B (4) (2011), 1643-1652.
2. M. Bombardelli, S. Varošanec, Properties of h-convex functions related to the Hermite–Hadamard–Fejér inequalities, Comput. Math. Appl. 58 (2009), 1869-1877.
3. W.W. Breckner, Stetigkeitsaussagen für eine Klasse verallgemeinerter konvexer funktionen in topologischen linearen Räumen, Publ. Inst. Math. 23 (1978), 13-20.
4. S.S. Dragomir, J. Pečarić, L.E. Persson, Some inequalities of Hadamard type, Soochow J. Math. 21 (1995), 335-341.
5. E.K. Godunova, V.I. Levin, Neravenstva dlja funkcii širokogo klassa, soderžaščego vypuklye, monotonnye i neko-torye drugie vidy funkcii, in: Vyčislitel. Mat. i. Mat. Fiz. Mežvuzov. Sb. Nauč. Trudov, pp. 138–142, MGPI, Moskva, 1985.
6. H. Hudzik and L. Maligranda, Some remarks on s-convex functions, Aequationes Math. 48(1) (1998), 100-111.
7. İ. İşcan, A new generalization of some integral inequalities and their applications, International Journal of Engineering and Applied sciences, 3(3) (2013), 17-27.
8. İ. İşcan, E. Set and M. Emin Özdemir, On new general integral inequalities for s-convex functions, Appl. Math. Comput. 246 (2014) 306-315.

9. İ. İşcan, E. Set, M. Emin Özdemir, Some new general integral inequalities for P-functions, Malaya J. Mat. 2(4)(2014) 510–516.
10. İ. İşcan, Some new general integral inequalities for h-convex and h-concave functions, Adv. Pure Appl. Math. 5(1) (2014), 21-29. Doi: 10.1515/apam-2013-0029.
11. U.S. Kirmaci, M.K. Bakula, M.E. Özdemir, J. Pečarić, Hadamard-type inequalities for s-convex functions, Appl. Math. Comput. 193 (2007), 26-35.
12. J. Park, Hermite-Hadamard-type inequalities for real α-star s-convex mappings, J. Appl. Math. & Informatics, 28(5-6) (2010), 1507-1518.
13. M.Z. Sarikaya , A. Saglam and H. Yıldırım, On some Hadamard–type inequalities for h-convex functions, J. Math. Inequal. 2(3) (2008), 335–341.
14. M. Tunç, Ostrowski-type inequalities via h-convex functions with applications to special means, J. Inequal. Appl. 2013(326) (2013), 1-10.
15. S. Varošanec, On h-convexity, J. Math. Anal. Appl. 326 (2007), 303-311.