On Hermite-Hadamard type integral inequalities for functions whose second derivative are nonconvex

Mehmet Zeki SARIKAYA; Hakan Bozkurt; Mehmet Eyüp  KIRIS

doi:10.26637/mjm203/016

Abstract

In this paper, we extend some estimates of the right hand side of a Hermite- Hadamard type inequality for nonconvex functions whose second derivatives absolute values are \(\varphi\)-convex, \(\log -\varphi\)-convex, and quasi-\(\varphi\) convex.

Keywords:

Hermite-Hadamard’s inequalities, ϕ-convex functions, log-ϕ-convex, quasi-ϕ-convex, Hölder’s inequality

Mathematics Subject Classification:

26D07, 26D10, 26D99

Authors Imprint References Funding Related Articles Downloads

Mehmet Zeki SARIKAYA Department of Mathematics, Faculty of Science and Arts, Düzce University, Düzce-TURKEY.
Hakan Bozkurt Department of Mathematics, Faculty of Science and Arts, Düzce University, Düzce-TURKEY.
Mehmet Eyüp KIRIS Department of Mathematics, Faculty of Science and Arts, Afyon Kocatepe University, Afyon-TURKEY.

Pages: 293-300
Date Published: 01-07-2014
Vol. 2 No. 03 (2014): Malaya Journal of Matematik (MJM)

M. Alomari, M. Darus and S. S. Dragomir, New inequalities of Hermite-Hadamard’s type for functions whose second derivatives absolute values are quasiconvex, Tamk. J. Math., 41(2010), 353-359. DOI: https://doi.org/10.5556/j.tkjm.41.2010.498

S.S.Dragomir andR.P.Agarwal, Two inequalities for differentiable mappings and applications to special means of real numbers and trapezoidal formula, Appl. Math. Lett., 11(5)(1998), 91–95. DOI: https://doi.org/10.1016/S0893-9659(98)00086-X

S. S. Dragomir and C. E. M. Pearce, Selected Topics on Hermite-Hadamard Inequalities and Applications, RGMIA Monographs, Victoria University, 2000.

S. Hussain, M. I. Bhatti and M. Iqbal, Hadamard-type inequalities for s-convex functions I, Punjab Univ. Jour. of Math., 41(2009), 51-60.

M. A. Noor, Some new classes of nonconvex functions, Nonl. Funct. Anal. Appl., 11(2006), 165-171.

M. A. Noor, On Hadamard integral inequalities involving two log-preinvex functions, J. Inequal. Pure Appl. Math., 8(3)(2007), 1-6.

M. A. Noor, Hermite-Hadamard integral inequalities for log-ϕ-convex functions, Nonlinear Analysis Forum, 13(2)(2008), 119–124.

M. A. Noor, On a class of general variotional inequalities, J. Adv. Math. Studies, 1(2008), 31-42.

K. I. Noor and M. A. Noor, Relaxed strongly nonconvex functions, Appl. Math. E-Notes, 6(2006), 259-267.

D.A. Ion, Some estimates on the Hermite-Hadamard inequality through quasi-convex functions, Annals of University of Craiova Math. Comp. Sci. Ser., 34(2007), 82-87.

C. E. M. Pearce and J. Peˇ cari´ c, Inequalities for differentiable mappings with application to special means and quadrature formulae, Appl. Math. Lett., 13(2)(2000), 51–55. DOI: https://doi.org/10.1016/S0893-9659(99)00164-0

J. Peˇ cari´ c, F. Proschan and Y. L. Tong, Convex functions, partial ordering and statistical applications, Academic Press, New York, 1991.

A. Saglam, M. Z. Sarikaya and H. Yildirim, Some new inequalities of Hermite-Hadamard’s type, Kyungpook Mathematical Journal, 50(2010), 399-410. DOI: https://doi.org/10.5666/KMJ.2010.50.3.399

M. Z. Sarikaya, A. Saglam and H. Yıldırım, New inequalities of Hermite-Hadamard type for functions whose second derivatives absolute values are convex and quasi-convex, International Journal of Open Problems in Computer Science and Mathematics ( IJOPCM), 5(3)(2012). DOI: https://doi.org/10.12816/0006114

M. Z. Sarikaya, A. Saglam and H. Yıldırım, On some Hadamard-type inequalities for h-convex functions, Journal of Mathematical Inequalities, 2(3)(2008), 335-341. DOI: https://doi.org/10.7153/jmi-02-30

M. Z. Sarikaya, M. Avci and H. Kavurmaci, On some inequalities of Hermite-Hadamard type for convex functions, ICMS Iternational Conference on Mathematical Science. AIP Conference Proceedings 1309, 852 (2010). DOI: https://doi.org/10.1063/1.3525218

M. Z. Sarikaya and N. Aktan, On the generalization some integral inequalities and their applications Mathematical and Computer Modelling, 54(9-10)(2011), 2175-2182. DOI: https://doi.org/10.1016/j.mcm.2011.05.026

M. Z. Sarikaya, E. Set and M. E. Ozdemir, On some new inequalities of Hadamard type involving h-convex functions, Acta Mathematica Universitatis Comenianae, Vol. LXXIX, 2(2010), 265-272.

M. Z. Sarikaya, H. Bozkurt and N. Alp, On Hadamard Type Integral Inequalities for nonconvex Functions, Mathematical Sciences And Applications E-Notes, in press, arXiv:1203.2282v1.

M. Z. Sarikaya, N. Alp and H. Bozkurt, On Hermite-Hadamard Type Integral Inequalities for preinvex and log-preinvex functions, Contemporary Analysis and Applied Mathematics, 1(2)(2013), 237-252. DOI: https://doi.org/10.1155/2013/126457