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ON HADAMARD TYPE INTEGRAL INEQUALITIES FOR NONCONVEX FUNCTIONS

Year 2013, Volume: 1 Issue: 2, 217 - 229, 01.12.2013

Abstract

In this paper, we extend some estimates of the right and left handside of a Hermite- Hadamard type inequality for nonconvex functions whosederivatives absolute values are ϕ-convex and quasi-ϕ-convex was introducedby Noor in [3]

References

  • S.S. Dragomir and R.P. Agarwal, Two inequalities for differentiable mappings and applica- tions to special means of real numbers and trapezoidal formula, Appl. Math. Lett., 11(5) (1998), 91–95.
  • S. S. Dragomir and C. E. M. Pearce, Selected Topics on Hermite-Hadamard Inequalities and Applications, RGMIA Monographs, Victoria University, 2000. [3] M. Aslam Noor, Some new classes of nonconvex
  • functions, Nonl.Funct.Anal.Appl.,11(2006),165-171
  • M. Aslam Noor, On Hadamard integral inequalities involving two log-preinvex functions, J. Inequal. Pure Appl. Math., 8(2007), No. 3, 1-6, Article 75.
  • M. Aslam Noor, Hermite-Hadamard integral inequalities for log-ϕ − convex functions, Nonl. Anal. Forum, (2009). [6] M. Aslam Noor, On a class of general variotional inequalities, J. Adv. Math. Studies, 1(2008), 31-42.
  • K. Inayat Noor and M. Aslam Noor, Relaxed strongly nonconvex functions, Appl. Math. E-Notes, 6(2006), 259-267.
  • U.S. Kırmacı, Inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula, Appl. Math. Comp., 147 (2004), 137-146.
  • U.S. Kırmacı and M.E. ¨Ozdemir, On some inequalities for differentiable mappings and ap- plications to special means of real numbers and to midpoint formula, Appl. Math. Comp., 153 (2004), 361-368. [10] U.S. Kırmacı, Improvement and further generalization of inequalities for differentiable map- pings and applications, Computers and Math. with Appl., 55 (2008), 485-493.
  • D.A. Ion, Some estimates on the Hermite-Hadamard inequality through quasi-convex func- tions, Annals of University of Craiova Math. Comp. Sci. Ser., 34 (2007) 82-87.
  • C.E.M. Pearce and J. Peˇcari´c, Inequalities for differentiable mappings with application to special means and quadrature formulae, Appl. Math. Lett., 13(2) (2000), 51–55.
  • J. Peˇcari´c, F. Proschan and Y.L. Tong, Convex functions, partial ordering and statistical applications, Academic Press, New York, 1991.
  • 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.
  • M. Z. Sarikaya, A. Saglam and H. Yıldırım, On some Hadamard-type inequalities for h-convex functions, Journal of Mathematical Inequalities, Volume 2, Number 3 (2008), 335-341.
  • 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 Confer- ence Proceedings 1309, 852 (2010).
  • M. Z. Sarikaya and N. Aktan, On the generalization some integral inequalities and their applications Mathematical and Computer Modelling, Volume 54, Issues 9-10, November 2011, Pages 2175-2182.
  • 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), pp. 265-272. [19] A. Saglam, M. Z. Sarikaya and H. Yildirim, Some new inequalities of Hermite-Hadamard’s type, Kyungpook Mathematical Journal, 50(2010), 399-410.
  • C¸ . Yıldız, M. G¨urb¨uz and A. O. Akdemir, The Hadamard type inequalities for m-convex functions, Konuralp Journal of Math. Volume 1 No. 1 pp. 40–47 (2013).
  • Department of Mathematics, Faculty of Science and Arts, D¨uzce University, D¨uzce- TURKEY
  • E-mail address: sarikayamz@gmail.com, insedi@yahoo.com and placenn@gmail.com

Year 2013, Volume: 1 Issue: 2, 217 - 229, 01.12.2013

Abstract

References

  • S.S. Dragomir and R.P. Agarwal, Two inequalities for differentiable mappings and applica- tions to special means of real numbers and trapezoidal formula, Appl. Math. Lett., 11(5) (1998), 91–95.
  • S. S. Dragomir and C. E. M. Pearce, Selected Topics on Hermite-Hadamard Inequalities and Applications, RGMIA Monographs, Victoria University, 2000. [3] M. Aslam Noor, Some new classes of nonconvex
  • functions, Nonl.Funct.Anal.Appl.,11(2006),165-171
  • M. Aslam Noor, On Hadamard integral inequalities involving two log-preinvex functions, J. Inequal. Pure Appl. Math., 8(2007), No. 3, 1-6, Article 75.
  • M. Aslam Noor, Hermite-Hadamard integral inequalities for log-ϕ − convex functions, Nonl. Anal. Forum, (2009). [6] M. Aslam Noor, On a class of general variotional inequalities, J. Adv. Math. Studies, 1(2008), 31-42.
  • K. Inayat Noor and M. Aslam Noor, Relaxed strongly nonconvex functions, Appl. Math. E-Notes, 6(2006), 259-267.
  • U.S. Kırmacı, Inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula, Appl. Math. Comp., 147 (2004), 137-146.
  • U.S. Kırmacı and M.E. ¨Ozdemir, On some inequalities for differentiable mappings and ap- plications to special means of real numbers and to midpoint formula, Appl. Math. Comp., 153 (2004), 361-368. [10] U.S. Kırmacı, Improvement and further generalization of inequalities for differentiable map- pings and applications, Computers and Math. with Appl., 55 (2008), 485-493.
  • D.A. Ion, Some estimates on the Hermite-Hadamard inequality through quasi-convex func- tions, Annals of University of Craiova Math. Comp. Sci. Ser., 34 (2007) 82-87.
  • C.E.M. Pearce and J. Peˇcari´c, Inequalities for differentiable mappings with application to special means and quadrature formulae, Appl. Math. Lett., 13(2) (2000), 51–55.
  • J. Peˇcari´c, F. Proschan and Y.L. Tong, Convex functions, partial ordering and statistical applications, Academic Press, New York, 1991.
  • 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.
  • M. Z. Sarikaya, A. Saglam and H. Yıldırım, On some Hadamard-type inequalities for h-convex functions, Journal of Mathematical Inequalities, Volume 2, Number 3 (2008), 335-341.
  • 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 Confer- ence Proceedings 1309, 852 (2010).
  • M. Z. Sarikaya and N. Aktan, On the generalization some integral inequalities and their applications Mathematical and Computer Modelling, Volume 54, Issues 9-10, November 2011, Pages 2175-2182.
  • 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), pp. 265-272. [19] A. Saglam, M. Z. Sarikaya and H. Yildirim, Some new inequalities of Hermite-Hadamard’s type, Kyungpook Mathematical Journal, 50(2010), 399-410.
  • C¸ . Yıldız, M. G¨urb¨uz and A. O. Akdemir, The Hadamard type inequalities for m-convex functions, Konuralp Journal of Math. Volume 1 No. 1 pp. 40–47 (2013).
  • Department of Mathematics, Faculty of Science and Arts, D¨uzce University, D¨uzce- TURKEY
  • E-mail address: sarikayamz@gmail.com, insedi@yahoo.com and placenn@gmail.com
There are 19 citations in total.

Details

Primary Language English
Journal Section Articles
Authors

Mehmet Zeki Sarıkaya This is me

Hakanbozkurt This is me

Necmettinalp This is me

Publication Date December 1, 2013
Submission Date March 9, 2015
Published in Issue Year 2013 Volume: 1 Issue: 2

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