ON SOME NEW HADAMARD TYPE INEQUALITIES FOR $(s, r)$-PREINVEX FUNCTIONS IN THE SECOND SENSE

Year 2017, Volume: 5 Issue: 1, 24 - 42, 01.04.2017

B. Meftah , K. Boukerrıoua , T. Chıheb

Abstract

In this paper the authors introduce a new class of preinvexity called $(s, r)$-preinvex functions in the second sense and establish some new Hadamard-type inequalities.

Keywords

Hermite-Hadamard inequality , preinvex functions , (s; r)- preinvex functions.

References

• 1] A.O. Akdemir and M. Tunc, On some integral inequalities for s-logarithmically convex functions and their applications, arXiv: 1212.1584v1[math.FA] 7 Dec 2012.
• [2] T. Antczak, r-preinvexity and r-invexity in mathematical programming. Comput. Math. Appl. 50 (2005), no. 3-4, 551{566.
• [3] M. Avriel, r-convex functions. Math. Programming 2 (1972), 309-323.
• [4] R. F. Bai, F. Qi, and B. Y. Xi, Hermite-Hadamard type inequalities for the m- and (alpha,m)- logarithmically convex functions. Filomat 27 (2013), no. 1, 1-7.
• [5] A. Barani, A.G. Ghazanfari and S. S. Dragomir, Hermite-Hadamard inequality for functions whose derivatives absolute values are preinvex, RGMIA Research Report Collection, 14 (2011), Article 64, 11 pp.
• [6] A. Barani, A. G. Ghazanfari and S. S. Dragomir, Hermite-Hadamard inequality through prequasi invex functions, RGMIA Research Report Collection, 14 (2011), Article 48, 7 pp.
• [7] A. Ben-Israel and B. Mond, What is invexity? J. Austral. Math. Soc. Ser. B 28 (1986), no. 1, 1-9.
• [8] W. W. Breckner, Stetigkeitsaussagen fur eine Klasse verallgemeinerter konvexer Funktionen in topologischen linearen Raumen. (German) Publ. Inst. Math. (Beograd) (N.S.) 23(37) (1978), 13-20.
• [9] S. S. Dragomir, J. E. Pecaric and L. E. Persson, Some inequalities of Hadamard type. Soochow J. Math. 21 (1995), no. 3, 335-341.
• [10] S. S. Dragomir and S. Fitzpatrik, The Hadamard's inequality for s-convex functions in the second sense, Demonstration Math. 32 (1999), no. 4, 687{696.
• [11] M. A. Hanson, On suciency of the Kuhn-Tucker conditions. J. Math. Anal. Appl. 80 (1981), no. 2, 545-550.
• [12] M. A. Latif and M. Shoaib, Hermite-Hadamard type integral inequalities for differentiable m-preinvex and (alpha;m)-preinvex functions. J. Egyptian Math. Soc. 23 (2015), no. 2, 236-241.
• [13] Li Jue-You, On Hadamard-type inequalities for s-preinvex func-tions. Journal of Chongqing Normal University: Natural Science, 27 (2010), no. 4, 5-8.
• [14] V.G. Mihesan, A generalization of the convexity, Seminar on Functional Equations, Approx. Convex, Cluj-Napoca, 1993 (Romania).
• [15] D. S. Mitrinovic, J. E. Pecaric and A. M. Fink, Classical and new inequalities in analysis. Mathematics and its Applications (East European Series), 61. Kluwer Academic Publishers Group, Dordrecht, 1993.
• [16] D. S. Mitrinovic and I. B. Lackovic, Hermite and convexity. Aequationes Math. 28 (1985), no. 3, 229{232.
• [17] S. R. Mohan and S. K. Neogy, On invex sets and preinvex functions. J. Math. Anal. Appl. 189 (1995), no. 3, 01-908.
• [18] N. P. G. Ngoc, N. V. Vinh and P. T. T. Hien, Integral inequalities of Hadamard type for r-convex functions. Int. Math. Forum 4 (2009), no. 33-36, 1723{1728.
• [19] M. A. Noor, Variational-like inequalities. Optimization 30 (1994), no. 4, 323-330.
• [20] M. A. Noor, Invex equilibrium problems. J. Math. Anal. Appl. 302 (2005), no. 2, 463-475.
• [21] M. A. Noor, On Hadamard integral inequalities involving two log-preinvex functions. JIPAM. J. Inequal. Pure Appl. Math. 8 (2007), no. 3, Article 75, 6 pp.
• [22] M. A. Noor, Hermite-Hadamard integral inequalities for log-preinvex functions. J. Math. Anal. Approx. Theory 2 (2007), no. 2, 126-131.
• 23] W. Orlicz, A note on modular spaces. I. Bull. Acad. Polon. Sci. Scr. Sci. Math. Astronom. Phys. 9 (1961) 157{162.
• [24] J. Park, On the Hermite-Hadamard-like type inequalities for co-ordinated (s; r)-convex mappings in the rst sense, Inter. J. of Pure and Applied Math.(IJPAM), 74 (2012), , No. 2, 251-263.
• [25] C. E. M. Pearce, J. Pecaric and V. Simic, Stolarsky means and Hadamard's inequality. J. Math. Anal. Appl. 220 (1998), no. 1, 99-109.
• [26] J. Pecaric, F. Proschan and Y. L. Tong, Convex functions, partial orderings, and statistical applications. Mathematics in Science and Engineering, 187. Academic Press, Inc., Boston, MA, 1992.
• [27] R. Pini, IInvexity and generalized convexity. Optimization 22 (1991), no. 4, 513-525.
• [28] F. Qi, Z. -L.Wei and Q. Yang, Generalizations and renements of Hermite-Hadamard's inequality. Rocky Mountain J. Math. 35 (2005), no. 1, 235-251.
• [29] A. Saglam, M. Z. Sarikaya and H. Yildirim, Some new inequalities of Hermite-Hadamard's type. Kyungpook Math. J. 50 (2010), no. 3, 399{410.
• [30] W. Ul-Haq, , and J. Iqbal, Hermite-Hadamard-type inequalities for $r$-preinvex functions. J. Appl. Math. 2013, Art. ID 126457, 5 pp.
• [31] Y. Wang - S. H. Wang - F. Qi, Simpson type integral inequalities in which the power of the absolute value of the rst derivative of the integrand is s-preinvex. Facta Univ. Ser. Math. Inform. 28 (2013), no. 2, 151{159.
• [32] S. Wang and X. Liu, New Hermite-Hadamard type inequalities for n-times differentiable and s-logarithmically preinvex functions. Abstr. Appl. Anal. 2014, Art. ID 725987, 11 pp.
• [33] T. Weir and B. Mond, Pre-invex functions in multiple objective optimization. J. Math. Anal. Appl. 136 (1988), no. 1, 29{38.
• [34] G. Zabandan, A. Bodaghi and A. Kilicman, The Hermite-Hadamard inequality for r-convex functions. J. Inequal. Appl. 2012, 2012:215, 8 pp.