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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

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.

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

Abstract

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.
There are 34 citations in total.

Details

Subjects Engineering
Journal Section Articles
Authors

B. Meftah

K. Boukerrıoua

T. Chıheb This is me

Publication Date April 1, 2017
Submission Date February 15, 2017
Acceptance Date November 16, 2016
Published in Issue Year 2017 Volume: 5 Issue: 1

Cite

APA Meftah, B., Boukerrıoua, K., & Chıheb, T. (2017). ON SOME NEW HADAMARD TYPE INEQUALITIES FOR $(s, r)$-PREINVEX FUNCTIONS IN THE SECOND SENSE. Konuralp Journal of Mathematics, 5(1), 24-42.
AMA Meftah B, Boukerrıoua K, Chıheb T. ON SOME NEW HADAMARD TYPE INEQUALITIES FOR $(s, r)$-PREINVEX FUNCTIONS IN THE SECOND SENSE. Konuralp J. Math. April 2017;5(1):24-42.
Chicago Meftah, B., K. Boukerrıoua, and T. Chıheb. “ON SOME NEW HADAMARD TYPE INEQUALITIES FOR $(s, r)$-PREINVEX FUNCTIONS IN THE SECOND SENSE”. Konuralp Journal of Mathematics 5, no. 1 (April 2017): 24-42.
EndNote Meftah B, Boukerrıoua K, Chıheb T (April 1, 2017) ON SOME NEW HADAMARD TYPE INEQUALITIES FOR $(s, r)$-PREINVEX FUNCTIONS IN THE SECOND SENSE. Konuralp Journal of Mathematics 5 1 24–42.
IEEE B. Meftah, K. Boukerrıoua, and T. Chıheb, “ON SOME NEW HADAMARD TYPE INEQUALITIES FOR $(s, r)$-PREINVEX FUNCTIONS IN THE SECOND SENSE”, Konuralp J. Math., vol. 5, no. 1, pp. 24–42, 2017.
ISNAD Meftah, B. et al. “ON SOME NEW HADAMARD TYPE INEQUALITIES FOR $(s, r)$-PREINVEX FUNCTIONS IN THE SECOND SENSE”. Konuralp Journal of Mathematics 5/1 (April 2017), 24-42.
JAMA Meftah B, Boukerrıoua K, Chıheb T. ON SOME NEW HADAMARD TYPE INEQUALITIES FOR $(s, r)$-PREINVEX FUNCTIONS IN THE SECOND SENSE. Konuralp J. Math. 2017;5:24–42.
MLA Meftah, B. et al. “ON SOME NEW HADAMARD TYPE INEQUALITIES FOR $(s, r)$-PREINVEX FUNCTIONS IN THE SECOND SENSE”. Konuralp Journal of Mathematics, vol. 5, no. 1, 2017, pp. 24-42.
Vancouver Meftah B, Boukerrıoua K, Chıheb T. ON SOME NEW HADAMARD TYPE INEQUALITIES FOR $(s, r)$-PREINVEX FUNCTIONS IN THE SECOND SENSE. Konuralp J. Math. 2017;5(1):24-42.
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