Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-27T06:50:36.983Z Has data issue: false hasContentIssue false
  • English
  • Français

EXTENSIONS OF THE HERMITE–HADAMARD INEQUALITY FOR $r$-PREINVEX FUNCTIONS ON AN INVEX SET

Part of: Inequalities

Published online by Cambridge University Press:  06 March 2017

DAH-YAN HWANG  and
SILVESTRU SEVER DRAGOMIR
DAH-YAN HWANG*
Affiliation:
Department of Information and Management, Taipei City University of Science and Technology, No. 2, Xueyuan Road, Beitou, 112, Taipei, Taiwan email dyhuang@tpcu.edu.tw
SILVESTRU SEVER DRAGOMIR
Affiliation:
Mathematics, School of Engineering and Science, Victoria University, PO Box 14428, Melbourne City, MC 8001, Australia School of Computational and Applied Mathematics, University of the Witwatersrand, Private Bag 3, Johannesburg 2050, South Africa email sever.dragomir@vu.edu.au
*
✉dyhuang@tpcu.edu.tw
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Necessary and sufficient conditions to characterise weakly $r$-preinvex functions on an invex set are obtained and used to establish generalisations of the Hermite–Hadamard inequality for such functions.

Keywords

extended meansinvex setr-convexr-preinvexHermite–Hadamard inequality

MSC classification

Primary: 26D15: Inequalities for sums, series and integrals
Secondary: 90C25: Convex programming
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 95 , Issue 3 , June 2017 , pp. 412 - 423
Copyright
© 2017 Australian Mathematical Publishing Association Inc. 

References

Antczak, T., ‘ r-preinvexity and r-invexity in mathematical programming’, Comput. Math. Appl. 50(3–4) (2005), 551–566.Google Scholar
Antczak, T., ‘A new method of solving nonlinear mathematical programming problems involving r-invex functions’, J. Math. Anal. Appl. 311(1) (2005), 313–323.Google Scholar
Antczak, T., ‘Mean value in invexity analysis’, Nonlinear Anal. 60 (2005), 1473–1484.CrossRefGoogle Scholar
Ben-Israel, A. and Mond, B., ‘What is invexity?’, J. Aust. Math. Soc. Ser. B 28 (1986), 1–9.Google Scholar
Hanson, M. A., ‘On sufficiency of the Kuhn–Tucker conditions’, J. Math. Anal. Appl. 80(2) (1981), 545–550.Google Scholar
Mitrinović, D. S., Pečarić, J. E. and Fink, A. M., Classical and New Inequalities in Analysis (Kluwer Academic, Dordrecht, 1993).Google Scholar
Mohan, S. R. and Neogy, S. K., ‘On invex sets and preinvex functions’, J. Math. Anal. Appl. 189 (1995), 901–908.Google Scholar
Noor, M. A., ‘Variational-like inequalities’, Optimization 30(4) (1994), 323–330.Google Scholar
Noor, M. A., ‘Invex equilibrium problems’, J. Math. Anal. Appl. 302(2) (2005), 463–475.CrossRefGoogle Scholar
Noor, M. A., ‘Hermite–Hadamard integral inequalities for log-preinvex functions’, J. Math. Anal. Approx. Theory 2(2) (2007), 126–131.Google Scholar
Ul-Haq, W. and Iqbal, J., ‘Hermite–Hadamard-type inequalities for r-preinvex functions’, J. Appl. Math. 2013 (2013), Article ID 126457, 5 pages.Google Scholar
Weir, T. and Mond, B., ‘Pre-invex functions in multiple objective optimization’, J. Math. Anal. Appl. 136(1) (1988), 29–38.Google Scholar
Yang, X. M., Yang, X. Q. and Teo, K. L., ‘Characterizations and applications of prequasi-invex functions’, J. Optim. Theory Appl. 110(3) (2001), 645–668.Google Scholar
Zhao, K.-Q., Long, P.-J. and Wan, X., ‘A characterization for r-preinvex function’, J. Chongqing Normal University (Natural Science) 28(2) (2011), 1–5.Google Scholar