SOME INEQUALITIES FOR DIFFERENTIABLE PREQUASIINVEX FUNCTIONS WITH APPLICATIONS

Year 2013, Volume: 1 Issue: 2, 17 - 29, 01.12.2013

Muhammed Amer Latıf

Abstract

In this paper, we present several inequalities of Hermite-Hadamardtype for differentiable prequasiinvex functions.Our results generalize those results proved in [2] and hence generalize those given in [7], [11] and [23]. Applications of the obtained results are given as well

Keywords

Hermite-Hadamard’s inequality , invex set , preinvex function , prequasiinvex , H¨older’s integral inequality

References

• T. Antczak, Mean value in invexity analysis, Nonl. Anal., 60 (2005), 1473-1484.
• M. Alomari , M. Darus, U.S. Kirmaci, Refinements of Hadamard-type inequalities for quasi- convex functions with applications to trapezoidal formula and to special means, Computers and Mathematics with Applications 59 (2010) 225 232.
• A. Barani, A.G. Ghazanfari, S.S. Dragomir, Hermite-Hadamard inequality through prequsi- invex functions, RGMIA Research Report Collection, 14(2011), Article 48, 7 pp.
• A. Barani, A.G. Ghazanfari, S.S. Dragomir, Hermite-Hadamard inequality for functions whose derivatives absolute values are preinvex, RGMIA Research Report Collection, 14(2011), Article 64, 11 pp.
• A. Ben-Israel and B. Mond, What is invexity?, J. Austral. Math. Soc., Ser. B, 28(1986), No. 1, 1-9.
• P.S. Bullen, Handbook of Means and Their Inequalities, Kluwer Academic Publishers, Dor- drecht, 2003.
• S. S. Dragomir, and R. P. Agarwal, Two inequalities for differentiable mappings and ap- plications to special means of real numbers and trapezoidal formula, Appl. Math. Lett., 11(5)(1998), 91–95.
• S. S. Dragomir, Two mappings in connection to Hadamard’s inequalities, J. Math. Anal. Appl., 167(1992), 42–56.
• D. -Y. Hwang, Some inequalities for differentiable convex mapping with application to weighted trapezoidal formula and higher moments of random variables, Appl. Math. Comp., 217(23)(2011), 9598-9605.
• M. A. Hanson, On sufficiency of the Kuhn-Tucker conditions, J. Math. Anal. Appl. 80 (1981) 545-550. [11] 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.
• U. S. Kırmacı, Inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula, Appl. Math. Comp., 147(1)(2004), 137-146.
• U. S. Kırmacı and M. E. ¨Ozdemir, On some inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula, Appl. Math. Comp., 153(2)(2004), 361-368.
• K. C. Lee and K. L. Tseng, On a weighted generalization of Hadamard’s inequality for G- convex functions, Tamsui-Oxford J. Math. Sci., 16(1)(2000), 91–104.
• A. Lupas, A generalization of Hadamard’s inequality for convex functions, Univ. Beograd. Publ. Elek. Fak. Ser. Mat. Fiz., 544-576(1976), 115–121.
• S. R. Mohan and S. K. Neogy, On invex sets and preinvex functions, J. Math. Anal. Appl. 189 (1995), 901–908.
• M. Aslam Noor, Hermite-Hadamard integral inequalities for log-preinvex functions, Preprint. [18] M. A. Noor, Variational-like inequalities, Optimization, 30 (1994), 323–330.
• M. A. Noor, Invex equilibrium problems, J. Math. Anal. Appl., 302 (2005), 463–475.
• 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. In- equal. Pure Appl. Math., 8(2007), No. 3, 1-14.
• R. Pini, Invexity and generalized convexity, Optimization 22 (1991) 513-525.
• 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.
• F. Qi, Z. -L.Wei and Q. Yang, Generalizations and refinements of Hermite-Hadamard’s in- equality, Rocky Mountain J. Math., 35(2005), 235–251.
• J. Peˇcari´c, F. Proschan and Y. L. Tong, Convex functions, partial ordering and statistical applications, Academic Press, New York, 1991.
• M. Z. Sarikaya, H. Bozkurt and N. Alp, On Hermite-Hadamard type integral inequalities for preinvex and log-preinvex functions, arXiv:1203.4759v1.
• M. Z. Sarıkaya and N. Aktan, On the generalization some integral inequalities and their applications Mathematical and Computer Modelling, 54(9-10)(2011), 2175-2182.
• M. Z. Sarikaya, M. Avci and H. Kavurmaci, On some inequalities of Hermite-Hadamard type for convex functions, ICMS International Conference on Mathematical Science, AIP Conference Proceedings 1309, 852(2010).
• M. Z. Sarikaya, O new Hermite-Hadamard Fej´er type integral inequalities, Stud. Univ. Babe¸s- Bolyai Math. 57(2012), No. 3, 377-386.
• A. Saglam, M. Z. Sarikaya and H. Yıldırım and, Some new inequalities of Hermite-Hadamard’s type, Kyungpook Mathematical Journal, 50(2010), 399-410.
• C. -L. Wang and X. -H. Wang, On an extension of Hadamard inequality for convex functions, Chin. Ann. Math., 3(1982), 567–570.
• S. -H. Wu , On the weighted generalization of the Hermite-Hadamard inequality and its applications, The Rocky Mountain J. of Math., 39(2009), no. 5, 1741–1749.
• T. Weir, and B. Mond, Preinvex functions in multiple bjective optimization, Journal of Mathematical Analysis and Applications, 136 (1998) 29-38.
• X. M. Yang and D. Li, On properties of preinvex functions, J. Math. Anal. Appl. 256 (2001), 229-241. [35] X.M. Yang, X.Q. Yang and K.L. Teo, Characterizations and applications of prequasiinvex functions, properties of preinvex functions, J. Optim. Theo. Appl. 110 (2001) 645-668.
• X. M. Yang, X. Q. Yang, K.L. Teo, Generalized invexity and generalized invariant monotonoc- ity, Journal of Optimization Theory and Applications 117 (2003) 607-625.
• College of Science, Department of Mathematics,, University of Hail, Hail 2440, Saudi Arabia E-mail address: m amer [e-mail]