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SOME INEQUALITIES FOR DIFFERENTIABLE PREQUASIINVEX FUNCTIONS WITH APPLICATIONS

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

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

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 latif@hotmail.com
Year 2013, Volume: 1 Issue: 2, 17 - 29, 01.12.2013

Abstract

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 latif@hotmail.com
There are 34 citations in total.

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Journal Section Articles
Authors

Muhammed Amer Latıf This is me

Publication Date December 1, 2013
Submission Date April 4, 2015
Published in Issue Year 2013 Volume: 1 Issue: 2

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APA Latıf, M. A. (2013). SOME INEQUALITIES FOR DIFFERENTIABLE PREQUASIINVEX FUNCTIONS WITH APPLICATIONS. Konuralp Journal of Mathematics, 1(2), 17-29.
AMA Latıf MA. SOME INEQUALITIES FOR DIFFERENTIABLE PREQUASIINVEX FUNCTIONS WITH APPLICATIONS. Konuralp J. Math. October 2013;1(2):17-29.
Chicago Latıf, Muhammed Amer. “SOME INEQUALITIES FOR DIFFERENTIABLE PREQUASIINVEX FUNCTIONS WITH APPLICATIONS”. Konuralp Journal of Mathematics 1, no. 2 (October 2013): 17-29.
EndNote Latıf MA (October 1, 2013) SOME INEQUALITIES FOR DIFFERENTIABLE PREQUASIINVEX FUNCTIONS WITH APPLICATIONS. Konuralp Journal of Mathematics 1 2 17–29.
IEEE M. A. Latıf, “SOME INEQUALITIES FOR DIFFERENTIABLE PREQUASIINVEX FUNCTIONS WITH APPLICATIONS”, Konuralp J. Math., vol. 1, no. 2, pp. 17–29, 2013.
ISNAD Latıf, Muhammed Amer. “SOME INEQUALITIES FOR DIFFERENTIABLE PREQUASIINVEX FUNCTIONS WITH APPLICATIONS”. Konuralp Journal of Mathematics 1/2 (October 2013), 17-29.
JAMA Latıf MA. SOME INEQUALITIES FOR DIFFERENTIABLE PREQUASIINVEX FUNCTIONS WITH APPLICATIONS. Konuralp J. Math. 2013;1:17–29.
MLA Latıf, Muhammed Amer. “SOME INEQUALITIES FOR DIFFERENTIABLE PREQUASIINVEX FUNCTIONS WITH APPLICATIONS”. Konuralp Journal of Mathematics, vol. 1, no. 2, 2013, pp. 17-29.
Vancouver Latıf MA. SOME INEQUALITIES FOR DIFFERENTIABLE PREQUASIINVEX FUNCTIONS WITH APPLICATIONS. Konuralp J. Math. 2013;1(2):17-29.
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