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Two Dimensional Cebysev Type Inequalities for Functions Whose Second Derivatives are Co-ordinated $\left( h_{1},h_{2}\right) $-Preinvex

Year 2018, Volume: 6 Issue: 1, 76 - 83, 15.04.2018

Badreddine Meftah

Abstract

In this paper, we extend the identity established in \cite{2} for preinvex functions. Using this novel identity we establish some new Cebysev  type inequalities involving functions of two independent variable whose mixed derivatives are co-ordinated $(h_{1},h_{2})$-preinvex.

Keywords

Cebysev type inequalities co-ordinated (h_{1};h_{2})-preinvex integral inequality

References

  • [1] F. Ahmad, N. S. Barnett and S. S. Dragomir, New weighted Ostrowski and Cˇ ebysˇev type inequalities. Nonlinear Anal. 71 (2009), no. 12, e1408–e1412.
  • [2] N. S. Barnett and S. S. Dragomir, An Ostrowski type inequality for double integrals and applications for cubature formulae. Soochow J. Math. 27 (2001), no. 1, 1–10.
  • [3] A. Ben-Israel and B. Mond, What is invexity? J. Austral. Math. Soc. Ser. B 28 (1986), no. 1, 1–9.
  • [4] P. L. Cˇ ebysˇev, Sur les expressions approximatives des inte´grales de´finies par les autres prises entre les meˆmes limites, Proc. Math. Soc. Charkov. 2 (1882), 93-98.
  • [5] A. Guezane-Lakoud and F. Aissaoui, New Cˇ ebysˇev type inequalities for double integrals. J. Math. Inequal. 5 (2011), no. 4, 453–462.
  • [6] M. A. Hanson, On sufficiency of the Kuhn-Tucker conditions. J. Math. Anal. Appl. 80 (1981), no. 2, 545–550.
  • [7] M. A. Latif and S.S. Dragomir, Some Hermite-Hadamard type inequalities for functions whose partial derivatives in absolute value are preinvex on the co-ordinates. Facta Univ. Ser. Math. Inform. 28 (2013), no. 3, 257–270.
  • [8] M. Matloka, On some Hadamard-type inequalities for (h1;h2)-preinvex functions on the co-ordinates. J. Inequal. Appl. 2013, 2013:227, 12 pp.
  • [9] B. Meftah and K. Boukerrioua, New Cˇ ebysˇev Type Inequalities for Functions whose Second Derivatives are (s1;m1)-(s2;m2)-convex on the coordinates. Theory Appl. Math. Comput. Sci. 5 (2015), no. 2, 116–125.
  • [10] B. Meftah and K. Boukerrioua, Cˇ ebysˇev type inequalities whose second derivatives are (s; r)-convex on the co-ordinates. J. Adv. Res. Appl. Math. 7 (2015), no. 3, 76–87.
  • [11] B. Meftah and K. Boukerrioua, On some Cˇ ebysˇev type inequalities for functions whose second derivatives are (h1;h2)-convex on the co-ordinates. Konuralp J. Math. 3 (2015), no. 2, 77–88.
  • [12] B. Meftah and R. Haouam, . On some Cˇ ebysˇev type inequalities for functions whose second derivatives are co-ordinated logarithmically convex. Int. J. Open Problems Compt. Math. 9 (2016), no. 4, 57-65.
  • [13] M. A. Noor, Variational-like inequalities. Optimization 30 (1994), no. 4, 323–330.
  • [14] M. A. Noor, Invex equilibrium problems. J. Math. Anal. Appl. 302 (2005), no. 2, 463–475.
  • [15] M. A. Noor, Hermite-Hadamard integral inequalities for log-preinvex functions. J. Math. Anal. Approx. Theory 2 (2007), no. 2, 126–131.
  • [16] M. A. Noor, Hadamard integral inequalities for product of two preinvex functions. Nonlinear Anal. Forum 14 (2009), 167–173.
  • [17] M. A. Noor, K. I. Noor, M. U. Awan and J. Li, On Hermite-Hadamard inequalities for h-preinvex functions. Filomat 28 (2014), no. 7, 1463–1474.
  • [18] M. A. Noor, K. I. Noor, M. U. Awan and F. Qi, Integral inequalities of Hermite-Hadamard type for logarithmically h-preinvex functions. Cogent Math. 2 (2015), Art. ID 1035856, 10 pp.
  • [19] B. G. Pachpatte, On Chebyshev type inequalities involving functions whose derivatives belong to Lp spaces. JIPAM. J. Inequal. Pure Appl. Math. 7 (2006), no. 2, Article 58, 6 pp.
  • [20] B. G. Pachpatte, On Cˇ ebysˇev-Gru¨ss type inequalities via Pecˇaric´’s extension of the Montgomery identity. JIPAM. J. Inequal. Pure Appl. Math. 7 (2006), no. 1, Article 11, 4 pp.
  • [21] R. Pini, Invexity and generalized convexity. Optimization 22 (1991), no. 4, 513–525.
  • [22] M. Z. Sarikaya, H. Budak and H. Yaldiz, Cˇ ebysev type inequalities for co-ordinated convex functions, Pure and Applied Mathematics Letters. 2 (2014), no. 8, 44-48.
  • [23] M. Z. Sarikaya, N. Alp and H. Bozkurt, On Hermite-Hadamard type integral inequalities for preinvex and log-preinvex functions, Contemporary Anal. Appl. Math., 1 (2013), 237–252.
  • [24] T. Weir and B. Mond, Pre-invex functions in multiple objective optimization. J. Math. Anal. Appl. 136 (1988), no. 1, 29–38.
  • [25] X. -M. Yang and D. Li, On properties of preinvex functions. J. Math. Anal. Appl. 256 (2001), no. 1, 229–241.

Year 2018, Volume: 6 Issue: 1, 76 - 83, 15.04.2018

Badreddine Meftah

Abstract

References

  • [1] F. Ahmad, N. S. Barnett and S. S. Dragomir, New weighted Ostrowski and Cˇ ebysˇev type inequalities. Nonlinear Anal. 71 (2009), no. 12, e1408–e1412.
  • [2] N. S. Barnett and S. S. Dragomir, An Ostrowski type inequality for double integrals and applications for cubature formulae. Soochow J. Math. 27 (2001), no. 1, 1–10.
  • [3] A. Ben-Israel and B. Mond, What is invexity? J. Austral. Math. Soc. Ser. B 28 (1986), no. 1, 1–9.
  • [4] P. L. Cˇ ebysˇev, Sur les expressions approximatives des inte´grales de´finies par les autres prises entre les meˆmes limites, Proc. Math. Soc. Charkov. 2 (1882), 93-98.
  • [5] A. Guezane-Lakoud and F. Aissaoui, New Cˇ ebysˇev type inequalities for double integrals. J. Math. Inequal. 5 (2011), no. 4, 453–462.
  • [6] M. A. Hanson, On sufficiency of the Kuhn-Tucker conditions. J. Math. Anal. Appl. 80 (1981), no. 2, 545–550.
  • [7] M. A. Latif and S.S. Dragomir, Some Hermite-Hadamard type inequalities for functions whose partial derivatives in absolute value are preinvex on the co-ordinates. Facta Univ. Ser. Math. Inform. 28 (2013), no. 3, 257–270.
  • [8] M. Matloka, On some Hadamard-type inequalities for (h1;h2)-preinvex functions on the co-ordinates. J. Inequal. Appl. 2013, 2013:227, 12 pp.
  • [9] B. Meftah and K. Boukerrioua, New Cˇ ebysˇev Type Inequalities for Functions whose Second Derivatives are (s1;m1)-(s2;m2)-convex on the coordinates. Theory Appl. Math. Comput. Sci. 5 (2015), no. 2, 116–125.
  • [10] B. Meftah and K. Boukerrioua, Cˇ ebysˇev type inequalities whose second derivatives are (s; r)-convex on the co-ordinates. J. Adv. Res. Appl. Math. 7 (2015), no. 3, 76–87.
  • [11] B. Meftah and K. Boukerrioua, On some Cˇ ebysˇev type inequalities for functions whose second derivatives are (h1;h2)-convex on the co-ordinates. Konuralp J. Math. 3 (2015), no. 2, 77–88.
  • [12] B. Meftah and R. Haouam, . On some Cˇ ebysˇev type inequalities for functions whose second derivatives are co-ordinated logarithmically convex. Int. J. Open Problems Compt. Math. 9 (2016), no. 4, 57-65.
  • [13] M. A. Noor, Variational-like inequalities. Optimization 30 (1994), no. 4, 323–330.
  • [14] M. A. Noor, Invex equilibrium problems. J. Math. Anal. Appl. 302 (2005), no. 2, 463–475.
  • [15] M. A. Noor, Hermite-Hadamard integral inequalities for log-preinvex functions. J. Math. Anal. Approx. Theory 2 (2007), no. 2, 126–131.
  • [16] M. A. Noor, Hadamard integral inequalities for product of two preinvex functions. Nonlinear Anal. Forum 14 (2009), 167–173.
  • [17] M. A. Noor, K. I. Noor, M. U. Awan and J. Li, On Hermite-Hadamard inequalities for h-preinvex functions. Filomat 28 (2014), no. 7, 1463–1474.
  • [18] M. A. Noor, K. I. Noor, M. U. Awan and F. Qi, Integral inequalities of Hermite-Hadamard type for logarithmically h-preinvex functions. Cogent Math. 2 (2015), Art. ID 1035856, 10 pp.
  • [19] B. G. Pachpatte, On Chebyshev type inequalities involving functions whose derivatives belong to Lp spaces. JIPAM. J. Inequal. Pure Appl. Math. 7 (2006), no. 2, Article 58, 6 pp.
  • [20] B. G. Pachpatte, On Cˇ ebysˇev-Gru¨ss type inequalities via Pecˇaric´’s extension of the Montgomery identity. JIPAM. J. Inequal. Pure Appl. Math. 7 (2006), no. 1, Article 11, 4 pp.
  • [21] R. Pini, Invexity and generalized convexity. Optimization 22 (1991), no. 4, 513–525.
  • [22] M. Z. Sarikaya, H. Budak and H. Yaldiz, Cˇ ebysev type inequalities for co-ordinated convex functions, Pure and Applied Mathematics Letters. 2 (2014), no. 8, 44-48.
  • [23] M. Z. Sarikaya, N. Alp and H. Bozkurt, On Hermite-Hadamard type integral inequalities for preinvex and log-preinvex functions, Contemporary Anal. Appl. Math., 1 (2013), 237–252.
  • [24] T. Weir and B. Mond, Pre-invex functions in multiple objective optimization. J. Math. Anal. Appl. 136 (1988), no. 1, 29–38.
  • [25] X. -M. Yang and D. Li, On properties of preinvex functions. J. Math. Anal. Appl. 256 (2001), no. 1, 229–241.
There are 25 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Research Article
Authors

Badreddine Meftah 0000-0002-0156-7864

Publication Date April 15, 2018
Submission Date July 22, 2017
Acceptance Date April 6, 2018
Published in Issue Year 2018 Volume: 6 Issue: 1

Cite

APA Meftah, B. (2018). Two Dimensional Cebysev Type Inequalities for Functions Whose Second Derivatives are Co-ordinated $\left( h_{1},h_{2}\right) $-Preinvex. Konuralp Journal of Mathematics, 6(1), 76-83.
AMA Meftah B. Two Dimensional Cebysev Type Inequalities for Functions Whose Second Derivatives are Co-ordinated $\left( h_{1},h_{2}\right) $-Preinvex. Konuralp J. Math. April 2018;6(1):76-83.
Chicago Meftah, Badreddine. “Two Dimensional Cebysev Type Inequalities for Functions Whose Second Derivatives Are Co-Ordinated $\left( H_{1},h_{2}\right) $-Preinvex”. Konuralp Journal of Mathematics 6, no. 1 (April 2018): 76-83.
EndNote Meftah B (April 1, 2018) Two Dimensional Cebysev Type Inequalities for Functions Whose Second Derivatives are Co-ordinated $\left( h_{1},h_{2}\right) $-Preinvex. Konuralp Journal of Mathematics 6 1 76–83.
IEEE B. Meftah, “Two Dimensional Cebysev Type Inequalities for Functions Whose Second Derivatives are Co-ordinated $\left( h_{1},h_{2}\right) $-Preinvex”, Konuralp J. Math., vol. 6, no. 1, pp. 76–83, 2018.
ISNAD Meftah, Badreddine. “Two Dimensional Cebysev Type Inequalities for Functions Whose Second Derivatives Are Co-Ordinated $\left( H_{1},h_{2}\right) $-Preinvex”. Konuralp Journal of Mathematics 6/1 (April2018), 76-83.
JAMA Meftah B. Two Dimensional Cebysev Type Inequalities for Functions Whose Second Derivatives are Co-ordinated $\left( h_{1},h_{2}\right) $-Preinvex. Konuralp J. Math. 2018;6:76–83.
MLA Meftah, Badreddine. “Two Dimensional Cebysev Type Inequalities for Functions Whose Second Derivatives Are Co-Ordinated $\left( H_{1},h_{2}\right) $-Preinvex”. Konuralp Journal of Mathematics, vol. 6, no. 1, 2018, pp. 76-83.
Vancouver Meftah B. Two Dimensional Cebysev Type Inequalities for Functions Whose Second Derivatives are Co-ordinated $\left( h_{1},h_{2}\right) $-Preinvex. Konuralp J. Math. 2018;6(1):76-83.
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