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

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. [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. [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. [3] A. Ben-Israel and B. Mond, What is invexity? J. Austral. Math. Soc. Ser. B 28 (1986), no. 1, 1–9.
4. [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. [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. [6] M. A. Hanson, On sufficiency of the Kuhn-Tucker conditions. J. Math. Anal. Appl. 80 (1981), no. 2, 545–550.
7. [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. [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. [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. [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. [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. [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. [13] M. A. Noor, Variational-like inequalities. Optimization 30 (1994), no. 4, 323–330.
14. [14] M. A. Noor, Invex equilibrium problems. J. Math. Anal. Appl. 302 (2005), no. 2, 463–475.
15. [15] M. A. Noor, Hermite-Hadamard integral inequalities for log-preinvex functions. J. Math. Anal. Approx. Theory 2 (2007), no. 2, 126–131.
16. [16] M. A. Noor, Hadamard integral inequalities for product of two preinvex functions. Nonlinear Anal. Forum 14 (2009), 167–173.
17. [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. [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. [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. [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. [21] R. Pini, Invexity and generalized convexity. Optimization 22 (1991), no. 4, 513–525.
22. [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. [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. [24] T. Weir and B. Mond, Pre-invex functions in multiple objective optimization. J. Math. Anal. Appl. 136 (1988), no. 1, 29–38.
25. [25] X. -M. Yang and D. Li, On properties of preinvex functions. J. Math. Anal. Appl. 256 (2001), no. 1, 229–241.