Improved Triple Integral Inequalities of Hermite-Hadamard Type

Authors: Leila Nasiri

Abstract:

In this paper, we present the concept of preinvex functions on the co-ordinates on an invex set and establish some triple integral inequalities of Hermite-Hadamard type for functions whose third order partial derivatives in absolute value are preinvex on the co-ordinates. The results presented here generalize the obtained results in earlier works for functions whose triple order partial derivatives in absolute value are convex on the co-ordinates on a rectangular box in R³.

Keywords: Co-ordinated preinvex functions, Hermite-Hadamard type inequalities, partial derivatives, triple integral.

References:

[1] T. Autczak, Mean value in invexity analysis, Nonlinear Analysis 60(2005), 1471-1484.
[2] M. Alomari and M. Darus, Hadamard-type inequalities for s−convex functions, Int. Math. Forum 3, 2008, no. 40, 1965-1975.
[3] A. Barani and F. Malmir, New Hermite-Hadamard type inequalities for convex functions on a rectangular box, Konuralp Journal of Mathematics, V. 4 No. 1 pp. 1-22(2016).
[4] D. Barrera, A. Guessab, M. J. Ibezan, O. Nouisser, Increasing the approximation order of spline quasi-interpolants, J. Comput. Appl. Math. 252 (2013), 27-39.
[5] S. S. Dragomir and R. P. Agarwal, Two inequalities for differentiable mappings and applications to special means of real numbers and to Trapezoidal formula, Appl. Math. Lett. 11(5) (1998), 91-95.
[6] S. S. Dragomir, On Hadamard,s inequality for convex functions on the co-ordinates in a rectangle from the plane, Taiw. J. Math., 4(2001), 775-788.
[7] A. Guessab, G. Schmeisser, Two Korovkin-type theorems in multivariate approximation, Banach J. Math. Anal. 2 (2008), no. 2, 121-128.
[8] A. Guessab, M. Moncayo, G. Schmeisser, A class of nonlinear four-point subdivision schemes, Adv. Comput. Math. 37 (2012), no. 2, 151-190.
[9] A. Guessab, Approximations of dierentiable convex functions on arbi trary convex polytopes, Appl. Math. Comput. 240 (2014), 326-338.
[10] D. Y. Hwang, K. L. Tseng and G. S. Yang, Some Hadamard,s inequalities for co-ordinated convex functions in a rectangle from the plane, Taiw. J. Math., 11(2007), 63-73.
[11] I. Iscan, Hermite-Hadamard,s inequalities for preinvex functions via fractional integrals and related fractional inequalities, arXiv:1204. 0272, submitted.
[12] M. A. Latif and M. Alomari, Hadamard-type inequalities for product two convex functions on the co-ordinates, Int. Math. Forum 4(47) (2009), 2327-2338.
[13] M. A. Latif and M. Alomari, On the Hadamard-type inequalities for h-convex functions on the co-ordinates, Int. J. Math. Analysis 3(33)(2009), 1645-1656.
[14] M. A. Latif and S. S. Dragomir, On some new inequalities for differentiable co-ordinated convex functions, J. Inequal. Appl., 2012, 2012:28 doi:10.1186/1029-242X-2012-28.
[15] M. A. Latif and S. S. Dragomir, Some Hermite- Hadamard type inequalities for whose partial derivatives in absolute value are preinvex on the co-ordinates, Facta University, Ser. Math. Inform., V. 28, No 3(2013), 257-270.
[16] SR. Mohan and SK. Neogy, On invex sets and preinvex functions, J. Math. Anal. Appl., 189(1995): 901-908.
[17] M. A. Noor, On Hadamard integral inequalities involving two log-preinvex functions, J. Inequal. Pure Appl. Math., 8 (3)(2007), Article 75, 6pp.
[18] M. A. Noor, Hermite-Hadamard integral inequalities for log-preinvex functions, Preprint.
[19] C. M. E. Pearce and J. E. Pecaric, Inequalities for differentiable mappings with applications to special means and quadrature formula, Appl. Math. Lett., 13(2)(2000), 51-55.
[20] M. Z. Sarikaya, H. Bozkurt and N. ALp, On Hermite-Hadamard type integral inequalities for preinvex and log-preinvex functions, arXiv:1203. 4759v1.
[21] M. Z. Sarikaya and E. Set, New some Hadamard,s type inequalities for co-ordinated convex functions, TOJIMS 28(2) (2012), 137-152.
[22] A. Saglam, H. Yidirim and M. Z. Sarikaya, Some new inequalities of Hermite-Hadamards type, Kynugpook Math. J. 50(2010). 399-410.
[23] M. E. Ozdemir, E. Set and M. Z. Sarikaya, Some new Hadamard,s type inequalities for coordinated m-convex and (α,m)− convex functions, Hacet. J. Math. Stat 40(2) (2011), 219-229.
[24] M. E. Ozdemir, H. Kavurmaci, A. O. Akdemir and M. Avci, Inequalities for convex and s− convex functions on Δ =
[a, b]×
[c, d], J. Inequal. Appl., 2012:21, doi:10.1186/1029-242X-2012-20.
[25] M. E. Ozdemir, M. A. Latif and A. O. Akdemir, On some Hadamard-type inequalities for product of two s−convex on the co-ordinates, J. Inequal. Appl., 2012:20, doi:10.1186/1029-242X-2012-21.
[26] M. E. Ozdemir, A. O. Akdemir and M. Tunc, On the Hadamard-type inequalities for co-ordinated convex functions, arXiv:1203. 4327v1.
[27] M. E. Ozdemir and A. O. Akdemir, On some Hadamard- type inequalities for convex functions on a rectangular box, V. 2011, year 2011 article ID jnaa-00101, 10 pages doi:1005899/2011/jnaa-00101.
[28] A. Weir and B. Mond, Preinvex functions in multiple objective optimization, J. Math. Anal. Appl., 136 (1988):29-38.
[29] A. Guessab, G. Schmeisser, Two Korovkin-type theorems in multivariate approximation, Banach J. Math. Anal. 2 (2008), no. 2, 121-128.
[30] A. Guessab, M. Moncayo, G. Schmeisser, A class of nonlinear four-point subdivision schemes, Adv. Comput. Math. 37 (2012), no. 2, 151-190.
[31] A. Guessab, Approximations of dierentiable convex functions on arbitrary convex polytopes, Appl. Math. Comput. 240 (2014), 326-338.
[32] D. Barrera, A. Guessab, M. J. Ibezan, O. Nouisser, Increasing the approximation order of spline quasi-interpolants, J. Comput. Appl. Math. 252 (2013), 27-39.