# American Institute of Mathematical Sciences

2014, 4(1): 1-8. doi: 10.3934/naco.2014.4.1

## On the Hermite--Hadamard inequality for convex functions of two variables

 1 Department of Mathematics, University of Macau, Macau, China

Received  December 2012 Revised  October 2013 Published  December 2013

Inspired by the results in [S. S. Dragomir and I. Gomm, Num. Alg. Cont. $\&$ Opt., 2 (2012), 271--278], we give some new bounds for two mappings related to the Hermite--Hadamard inequality for convex functions of two variables, and apply them to special functions to get some results for the $p$-logarithmic mean. We also apply the Hermite--Hadamard inequality to matrix functions in this paper.
Citation: Shu-Lin Lyu. On the Hermite--Hadamard inequality for convex functions of two variables. Numerical Algebra, Control & Optimization, 2014, 4 (1) : 1-8. doi: 10.3934/naco.2014.4.1
##### References:
 [1] M. Alomari and M. Darus, Co-ordinated s-convex function in the first sense with some Hadamard-type inequalities,, Int. J. Contemp. Math. Sci., 3 (2008), 1557. [2] Y. Ding, Two classes of means and their applications,, Math. Pract. Theory, 25 (1995), 16. [3] S. S. Dragomir, On the Hadamard's inequality for convex functions on the co-ordinates in a rectangle from the plane,, Taiwanese J. Math., 5 (2001), 775. [4] S. S. Dragomir and I. Gomm, Some new bounds for two mappings related to the Hermite-Hadamard inequality for convex functions,, Num. Alg. Cont. & Opt., 2 (2012), 271. doi: 10.3934/naco.2012.2.271. [5] S. S. Dragomir and C. E .M. Pearce, Selected Topics on Hermite-Hadamard Inequalities and Applications,, RGMIA Monographs, (2000). [6] S. S. Dragomir and S. Wang, An inequality of Ostrowski-Grüss' type and its applications to the estimation of error bounds for some special means and for some numerical quadrature rules,, Computers. Math. Applic., 33 (1997), 15. doi: 10.1016/S0898-1221(97)00084-9. [7] S. S. Dragomir and S. Wang, Applications of Ostrowski's inequality to the estimation of error bounds for some special means and for some numerical quadrature rules,, Appl. Math. Lett., 11 (1998), 105. doi: 10.1016/S0893-9659(97)00142-0. [8] N. J. Higham, Functions of Matrices: Theory and Computation,, SIAM, (2008). doi: 10.1137/1.9780898717778. [9] 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,, Taiwanese J. Math., 11 (2007), 63. [10] M. A. Latif and M. Alomari, On Hadamard-type inequalities for h-convex functions on the co-ordinates,, Int. J. Math. Anal., 3 (2009), 1645. [11] M. A. Latif and S. S. Dragomir, On some new inequalities for differentiable co-ordinated convex functions,, J. Inequal. Appl., 1 (2012), 1. doi: 10.1186/1029-242X-2012-28. [12] M. E. Özdemir, E. Set and M. Z. Sarikaya, Some new Hadamard type inequalities for co-ordinated m-convex and (α, m)-convex functions,, Hacet. J. Math. Stat., 40 (2011), 219. [13] L. Pei, Typical Problems and Methods in Mathematical Analysis,, 2nd edition, (2006). [14] G. Pólya and G. Szegö, Isoperimetric Inequalities in Mathematical Physics,, Princeton University Press, (1951). [15] M. Z. Sarikaya, On the Hermite-Hadamard type inequalities for co-ordinated convex function via fractional integrals,, Integr. Transf. Spec. F., 25 (2014), 134. doi: 10.1080/10652469.2013.824436. [16] M. Z. Sarikaya, E. Set, M. E. Özdemir and S. S.Dragomir, New some Hadamard's type inequalities for co-ordinated convex functions,, Tamsui Oxf. J. Inf. Math. Sci., 28 (2012), 137. [17] W. Xu and H. Xu, A generalization of convex functions,, Journal of Guyuan Teachers College (Natural Science Edition), 24 (2003), 27.

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##### References:
 [1] M. Alomari and M. Darus, Co-ordinated s-convex function in the first sense with some Hadamard-type inequalities,, Int. J. Contemp. Math. Sci., 3 (2008), 1557. [2] Y. Ding, Two classes of means and their applications,, Math. Pract. Theory, 25 (1995), 16. [3] S. S. Dragomir, On the Hadamard's inequality for convex functions on the co-ordinates in a rectangle from the plane,, Taiwanese J. Math., 5 (2001), 775. [4] S. S. Dragomir and I. Gomm, Some new bounds for two mappings related to the Hermite-Hadamard inequality for convex functions,, Num. Alg. Cont. & Opt., 2 (2012), 271. doi: 10.3934/naco.2012.2.271. [5] S. S. Dragomir and C. E .M. Pearce, Selected Topics on Hermite-Hadamard Inequalities and Applications,, RGMIA Monographs, (2000). [6] S. S. Dragomir and S. Wang, An inequality of Ostrowski-Grüss' type and its applications to the estimation of error bounds for some special means and for some numerical quadrature rules,, Computers. Math. Applic., 33 (1997), 15. doi: 10.1016/S0898-1221(97)00084-9. [7] S. S. Dragomir and S. Wang, Applications of Ostrowski's inequality to the estimation of error bounds for some special means and for some numerical quadrature rules,, Appl. Math. Lett., 11 (1998), 105. doi: 10.1016/S0893-9659(97)00142-0. [8] N. J. Higham, Functions of Matrices: Theory and Computation,, SIAM, (2008). doi: 10.1137/1.9780898717778. [9] 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,, Taiwanese J. Math., 11 (2007), 63. [10] M. A. Latif and M. Alomari, On Hadamard-type inequalities for h-convex functions on the co-ordinates,, Int. J. Math. Anal., 3 (2009), 1645. [11] M. A. Latif and S. S. Dragomir, On some new inequalities for differentiable co-ordinated convex functions,, J. Inequal. Appl., 1 (2012), 1. doi: 10.1186/1029-242X-2012-28. [12] M. E. Özdemir, E. Set and M. Z. Sarikaya, Some new Hadamard type inequalities for co-ordinated m-convex and (α, m)-convex functions,, Hacet. J. Math. Stat., 40 (2011), 219. [13] L. Pei, Typical Problems and Methods in Mathematical Analysis,, 2nd edition, (2006). [14] G. Pólya and G. Szegö, Isoperimetric Inequalities in Mathematical Physics,, Princeton University Press, (1951). [15] M. Z. Sarikaya, On the Hermite-Hadamard type inequalities for co-ordinated convex function via fractional integrals,, Integr. Transf. Spec. F., 25 (2014), 134. doi: 10.1080/10652469.2013.824436. [16] M. Z. Sarikaya, E. Set, M. E. Özdemir and S. S.Dragomir, New some Hadamard's type inequalities for co-ordinated convex functions,, Tamsui Oxf. J. Inf. Math. Sci., 28 (2012), 137. [17] W. Xu and H. Xu, A generalization of convex functions,, Journal of Guyuan Teachers College (Natural Science Edition), 24 (2003), 27.
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