# 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), 1557-1567.  Google Scholar [2] Y. Ding, Two classes of means and their applications, Math. Pract. Theory, 25 (1995), 16-20. (Chinese) Google Scholar [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), 775-788.  Google Scholar [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), 271-278. doi: 10.3934/naco.2012.2.271.  Google Scholar [5] S. S. Dragomir and C. E .M. Pearce, Selected Topics on Hermite-Hadamard Inequalities and Applications, RGMIA Monographs, Victoria University, 2000. Google Scholar [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), 15-20. doi: 10.1016/S0898-1221(97)00084-9.  Google Scholar [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), 105-109. doi: 10.1016/S0893-9659(97)00142-0.  Google Scholar [8] N. J. Higham, Functions of Matrices: Theory and Computation, SIAM, 2008. doi: 10.1137/1.9780898717778.  Google Scholar [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), 63-73.  Google Scholar [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), 1645-1656.  Google Scholar [11] M. A. Latif and S. S. Dragomir, On some new inequalities for differentiable co-ordinated convex functions, J. Inequal. Appl., 1 (2012), 1-13. doi: 10.1186/1029-242X-2012-28.  Google Scholar [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), 219-229.  Google Scholar [13] L. Pei, Typical Problems and Methods in Mathematical Analysis, 2nd edition, Higher Education Press, Beijing, 2006. (Chinese) Google Scholar [14] G. Pólya and G. Szegö, Isoperimetric Inequalities in Mathematical Physics, Princeton University Press, Princeton, 1951.  Google Scholar [15] M. Z. Sarikaya, On the Hermite-Hadamard type inequalities for co-ordinated convex function via fractional integrals, Integr. Transf. Spec. F., 25 (2014), 134-147. doi: 10.1080/10652469.2013.824436.  Google Scholar [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), 137-152.  Google Scholar [17] W. Xu and H. Xu, A generalization of convex functions, Journal of Guyuan Teachers College (Natural Science Edition), 24 (2003), 27-30. (Chinese) Google Scholar

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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), 1557-1567.  Google Scholar [2] Y. Ding, Two classes of means and their applications, Math. Pract. Theory, 25 (1995), 16-20. (Chinese) Google Scholar [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), 775-788.  Google Scholar [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), 271-278. doi: 10.3934/naco.2012.2.271.  Google Scholar [5] S. S. Dragomir and C. E .M. Pearce, Selected Topics on Hermite-Hadamard Inequalities and Applications, RGMIA Monographs, Victoria University, 2000. Google Scholar [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), 15-20. doi: 10.1016/S0898-1221(97)00084-9.  Google Scholar [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), 105-109. doi: 10.1016/S0893-9659(97)00142-0.  Google Scholar [8] N. J. Higham, Functions of Matrices: Theory and Computation, SIAM, 2008. doi: 10.1137/1.9780898717778.  Google Scholar [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), 63-73.  Google Scholar [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), 1645-1656.  Google Scholar [11] M. A. Latif and S. S. Dragomir, On some new inequalities for differentiable co-ordinated convex functions, J. Inequal. Appl., 1 (2012), 1-13. doi: 10.1186/1029-242X-2012-28.  Google Scholar [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), 219-229.  Google Scholar [13] L. Pei, Typical Problems and Methods in Mathematical Analysis, 2nd edition, Higher Education Press, Beijing, 2006. (Chinese) Google Scholar [14] G. Pólya and G. Szegö, Isoperimetric Inequalities in Mathematical Physics, Princeton University Press, Princeton, 1951.  Google Scholar [15] M. Z. Sarikaya, On the Hermite-Hadamard type inequalities for co-ordinated convex function via fractional integrals, Integr. Transf. Spec. F., 25 (2014), 134-147. doi: 10.1080/10652469.2013.824436.  Google Scholar [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), 137-152.  Google Scholar [17] W. Xu and H. Xu, A generalization of convex functions, Journal of Guyuan Teachers College (Natural Science Edition), 24 (2003), 27-30. (Chinese) Google Scholar
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