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

Shu-Lin  Lyu

doi:10.3934/naco.2014.4.1

Article Contents

2014, Volume 4, Issue 1: 1-8. Doi: 10.3934/naco.2014.4.1

This issue Previous Article Next Article The Toland-Fenchel-Lagrange duality of DC programs for composite convex functions

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

Shu-Lin Lyu^1,

1.
Department of Mathematics, University of Macau, Macau

Received: December 2012

Revised: October 2013

Published: December 2013

Abstract / Introduction Related Papers Cited by

Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 26D15, 26B25, 26E60; Secondary: 65F60.

Citation:

Related Papers

Cited by

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.
[2]	Y. Ding, Two classes of means and their applications, Math. Pract. Theory, 25 (1995), 16-20. (Chinese)
[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.
[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.
[5]	S. S. Dragomir and C. E .M. Pearce, Selected Topics on Hermite-Hadamard Inequalities and Applications, RGMIA Monographs, Victoria University, 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), 15-20.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), 105-109.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), 63-73.
[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.
[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.
[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.
[13]	L. Pei, Typical Problems and Methods in Mathematical Analysis, 2nd edition, Higher Education Press, Beijing, 2006. (Chinese)
[14]	G. Pólya and G. Szegö, Isoperimetric Inequalities in Mathematical Physics, Princeton University Press, Princeton, 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), 134-147.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), 137-152.
[17]	W. Xu and H. Xu, A generalization of convex functions, Journal of Guyuan Teachers College (Natural Science Edition), 24 (2003), 27-30. (Chinese)