Multi-dimensional Hadamard's inequalities

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Published: Mar 31, 2012
DOI: https://doi.org/10.5556/j.tkjm.43.2012.677
Keywords:
Hadamard's inequalities convex function regular polygon

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Yin Chen
Department ofMathematical Sciences, Lakehead University, Thunder Bay, Ontario, Canada, P7B 5E1.

Abstract

In this paper, Hadamard's inequalities are extended to a convex function on a convex set in $R^2$ or $R^3$. In particular, it is proved that the average of convex function on a ball of radius $r$ is between the average of the function on the circle of radius r and that on the circle of $\frac{2r}{3}$

Article Details

How to Cite
Chen, Y. (2012). Multi-dimensional Hadamard’s inequalities. Tamkang Journal of Mathematics, 43(1), 1–10. https://doi.org/10.5556/j.tkjm.43.2012.677
Issue
Vol. 43 No. 1 (2012)
Section
Papers
Author Biography

Yin Chen, Department ofMathematical Sciences, Lakehead University, Thunder Bay, Ontario, Canada, P7B 5E1.

Associate Professor
Department ofMathematical Sciences,
Lakehead University

References

M. Bessenyei, The Hermite-Hadamard inequality on simplices, Amer. Math. Monthly, 115 (2008),No.4, 339-345.

Y. Chen, Hadamard's inequality on a polygon, Tamkang J. Math., 35 (2004), No.3, 247-254.

S. S. Dragomir, On Hadamard's inequality on a disk, JIPAM. J. Inequal. Pure Appl. Math., 1 (2000), no. 2, Article 2, 11 pp.

B. Gavrea, On Hadamard's inequality for the convex mappings defined on a convex domain in the space, J. Inequal. Pure Appl. Math., 1 (2000), Article 9.

J. Hadamard, Etude sur les propriet es des fonctions entie res et en particulier d'une fonction consideree par Riemann, J. Math. Pures Appl., 58 (1893), 171-215.

Ch. Hermite, Sur deux limites d'une integrale define, Mathesis, 3 (1883), 82.

L. Matejicka, Elementary proof of the left multidimensional Hermite-Hadamard inequality on certain convex sets, J. Math. Inequal., 4 (2010), no. 2, 259-270.

D. S. Mitrinovic and I. B. Lackovic, Hermite and convexity, Aequationes Math., 28 (1985), 229-232.

C. P. Niculescu, The Hermite-Hadamard inequality for convex functions of a vector variable, Math. Inequal. Appl., 5 (2002), 619-623.

C. P. Niculescu and L.-E. Persson, Old and new on the Hermite-Hadamard inequality, Real Anal. Exchange, 29 (2003/2004) 619-623.

T. Ransford, Potential Theory In The Complex Plane, Cambridge University Press, 1995.

A. W. Robert and D. E. Varberg, Convex Functions, Academic Press, New York, 1973.

T. Trif, Characterizations of convex functions of a vector variable via Hermite-Hadamard's inequality, J. Math. Inequal., 1 (2008), No.2, 37-44.