Comparison of differences between arithmetic and geometric means

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Published: Dec 31, 2011
DOI: https://doi.org/10.5556/j.tkjm.42.2011.747
Keywords:
Self-improvement Arithmetic-Geometric inequality

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J. M. Aldaz
Departamento deMatemáticas, Universidad Autónoma deMadrid, Cantoblanco 28049, Madrid, Spain.

Abstract

We complement a recent result of S. Furuichi, by showing that the differences $\sum_{i=1}^n \alpha_i x_i - \prod_{i=1}^n x_i^{\alpha_i}$ associated to distinct sequences of weights are comparable, with constants that depend on the smallest and largest quotients of the weights.

Article Details

How to Cite
Aldaz, J. M. (2011). Comparison of differences between arithmetic and geometric means. Tamkang Journal of Mathematics, 42(4), 453–462. https://doi.org/10.5556/j.tkjm.42.2011.747
Issue
Vol. 42 No. 4 (2011)
Section
Papers
Author Biography

J. M. Aldaz, Departamento deMatemáticas, Universidad Autónoma deMadrid, Cantoblanco 28049, Madrid, Spain.

Professor at the Math. Department.

References

J. M. Aldaz, Self-improvement of the inequality between arithmetic and geometric means}, Journal of Mathematical Inequalities, 3, (2009), 213--216. arXiv:0807.1788.

J. M. Aldaz, A refinement of the inequality between arithmetic and geometric means, Journal of Mathematical Inequalities, 2(2008), 473--477. arXiv:0811.3145.

J. M. Aldaz, Concentration of the ratio between the geometric and arithmetic means, Journal of TheoreticalProbability, 23,498--508, DOI:10.1007/s10959-009-0215-9. arXiv:0807.4832.

J. M. Aldaz, A stability version of H"older's inequality, Journal of Mathematical Analysis and Applications, 343(2008), 842-852. doi:10.1016/j.jmaa.2008.01.104. arXiv:0710.2307.

Sever S. Dragomir, Bounds for the normalised Jensen functional, Bull. Austral. Math. Soc., 74(2006),471--478.

Lawrence C. Evans and Ronald F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics. CRC Press, Boca Raton, FL,1992.

Herbert Federer, Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften, Band 153 Springer-Verlag New York Inc., New York 1969.

Shigeru Furuichi, A refinement of the arithmetic-geometric mean inequality, arXiv:0912.5227.

Ronald E. Glaser, The ratio of the geometric mean to the arithmetic mean for a random sample from a gamma distribution, J. Amer. Statist. Assoc.,71(1976), 480--487.

Flavia Corina Mitroi, About the precision in Jensen-Steffensen inequality, An. Univ. Craiova Ser. Mat. Inform., 37(2010), 73--84.

J. Michael Steele, The Cauchy-Schwarz master class. An introduction to the art of mathematical inequalities, MAA Problem Books Series. Mathematical Association of America, Washington, DC; Cambridge University Press, Cambridge, 2004.