Hostname: page-component-7c8c6479df-r7xzm Total loading time: 0 Render date: 2024-03-29T02:36:30.534Z Has data issue: false hasContentIssue false

EXACT UPPER AND LOWER BOUNDS ON THE DIFFERENCE BETWEEN THE ARITHMETIC AND GEOMETRIC MEANS

Published online by Cambridge University Press:  04 May 2015

IOSIF PINELIS*
Affiliation:
Department of Mathematical Sciences, Michigan Technological University, Houghton, Michigan 49931, USA email ipinelis@mtu.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Exact upper and lower bounds on the difference between the arithmetic and geometric means are obtained. The inequalities providing these bounds may be viewed, respectively, as a reverse Jensen inequality and an improvement of the direct Jensen inequality, in the case when the convex function is the exponential.

Type
Research Article
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

References

Bhatia, R. and Davis, C., ‘A better bound on the variance’, Amer. Math. Monthly 107(4) (2000), 353357.CrossRefGoogle Scholar
Dharmadhikari, S. W. and Joag-Dev, K., ‘Upper bounds for the variances of certain random variables’, Comm. Statist. Theory Methods 18(9) (1989), 32353247.CrossRefGoogle Scholar
Dragomir, S. S., ‘Some reverses of the Jensen inequality with applications’, Bull. Aust. Math. Soc. 87(2) (2013), 177194.CrossRefGoogle Scholar
Karlin, S. and Studden, W. J., Tchebycheff Systems: With Applications in Analysis and Statistics, Pure and Applied Mathematics, XV (Interscience–John Wiley, New York–London–Sydney, 1966).Google Scholar
Klurman, O., ‘Problem 11800’, Amer. Math. Monthly 121(8) (2014), 739.Google Scholar
Kreĭn, M. G. and Nudel’man, A. A., The Markov Moment Problem and Extremal Problems (American Mathematical Society, Providence, RI, 1977), [Ideas and problems of P. L. Čebyšev and A. A. Markov and their further development. Translated from the Russian by D. Louvish, Translations of Mathematical Monographs, Vol. 50].CrossRefGoogle Scholar
Pinelis, I., ‘Tchebycheff systems and extremal problems for generalized moments: a brief survey’, arXiv:1107.3493, 2011.Google Scholar
Pinelis, I., ‘An asymptotically Gaussian bound on the Rademacher tails’, Electron. J. Probab. 17 (2012), 122.CrossRefGoogle Scholar