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

Part of: Distribution theory - Probability Inequalities

Published online by Cambridge University Press:  04 May 2015

IOSIF PINELIS
IOSIF PINELIS*
Affiliation:
Department of Mathematical Sciences, Michigan Technological University, Houghton, Michigan 49931, USA email ipinelis@mtu.edu
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Abstract

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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.

Keywords

arithmetic meangeometric meanJensen inequalityreverse Jensen inequalityexact boundsTchebycheff–Markoff systems

MSC classification

Primary: 60E15: Inequalities; stochastic orderings
Secondary: 26D15: Inequalities for sums, series and integrals 90C46: Optimality conditions, duality
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 92 , Issue 1 , August 2015 , pp. 149 - 158
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

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