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On Inequalities Connecting Arithmetic Means and Geometric Means of Two Sets of Three Positive Numbers

Published online by Cambridge University Press:  03 November 2016

Alexander Oppenheim
Alexander Oppenheim*
Affiliation:
University of Malaya
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Suppose that a 1, …, an and c 1, …, cn are two sets of n positive numbers. If the arithmetic mean of the ci is at least equal to that of the ai , when can we say non-trivially that the geometric mean of the ci is at least equal to that of the ai and that equality will require equality (in some order) of the ai and ci ?

A well-known answer of course occurs when each c is equal to the arithmetic mean of the ai . When the two sets each consist of three numbers I can give an answer including this important case. To obtain this result it is sufficient to confine the ci to the interval from the least to the greatest of the ai .

Information

Type
Research Article
Information
The Mathematical Gazette , Volume 49 , Issue 368 , May 1965 , pp. 160 - 162
Copyright
Copyright © Mathematical Association 1965

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