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Some inequalities for arithmetic and geometric means

Published online by Cambridge University Press:  14 November 2011

Horst Alzer
Horst Alzer
Affiliation:
Morsbacher Strasse 10, 51545 Waldbröl, Germany
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Abstract

Let An and Gn (respectively, A′n and G′n) be the weighted arithmetic and geometric means of x1, …, xn (respectively, 1 – x1, …, 1 – xn). We present sharp upper and lower bounds for the differences and . And we determine the best possible constants r and s such that

holds for all xi ∈ [a, b] (i = 1, …, n; 0 < a < b < 1). Our theorems extend and sharpen results of Fan, Cartwright and Field, McGregor and the author.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 129 , Issue 2 , 1999 , pp. 221 - 228
Copyright
Copyright © Royal Society of Edinburgh 1999

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References

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