Hostname: page-component-77c89778f8-m42fx Total loading time: 0 Render date: 2024-07-21T07:26:23.626Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Alzer, H.. Ungleichungen für geometrische und arithmetische Mittelwerte. Proc. Kon. Nederl. Akad. Wetensch. 91 (1988), 365–374.CrossRefGoogle Scholar
2Alzer, H.. The inequality of Ky Fan and related results. Ada Appl. Math. 18 (1995), 305–354.Google Scholar
3Beckenbach, E. F. and Bellman, R.. Inequalities (Berlin: Springer, 1961).CrossRefGoogle Scholar
4Cartwright, D. I. and Field, M. J.. A refinement of the arithmetic mean-geometric mean inequality. Proc. Am. Math. Soc. 71 (1978), 36–38.CrossRefGoogle Scholar
5McGregor, M. T.. Short proofs of some inequalities of Horst Alzer. Arch. Math. (Brno) 29 (1993), 167–168.Google Scholar