On some inequalities involving \((n!)^{1/n}\). II (Q1338467)
From MaRDI portal
 | This is the item page for this Wikibase entity, intended for internal use and editing purposes. Please use this page instead for the normal view: On some inequalities involving \((n!)^{1/n}\). II |
scientific article; zbMATH DE number 698619
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | On some inequalities involving \((n!)^{1/n}\). II |
scientific article; zbMATH DE number 698619 |
Statements
On some inequalities involving \((n!)^{1/n}\). II (English)
0 references
1 April 1996
0 references
Let \(G(n)\) be the geometric mean of the first \(n\) positive integers, that is, \(G(n)= (n!)^{1/n}\). Then \[ 1< 1+ {G(n)\over G(n- 1)}- {G(n+ 1)\over G(n)}< 1+ {1\over n}- {1\over n+ 1}< n {G(n+ 1)\over G(n)}- (n- 1) {G(n)\over G(n- 1)} \] holds for all \(n\geq 3\). [ For Part I see Rocky Mt. J. Math. 24, No. 3, 867-873 (1994; Zbl 0826.26005)].
0 references
inequalities
0 references
geometric mean
0 references
positive integers
0 references
0.97671753
0 references
0.9077319
0 references
