NOTE ON AN INEQUALITY INVOLVING $(n!)^1/n$
H. ALZER
Abstract.
We prove: If $G(n)=(n!)^1/n$ denotes the geometric mean of the first $n$ positive integers, then \frac1e^2<(G(n))^2-G(n-1)G(n+1) holds for all $n\geq 2$. The lower bound $\frac1e^2$ is best possible.
Compressed Postscript 
