Let
![{a_1,a_2,\dots,a_n}](/media/m/0/0/1/00194ff24660ff13f546c5de374919cb.png)
be positive real numbers,
![{n>1}](/media/m/2/0/c/20cd7d1dc55ea9b07802f153a5a547e8.png)
. Denote by
![g_n](/media/m/b/c/e/bce5ee3ee34a4ee847576458ee9d0a8b.png)
their geometric mean, and by
![A_1,\,A_2,\,\dots,\,A_n](/media/m/1/5/a/15a62e84f37058e72ebcb4b49fb0da2e.png)
the sequence of arithmetic means defined by
![A_k=\frac{a_1+a_2+\cdots+a_k}{k},\qquad k=1,2,\dots,n.](/media/m/8/c/2/8c2d7a0c2a5c92de927c375f5bea7c2d.png)
Let
![G_n](/media/m/4/f/4/4f4dc1ef5a9ba3e85496b85f991228f1.png)
be the geometric mean of
![A_1,A_2,\dots,A_n](/media/m/7/3/d/73db0a7ef4d63a1911d322e68a649551.png)
. Prove the inequality
![n \root n\of{\frac{G_n}{A_n}}+ \frac{g_n}{G_n}\le n+1](/media/m/c/e/d/cedd02e8e16748140fb1f1de1912b09e.png)
and establish the cases of equality.
%V0
Let ${a_1,a_2,\dots,a_n}$ be positive real numbers, ${n>1}$. Denote by $g_n$ their geometric mean, and by $A_1,\,A_2,\,\dots,\,A_n$ the sequence of arithmetic means defined by
$$A_k=\frac{a_1+a_2+\cdots+a_k}{k},\qquad k=1,2,\dots,n.$$
Let $G_n$ be the geometric mean of $A_1,A_2,\dots,A_n$. Prove the inequality $$n \root n\of{\frac{G_n}{A_n}}+ \frac{g_n}{G_n}\le n+1 $$ and establish the cases of equality.