MathDB

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.
Proposed by Finbarr Holland, Ireland

Problem Statement