2004 IMO Shortlist S21

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*

令 ${a_1,a_2,\dots,a_n}$ 为正实数， ${n>1}$ 。 $g_n$ 表示它们的几何平均值，$A_1,A_2,\dots,A_n$ 表示由 $$A_k=\frac{a_1+a_2+\cdots+a_k}{k},\qquad k=1,2,\dots,n 定义的算术平均值序列。$$ 设 $G_n$ 为 $A_1,A_2,\dots,A_n$ 的几何平均值。证明不等式$$

n \root n\of{\frac{G_n}{A_n}}+ \frac{g_n}{G_n}\le n+1 $$ 并建立相等的情况。

*由爱尔兰 Finbarr Holland 提议*