Help with this proof of inequality

Question

Prove that n($a_1a_2.....a_n$)$\leq$($a_1^n + a_2^n + a_3^n.....a_n^n$) I tried using AM GM inequality but I got $n^n$ and $(a_1+a_2....a_n)^n$. I got these two terms If anyone could help me or hint me I would appreciate it.

Answer 1

Assuming $a_i\ge 0$, this is a direct application of AM-GM indeed

$$\frac{\sum a_i^n}n\ge \sqrt[n]{\prod a_i^n}= \sqrt[n]{\left(\prod a_i\right)^n}=\prod a_i$$

otherwise, as noticed, it is not true.

Answer 2

I think it's impossible to prove this inequality.

Try, $n=3$, $a_1=a_2=1$ and $a_3=-3$.

Answer 3

This is a direct application of AM-GM inequality: $$\ \frac{\sum_{i=1}^{n} {{a}_{i}}^{n}}{n} \geq {(\prod_{i=1}^{n} {{a}_{i}}^{n})}^{\frac{1}{n}}=\prod_{i=1}^{n}{a}_{i}$$