How find the $\inf\sqrt[r]{n_{1}n_{2}\cdots n_{r}}$

Question

Let $r,n$ be given positive integers,and $n_{r}$ are positive integers, such $$n_{1}+n_{2}+\cdots+n_{r}=n$$ find the $$\inf\sqrt[r]{n_{1}n_{2}\cdots n_{r}}$$

I known use AM-GM $$n_{1}+n_{2}+\cdots+n_{r}\ge r\sqrt[r]{n_{1}n_{2}\cdots n_{r}}$$ so we have find the sup is $$\dfrac{n}{r}?$$

and How to find the inf?

Answer 1

Let $x>y$, where $\{x,y\}\subset\mathbb N$.

Hence, since $xy\geq(x+1)(y-1)$, we get the minimum for $n_1=n_2=...=n_{r-1}=1$.