Let $x\_1, x\_2, \ldots x\_n$ be postive integers with $x\_i\ge 2(1\le i\le n)$. Can we have the following inequality:
$$(1+x\_1)(1+x\_2)\cdots(1+x\_n)\le f(n)x\_1\cdots x\_n,$$
where $f(n)$ is a term dependent on $n$ and I hope it is increasing in Polynomials or lower order. Does anyone can do it? Many thanks!