MathNet 2015-part-02 — 191 problems (EN–VI) Question 1

Let $\mathbb{N} = \{1, 2, 3, \dots\}$ be the set of positive integers. Find all functions $f$, defined on $\mathbb{N}$ and taking values in $\mathbb{N}$, such that $(n - 1)^2 \lt f(n)f(f(n)) \lt n^2 + n$ for every positive integer $n$.