Let $f:\mathbb{N}\rightarrow\mathbb{R}$ be a function such that $f(n)\in o(n)$. Is it always possible to find a function $g:\mathbb{N}\rightarrow\mathbb{R}$ and a constant $r>1$ such that $g(n)\in\omega(n^r)$ but $g(f(n))\in o(n^{r-1})$?
(See the definitions of $o$ and $\omega$ here.)
$f(n)= n/\log n\in o(n)$, $g(n)\in \omega (n^r)$ implies $|g(n)| \geq Mn^r$ but $|g(f(n))| \geq M n^r/\log^rn\notin o(n^{r-1})$.