Given n, the set of integers m coprime to n has nonzero asymptotic density, thus so does the set mn. But then phi(n)/n > phi(mn)/mn , so the lim inf of phi(n)/n will remain the same off of a set of zero asymptotic density. So there is no such f() for asymptotic density.  (I assume you mean + epsilon in your formulation.)

Given $n$, the set of integers $m$ coprime to $n$ has nonzero asymptotic density, thus so does the set $mn$. But then $\phi(n)/n > \phi(mn)/mn$ , so the $\liminf_n\phi(n)/n$ will remain the same off of a set of zero asymptotic density. So there is no such $f(\cdot)$ for asymptotic density. (I assume you mean + epsilon in your formulation.)