I need to test a condition on a function that must grow faster than $\Theta(n^2)$ but slower than $\Theta(n^2H_{n^2})$ when $n\to\infty$($H_{n}$ is the Harmonic Number, so it can be replaced by its asymptotic equivalent $\log(n)+\gamma$). For this purpose, I tried with the following form:
$$f(n)=n^2(H_{n^2})^\xi\quad\colon\xi\in[0,1]$$
As seen, when the parameter $\xi$ ranges from 0 to 1, it spans a set of functions $S$ that match the restriction $s(n)=O(n^2H_{n^2})-o(n^2)\enspace\forall s(n)\in S$. However, if we consider the set of all existing real functions whose growth lies in the set $O(n^2H_{n^2})-o(n^2)$, there might be more functions than the spanned from $f(n)$ with by varying its parameter.
Thus, is there any way to prove that $f(n)$ spans every function with a growth faster than $\Theta(n^2)$ but slower than $\Theta(n^2H_{n^2})$?