Comparing the growth/decay rates of $f(n)$ and $\frac{n-1}{n^2}$.

Question

Let $f:[0,\infty)\to[0,\infty)$ be a smooth, strictly monotonically increasing function. Consider the inequality $$ \qquad \quad f(n) \cdot \frac{n-1}{n^2} > a \qquad (n\in\{2,3,4,...\};\ a>0). \qquad (*) $$

Assume that inequality $(*)$ holds for $n=2$. Notice that $\frac{n-1}{n^2}$ is strictly monotonically decreasing when $n\geq 2$.

I am interested in characterizing the following:

1. the necessary and sufficient condition(s) on $f$ such that $(*)$ holds $\forall n\in\{2,3,4,...\}$;
2. the necessary and sufficient condition(s) on $f$ such that for some $\tilde{n}\in\{2,3,4,...\}$, $(*)$ holds $\forall n\in\{2,...,\tilde{n}\}$ and does not hold $\forall n>\tilde{n}$.

How can this be accomplished? Thank you very much to anyone who considers my question.