I have a family of positive functions $f_\alpha$, parametrized by $\alpha>0$, for which I can prove that for all $n>0$ and all $0\leq\eta <1$

$$f_\alpha(n) \leq n^{-\eta\,\alpha}$$

This is just so not enough to conclude that for all $\alpha>0$

$$ f_\alpha(n) \in O(n^{-\alpha}) \quad (n \to \infty), $$

but it is very close.

Is there a compact standard notation for situations like these, i.e., for when a function is just so not bounded by some algebraic function, but by any other algebraic function that decays just an arbitrarily little slower?


I've seen the notation $n^{-\alpha+o(1)}$ used for similar notions in the past; note that this engulfs the outer constant term and so there's generally no need to use e.g. $O(n^{-\alpha+o(1)})$.


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