Level $k$ bounds in Analysis of Boolean functions In Ryan O'Donnell's book Analysis of Boolean functions, following Corollary 9.25 the following appears:

If $f\colon \{-1,1\}^n \to \{0,1\}$, and we have $\mathbb{E}[f] = \alpha$, then for any integer $k$ at most $2\ln \frac{1}{\alpha}$, we have $W^{\leq k}[f] \leq \rho^{-k}\alpha^{2(1-\rho)}$. He then deduces the following corollary:


  If $\epsilon &gt; 0$ and $k_\epsilon = 2(1-\epsilon)\ln \frac{1}{\alpha}$, then 
  $W^{\leq k_\epsilon}[f] \leq \alpha^{\epsilon^2}$


My question is isn't this bound meaningless?

Indeed, we have

$W^{\leq k_\epsilon}[f] \leq \lVert f\rVert_2^2 = \alpha \ll \alpha^{\epsilon^2}$
Just posting an answer in order to close the question.

See the helpful comments to the question which contain all the info;


Yes it is vacuous, but if you use the stronger bound you get something nontrivial for large $\epsilon$, of course for fixed $k$ you also have interesting things, it's just the estimates in the book lose too much.
Better bounds are known (thanks ot Yuval) : http://drops.dagstuhl.de/opus/volltexte/2019/10829/pdf/LIPIcs-CCC-2019-7.pdf