I want to express the following statement about a function $f(n)$: there exists $f_\Omega\in\Omega(h(n))$ such that $f\in O(g(f_\Omega(n))$. What's the correct notation for this? Is it $f\in O(g(\Omega(h(n))))$? Or does that notation mean that $f\in O(g(f_\Omega(n))$ for all $f_\Omega\in\Omega(h(n))$?
For example, take $f(n)=2^{-\alpha n}(n^4+n^2+\log(n))$ for $\alpha>0$. Such a function has that there exists $f_\Omega\in\Omega(n)$ (take $f_\Omega(n)=\alpha n/2$ for example) such that $f\in O(2^{-f_\Omega(n)})$. I want a nice succinct way to say this that stresses the important part (the exponential) and without mentioning the constant $\alpha$ as it is subject to change. I was thinking that this does it: $2^{-5n}(n^4+n^2+\log(n))\in O(2^{-\Omega(n)})$. But I'm not sure if this is "there exists $f_\Omega$" or "for all $f_\Omega$" (which would mean $f$ is eventually zero?) and if it's the latter then what's the right way of writing this?