Consider the folowing subsets of possible $f(x)$s:
- Group 1: There exists a constant $c<1$ such that $f(x)=\mathcal{O}(e^{c x})$.
- Group 2: If for all $c<1$, $f(x)=\omega(e^{c x})$.
Is there a simpler way of grouping them in terms of asymptotic notations such as of Big-O, little-o, etc.?
This is a part of a bigger problem. To solve the problem I had two solutions, each one for one of the groups. The grouping I have above is a little strange, but that's what I could come up with. Is there a simpler grouping? I thought it might be equivalent to:
- Group 1: $f(x)=o(e^{c x})$.
- Group 2: $f(x)=\Omega(e^{c x})$.
Then, I realized it is not correct. Any suggestions for a simpler notation for such a grouping?