# Are these two bounds close to each other?

Recently I proved a some bound about something. The bound is (details : come soon)

Upper bound $f(k)< k^{k^{O(k)}}$.

Lower bound $f(k)< k^{k^2-o(k)}$

My question is

1. Are these two bounds close? For general meaning.

2. What should I call the lower bound? An exponential function? Or something other looks like a litter larger. Clearly it is not as large as double exponential.

3. Should I need to stress that the exponent in the lower bound is $k^2$.

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For large, and even medium-sized $k$, the upper bound is hugely bigger than the lower bound. But for $k$ not too large, the two bounds might be not far from each other if the implied constant in the $O(k)$ is very close to $0$.
The $O(k)$ means that for the upper bound, all we know is that the exponent of $k$ is, for large $k$, less than $k^{ck}$ for some constant $c$. The lower bound is not quite put in lower bound language: a lower bound means a floor below which a function does not go, so I assume that what is meant is $f(k)\gt k^{k^2-o(k)}$. Thus the exponent of $k$ is of size roughly $k^2$.
So already for the upper bound, the exponent of $k$ is a fast growing function, while the exponent in the lower bound grows relatively slowly.