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 


*

*Are these two bounds close? For general meaning. 

*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.

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

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.
In each case, exponentiating results in very fast growing functions. However, the upper bound grows enormously faster. 
