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 $k^2$.