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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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