I am looking for a reference to results on sharp upper and lower bounds for the $2n$th power towers $$x^x,x^{x^{x^x}},x^{x^{x^{x^{x^x}}}},x^{x^{x^{x^{x^{x^{x^x}}}}}},\cdots$$ over the intervals $x\in[0,1]$ and/or $[1,\infty)$ for any natural number $n$, such that the bounds contain exponent terms stacked no higher than the form $a^b$.
An example would be the inequality — valid for $[0,\infty)$: $$\frac12x^2+\frac12\le x^{x^{x^{x^{x^x}}}}$$ that has been shown on MSE; however, this is for a specific $n=3$. I believe the case $x^x$ with $n=1$ is a bit better established, where we can derive lower bounds using Padé approximants, for instance.
Are there any papers that investigate this problem for this set of power towers?
