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I have asked the question "Which is bigger, $n!^{n!}$ or $(n^{n})!$ ?", which there is an elementary proof. So now my question is:

For an increasing function $f: \mathbb{R} \rightarrow \mathbb{R}$ do we always have $f(n)^{f(n)} \leq f(n^{n})$?

If $f = 1_{\mathbb{R}}$ this is trivial. If $f$ is a line $mx + b$, then can we say that $(mn+b)^{mn+b} \leq m(n^{n})+b$ ?

I would mostly appreciate hints, as I am not experienced in these kinds of problems. But full solutions are welcome of course.

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Certainly not. Note that $f(n) = n^2$ is an increasing function(on positive real numbers), but $f(n)^{f(n)} = (n^2)^{n^2} = n^{2n^2}$, while $f(n^n) = (n^n)^2 = n^{2n}$. For $n>1$, $2n^2>2n$, so $n^{2n^2} > n^{2n}$, contradicting your statement.

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A simple counterexample: let $f$ be the constant function $f(x)=2$.

Then $f(n)^{f(n)}=4$ but $f(n^n)=2$

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