Is the function $\lceil\lg \lg n\rceil!$ polynomially bounded?

Question

I'm totally lost so please be really explicit in your answers. Thanks for the help.

Polynomially Bounded: $f(x)$ is polynomially bounded if for some constants $c$, $a$ and $x_0$, $$f(x) \le cx^a$$, for all $x > x_0$

Answer 1

Note that $$f(n^2)=\lceil\lg\lg( n^2)\rceil!=\lceil\lg(2\lg n)\rceil!=\lceil1+\lg\lg n\rceil!=(1+\lg\lg n)\cdot f(n),$$ that is $\frac{f(n^2)}{f(n)}$ grows slower than $n$. On the other hand, with $g(n) = c n^a$, we have $\frac{g(n^2)}{g(n)}=n^a$, so this suggests that $a=1$ should suffice (or in fact any $a>0$). (To make this stringent, note that $f$ is monotonic).