Is $(\log(n))!$ a polynomially bounded function? Is the following statement true? How would you prove it?


i.e. Is it a polynomially bounded?
$$
\lceil \lg(n) \rceil ! \in O(n^k)
$$
How about
$$
\lceil \lg \lg(n) \rceil ! \in O(n^k)
$$
Thanks a lot!

A start: By considering $(1\cdot 2\cdot 3\cdots \cdot w)(w\cdot (w-1) \cdot (w-2)\cdots \cdot 1)$, we can show that 
$(w!)^2\ge w^{w}$, and therefore
$w!\ge w^{w/2}$. 
Let $w=\log n$. 
For the second problem, the upper bound $w!\le w^w$ (for $w\gt 0$) is enough. 
