Hello Everyone!
I have a couple of questions:
(1) is $\displaystyle (\log n!)^3 = O(3n^3+3)$ true?
(2) is $\displaystyle n^{\log _2 n} = O(2^{\sqrt{n}})$ true?
I'm going for yes for (2), but have no clue regarding (1)
For the first one,
if $\displaystyle (\log n!)^3 \in O(3n^3+3) \subseteq O(n^3)$
then $\displaystyle (\log n!)^3 \leq C^3n^3$ for some $\displaystyle C$ and $\displaystyle n$ sufficiently large.
So $\displaystyle \log n! \leq Cn$ or $\displaystyle n! \leq a^{n} $ which is not true for $\displaystyle n$ much larger than $\displaystyle a$. Contradiction.
For the second one,
$\displaystyle n^{\log _2 n} = 2^{(\log_2 n)^2}$
Now show that $\displaystyle (\log_2 n)^2 \in O(\sqrt{n})$.