Solving Recurrence relation with master method?
You are claiming that "we can prove that $af(n/b) \leq cf(n)$". There are two problems with this claim:


This is now what you actually need in order to apply the master theorem. The master theorem needs you to show that there exists $c &lt; 1$ for which $af(n/b) \leq c f(n)$ for all large enough $n$.
You haven't proved your claim.


Reiterating, to prove the claim you need to come up with a value of $c &lt; 1$ such that the following inequality holds for large enough $n$:
$$
9 (n/3)^3  \log(n/3) \leq cn^3 \log n.
$$
This holds for $c = 1/3$, since
$$
9(n/3)^3 \log(n/3) &lt; 9(n/3)^3 \log n = \frac{1}{3} n^3 \log n.
$$