I want to know if the recurrence equation $T(n) = 9T(\frac{n}{3}) + nlogn$, can or cannot be solved using master theorem.

At first, I naively went for $O(n^2)$ applying case 1 of master theorem. But then someone pointed out that $f(n)=nlogn$ and $n^{\log_3 9}$ do not have polynomial difference and therefore master theorem cannot be used.

Then I found this proof for master theorem. Proof of case 1 is on page 2 of the document right on top of the page. The proof is based on the inequality:

$f(\frac{n}{b^i}) \leq(\frac{n}{b^i})^{\log_a b -\epsilon}$

which allows the right side of the above inequality to replace f(n) and the proof goes on from there.

As I see it, the above inequality holds for $f(n)=nlogn$ so I think master theorem can be applicable here. I want to know if I am right and if not, why?