I am given an example of a case where the master theorem does not apply, but it seems like it should apply.
This was the reasoning:
$T(n) = 3T(n/3) + n \log n$ with $ a = 3, b=3, f(n) = n\log n$ and $n\log_b a = n\log_3 3 = n$
$f(n)$ is asymptotically larger than $n\log_b (a)$ , but not polynomially larger. The ratio $n \log n / n = \log n$ is asymptotically less than $n^\epsilon$ for any positive $\epsilon$. Thus, the Master Theorem doesn’t apply here
However, it seems like case 2 of the master theorem should apply here. We have $f(n) = n\log n$, and thus $f(n) = \theta(n^{\log_3 3} \log n$)
Then: $T(n) = \theta(n^{\log_a b} \log^{k+1}n) = \theta(n\log^2 n)$
Which is correct?