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Is there a term for $O( (m+n) \log{mn})$? I remember seeing it often in some context but can't remember.

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Note that it's the same class as $O((m+n)\log(m+n))$ which is just $O(n\log n)$ with $n_{\rm new} = m+n_{\rm old}$. –  Henning Makholm Nov 14 '12 at 21:25
    
Could show why $O(\log{m}+\log{n}) = O(\log{(m+n)})$ in an answer? –  Jacob Nov 14 '12 at 23:42
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Note that $O(\log mn)$ is the same class as $O(\log(m+n))$.

Assume $m,n\ge 1$ (since only the behavior for large $m$ and $n$ interests us). Then in one direction we have $$\log mn = \log m + \log n \le 2 \max(\log m,\log n) \le 2\log\max(m,n) \le 2\log(m+n)$$ and in the other direction $$\log(m+n) \le \log(2\max(m,n)) = 1+\max(\log m,\log n) \le 1 + (\log m + \log n) = 1+\log mn$$

Therefore $O((m+n)\log mn)$ is just $O(k\log k)$ for $k=m+n$.

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