Usually when I face solving recursions I use generating functions but I'm not aware of any "tools" to use when min/max expressions are involved.
For example, I have the following recursive term: $$T(m,n) = max \{\;T(m/2,m/2)+2T(m/2,n-m/2),\;\; 2T(m,n/2)\;\}$$
(Assume $T(m,n)=1$ for $m \leq 1$ or $n\leq 1$, I don't care about constants)
It is clear that $mn$ is an upper bound, is there a better (more than by a constant) upper bound or a matching lower bound? Is there some general approach to this kind of stuff?
Thanks!