Simplify $O(\min(m, n))$?
Based on definition for non negative functions
$$O(f)=\{g: \exists C&gt;0, \exists N \in \mathbb{N}, \forall n&gt;N, g(n) \leqslant Cf(n)\}$$
I prefer to consider your question as about $O(\min(f,g))$.
Now, if from one hand $O(f) \cap O(g) \subset O(\min(f,g))$: $\phi \leqslant C_1 \cdot f$ and $\phi \leqslant C_2 \cdot g$ we obtain $\phi \leqslant C \cdot \min(f,g)$. Reverse can be show in same way, so we can be sure that
$$O(f) \cap O(g) = O(\min(f,g))$$