Suppose we have the following lower and upper bounds for an invariant $\chi(G_N)$, where $G_N$ is a graph on $N$ vertices, $N=f(k,n,m) $ and $N,k,n,m\in \mathbb{N}$:
$$ \alpha(1-O\left({k}/{n}\right)) \le \chi(G_N) \le \alpha, $$
and
$$ \alpha(1-O\left({m}/{n}\right)) \le \chi(G_N) \le \alpha. $$
Based on the values of parameters $k,n,m$, either of the lower bounds above can be stronger than the other. How can I represent the lower bounds using one term? Is it mathematically correct to set $ \epsilon = \min\{ O\left({k}/{n}\right),O\left({m}/{n}\right) \} $ and say $ \alpha(1-\epsilon) \le \chi(G_N) \le \alpha $?
Is it correct to set $ g=\min\{ {m}/{n}, k/n\}$ and say $ \alpha(1-O(g)) \le \chi(G_N) \le \alpha $?