Writing this in the big O notation?

Question

I have two real variables $\sigma_1$ and $\sigma_2$ where $0 < \sigma_1, \sigma_2 << 1$.

I am getting maximal errors of the form of products of linear combinations of the two variables. For example, I have a second order error of $$\sigma_1 (\sigma_1 - 2 \sigma_2).$$

Is this the same as writing $$\sigma_1 (\sigma_1 - 2 \sigma_2) = O((\sigma_1+\sigma_2)^2)?$$ How does it work with linear combinations?

Thanks!

Answer 1

If $0 < \sigma_1,\sigma_2$ then it is indeed true that

$$ \sigma_1 (\sigma_1 - 2\sigma_2) = O((\sigma_1+\sigma_2)^2) $$

since

$$ -\frac{1}{3} \leq \frac{\sigma_1 (\sigma_1 - 2\sigma_2)}{(\sigma_1+\sigma_2)^2} \leq 1. $$