I have two $\mathbb{R}$-valued functions $f,g : G \to \mathbb{R}$ on a Lie group $G$. For example, $G = SO(3)$. The two functions $f,g$ are ugly and it is hard to compute the exact value of $f(M), g(M)$ for $M \in G$. I would like to see if one dominates the other, i.e. $f(M) \leq g(M)$ for all $M \in G$ or vise versa. Is there any theorem or result regarding this kind problem?
So far, I have tried some simulation:
1) randomly choose $M_1, ..., M_k \in G$
2) compute the approximate values of $f(M_i)$, $g(M_i)$, denoted $\hat{f}(M_i)$ and $\hat{g}(M_i)$
3) the result is that $\hat{f}(M_i) < \hat{g}(M_i)$ for all $i$
I don't think the simulation result is valid as a proof. So I am wondering if there is a specific mathematical discipline that focuses on this kind of problem.
Thanks in advance for any helpful response!