Sometimes sophisticated algorithms have very complicated looking expressions for the run time complexity which got me to wondering.
Let $F$ be the set of functions containing, constant functions, polynomial functions, the exponential function, the log function, and closed over composition.
Is there an algorithm that can compare the big $\mathcal{O}$ complexity of two arbitrary functions $f_1,f_2 \in F$ and determine if $\mathcal{O}(f_1) < \mathcal{O}(f_2)$