If we know that a polynomial $p(t)$ of form $c_1 + c_2 \cdot t + ... c_i t^{i}$ is bounded by the functions $c\cdot t^k$ and $d \cdot t^{\ln{t}}$ , where $c,d >0$:
Therefore we have: $p(t) \leq c\cdot t^k$ and $p(t) \leq d\cdot t^{\ln{t}}$ and if we take the difference of those two inequalities we get $0 \leq c\cdot t^k - d \cdot t^{\ln{t}}$ how can we prove that $d \cdot t^{\ln{t}} \leq c \cdot t^k$? For a certain $t > t_0$
Is there an easy way of proving this inequality?
You cannot take the difference between two inequalitties:
$$1<2$$ $$1<3$$
but you don't have
$$0<2-3 \rightarrow 3<2$$
I have to add that we are in the easy case where everything is superior to 0. Otherwise it becomes even more messy.