This is a follow up on my previous linear recurrence inequality question.
I have some matrices which satisfy a linear recurrence formula of the form
$$
A_{n+1} = \alpha A_{n} + \beta A_{n-1},\qquad n\geq 1,
$$
with $A_0 \in \mathbb{R}^{d\times d}$ given. Assume that $\alpha,\beta \geq 0$. Defining the sequence of norms $\{ f_n = ||A_n||\}_{n=0}^{\infty}$, we come up with a *linear recurrence inequality*,
$$
f_{n+1} \leq \alpha f_n + \beta f_{n-1},\qquad n\geq 1,
$$
and real non-negative $\alpha, \beta$, with the property that $f_n\geq 0$ for all $n$. The question is: is it correct to solve this by the standard generating functions method? (if the coefficients are allowed to be negative then this is not true, see the answer to my previous question).

So if we let $F(x) = \sum_{x=0}^\infty f_n x^n$, then (maybe assuming $x\in (0,\mu)\in \mathbb{R}^+$) we obtain $$ F(x) \leq \dfrac{f_0 + xf_1 - \alpha x f_0}{1-\alpha x - \beta x^2}. $$ The question is if we are allowed to compare the Taylor coefficients on the left and on the right hand sides of this inequality, $$ f_n = [x^n] F(x) \overset{?}{\leq} [x^n] \dfrac{f_0 + xf_1 - \alpha x f_0}{1-\alpha x - \beta x^2}. $$