Given two real analytic functions, $g(x)$ and $f(x)$, on an open interval $I\subset \mathbb{R}$, it is obvious that $g(x) \leq f(x)$ does *not* imply $g_n \leq f_n$ (here $g_n = [x^n] g(x)$ denotes the $n$-th Taylor coefficient of $g$). However, does the inequality in the coefficients hold if we restrict to the class of univariate holonomic functions? Let me illustrate with a simple example:

Suppose we are given a linear recurrence inequality, $$ f_{n+1} \leq \alpha f_n + \beta f_{n-1},\qquad n\geq 1, $$ and real $\alpha, \beta$, and $f_0, f_1$ are given. We approach this with the standard generating function $F(x) = \sum_{x=0}^\infty f_n x^n$, 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, $$ f_n = [x^n] F(x) \overset{?}{\leq} [x^n] \dfrac{f_0 + xf_1 - \alpha x f_0}{1-\alpha x - \beta x^2}. $$