Suppose that we have two generating functions $f(z)=\sum_{n \ge 0} f_n z^n$ and $g(z)=\sum_{n \ge 0} g_n z^n$ with non-negative coefficients. Assume that the radius of convergence is $\infty$ for both functions and that $$ f(z) \le g(z) $$ for all $z \ge 0$. Then, do we have $f_n \le g_n$ for all $n \in \mathbb N$?
I realized, this it to ask if $f(z)=\sum_{n \ge 0} f_n z^n \ge 0$ on $[0,\infty)$ implies all coefficients are non-negative. I.e., does non-negativity of generating function implies non-negativity of coefficients?