Given two power series $g (x)=\sum a_n x^n $ and $h (x)=\sum b_n x^n$ with radius of convergence $R_1$ and $R_2$ such that $g(x) \leq h (x)$ for all $x \in \{|x| \leq min \{R_1,R_2\}\}$, Does this imply that $R_1 \leq R_2$ ?
I would be grateful for any hints
Assuming you take only real $x$ and $a_i$ and $b_i$ are real (inequality isn't defined on $\mathbb{C}$), it doesn't. Take, for example $g(x) = \frac{1}{2 + x^2}$ and $h(x) = \frac{1}{1 + x^2}$. Then $g(x) < h(x)$, but $R_1 = \sqrt{2}$ and $R_2 = 1$. To get example for the other side (when smaller function has smaller radius of convergence), take $\frac{100}{2 + x^2}$ and $\frac{1}{1 + x^4}$.