# Radius of convergence comparison for power series

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 $$\frac{1}{1 + x^4}$$.