I have the following question:
Given,
$f_1(a), f_2(a),\ldots, f_n(a)$ and $g_1(a), g_2(a),\ldots, g_n(a)$ are strictly increasing positive "polynomial" functions of $a$.
It is also known that
$$\frac{f_1(a)}{g_1(a)}, \frac{f_2(a)}{g_2(a)},\ldots,\frac{f_n(a)}{g_n(a)}$$ are strictly increasing functions of $a$.
Does
\begin{equation} \frac{f_1(a)+f_2(a)+\cdots+f_n(a)}{g_1(a)+g_2(a)+\cdots+g_n(a)} \end{equation}
EVENTUALLY increases with $a$?
No, take for counter-example these four function: $$f_1(a) = a + 0.5$$ $$g_1(a) = 2a + 1.01$$ $$f_2(a) = 1.01a + 2$$ $$g_2(a) = 0.5a + 1$$
Here all the function and both $f_i(a)/g_i(a)$ are increasing for each $a \in \mathbb{R_+}$, but $\sum f_i / \sum g_i$ decrease for each $a \in \mathbb{R_+}$.