although it seems very simple and obvious, I have no idea how to give an analytical proof for this problem. I will be very happy if there are some smart ideas...
Given,
$f_1(a), f_2(a),..., f_n(a)$ and $g_1(a), g_2(a),..., g_n(a)$ are strictly increasing positive functions of $a$.
It is also known that
$\frac{f_1(a)}{g_1(a)}$, $\frac{f_2(a)}{g_2(a)}$,...,$\frac{f_n(a)}{g_n(a)}$ are strictly increasing functions of a.
I want to know if
\begin{equation} \frac{f_1(a)+f_2(a)+...+f_n(a)}{g_1(a)+g_2(a)+...+g_n(a)} \end{equation}
is also an increasing function of $a$.