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 $a$.
Best Answer
False.
As example shows decrease.