The ratio of two strictly increasing functions 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 $a$.
 A: No. A counterexample is


*

*$f_1(a) = 2a+a^2$

*$f_2(a) = a^3$

*$g_1(a) = a$

*$g_2(a) = a^2$


Where
$$ \frac{2a+a^2+a^3}{a+a^2} = \frac{2+a+a^2}{1+a}$$
decreases from $a=0.1$ to $a=0.2$.
Intuitively, the problem is that $\frac{f_1+f_2}{g_1+g_2}=\frac{f_1}{g_1}\frac{g_1}{g_1+g_2} + \frac{f_2}{g_2}\frac{g_2}{g_1+g_2}$ is a weighted average of $\frac{f_1}{g_2}$ and $\frac {f_2}{g_2}$, where we don't have any applicable hypotheses about how the weights evolve. If $g_2$ grows faster than $g_1$, then the weight shifts from $\frac{f_1}{g_1}$ for $\frac{f_2}{g_2}$, which can make the average drop of the latter is smaller and changes slowly compared to the relative change between $g_1$ and $g_2$.
A: False.
As example shows decrease.
A: Hint:
For $a,b,c,d > 0$ we have something I call naive fraction addition rule:
$$\frac{a}{b} \leq \frac{c}{d} \Rightarrow \frac{a}{b} \leq \frac{a+c}{b+d} \leq \frac{c}{d}$$
Edit:
Ok, I made a mistake in my calculations, checking it for the second time I created a counter-example, so your conjecture is false:
Let $f_1(x) = 1+x$, $f_2(x) = x^3$, $g_1(x) = 2+x$, $g_2(x) = 100x^2$, then
$$h(x) = \frac{f_1(x)+f_2(x)}{g_1(x)+g_2(x)} = \frac{1+x+x^3}{2+x+100x^2}$$
is not increasing, in fact $h(0) = \frac{1}{2}$ and $h(1) = \frac{3}{103} &lt \frac{1}{2}$. I know that those functions are not strictly positive and increasing on whole $\mathbb{R}$, but I guess it is easy enough to see how to fix those minor issues (and writing it explicitly here would make the example unreadable).
