Proving monotonicity of a ratio of two sums

[raphile]

Hi everyone. In a proof I'm working on, I have a ratio of two sums of functions in the following form:

$\frac{f_1(x)+f_2(x)+...f_n(x)}{g_1(x)+g_2(x)+...+g_n(x)}$

I want to prove this ratio is monotonically increasing in $x$. All of the functions $f_i(x)$ and $g_i(x)$ are positive and also (importantly) I know that for all $i=1,2,...,n$, the ratio $f_i(x)/g_i(x)$ is monotonically increasing in $x$, i.e. $f_1(x)/g_1(x)$ is increasing in $x$, $f_2(x)/g_2(x)$ is increasing in $x$, etc.

Is there a simple way to prove this without requiring further information about these functions? I've been stuck on it for a while. Does it have to be true that the ratio of the sums is increasing? If anyone can suggest a straightforward approach (or tell me if it's not possible without further information) I'd be very grateful, thanks!

 

[Bipolarity]

Hi everyone. In a proof I'm working on, I have a ratio of two sums of functions in the following form:

$\frac{f_1(x)+f_2(x)+...f_n(x)}{g_1(x)+g_2(x)+...+g_n(x)}$

I want to prove this ratio is monotonically increasing in $x$. All of the functions $f_i(x)$ and $g_i(x)$ are positive and also (importantly) I know that for all $i=1,2,...,n$, the ratio $f_i(x)/g_i(x)$ is monotonically increasing in $x$, i.e. $f_1(x)/g_1(x)$ is increasing in $x$, $f_2(x)/g_2(x)$ is increasing in $x$, etc.

Is there a simple way to prove this without requiring further information about these functions? I've been stuck on it for a while. Does it have to be true that the ratio of the sums is increasing? If anyone can suggest a straightforward approach (or tell me if it's not possible without further information) I'd be very grateful, thanks!

Have you tried induction? It is usually the first thing I think of when solving problems like this.

BiP

 

[haruspex]

Is it even true?
Consider f1(x) = x +x^2, g1(x) = 10x, f2(x) = 10+x^2, g2(x) = 1.
When x v small, (f1+f2)/(g1+g2) ~ 10. At x = 1, ratio is 13/11.

 

[raphile]

Many thanks for the help. Sorry for the late reply - I'm still working on the problem and trying things out.

At least now I'm convinced that the condition that $f_i(x)/g_i(x)$ is monotonically increasing for all $i=1,2,...,n$ is not sufficient for the overall ratio to be increasing, which I wasn't sure about before. I'm trying some things based on induction which rely on some other properties of these functions.