Proving monotonicity of a ratio of two sums

1. Nov 4, 2012

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!

2. Nov 4, 2012

Bipolarity

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

BiP

3. Nov 5, 2012

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.

4. Nov 12, 2012

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.