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:

[itex]\frac{f_1(x)+f_2(x)+...f_n(x)}{g_1(x)+g_2(x)+...+g_n(x)}[/itex]

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

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 [itex]f_i(x)/g_i(x)[/itex] is monotonically increasing for all [itex]i=1,2,...,n[/itex] 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.