# Proving monotonicity of a ratio of two sums

by raphile
Tags: monotonicity, proving, ratio, sums
 P: 23 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!
P: 758
 Quote by 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!
Have you tried induction? It is usually the first thing I think of when solving problems like this.

BiP
 HW Helper Sci Advisor Thanks P: 7,850 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.
P: 23

## Proving monotonicity of a ratio of two sums

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.

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