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.

 

[blue_raver22]

Hello,

I can suggest a straightforward approach to prove the monotonicity of the ratio of two sums in this case. Since we know that each individual ratio f_i(x)/g_i(x) is monotonically increasing in x, we can use this information to prove the monotonicity of the overall ratio of two sums.

First, let's consider the numerator of the ratio, which is the sum of the functions f_i(x). Since each individual function is positive and monotonically increasing in x, it follows that the sum of these functions is also positive and monotonically increasing in x. This means that as x increases, the numerator of the ratio will also increase.

Next, let's consider the denominator of the ratio, which is the sum of the functions g_i(x). Similar to the numerator, since each individual function is positive and monotonically increasing in x, the sum of these functions will also be positive and monotonically increasing in x. This means that as x increases, the denominator of the ratio will also increase.

Therefore, as both the numerator and denominator of the ratio are monotonically increasing in x, the overall ratio will also be monotonically increasing in x. This proves that the ratio of the two sums is monotonically increasing in x.

In conclusion, we can prove the monotonicity of the ratio of two sums without requiring further information about the individual functions. However, this proof relies on the assumption that each individual ratio f_i(x)/g_i(x) is monotonically increasing in x. If this assumption is not true, then it is not possible to prove the monotonicity of the overall ratio without further information about the functions. I hope this helps.