Proving monotonicity of a ratio of two sums


by raphile
Tags: monotonicity, proving, ratio, sums
raphile
raphile is offline
#1
Nov4-12, 05:21 PM
P: 23
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!
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Bipolarity
Bipolarity is offline
#2
Nov4-12, 06:19 PM
P: 783
Quote Quote by raphile View Post
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!
Have you tried induction? It is usually the first thing I think of when solving problems like this.

BiP
haruspex
haruspex is offline
#3
Nov5-12, 01:36 AM
Homework
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Thanks ∞
P: 9,158
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
raphile is offline
#4
Nov12-12, 06:48 AM
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 [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.


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