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Proving monotonicity of a ratio of two sums 
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#1
Nov412, 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! 


#2
Nov412, 06:19 PM

P: 783

BiP 


#3
Nov512, 01:36 AM

Homework
Sci Advisor
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Thanks
P: 9,787

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
Nov1212, 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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