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!
 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!
 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.