Monotonicity of ratio of sums $\frac{\sum f_i}{\sum g_i}$ when each $\frac{f_i(x)}{g_i(x)}$ is increasing

Asked Sep 14 '21 at 14:08

Active Sep 15 '21 at 10:18

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Suppose that I have a sequence of functions $f_i$ and $g_i$, and I know that for each $i, \frac{f_i}{g_i}$ is increasing in its argument.

Is it then also true that $\frac{\sum_i^N f_i(x)}{\sum_i^N g_i(x)}$ is increasing in $x$?

Additionally I know that for all $i$: $f_i,g_i >0$, $g'_i<0$, $g_i>g_{i+1}$, $\frac{f_i(x)}{g_i(x)}<\frac{f_{i+1}(x)}{g_{i+1}(x)}$. But I don't believe I need that.

I could only show this for special cases -- simply by taking the derivative and looking at the sign of $\sum f'_i(x) \sum g_i(x) - \sum f_i(x) \sum g'_i(x)$ and making assumptions. I played around with $N=2$. However, I could not rearrange it to prove my conjecture in general. Is it even true? I could not find a counter example.

edited Sep 15 '21 at 10:18

asked Sep 14 '21 at 14:08

Bayesian

2

You definitely need $g_i>0$, as easy counterexample $(x+(-x^3))/(1+(-\frac12))$ shows. – user10354138 Sep 14 '21 at 14:27
Where is the $i$ in this example ? without it it serves no purpose. – Medo Sep 14 '21 at 14:28
I see, you mean $f_1(x)=x, g_1(x)=1, f_2(x)=-x^3, g_2(x)=-1/2$. – Bayesian Sep 14 '21 at 14:33

Monotonicity of ratio of sums $\frac{\sum f_i}{\sum g_i}$ when each $\frac{f_i(x)}{g_i(x)}$ is increasing

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