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although it seems very simple and obvious, I have no idea how to give an analytical proof for this problem. I will be very happy if there are some smart ideas...

Given,

$f_1(a), f_2(a),..., f_n(a)$ and $g_1(a), g_2(a),..., g_n(a)$ are strictly increasing positive functions of $a$.

It is also known that

$\frac{f_1(a)}{g_1(a)}$, $\frac{f_2(a)}{g_2(a)}$,...,$\frac{f_n(a)}{g_n(a)}$ are strictly increasing functions of a.

I want to know if

\begin{equation} \frac{f_1(a)+f_2(a)+...+f_n(a)}{g_1(a)+g_2(a)+...+g_n(a)} \end{equation}

is also an increasing function of $a$.

  • 0
    I asked them to close it there. Thanks alot. They suggested me to come to this web-site.2012-04-25

3 Answers 3

0

Hint:

For $a,b,c,d > 0$ we have something I call naive fraction addition rule:

$\frac{a}{b} \leq \frac{c}{d} \Rightarrow \frac{a}{b} \leq \frac{a+c}{b+d} \leq \frac{c}{d}$

Edit:

Ok, I made a mistake in my calculations, checking it for the second time I created a counter-example, so your conjecture is false:

Let $f_1(x) = 1+x$, $f_2(x) = x^3$, $g_1(x) = 2+x$, $g_2(x) = 100x^2$, then $h(x) = \frac{f_1(x)+f_2(x)}{g_1(x)+g_2(x)} = \frac{1+x+x^3}{2+x+100x^2}$ is not increasing, in fact $h(0) = \frac{1}{2}$ and h(1) = \frac{3}{103} < \frac{1}{2}. I know that those functions are not strictly positive and increasing on whole $\mathbb{R}$, but I guess it is easy enough to see how to fix those minor issues (and writing it explicitly here would make the example unreadable).

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    @SeyhmusGüngören actually, it is enough to ta$k$e [example by scineram](http://www.wolframalpha.com/input/?i=$P$lot%5B%28x%5E1.5%2Bx%5E0.05%29%2F%28x%2Bx%5E0.045%29%2C%7Bx%2C0.1%2C0.2%7D%5D) and [rescale the domain](http://www.wolframalpha.com/input/?i=Plot%5B%28%280.1tanh%5Bx%5D%2B0.15%29%5E1.5%2B%280.1tanh%5Bx%5D%2B0.15%29%5E0.05%29%2F%28%280.1tanh%5Bx%5D%2B0.15%29%2B%280.1tanh%5Bx%5D%2B0.15%29%5E0.045%29%2C%7Bx%2C-3%2C3%7D%5D).2012-04-25
2

No. A counterexample is

  • $f_1(a) = 2a+a^2$
  • $f_2(a) = a^3$
  • $g_1(a) = a$
  • $g_2(a) = a^2$

Where $ \frac{2a+a^2+a^3}{a+a^2} = \frac{2+a+a^2}{1+a}$ decreases from $a=0.1$ to $a=0.2$.

Intuitively, the problem is that $\frac{f_1+f_2}{g_1+g_2}=\frac{f_1}{g_1}\frac{g_1}{g_1+g_2} + \frac{f_2}{g_2}\frac{g_2}{g_1+g_2}$ is a weighted average of $\frac{f_1}{g_2}$ and $\frac {f_2}{g_2}$, where we don't have any applicable hypotheses about how the weights evolve. If $g_2$ grows faster than $g_1$, then the weight shifts from $\frac{f_1}{g_1}$ for $\frac{f_2}{g_2}$, which can make the average drop of the latter is smaller and changes slowly compared to the relative change between $g_1$ and $g_2$.

2

False.

As example shows decrease.

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    Thank you very much. It is false. It is neither incereasing nor decreasing and no such generalizations can be made.2012-04-25