Is it possible to prove this inequality?

Question

Suppose I have the following pairs of variables:

• $a$ , $\hat{a}$
• $b$ , $\hat{b}$
• $c$ , $\hat{c}$
• $d$ , $\hat{d}$

Now, suppose I have the following functions:

• Function 1:

$$ F(A, B, C, D) = \frac{\frac{A}{B}}{\frac{C}{D}}$$

• Function 2:

$$ G(A, B, C, D) = \frac{\frac{A}{A+B}}{\frac{C}{C+D}}$$

My Question: When $a >> \hat{a}$ and $c >> \hat{c}$, us it possible to mathematically prove that : $$|F(a, b, c, d) - F(\hat{a}, \hat{b}, \hat{c}, \hat{d})| < |G(a, b, c, d) - G(\hat{a}, \hat{b}, \hat{c}, \hat{d})|$$

That is: if we fix some values of $b$ and $d$ - and then, when $(a,\hat{a})$ differ by a lot and $(c,\hat{c})$ differ by a lot, is the above inequality correct?

Thanks!

Answer 1

If $a = 10^6$, $b = c = d = 1$, $\hat{a} = 0$, $\hat{b} = 1 = \hat{d}$, $\hat{c} = -10^6$, we have $a >> \hat{a}$, and $c >> \hat{c}$, but

\begin{align} |F(10^6,1,1,1) - F(0,1,-10^6,1)| = 10^6 \end{align}

and

\begin{align} |G(10^6,1,1,1) - G(0,1,-10^6,1)| = \frac{2(10^6)}{10^6 + 1} \approx 2 \end{align}