Suppose I have the following pairs of variables:
- $a$ , $\hat{a}$
- $b$ , $\hat{b}$
- $c$ , $\hat{c}$
- $d$ , $\hat{d}$
With the following constraints such that $a, \hat{a}, b, \hat{b}, c, \hat{c}, d, \hat{d} \in \mathbb{N}$
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!
Note: A related question I asked without constraints Is it possible to prove this inequality?