This is a bit of a soft question. Suppose we consider a function $f\,\colon\,\Omega \to \mathbb{R}_+$, where
$$ \Omega := \{(x,y)\in \mathbb{R}^2\;|\; x,y \geq 1\}, $$ for which it is known that for $x,y$ large enough there exist positive constants $C_1$ and $C_2$ such that
$$ \text{(1) }f(x,y) \leq C_1 x^{-3}y^{-1}\quad\text{ and }\quad \text{(2) }f(x,y) \leq C_2 x^{-1}y^{-3}. $$
My initial silly intuition was that this in fact implies that for $x,y$ large enough there exists $C_3>0$ such that
$$ \text{(3) }f(x,y) \leq C_3 x^{-3}y^{-3}. $$ It's clear that (3) implies both (1) and (2), but what about the other way round? It's definitely safe to divide (1) and (2) by 2 and add them up to conclude that there exists $C_4>0$ such that for $x,y$ large enough we have
$$ \text{(4) }f(x,y) \leq C_4\left(x^{-3}y^{-1} + x^{-1}y^{-3}\right) =:g(x,y). $$ But then $g$ satisfies neither (1) nor (2) and so statement (4) is weaker than (1)&(2).
Can you please tell me whether (1)&(2) implies (3) and if not then why (and ideally an example of a function $f$)? Also, if (3) is not true then can we somehow improve on (4)?
This seemed trivial and silly at first but the more I think about the more confused I am. I would greatly appreciate any help.