Say that a representation $n=a^2+b^2$ (where $n,a,b$ are integers) is normalized if $0\leq a \leq b$. Among those normalized representations, there is a unique one minimizing $a$ (or equivalently maximizing $b$) ; we call it the canonical representation of $n$ (when it exists), and we denote it by $n=x(n)^2+y(n)^2$.
It is true that $y$ is sup-multiplicative, i.e.
$$ y(nm)\geq y(n)y(m) \tag{1} $$
whenever $y(n)$ and $y(m)$ exist ? I have checked with a computer that (1) holds for $0\leq n,m \leq 1000$.
if $0 \leq x \leq y$ and $0 \leq u \leq v,$ then $m = x^2 + y^2$ and $n = u^2 + v^2,$ we get your inequality from $$ mn = |xv - yu|^2 + (xu + yv)^2, $$ and $$ xu + yv \geq yv $$