Considering $0 < x <1$ and $0 < y < 1$, I want to prove that the following expression defined by the quotient
$$ \frac{(1-x^n y^n)^2}{(1-x^{2n})(1-y^{2n})} $$
is decreasing in $n$.
Or, equivalently, that the following expression is decreasing in $n$
$$ \frac{\left( \sum_{i=1}^n x^i y^i \right)^2}{\sum_{i=1}^n x^{2i} \sum_{i=1}^{n} y^{2i}}. $$
I get nothing by studying the sign of the first derivative.