I am working in data science and I have to deal with the following problem for which I would like to find a simplification:
We call a function almost positive if $f(x_1,y_1)f(x_2,y_2)-f(x_1,y_2)f(x_2,y_1) \ge 0$ for all $0< x_1\le x_2 < \infty$ and $0 < y_1\le y_2 < \infty.$
I would like to know: Are there any sufficient and necessary criteria for a function $f$ to be almost positive?
Background: The problem is that I often have a positive smooth function $f$ which I need to check for almost positivity. Those functions $f$ are usually cumbersome expressions such that checking
$f(x_1,y_1)f(x_2,y_2)-f(x_1,y_2)f(x_2,y_1) \ge 0$ is almost impossible analytically, because one has to compare infinitely many variables which each other and unless one can simplify the expression in a clever way, checking this condition is hopeless.
I am therefore asking whether there is an equivalent criterion to the almost positivity condition which I can check in a more direct way? Ideally there would exist an "intrinsic" criterion for functions $f$ which implies this property.
If there is nothing equivalent to almost positivity, perhaps there exist rather general sufficient conditions which imply almost positivity?