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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?

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  • $\begingroup$ What is an example of such a function? Have you computed the local conditions? $\endgroup$ – Igor Rivin Sep 1 '18 at 22:44
  • $\begingroup$ Also, can you compute some level sets, especially where the function f has value zero? Gerhard "Maybe Special Cases Can Help" Paseman, 2018.09.01. $\endgroup$ – Gerhard Paseman Sep 1 '18 at 22:49
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You say your function is smooth, so letting $x_1 = x, y_1 = y, x_2 = x + \Delta x, y_2 = y+\Delta y,$ we get in the limit as the deltas go to zero, if we ignore the second order terms, then

$$ \dfrac{\partial f}{\partial x} \dfrac{\partial f} {\partial x} d x d y < 0.$$ This indicates that we cannot ignore the second order terms, and when the smoke clears, we seem to get

$$f\dfrac{\partial^2 f}{\partial x \partial y} - \dfrac{\partial f}{\partial x} \dfrac{\partial f} {\partial x} \geq 0,$$ which is nonlinear and hyperbolic. Changing it to an equation may or may not be enlightening.

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To follow up @igor answer, the equation can be written as: $$ \dfrac{\partial^2 \ln f}{\partial x \partial y} \geq 0. $$ Moreover, the inequality is not only necessary, bu sufficient, since: $$ \ln \left( \frac{f(x_2,y_2)f(x_1,y_1)}{f(x_2,y_1)f(x_1,y_2)} \right) = \int_{x_1}^{x_2} \int_{y_1}^{y_2} \dfrac{\partial^2 \ln f}{\partial x \partial y} dy \ dx. $$

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  • $\begingroup$ The logarithm of a positive expression $$\frac{f(x_2,y_2)f(x_1,y_1)}{f(x_2,y_1)f(x_1,y_2)} $$ may be negative, is not so? $\endgroup$ – user64494 Sep 2 '18 at 9:20
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    $\begingroup$ It may be negative. The logarithm of such an expression (or any number) is positive if and only if the expression is greater than or equal to 1. $\endgroup$ – VictorZurkowski Sep 2 '18 at 9:24
  • $\begingroup$ Looks like an easily checkable condition... $\endgroup$ – Igor Rivin Sep 7 '18 at 19:49

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