Functions that are log-supermodular and submodular do they have some special name? Or any special properties? Any references? Is the function below log-super-modular and sub-modular?
Consider $\mathbb{R}^2_{+}$ as a lattice with the coordinate-wise order, $(a,b)\ge (c,d)$ iff $a\ge c$ and $b\ge d$.
Remember a function is super-modular if $f(x\vee y)+f(x\wedge y) \ge f(x) + f(y)$ where $\vee$ is the coordinate-by-coordinate maximum and $\wedge$ is the coordinate-by-coordinate minimum.
It is sub-modular iff $-f$ is super-modular. It is log super-modular if $\log(f)$ is super-modular.
If $f$ is twice differentiable, it is super-modular if and only if it has positive cross-partial derivatives: $\frac{\partial f}{\partial x_j\partial x_k}\ge 0$ for all $j\neq k$.
Consider the function $f:\mathbb{R}^2_{++}\rightarrow \mathbb{R}$ defined by:
$$ f(x,y)=\dfrac{y^2}{x(x+y)(x+2y)} $$
We have that $$\partial_{xy}f=-\dfrac{2 (3 x^2 y + 9 x y^2 + 7 y^3)}{(x + y)^3 (x + 2 y)^3}<0$$ and also $$\partial_{xy}\log(f)=\dfrac{3 x^2 + 8 x y + 6 y^2}{(x + y)^2 (x + 2 y)^2}>0$$.
So $f$ is both submodular and log-supermodular in its domain (the positive orthant).