I am considering the following Pseudo-Boolean function $f:\{0,1\}^{3}\rightarrow \mathbb{R}$, where f is
$$ f(I_{1},I_{2},I_{3}) = \left(I_{1}+I_{2}+I_{3}\right)^{\beta} $$
with $\beta>0$.
after some manipulations I arrive to the following expression,
$$\left[\left(I_{1}+I_{2}+I_{3}\right)^{\beta}-\left(I_{1}+I_{2}\right)^{\beta}-\left(I_{1}+I_{3}\right)^{\beta}-\left(I_{2}+I_{3}\right)^{\beta}+I_{1}^{\beta}+I_{2}^{\beta}+I_{3}^{\beta}\right]>0$$
One can show that this inequality holds whenever $\beta<1$ or $\beta>2$. To me, this is reminiscent of the first derivative being convex in the scalar case.
Also, a sufficient condition for this inequality hold is that for any combination of $i,j,k$ the following holds
$$\left(I_{i}+I_{j}+I_{k}\right)^{\beta}-\left(I_{i}+I_{k}\right)^{\beta}-\left(I_{j}+I_{k}\right)^{\beta}+I_{k}^{\beta}>\left(I_{i}+I_{j}\right)^{\beta}-(I_{i}^{\beta}+I_{j}^{\beta})$$
that sort of says that the double difference is larger than the first difference of something like that.
Therefore, my question is whether this type of function has a precise characterization and/or it has a name and further properties that one can exploit. I do not know if this rings a bell to any of you, but any reference/intuition or precise characterization would be very helpful.