Consider the case of vectors $l, x, u \in \mathbb{R}^{p+}$ such that for each element of the vectors it holds that $l \le x \le u.$
This ordering holds for example for $l= [1,2,3]^T$, $x=[4,5,6]^T$, $u=[7,8,9]^T$, because $1<4<7, \ 2<5<8, \ 3<6<9$.
I am interested whether this ordering also holds after applying the function $f$ on the vectors, in particular the summation of all elements: $f(y)=e^Ty$, where $e$ the unit vector.
In the example we have $6<15<24$, so the order holds after applying $f$.
Is this problem known under a particular name and does it have a general answer? Note again we are in $\mathbb{R}^{+}$, but I am not sure if this is relevant.