Consider the function $\phi:\mathbb{R}_+\to \mathbb{R}_+$ which is convex and non decreasing. Also let $\ell_i:\mathbb{R}_+\to \mathbb{R}_+$ be convex and non decreasing for $i\in \{1,\dots,n\}$. Let $\mathcal{S}$ denote a collection of $p$ subsets of $\{1,\dots,n\}$. For $x\in \mathbb{R}_+^p$ and consider the function $f(x)\equiv\sum_{S\in \mathcal{S}} x_S \phi(\sum_{i\in S} \ell_i(\sum_{S'\ni i} x_{S'}))$.
It can be shown that the convexity holds if the vector function $\{\phi_S(x)\equiv\phi(\sum_{i\in S} \ell_i(\sum_{S'\ni i} x_{S'}))\}_{S\in\mathcal{S}} $ is monotone.
Under which condition does the monotone property hold? More generally when is $f(x)$ convex?