Let $f:\mathbb{R}^n_+ \rightarrow \mathbb{R}$ and $F:\mathbb{R}^n_+ \rightarrow \mathbb{R}$ be both increasing, differentiable, homogeneous of degree one and Schur-concave.
For all $X\in \mathbb{R}_+^{n\times n}$ define $W(X)=F\left[f\left(x_1\right),...,f\left(x_n\right) \right]$, where $x_j$ is the j-th row of $X$.
Denote with $\mathbf{W}$ the set of those $W$ such that $F$ is more inequality averse than $f$, meaning that, for all $y\in \mathbb{R}_+^{n}$, $f(y)=f(a,...,a)$ and $F(b,...,b)=F(y)$ for some real numbers $a\geq b$.
Given two matrices $X,Y\in \mathbb{R}_+^{n\times n}$ I would like to establish necessary and sufficient conditions for $W(X)\geq W(Y)$ for all $W\in \mathbf{W}$.
Suppose I focus on $\mathbf{W}_g\subset \mathbf{W}$, which is the set of $W$ composed of $f$ functions at least as inequality averse as $g$.
I think that if $F$ is additive, then - assuming $f(x_1)\leq f(x_2)\leq ... f(x_n)$ and $f(y_1)\leq f(y_2)\leq ... f(y_n)$ - the following two conditions are quivalent:
(i) $\sum_{i=1}^k f(x_i) \geq \sum_{i=1}^k f(y_i) \qquad \forall k=1,...,n$
(ii) $W(X)\geq W(Y) \qquad \forall W\in \mathbf{W}_f$.
The reason being that, when $F$ is additive, I can write $W(X)=\sum_{i=1}^n u\left(f(x_i) \right) $ for an increasing and concave $u:\mathbb{R}_+\rightarrow \mathbb{R}_+$, and apply the result in Meyer, Jack. "Second degree stochastic dominance with respect to a function." International Economic Review (1977): 477-487.
Do you think the above condition extends to the more general case of $F$ non-additive?
Given the link between (i) and the concept of majorization, I though some of you may be able to help me with this. Thanks a lot!