when does schur convex function imply weak majorization

Question

I am studying about majorization and Schur convexity. And would appreciate a help.

Assume that you have a two vectors say $x,y \in R^d$, further assume that we have a Schur Convex function $\phi(x)$, we know that if $y$ weakly majorizes $x$ then $\phi(x)\leq \phi(y)$. Is there any result that we have the above hold for on the other way around, i.e., if and only if.

Thanks

Answer 1

A vector $y$ weakly majorizes a vector $x$ if and only if $\sum_{i=1}^d h(y_i)\geq \sum_{i=1}^d h(x_i)$ for every (continuous) convex function $h: \mathbf{R} \rightarrow \mathbf{R}$.