Let $f:\mathbb{R}^n_+ \rightarrow \mathbb{R}$ be a continuous, strictly increasing and Schur-concave function homogeneous of degree one. Let $x,y\in \mathbb{R}^n_+$ be decreasingly ordered vectors such that $x \geq y$ and denote $\delta_k \in \mathbb{R}^n_+$ a vector of zeroes with $k$-th element equal to $\delta>0$. I would like to check if, for all $k=1,...,n-1$,
$f \left( x + \delta_k - \delta_{k+1} \right) - f \left( x + \delta_k \right) \geq f \left( y - \delta_{k+1} \right) - f \left( y \right)$
I managed to prove this for the case of $f$ concave but the class of Schur-concave functions contains also non concave functions. Any suggestion? Thanks a lot.