I am looking for an example of a metric/distance function $D(\rho,\sigma)$ which is Schur-concave apart from fidelity. In particular I am interested in the relation
$D(\rho, \sum_i p_i \sigma_i) \leq_{?} D(\rho, \sum_i q_i \sigma_i)$,
when $\vec{p}$ majorizes $\vec{q}$.
