  A well-known result due to Fannes is a certain upper bound on the modulus of
continuity of the von Neumann entropy with respect to the trace distance
between density matrices; this distance is the maximum probability of
distinguishing between the corresponding quantum states. Much more recently,
Audenaert obtained an exact expression of this modulus of continuity.
  In the present note, Audenaert's result is extended to a broad class of
entropy functions indexed by arbitrary continuous convex functions $f$ in place
of the Shannon--von Neumann function $x\mapsto x\log_2x$. The proof is based on
the Schur majorization.
