  Several relations between the Holevo capacity and the entanglement-assisted
classical capacity of a quantum channel are proved, necessary and sufficient
conditions for their coincidence are obtained. In particular, it is shown that
these capacities coincide if (correspondingly, only if) the channel
(correspondingly, the $\chi$--essential part of the channel) belongs to the
class of classical-quantum channels (the $\chi$-essential part is a restriction
of a channel obtained by discarding all states useless for transmission of
classical information). The obtained conditions and their corollaries are
generalized to channels with linear constraints. By using these conditions it
is shown that the question of coincidence of the Holevo capacity and the
entanglement-assisted classical capacity depends on the constraint (even for
classical-quantum channels).
  Properties of the difference between the quantum mutual information and the
$\chi$-function (constrained Holevo capacity) of a quantum channel are
explored.
