  We establish an information inequality that is intimately connected to the
evaluation of the sum rate given by Marton's inner bound for two receiver
broadcast channels with a binary input alphabet. This generalizes a recent
result where the inequality was established for a particular channel, the
binary skew-symmetric broadcast channel. The inequality implies that randomized
time-division strategy indeed achieves the sum rate of Marton's inner bound for
all binary input broadcast channels.
