On the mutual information of memoryless channels

Let
x_1, \cdots ,x_n
and
y_1, \cdots, y_n
be input and output sequences of a channel. In the case of memoryless input sources, the following inequality on mutual information is well known: \begin{equation} I((x_1, \cdots ,x_n),(y_1, \cdots ,y_n)) \geq \sum I(x_i, y_i). \end{equation} It is straightforward to show that the inequality sign is reversed if the channel instead of the input source is memoryless. In this paper we establish these inequalities when the input and output are functions instead of sequences.