Past exam of the communication Theory course of the University of Oxford / 2007 / 3 / b / Solution

Gibbs inequality makes mutual information nonnegative, so $C\ge 0$. The two conditional-Shannon entropy expressions giveEach Shannon entropy is bounded by the base-two logarithm of its alphabet size. Maximizing over the input distribution therefore gives $\boxed{0\le C\le {\log }_{2}m}$ for the smaller size $m$.