Solution
= Solution
<Gibbs inequality> makes <mutual information> nonnegative, so $C\ge0$. The two conditional-<Shannon entropy> expressions give
$$
I(X;Y)=H(X)-H(X\mid Y)=H(Y)-H(Y\mid X)\le\min(H(X),H(Y)).
$$
Each <Shannon entropy> is bounded by the base-two logarithm of its alphabet size. Maximizing over the input distribution therefore gives $\boxed{0\le C\le\log_2m}$ for the smaller size $m$.