Extremal relations between shannon entropy and ℓα-norm

The paper examines the relationships between the Shannon entropy and the ℓ
_α
-norm for n-dimensional probability vectors, n ≤ 2. More precisely, we investigate the sharp bounds on the ℓ
_α
-norm with a fixed Shannon entropy, and vice versa. As applications of the results, we derive the sharp bounds between the Shannon entropy and several information measures which are determined by the ℓ
_α
-norm. Moreover, we apply these results to uniformly focusing channels. Then, we show the sharp bounds on Gallager's reliability functions E
₀
with a fixed mutual information under a uniform input distribution.