Extremal Relations Between Shannon Entropy and ℓα-Norm

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The paper examines relationships between the Shannon entropy and the ℓα-norm for n-ary probability vectors, n ≥ 2. More precisely, we investigate the tight bounds of the ℓα-norm with a fixed Shannon entropy, and vice versa. As applications of the results, we derive the tight bounds between the Shannon entropy and several information measures which are determined by the ℓα-norm, e.g., Rényi entropy, Tsallis entropy, the R-norm information, and some diversity indices. Moreover, we apply these results to uniformly focusing channels. Then, we show the tight bounds of Gallager's E₀ functions with a fixed mutual information under a uniform input distribution.