Extremal Relations Between Shannon Entropy and ${\ell }_{\alpha }$-Norm

Abstract

The paper examines relationships between the Shannon entropy and the ${\ell }_{\alpha }$-norm for $n$-ary probability vectors, $n\ge 2$. More precisely, we investigate the tight bounds of the ${\ell }_{\alpha }$-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 ${\ell }_{\alpha }$-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 $E_0$ functions with a fixed mutual information under a uniform input distribution.

arXiv commenta short version was submitted to ISIT'2016