Sharp Bounds on Arimoto's Conditional R\'{e}nyi Entropies Between Two Distinct Orders

This study examines sharp bounds on Arimoto's conditional R\'enyi entropy of order $\beta$ with a fixed another one of distinct order $\alpha \neq \beta$. Arimoto inspired the relation between the R\'enyi entropy and the $\ell_{r}$-norm of probability distributions, and he introduced a conditional version of the R\'enyi entropy. From this perspective, we analyze the $\ell_{r}$-norms of particular distributions. As results, we identify specific probability distributions whose achieve our sharp bounds on the conditional R\'enyi entropy. The sharp bounds derived in this study can be applicable to other information measures, e.g., the minimum average probability of error, the Bhattacharyya parameter, Gallager's reliability function $E_{0}$, and Sibson's $\alpha$-mutual information, whose are strictly monotone functions of the conditional R\'enyi entropy.