  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.
