  Many axiomatic definitions of entropy, such as the R\'enyi entropy, of a
random variable are closely related to the $\ell_{\alpha}$-norm of its
probability distribution. This study considers probability distributions on
finite sets, and examines the sharp bounds of the $\ell_{\beta}$-norm with a
fixed $\ell_{\alpha}$-norm, $\alpha \neq \beta$, for $n$-dimensional
probability vectors with an integer $n \ge 2$. From the results, we derive the
sharp bounds of the R\'enyi entropy of positive order $\beta$ with a fixed
R\'enyi entropy of another positive order $\alpha$. As applications, we
investigate sharp bounds of Ariomoto's mutual information of order $\alpha$ and
Gallager's random coding exponents for uniformly focusing channels under the
uniform input distribution.
