Sharp Bounds on Arimoto's Conditional Renyi Entropies Between Two Distinct Orders

This study examines sharp bounds on Arimoto's conditional Renyi entropy of order beta with a fixed another one of distinct order alpha not equal beta. Arimoto inspired the relation between the Renyi entropy and the l(r)-norm of probability distributions, and he introduced a conditional version of the Renyi entropy. From this perspective, we analyze the l(r)-norms of particular distributions. As results, we identify specific probability distributions which achieve our sharp bounds on the conditional Renyi entropy. The sharp bounds derived in this study can be applicable to other information measures which are strictly monotone functions of the conditional Renyi entropy.